---

Python Code Version 3.0 for

Climate Mathematics: Theory and Applications

---

Samuel S.P. Shen and Richard C.J. Somerville

---

This Version 3.0 is authored by Briana Ramirez, edited by Samuel Shen. Liu Yang, Sandra Villamar, and Joaquin Stawsky contributed codes to this version.

Video tutorial for the python code can be found at the following URL: https://www.youtube.com/channel/UC7D9i0kBMzPTHyEhU0h6W9g

This version is based upon the previous version described in the following box.

######################################################################################################################
#This Python Code for Climate Science is written for the book entitled "Climate Mathematics: Theory and Applications"#
#A Cambridge University Press book authored by SSP Shen and RCJ Somerville in July 2019                              #
#The Python codes were based on the R codes written by Samuel Shen Distinguished Professor,                          #
#San Diego State University, USA and were translated from R by Louis Selstad, Stephen Shen,                          #
#Gregori Clarke, and Dakota Newmann and edited by Samuel Shen.                                                       #
######################################################################################################################

#FIRST TIME Python users*****
#These package need to be installed (on the terminal or anaconda interface) before importing them below. 

#Follow this tutorial for package installation before
# https://towardsdatascience.com/importerror-no-module-named-xyz-45e4a5339e1b

#Change your file path to the folder where your downloaded data is stored
#MAC HELP: https://support.apple.com/guide/mac-help/go-directly-to-a-specific-folder-on-mac-mchlp1236/mac
#PC HELP: https://www.sony.com/electronics/support/articles/00015251
import os
# os.chdir("/Users/sshen/climmath/data")
os.chdir('/Users/HP/Documents/sshen/climmath/data')

%%javascript
IPython.OutputArea.prototype._should_scroll = function(lines) {
    return false;
}

#Style Dictionary to standardize plotting scheme between different python scripts 
import matplotlib.pyplot as plt

styledict = {'xtick.labelsize':20,
             'xtick.major.size':9,
             'xtick.major.width':1,
             'ytick.labelsize':20,
             'ytick.major.size':9,
             'ytick.major.width':1,
             'legend.framealpha':0.0,
             'legend.fontsize':15,
             'axes.labelsize':20,
             'axes.titlesize':25,
             'axes.linewidth':2,
             'figure.figsize':(12,8),
             'savefig.format':'jpg'}
plt.rcParams.update(**styledict)

#Function that creates personalized discrete Colormap
import numpy as np 
from matplotlib import cm as cm1
from matplotlib.colors import ListedColormap, to_rgba

def newColMap(colors):
    """
    This function creates a new color map from a list of colors given
    as a parameter. Recommended length of list of colors is at least 6.
    """
    first = np.repeat([to_rgba(colors[0])], 2, axis = 0)
    last = np.repeat([to_rgba(colors[-1])], 2, axis = 0)
    v = cm1.get_cmap('viridis', 16*(len(colors)-2))
    newcolors = v(np.linspace(0, 1, 16*(len(colors)-2)))
    for (i, col) in enumerate(colors[1:-1]):
        newcolors[16*i : 16*(i+1), :] = to_rgba(col)
    return ListedColormap(np.append(np.append(first,newcolors, axis=0), last, axis=0))

---

Chapter 2: Basics of Python Programming

---

import numpy as np 
import matplotlib.pyplot as plt
from matplotlib.colors import rgb2hex
from matplotlib.patches import Polygon
import matplotlib.mlab as mlab
import pandas as pd

from cartopy import crs, mpl
import matplotlib.ticker as mticker
from cartopy.mpl.gridliner import LONGITUDE_FORMATTER, LATITUDE_FORMATTER

from mpl_toolkits.basemap import Basemap
# Having an issue with the import “mpl_toolkits.basemap” package? 
# Unlike other package installations, this package requires additional steps to install. 
#    1.Download basemap package from this website
#      https://www.lfd.uci.edu/~gohlke/pythonlibs/#basemap
#          *Note the package version needs to be the same as Python version
#    2.Run the following line in your terminal
#      pip install basemap-1.2.2-cp37-cp37m-win_amd64.whl

from mpl_toolkits import mplot3d
import netCDF4
from netCDF4 import Dataset as ds
from urllib import request
import scipy as sp
from scipy import stats as stt
from sklearn import datasets, linear_model
from sklearn.neighbors import KernelDensity as kd
from sklearn.linear_model import LinearRegression
from random import randint
from scipy.stats import norm
import statsmodels
import sklearn
import math
import statistics
import sympy as sy
from sympy import symbols, diff
import statsmodels.api as sm
from datetime import date

2.2: Python Tutorial

Python As a Smart Calculator

# Basic arithmetic 
1+4

5

# Basic arithmetic 
2 + np.pi/4 - 0.8

1.9853981633974482

# Variable Assignment
x = -1
y = 2
z = 4

# x**y means x^y: x to the power of y
t = 2*x**y-z  
t

-2

# Variable Assignment
u = 2
v = 3
u+v

5

# np.sin is the sine function in Python
np.sin(u*v)  

-0.27941549819892586

Define a Sequence in Python

# Enter the temperature data
tmax = [77,72,75,73,66,64,59] 

# Show the temperature data
tmax 

[77, 72, 75, 73, 66, 64, 59]

# A Python sequence starts from 0 while R starts from 1
np.arange(9) 
# A sequence is called an array in Python 

array([0, 1, 2, 3, 4, 5, 6, 7, 8])

# np.arange(x,y) creates an array of numbers incremented by 1 between integer x and integer (y-1)
np.arange(1, 9) 

array([1, 2, 3, 4, 5, 6, 7, 8])

# np.arange(x,y) can take negative and real numbers with a default increment of 1
np.arange(-5.5, 2) 

array([-5.5, -4.5, -3.5, -2.5, -1.5, -0.5,  0.5,  1.5])

Define a Function in Python

# Define the x^2 function and call it samfctn(x)
def samfctn(x):
    return x * x
samfctn(4)

16

#Fig 2.3
x = np.linspace(1,8,7)
tmax = [77,72,75,73,66,64,59] 

plt.plot(x, tmax, "o")
plt.grid()

# Define a multivariate function "x+y-z/2" 
def fctn2(x, y, z):
    return x + y - z / 2;
fctn2(1, 2, 3)

1.5

Plot with Python

Plot the curve of y = sin(x) from -pi to 2*pi

# math.pi is the same as np.pi and scipy.pi, just using different packages
t = np.arange(-math.pi, 2*math.pi, 0.1) 
plt.plot(t, np.sin(t), "o")
plt.grid()
plt.show()

Plot x-y data based on a formula

# Uniform division using 20 points
t = np.linspace(-3, 2, 20) 
plt.plot(t, t ** 2) 
plt.grid()
plt.show()

Plot 3D Surface

x = np.linspace(-1, 1, 100)
y = np.linspace(-1, 1, 100)

# Defines a formula for Z
def f(x, y):
    return 1 - x**2 - y**2
 
X, Y = np.meshgrid(x, y)

# Renders z function on the x, y grid
Z = f(X, Y) 

# Creates the 3D plot
fig = plt.figure()
ax = plt.axes(projection = '3d')
ax.contour3D(X, Y, Z, 30, cmap = 'binary')
plt.show()

Contour Plot

#Fig 2.4
x = np.linspace(-1, 1, 30)
y = np.linspace(-1, 1, 30)
 
X, Y = np.meshgrid(x, y)
Z = f(X, Y)

# Creates and colors the contour plot
plt.contourf(X, Y, Z, 20, cmap = 'cool') #cmap='color-scheme'

# Adds colorbar for color map of contours
plt.colorbar()
plt.show()

Plot 3D Contour Plot

x = np.linspace(-1, 1, 100)
y = np.linspace(-1, 1, 100)

# Defines a formula for Z
def f(x, y):
    return 1 - x**2 - y**2
 
X, Y = np.meshgrid(x, y)

# Renders z function on the x, y grid
Z = f(X, Y)

# Creates the 3D plot
fig = plt.figure()
ax = plt.axes(projection = '3d')
ax.plot_surface(X, Y, Z, rstride=1, cstride=1,
                cmap='viridis', edgecolor='none') #cmap = 'color'

# Labels the plot
ax.set_title('Contour surface')
ax.set_xlabel('x', labelpad=15)
plt.xticks(fontsize=14)
ax.set_ylabel('y', labelpad=15)
plt.yticks(fontsize=14)
ax.set_zlabel('z', labelpad=15)
ax.set_zticklabels([-1,-.75,-.5,-.25,0,.25,.5,.75,1],Fontsize=14)
ax.view_init(30)
plt.show()

<ipython-input-24-723a2fbd47ff>:26: UserWarning: FixedFormatter should only be used together with FixedLocator
  ax.set_zticklabels([-1,-.75,-.5,-.25,0,.25,.5,.75,1],Fontsize=14)
<ipython-input-24-723a2fbd47ff>:26: MatplotlibDeprecationWarning: Case-insensitive properties were deprecated in 3.3 and support will be removed two minor releases later
  ax.set_zticklabels([-1,-.75,-.5,-.25,0,.25,.5,.75,1],Fontsize=14)

Symbolic Calculations with Python

# Makes 'x' and 'y' symbols used for derivatives
x = symbols('x')
y = symbols('y')
# 'diff' command takes the derivative of x^2 w.r.t. x
diff(x ** 2, x)

$\displaystyle 2 x$

# Creates a function of x^2 
fx = x ** 2

# Takes the derivative of function w.r.t. x
rslt = diff(fx, x)
print('The derivative of the function fx w.r.t. x is: ', rslt)

The derivative of the function fx w.r.t. x is:  2*x

# Creates a function of x^2 * sin(x)
fx = (x ** 2) * sy.sin(x)

# Take the derivative of the function w.r.t. x
rslt = diff(fx, x)
print('The derivative of the function w.r.t. x is: ', rslt)

The derivative of the function w.r.t. x is:  x**2*cos(x) + 2*x*sin(x)

# Create a multivariate function: x^2+y^2 
fxy = x ** 2 + y ** 2

# Take partial derivative of function w.r.t. x
rslt_x = diff(fxy, x)

# Take partial derivative of function w.r.t. y
rslt_y = diff(fxy, y)
print('The partial derivative of the function w.r.t. x is: ', rslt_x)
print('The partial derivative of the function w.r.t. y is: ', rslt_y)

The partial derivative of the function w.r.t. x is:  2*x
The partial derivative of the function w.r.t. y is:  2*y

# Integrates x^2 from 0 to 1
sy.integrate(x ** 2, (x, 0, 1))

$\displaystyle \frac{1}{3}$

# Integrates cos(x) from 0 to pi/2
sy.integrate(sy.cos(x), (x, 0, math.pi / 2))

$\displaystyle 1.0$

Vectors and Matrices

# Creates a 5X1 matrix
np.mat([[1], [6], [3], [np.pi], [-3]])

matrix([[ 1.        ],
        [ 6.        ],
        [ 3.        ],
        [ 3.14159265],
        [-3.        ]])

# Creates a sequence from 2 to 6
np.arange(2, 7)

array([2, 3, 4, 5, 6])

# Creates a sequence incremented by 2 from 1 to 10 
np.arange(1, 10, 2)

array([1, 3, 5, 7, 9])

# Creates a 4X1 matrix
x = np.mat([[1], [-1], [1], [-1]])

# Adds number 1 to each element of x
print(x + 1)

[[2]
 [0]
 [2]
 [0]]

# Multiplies number 2 to each element of x
print(x * 2)

[[ 2]
 [-2]
 [ 2]
 [-2]]

# Divides each element of x by 2
print(x / 2)

[[ 0.5]
 [-0.5]
 [ 0.5]
 [-0.5]]

# Creates a 1X4 vector with a sequence incremented by 1 from 1 to 4
y = np.mat(np.arange(1, 5))
# Computes element-wise multiplication and results in a 4X1 vector
np.multiply(x, y.T)

matrix([[ 1],
        [-2],
        [ 3],
        [-4]])

# Dot product and results in 1X1 matrix
np.dot(x.T, y.T)

matrix([[-2]])

# Transforms x into a row 1X4 vector
x.T

matrix([[ 1, -1,  1, -1]])

#The dot product of two vectors  
np.dot(y, x)

matrix([[-2]])

# This column-times-row yields a 4X4 matrix
# Note: For detailed difference between np.multiply and np.dot visit the following URL:
#    https://www.geeksforgeeks.org/difference-between-numpy-dot-and-operation-in-python/

np.matmul(x, y)

matrix([[ 1,  2,  3,  4],
        [-1, -2, -3, -4],
        [ 1,  2,  3,  4],
        [-1, -2, -3, -4]])

# Convert a vector into a matrix of the same number of elements
# y.reshape() splits the elements in the vector by rows
# y.reshape().transpose() splits in order of columns
my = y.reshape(2, 2).transpose()
print(my)

[[1 3]
 [2 4]]

# Finds dimensions of a matrix
my.shape

(2, 2)

# Converts a matrix to a vector, via row-major order
my.flatten()

matrix([[1, 3, 2, 4]])

# Reshapes x to a 2X2 matrix
mx = x.reshape(2, -1).transpose()
# Multiplication between each pair of elements
np.multiply(mx, my)

matrix([[ 1,  3],
        [-2, -4]])

# Division between each pair of elements
np.divide(mx, my)

matrix([[ 1.        ,  0.33333333],
        [-0.5       , -0.25      ]])

# Subtract 2 times a matrix from another
np.subtract(mx, 2 * my)

matrix([[-1, -5],
        [-5, -9]])

# This is matrix multiplication in linear algebra
np.matmul(mx, my)

matrix([[ 3,  7],
        [-3, -7]])

# Determinant of a matrix
x = np.linalg.det(my)
print(x)

-2.0

# Inverse of a matrix
z = np.linalg.inv(my)
print(z)

[[-2.   1.5]
 [ 1.  -0.5]]

#Verifies that the inverse of the matrix is correct by multiplying 
# the original matrix and the inverse matrix
# Note: np.multiply is element-wise multiplication and np.matmul is matrix multiplication
a = np.matmul(z, my)
print(a)

[[1. 0.]
 [0. 1.]]

# Returns a left-to-right diagonal vector of a matrix
np.diagonal(my)

array([1, 4])

# Returns the eigenvalues AND eigenvectors of the "my" matrix
myeigvect = np.linalg.eig(my)
print(myeigvect)

(array([-0.37228132,  5.37228132]), matrix([[-0.90937671, -0.56576746],
        [ 0.41597356, -0.82456484]]))

# Returns the singular value decomposition (SVD) of the "my" matrix
mysvd = np.linalg.svd(my)
print(mysvd)

(matrix([[-0.57604844, -0.81741556],
        [-0.81741556,  0.57604844]]), array([5.4649857 , 0.36596619]), matrix([[-0.40455358, -0.9145143 ],
        [ 0.9145143 , -0.40455358]]))

# Returns a solved linear matrix equation of the "my" matrix and array [1,3]
ysolve = np.linalg.solve(my, [1,3])
print(ysolve)

[ 2.5 -0.5]

# Verifies the solution is correct by multiplying the "my" matrix with "ysolve" 
n = np.matmul(my, ysolve)
print(n)

[[1. 3.]]

Simple Statistics with Python

# Generates 10 normally distributed numbers of N(0,1^2)
random = np.random.normal(0, 1, 10)
print(random)

[ 0.13676212 -0.32058394  2.30943812  0.15169141  0.44322037 -1.97078952
 -0.47919745 -0.74000039  0.48480538  0.84429359]

# Returns the mean of that random sample
mean = np.mean(random)
print(mean)

0.08596396863781804

# Returns the variance of that random sample
var = np.var(random)
print(var)

1.1209648938663797

# Returns the standard deviation of that random sample
std = np.std(random)
print(std)

1.0587562957859469

# Returns the median of that random sample
median = np.median(random)
print(median)

0.1442267654433897

# Returns quantile values in [0, 0.25, 0.5, 0.75, 1]
# quantile = pd.DataFrame(random).quantile([0, 0.25, 0.5, 0.75, 1]) # another way
quantile = np.quantile(random, [0, 0.25, 0.5, 0.75, 1])
print(quantile)

[-1.97078952 -0.43954407  0.14422677  0.47440912  2.30943812]

# Returns range of values (max - min) in random sample
rang = np.ptp(random)
print(rang)

4.280227637673928

# Returns the min value in random sample
minimum = np.min(random)
print(minimum)

# Returns the max value in random sample
maximum = np.max(random)
print(maximum)

-1.9707895168492275
2.3094381208247006

# Yields the box plot of random values
fig = plt.figure(1, figsize=(8, 10))
ax = fig.add_subplot(111)
bp = ax.boxplot(random)
plt.show()

# Yields statistical summary of the data sequence
random = np.random.normal(0, 1, 12)
summ = pd.DataFrame(random).describe()
print(summ)

               0
count  12.000000
mean   -0.249608
std     1.041922
min    -2.312685
25%    -0.731114
50%    -0.270111
75%     0.122854
max     1.540865

# Yields the histogram of 1000 random numbers with a normal distribution
w = np.random.normal(0, 1, 1000)
hist = plt.hist(w, facecolor='black')
plt.show()

#Fig 2.5
# Linear Regression and linear trend line
# 2007-2016 data of global temperature anomalies
# Source: NOAAGlobalTemp data
from sklearn.linear_model import LinearRegression

t = np.linspace(2007, 2016, 10)
T = [.36,.3,.39,.46,.33,.38,.42,.5,.66,.7]
lm = LinearRegression()
lm.fit(t.reshape(-1,1), T)
predictions = lm.predict(t.reshape(-1, 1))

plt.plot(t, predictions, '-r')
plt.plot(t,T,"-o", color="black")
plt.title("2007-2016 Global Temperature Anomalies")
plt.xlabel("Year")
plt.ylabel("Temperature")
plt.grid()
plt.show()  

Input Data by reading a file

# CSV file 
file = "NOAAGlobalT.csv"
data = pd.read_csv(file)

# TXT file 
file = 'http://shen.sdsu.edu/data/' + 'aravg.mon.land_ocean.90S.90N.v4.0.1.201703.asc.txt'
data = pd.read_table(file, header = None, delim_whitespace = True)

# ASC file 
file = open('NOAAGlobalTemp.gridded.v4.0.1.201701.asc', "r")
# read float numbers from the file
data = []
for line in file:
    x = line.strip().split()
    for f in x:
        data.append(float(f))

# NetCDF file 
file = "air.mon.mean.nc"
data = ds(file,"r+")

---

Chapter 3: Basic Statistical Methods for Climate Data Analysis

---

import numpy as np 
import matplotlib.pyplot as plt
from matplotlib.colors import rgb2hex
from matplotlib.patches import Polygon
import matplotlib.mlab as mlab
import pandas as pd
from mpl_toolkits.basemap import Basemap
from mpl_toolkits import mplot3d
from netCDF4 import Dataset as ds
from urllib import request
import scipy as sp
from scipy import stats as stt
from sklearn import datasets, linear_model
from sklearn.neighbors import KernelDensity as kd
from sklearn.linear_model import LinearRegression
from random import randint
from scipy.stats import norm
import statsmodels
import sklearn
import math
import statistics
import sympy as sy
from sympy import symbols, diff
import statsmodels.api as sm
from datetime import date
from scipy.stats import norm, kurtosis
from sklearn.linear_model import LinearRegression
import seaborn as sns
import statsmodels.api as sm 
import pylab as py 
from scipy.stats import norm, t
from scipy.integrate import quad
import warnings
warnings.filterwarnings("ignore")

3.1: Statistical Indices from Global Temperature Data from 1880 to 2015

# Download the NCEI spatial average time series of monthly data
# by internet search using the file name.
aveNCEI = pd.read_table('http://shen.sdsu.edu/data/' + 
                        'aravg.mon.land_ocean.90S.90N.v4.0.1.201703.asc.txt', 
                        header = None, delim_whitespace = True)
aveNCEI = aveNCEI.values
aveNCEI

array([[ 1.88000e+03,  1.00000e+00, -2.54882e-01, ...,  1.25049e-01,
         1.18830e-01,  1.50000e-02],
       [ 1.88000e+03,  2.00000e+00, -3.90030e-01, ...,  1.69170e-01,
         8.25840e-02,  8.65860e-02],
       [ 1.88000e+03,  3.00000e+00, -4.14509e-01, ...,  1.28022e-01,
         8.60760e-02,  4.19460e-02],
       ...,
       [ 2.01700e+03,  1.00000e+00,  6.36712e-01, ...,  1.25049e-01,
         1.28754e-01,  1.50000e-02],
       [ 2.01700e+03,  2.00000e+00,  6.96222e-01, ...,  1.69170e-01,
         1.81365e-01,  1.50000e-02],
       [ 2.01700e+03,  3.00000e+00,  7.69576e-01, ...,  1.28022e-01,
         1.61637e-01,  3.36150e-02]])

Mean, Variance, Standard Deviation, Skewness, Kurtosis, and Quantiles

# Yields the dimensions of data
aveNCEI.shape 

(1647, 10)

# Take only the second column of this data matrix 
tmean15 = aveNCEI[:,2]

# Yields the first 6 values
tmean15[:6]

array([-0.254882, -0.39003 , -0.414509, -0.314104, -0.327618, -0.421961])

# Yields the sample Mean
np.mean(tmean15)

-0.19504468609593198

# Yields the sample Standard Deviation
np.std(tmean15)

0.3289373979136597

# Yields the sample Variance
np.var(tmean15)

0.10819981174620931

# Yields the sample Skewness
# pd.DataFrame(tmean15).skew() # another way
stt.skew(tmean15)

0.6570797310887514

# Yields the sample Kurtosis
kurtosis(tmean15)

-0.09874723968157051

# Yields the sample Median
np.median(tmean15)

-0.263387

# Yields desired quantiles
pd.DataFrame(tmean15).quantile([0, 0.25, 0.5, 0.75, 1])

	0
0.00	-0.934475
0.25	-0.440763
0.50	-0.263387
0.75	0.006320
1.00	0.949248

Correlation, Covariance, and Linear Trend

# Creating a year sequence 
timemo = np.linspace(1880, 2015, 1647)

# Creating Linear Regression 
lm = LinearRegression()

# Training the model
lm.fit(timemo.reshape(-1, 1), tmean15)

# Using the model to make predictions
predictions = lm.predict(timemo.reshape(-1, 1))

print("Coeff. of Linear Model:", lm.coef_)
print("Intercept of Linear Model: ", lm.intercept_)

Coeff. of Linear Model: [0.00698074]
Intercept of Linear Model:  -13.790040882746423

#Fig 3.1
# Linear Model Visualization
plt.close()

plt.plot(timemo, tmean15, '-', color="black")
plt.title('Global Annual Mean Land and Ocean Surface \n Temperature Anomalies: 1880-2015')
plt.xlabel('Year')
plt.ylabel('Temperature Anomaly [$\degree$C]')

plt.plot(timemo, predictions, '-', color="red")
font = {'family': 'serif',
        'color':  'red',
        'weight': 'normal',
        'size': 18,
        }
plt.text(1885, .52, 'Linear trend: 0.70 [$\degree$C] per Century', fontdict = font)
plt.grid()
plt.show()  

# Yields the covariance
np.cov(timemo,tmean15)[0][1]

10.621333849840694

# Yields the correlation
np.corrcoef(timemo,tmean15)[0][1]

0.8275518334843865

# Verify the correlation result 
x = timemo
y = tmean15
sdx = np.std(x)
sdy = np.std(y)
cxy = np.cov(x,y)
rxy = cxy / (sdx*sdy)
rxy[0][1]

0.8280545988753246

# Verify the trend result
rxy*sdy/sdx 

array([[1.00060753e+00, 6.98498363e-03],
       [6.98498363e-03, 7.11994446e-05]])

# Verify the trend result
cxy/(sdx**2)

array([[1.00060753e+00, 6.98498363e-03],
       [6.98498363e-03, 7.11994446e-05]])

3.2: Commonly used Statistical Plots

Histogram

#Fig 3.2
# Plot a histogram of the tmean15 data
h, mids = np.histogram(tmean15)
xfit = np.linspace(np.min(tmean15),np.max(tmean15),30)

# Set histograms bin width 
areat = (np.diff(mids[0:2])*np.size(tmean15))

# Create the fitter y-values for the normal curve
yfit = areat*(norm.pdf(xfit,np.mean(tmean15),np.std(tmean15)))

plt.figure(figsize=(10,8))

plt.hist(tmean15,facecolor='w',edgecolor="k")

# Plot the normal fit on the histogram
plt.plot(xfit,yfit,color = "blue")

plt.title("Histogram of 1880-2015 Temperature Anomalies")
plt.xlabel("Temperature Anomalies [$\degree$C]")
plt.ylabel("Frequency")

plt.show()

Box Plot

#Fig 3.3
# Boxplot of the global annual mean temperature anomalies from 1880-2015
plt.figure(figsize=(7, 9))
plt.boxplot(tmean15)
plt.ylabel('Temperature Anomalies [$\degree$C]')
plt.show()

Scatter Plot

#Fig 3.4
# Retrieving all necessary data
ust = pd.read_csv('USJantemp1951-2016-nohead.csv'
                  ,header=None)
soi = pd.read_csv('soi-data-nohead.csv'
                  ,header=None)
soid = np.array(soi.iloc[:,1])
soim = np.reshape(soid,(66,12))

# Make the SOI into a matrix with each month as a column 
# Take the first column for January SOI
soij = soim[:,0]

# Take the third column for January US temperature data
ustj = np.array(ust.iloc[:,2])

# Set up figure
plt.figure(figsize=(12,7))

# Create scatter plot
plt.scatter(soij,ustj,color='k')

# Create a linear trendline for the data
lin = np.polyfit(soij, ustj, 1)
linreg = np.poly1d(lin)

plt.plot(soij,linreg(soij),'r')

plt.ylabel('US Temp. Anomalies [$\degree$F]')
plt.xlabel('SOI [dimensionless]')
plt.title("Scatter plot between Jan. SOI and US Temp.", pad = 10)
plt.grid()
plt.show()

Q-Q Plot

#Fig 3.5
# Q-Q plot for the standardized temperature anomalies 
tstand = (tmean15-np.mean(tmean15))/np.std(tmean15)

# Simulate 136 points that follow a norm(0,1) distribution
qn = np.random.normal(0,1,136)

# Sort the points
qns = np.sort(qn)

# Create the Q-Q line
# pp = sm.ProbPlot(qns, fit=True)
qq = sm.qqplot(tstand, line ='45', marker = 'o', markerfacecolor = 'w', markeredgecolor = 'k')

# qq = pp.qqplot(marker='.', markerfacecolor='k', markeredgecolor='k', alpha=0.3)
# sm.qqline(qq.axes[0], line='45', fmt='k--')
# Create the Q-Q plot 
py.scatter(qns, -qns[::-1], color = "purple")

py.title("Q-Q Plot for the Standardized Global Temp. Anomalies vs N(0,1)", size = 20)
py.ylabel("Quantile of Temperature Anomalies", size = 19)
py.xlabel("Quantile of N(0,1)", size = 19)
py.grid()
py.show()

3.3: Probability Distributions

#Fig 3.6
# Display the different cloudiness climates of three cities
labels = ['Las Vegas', 'San Diego', 'Seattle']
clear = [0.58,0.40,0.16]
cloudy = [0.42,0.6,0.84]

# Create the label locations
x = np.arange(len(labels))  
# Set the width of the bars
width = 0.35  

# Prepare figure
fig, ax = plt.subplots(figsize=(12,10))

# Adding data to figure 
rects1 = ax.bar(x - width/2, clear, width, label='Clear')
rects2 = ax.bar(x + width/2, cloudy, width, label='Cloudy')

# Add some text for labels, title and custom x-axis tick labels, etc.
ax.set_ylabel('Probability')
ax.set_title('Probability Distribution of Cloudiness')
ax.set_xticks(x)
ax.set_xticklabels(labels)
ax.legend()


# Creating a function to label each rectange with its respective frequency
def autolabel(rects):
    """Attach a text label above each bar in *rects*, displaying its height."""
    for rect in rects:
        height = rect.get_height()
        ax.annotate('{}'.format(height),
                    xy=(rect.get_x() + rect.get_width() / 2, height),
                    xytext=(0, 3),  # 3 points vertical offset
                    textcoords="offset points",
                    ha='center', va='bottom', fontsize=15)

# Labeling each rectangle 
autolabel(rects1)
autolabel(rects2)

plt.show()

PDF of the Standard Normal Distribution

#Fig 3.7
# Set mu and sigma 
mu = 0
sigma = 1

# Create normal probability function
def intg(x):
    return 1/(sigma * np.sqrt(2 * np.pi)) *np.exp( - (x - mu)**2 / (2 * sigma**2))

x = np.linspace(-3, 3)
y = intg(x)

# Set up figure
fig, ax = plt.subplots(figsize=(12,10))
ax.plot(x, y, 'black', linewidth=1)
ax.set_ylim(bottom=0)

# Set desired quantities
a_ , b_ = -3,-1.5
a, b = -1.2, 3

# Make the shaded region
ix_ = np.linspace(a_,b_)
ix = np.linspace(a, b)

# Using Integrating function
iy_ = intg(ix_)
iy = intg(ix)

# Creating desired vertical separations 
verts_ = [(a_, 0), *zip(ix_, iy_), (b_, 0)]
verts = [(a, 0), *zip(ix, iy), (b, 0)]

# Creating desired polygons
poly_ = Polygon(verts_, facecolor='lightblue', edgecolor='black', linewidth=0.5)
poly = Polygon(verts, facecolor='lightblue', edgecolor='black', linewidth=0.5)

# Adding polygon
ax.add_patch(poly_)
ax.add_patch(poly)

ax.text(-0.4,0.25,"Area = 1", fontsize=20)
ax.text(-1.78, .05, "$f(x)$", fontsize=15)
ax.text(-1.62, .01, "$x$", fontsize=15)
ax.text(-1.45, .08, "$dx$", fontsize=15)
ax.text(-1.2, .01, "$x+dx$", fontsize=15)
ax.text(-0.4, .09, '$\int^{\infty}_{-\infty}f(x)dx=1$', fontsize=15)

# Creating desired arrow
plt.arrow(-2,.2,0.65,-0.1)
ax.text(-2.3,0.2,"$dA=f(x)dx$", fontsize=15, fontweight="bold")

# Label figure
plt.title("PDF of Standard Normal Distribution")
plt.xlabel("Random Variable x")
plt.ylabel("Probability Density")

plt.show()

Normal Distribution

#Fig 3.8
# Creation of five different normal distributions 
bins = np.linspace(-8,8,200)
mu, sigma = 0,1
y1 = 1/(sigma * np.sqrt(2 * np.pi)) *np.exp( - (bins - mu)**2 / (2 * sigma**2) )
mu, sigma = 0,2
y2 = 1/(sigma * np.sqrt(2 * np.pi)) *np.exp( - (bins - mu)**2 / (2 * sigma**2) )
mu, sigma = 0,1/2
y3 = 1/(sigma * np.sqrt(2 * np.pi)) *np.exp( - (bins - mu)**2 / (2 * sigma**2) )
mu, sigma = 3,1
y4 = 1/(sigma * np.sqrt(2 * np.pi)) *np.exp( - (bins - mu)**2 / (2 * sigma**2) )
mu, sigma = -4,1
y5 = 1/(sigma * np.sqrt(2 * np.pi)) *np.exp( - (bins - mu)**2 / (2 * sigma**2) )


df = pd.DataFrame({'x': bins, 'y1': y1, 'y2': y2, 'y3': y3, 'y4': y4, 'y5': y5})

plt.figure(figsize=(12,10))

# Multiple line plot
plt.plot( 'x', 'y1', data=df, marker='', markerfacecolor='red', markersize=8, color='red'
         , linewidth=4, label="$\mu = 0$ and $\sigma = 1$")
plt.plot( 'x', 'y2', data=df, marker='', color='blue', linewidth=2, label="$\mu = 0$ and $\sigma = 2$")
plt.plot( 'x', 'y3', data=df, marker='', color='black', linewidth=2, label="$\mu = 0$ and $\sigma = 1/2$")
plt.plot( 'x', 'y4', data=df, marker='', color='purple', linewidth=2, label="$\mu = 3$ and $\sigma = 1$")
plt.plot( 'x', 'y5', data=df, marker='', color='green', linewidth=2,  label="$\mu = -4$ and $\sigma = 1$")
plt.legend(loc = 'upper right')
plt.title("Normal Distribution N$(\mu,\sigma^2)$")
plt.xlabel("Random Variable X")
plt.ylabel("Probability Density")
plt.grid()

# Verifying the normal equation
mu = 0
sigma = 1
def intg(x):
    return 1/(sigma * np.sqrt(2 * np.pi)) *np.exp( - (x - mu)**2 / (2 * sigma**2) )

i = quad(intg, -2, 2)
print("The integral evaluated to:", round(i[0],6), "and the absolute error is:", i[1])

The integral evaluated to: 0.9545 and the absolute error is: 1.8403560456416157e-11

Student's t-distribution

#Fig 3.9
x = np.linspace(-4,4,200)
# The t function in python does not allow for using infinity as a parameter, so we will use 100 million to 
# simulate the t-distribution with infinity degrees of freedom (dof). 
myinf = 100000000

plt.figure(figsize=(12,10))

# Plot the t-distribution with various degrees of freedom
plt.plot(x, t(3).pdf(x),linewidth=4,color="red", label="$df=3$")
plt.plot(x, t(1).pdf(x),linewidth=2,color="blue", label="$df=1$")
plt.plot(x, t(2).pdf(x),linewidth=2,color="black", label="$df=2$")
plt.plot(x, t(6).pdf(x),linewidth=2,color="purple", label="$df=6$")
plt.plot(x, t(myinf).pdf(x),linewidth=2,color="green", label="$df=\infty$")

# Creating legend
plt.legend()

# Labeling plot 
plt.xlabel("Random Variable t")
plt.ylabel("Probability Density")
plt.title("Student t-distribution T$(t,df)$")
plt.grid()
plt.show()

3.4: Estimate and Its Error

# Confidence interval simulation 
# True mean
mu = 14 

# True standard deviation
sig = 0.3 

# Sample size
n = 50 

d = 1.96*sig/n**(1/2)
lowerlim = mu-d
upperlim = mu+d

# Number of simulations
ksim = 10000  

# Simulation counter
k = 0 

xbar = []
for i in range(ksim):
    x = np.mean(np.random.normal(mu,sig,n))
    xbar.append(x)
    if (x >= lowerlim) & (x <= upperlim):
        k = k+1
        
print("Simulation Counter:", k,"and Total Number of Simulations:", ksim)    
print("Lower CI:", round(lowerlim,2),"and Upper CI:", round(upperlim,2))   

Simulation Counter: 9482 and Total Number of Simulations: 10000
Lower CI: 13.92 and Upper CI: 14.08

#Fig 3.10
# Confidence Interval Simulation 
np.random.seed(19680801)
num_bins = 51

# Create histogram with parameters
fig = plt.subplots(figsize=(12,10))
n, bins,patches = plt.hist(xbar,num_bins, color='blue', alpha=0.5, histtype='bar', ec='black')

# Label histogram
plt.title("Histogram of the Simulated Sample Mean Temperatures")
plt.ylabel("Frequency")
plt.xlabel("Temperature [$\degree$C]")

# Show the plot with extra information
print("95% Confidence Interval (13.92,14.08)")
plt.show()

95% Confidence Interval (13.92,14.08)

#Fig 3.11
# Plot confidence intervals and tail probabilities 

# Set mu and sigma 
mu = 0
sigma = 1

# Create normal probability function
def intg(x):
    return 1/(sigma * np.sqrt(2 * np.pi)) *np.exp( - (x - mu)**2 / (2 * sigma**2) )

# Set desired quantities
a_ , b_ = -3,3
a, b = -1.96, 1.96
a1, b1 = -1.0, 1.0 # integral limits

x = np.linspace(-3, 3)
y = intg(x)

# Set up figure
fig, ax = plt.subplots(figsize=(12,10))
ax.plot(x, y, 'black', linewidth=1)
ax.set_ylim(bottom=0)

# Make the shaded region
ix_ = np.linspace(a_,b_)
ix = np.linspace(a, b)
ix1 = np.linspace(a1, b1)

# Using Integrating function
iy_ = intg(ix_)
iy = intg(ix)
iy1 = intg(ix1)

# Creating desired vertical seperations 
verts_ = [(a_, 0), *zip(ix_, iy_), (b_, 0)]
verts = [(a, 0), *zip(ix, iy), (b, 0)]
verts1 = [(a1, 0), *zip(ix1, iy1), (b1, 0)]

# Creating desired polygons
poly_ = Polygon(verts_, facecolor='red', edgecolor='black', linewidth=0.5)
poly = Polygon(verts, facecolor='lightblue', edgecolor='black')
poly1 = Polygon(verts1, facecolor='white', edgecolor='black')

# Adding polygon
ax.add_patch(poly_)
ax.add_patch(poly)
ax.add_patch(poly1)

# Adding appropriate text
ax.text(0.5 * (a + b), 0.24, "Probability = 0.68",
        horizontalalignment='center', verticalalignment="center", fontsize=15, fontweight="bold")
ax.text(-1.55, .02, "SE", fontsize=15)
ax.text(1.40, .02, "SE", fontsize=15)
ax.text(-0.9, .02, "SE", fontsize=15)
ax.text(0.65, .02, "SE", fontsize=15)
ax.text(-0.05, .02, r'$\bar{x}$', fontsize=15)

# Creating desired arrows
plt.arrow(-2.2,.02,-0.5,0.05)
ax.text(-3,0.08,"Probability \n = 0.025", fontsize=15, fontweight="bold")

# Label the figure
plt.ylabel("Probability Density")
plt.xlabel("True Mean as a Random Variable")
plt.title("Confidence Intervals and Confidence Levels")

ax.set_xticks((a,a1,0,b1,b))
ax.set_xticklabels(('','','','',''))

plt.show()

Confidence interval for NOAAGlobalTemp 1880-2015

# Estimate the mean and error bar for a large sample 
dat1 = pd.read_table('http://shen.sdsu.edu/data/' + 
                        'aravg.mon.land_ocean.90S.90N.v4.0.1.201703.asc.txt', 
                        header = None, delim_whitespace = True)
dat1 = dat1.values
tmean15 = dat1[:,2]
MeanEst = np.mean(tmean15)
sd1 = np.std(tmean15)
StandErr = sd1/(len(tmean15))**(1/2)
ErrorMar = 1.96*StandErr
print("Mean Estimate:", MeanEst, "\n Lower bound:", MeanEst-ErrorMar, "\n Upper bound:", MeanEst+ErrorMar)

Mean Estimate: -0.19504468609593198 
 Lower bound: -0.21093097749081338 
 Upper bound: -0.17915839470105058

Statistical inference for xbar using a z-score

#Fig 3.12
# Setting mu and sigma
mu = 0
sig = 1

# Creating a normal probability density function 
def intg(x):
    return 1/(sig * np.sqrt(2 * np.pi)) *np.exp( - (x - mu)**2 / (2 * sigma**2) )

# Creating desired quantities 
a_ , b_ = -3,-2.4
a, b = -1.96, 3

x = np.linspace(-3, 3)
y = intg(x)

# Creating figure
fig, ax = plt.subplots(figsize=(12,10))
ax.plot(x, y, 'black', linewidth=1)
ax.set_ylim(bottom=0)

# Make the shaded region
ix_ = np.linspace(a_,b_)
ix = np.linspace(a, b)

# Integration between appropriate values
iy_ = intg(ix_)
iy = intg(ix)

# Creating array of vertical separations
verts_ = [(a_, 0), *zip(ix_, iy_), (b_, 0)]
verts = [(a, 0), *zip(ix, iy), (b, 0)]

# Creating shaded polygons
poly_ = Polygon(verts_, facecolor='lightblue', edgecolor='black', linewidth=0.5)
poly = Polygon(verts, facecolor='#A4C9A4', edgecolor='black')

# Adding desired polygons
ax.add_patch(poly_)
ax.add_patch(poly)

# Adding text
ax.text(-0.05, 0.1, "$H_0$ Probability 0.975",
        horizontalalignment='center', verticalalignment="center", fontsize=15)

# Creating appropriately placed arrow
plt.arrow(-2.50,.02,-0.45,0.05)
ax.text(-3,0.08,"p-value", fontsize=15)

# Label plot
plt.ylabel("Probability Density")
plt.xlabel("x: Standard Normal Random Variable")
plt.title("z-score, p-value, and Significance Level")

plt.ylim((0,.5))

# Set various parameters
ax.set_xticks((b_,a))
ax.set_xticklabels(('z','$z_{0.025}$'),fontsize=15)

ax.text(1.15,0.3,"Where z is your z-score \n and $z_{0.025}=-1.96$", fontsize=15)

ax.set_yticks(())
ax.set_yticklabels((""))

ax.axvline(x=-1.96, linewidth=2, color='r')

plt.arrow(-3,0.43,0.9,0, color="lightblue", linewidth=5, head_width=0.002, head_length=0.02)
plt.text(-3,0.46,"$H_1$ region", fontsize=15)
plt.arrow(3,0.43,-4.8,0, color='#A4C9A4', linewidth=5, head_width=0.002, head_length=0.02)
plt.text(1.75,0.46,"$H_0$ region", fontsize=15)

plt.show()

Mean of a Small Sample Size t-Test

# Hypothesis testing for NOAAGlobalTemp 2006-2015
dat1 = pd.read_table('http://shen.sdsu.edu/data/' + 
                        'aravg.mon.land_ocean.90S.90N.v4.0.1.201703.asc.txt', 
                        header = None, delim_whitespace = True)
dat1 = dat1.values
tm0615 = dat1[127:136,2]
MeanEst = np.mean(tm0615)
sd1 = np.std(tm0615)
n = 10
t_score = (MeanEst-0)/ (sd1/np.sqrt(n))
print(" Mean:", MeanEst, "\n","SD:",sd1, "\n","t-score:", t_score)

 Mean: -0.5753087777777778 
 SD: 0.08889553942037536 
 t-score: -20.465437383334567

# Yields the p-value
t.pdf(t_score,df=n-1)

1.5984763823270562e-09

# Yields the critical t-score
t.ppf(1-0.025,n-1)

2.2621571627409915

Comparing Temperatures of Two Short Periods

# Hypothesis testing for Global Temp for 1981-1990 and 1991-2000
dat1 = pd.read_table('http://shen.sdsu.edu/data/' + 
                        'aravg.mon.land_ocean.90S.90N.v4.0.1.201703.asc.txt', 
                        header = None, delim_whitespace = True)
dat1 = dat1.values

# Seperating data into 2 sets
tm8190 = dat1[101:111,2]
tm9100 = dat1[112:121,2]

# Finding mean of data 
barT1 = np.mean(tm8190)
barT2 = np.mean(tm9100)

# Finding standard deviation 
S1sd = np.std(tm8190)
S2sd = np.std(tm9100)

# Setting sample size
n1 = n2 = 10
Spool = np.sqrt((n1-1)*S1sd**2 + (n2-1)*S2sd**2)/(n1+n2-2)

# Yields t-score and boundary conditions for critical value 
T = (barT2-barT1)/(Spool*np.sqrt(1/n1 + 1/n2))
tlow = t.ppf(0.025,n1+n2-2)
tup = t.ppf(1-0.025,n1+n2-2)
print(" t-score:", round(T,5), "\n",
     "tlow =", tlow, "\n",
     "tup=", tup)

 t-score: -15.95129 
 tlow = -2.10092204024096 
 tup= 2.10092204024096

# Yields the p-value
print("p-value:",t.pdf(T,n1+n2-2))

p-value: 2.4257289463517674e-12

# Prints means of the separated data 
print(" 1981-1990 temp =", barT1, "\n", "1991-2000 temp =", barT2)

 1981-1990 temp = -0.2861512 
 1991-2000 temp = -0.42506244444444446

# Yields the difference between 
barT2-barT1

-0.13891124444444447

3.5: Statistical Inference of a Linear Trend

dat1 = pd.read_table('http://shen.sdsu.edu/data/' + 
                        'aravg.mon.land_ocean.90S.90N.v4.0.1.201703.asc.txt', 
          header = None, delim_whitespace = True)
dat1 = dat1.values
tm = dat1[:,2]
x = np.linspace(1880,2015,len(tm))
lm = LinearRegression()
lm.fit(x.reshape(-1, 1), tm)
predictions = lm.predict(timemo.reshape(-1, 1))
print("Coeff. of Linear Model:", lm.coef_)
print("Intercept of Linear Model: ", lm.intercept_)
print("The r-squared of Model: ", lm.score(x.reshape(-1, 1),tm))

Coeff. of Linear Model: [0.00698074]
Intercept of Linear Model:  -13.790040882746423
The r-squared of Model:  0.6848420371033697

---

Chapter 4: Climate Data Matrices and Linear Algebra

---

# import numpy as np 
# import matplotlib.pyplot as plt
# from matplotlib.colors import rgb2hex
# from matplotlib.patches import Polygon
# import matplotlib.mlab as mlab
# import pandas as pd
# # from mpl_toolkits.basemap import Basemap

# from cartopy import crs, mpl
# import matplotlib.ticker as mticker
# from cartopy.mpl.gridliner import LONGITUDE_FORMATTER, LATITUDE_FORMATTER

# from mpl_toolkits import mplot3d
# from netCDF4 import Dataset as ds
# from urllib import request
# import scipy as sp
# from scipy import stats as stt
# from sklearn import datasets, linear_model
# from sklearn.neighbors import KernelDensity as kd
# from sklearn.linear_model import LinearRegression
# from random import randint
# from scipy.stats import norm
# import statsmodels
# import sklearn
# import math
# import statistics
# import sympy as sy
# from sympy import symbols, diff
# import statsmodels.api as sm
# from datetime import date
# from scipy.stats import norm, kurtosis
# from sklearn.linear_model import LinearRegression
# import seaborn as sns
# import pylab as py 
# from scipy.stats import norm, t
# import warnings
# warnings.filterwarnings("ignore")

4.2: Matrix Algebra

# Creating a Matrix 
np.mat([[1,2], [3,4]])

matrix([[1, 2],
        [3, 4]])

# Matrix Subtraction
np.mat([[1,1], [1,-1]]) - np.mat([[1,2], [3,0]])

matrix([[ 0, -1],
        [-2, -1]])

# Matrix Multiplication 
np.mat([[1,1], [1,-1]]) * np.mat([[1,3], [2,4]])

matrix([[ 3,  7],
        [-1, -1]])

# Matrix Multiplication 
np.mat([[1,3], [2,4]]) * np.mat([[1,1], [1,-1]]) 

matrix([[ 4, -2],
        [ 6, -2]])

# Yields matrix inverse 
np.linalg.inv(np.mat([[1,1], [1,-1]]))

matrix([[ 0.5,  0.5],
        [ 0.5, -0.5]])

# Verify the result of inverse
np.mat([[0.5,0.5], [0.5,-0.5]]) * np.mat([[1,1], [1,-1]])

matrix([[1., 0.],
        [0., 1.]])

# Yields the solved linear system 
np.linalg.solve(np.mat([[1,1], [1,-1]]),[20,4])

array([12.,  8.])

# Creating Matrix A
A = np.mat([[1,2,3], [-1,0,1]])
A

matrix([[ 1,  2,  3],
        [-1,  0,  1]])

# Yields the Covariance Matrix of A
covm = (1/A.shape[1])*A*A.T
covm

matrix([[4.66666667, 0.66666667],
        [0.66666667, 0.66666667]])

# Initializing arbitrary vector u
u = np.mat([[1,-1]]).T

# Yields new vector v in a different direction
v = covm*u
v

matrix([[4.],
        [0.]])

# Yields eigenvalues and eigenvectors in one output
ew = np.linalg.eig(covm)

# Yields the eigenvalues of covm 
ew[0] 

array([4.77485177, 0.55848156])

# Yields the eigenvectors
ew[1] 

matrix([[ 0.98708746, -0.16018224],
        [ 0.16018224,  0.98708746]])

# Verify the eigenvalue and eigenvectors 
covm * ew[1] / ew[0]
# The first column corresponds to the first eigenvalue

matrix([[ 0.98708746, -0.16018224],
        [ 0.16018224,  0.98708746]])

4.5: SVD Representation Model for Space-Time Data

# Develop a 2-by-3 space time data matrix for SVD
A = np.mat([[1,2,3],[-1,0,1]])
A

matrix([[ 1,  2,  3],
        [-1,  0,  1]])

# Perform SVD calculation
msvd = np.linalg.svd(A)

# Yields the D vector in SVD 
d = msvd[1]
d

array([3.78477943, 1.29438969])

# Yields the U matrix in SVD 
u = msvd[0].T
u

matrix([[-0.98708746, -0.16018224],
        [-0.16018224,  0.98708746]])

# Yields the V matrix of SVD
v = msvd[2]
v = np.delete(v,2,0).T
v

matrix([[-0.21848175, -0.88634026],
        [-0.52160897, -0.24750235],
        [-0.8247362 ,  0.39133557]])

# Verify that A = UDV', where V' is the transpose of V.
# Yields Diagonal of D
diagd = np.diag(d)

# Yields the Transpose of V
vT = v.T

# Verify the SVD result 
verim = u*diagd*vT
np.matrix.round(verim)
# This is the original matrix A 

matrix([[ 1.,  2.,  3.],
        [-1.,  0.,  1.]])

# Yields the covariance of matrix A
covm = (1/A.shape[1])*A*A.T

# Yields the eigenvectors and eigenvalues of matrix A
eigcov = np.linalg.eig(covm)

# The first element of eigcov are the eigenvalues of matrix A
eigcov[0]

array([4.77485177, 0.55848156])

# Yields the eigenvectors of matrix A
eigcov[1]

matrix([[ 0.98708746, -0.16018224],
        [ 0.16018224,  0.98708746]])

# Verify eigenvalues with SVD 
((d**2)/A.shape[1])

array([4.77485177, 0.55848156])

# Yields true eigenvalues of matrix A
eigcov[0]

array([4.77485177, 0.55848156])

4.6: SVD Analysis of Southern Oscillation Index

Download pressure data for Tahiti from https://shen.sdsu.edu/data/PSTANDtahiti into your directory

Plot the SOI and the standardized SLP data at Darwin and Tahiti

# Set up File path
file = 'PSTANDtahiti.txt'
Pta = np.loadtxt(file,skiprows=0)
PTA = np.reshape(Pta[:,1:],(780))

# Make an array of years for our x-axis
xtime = np.linspace(1951,2016,(65*12)+1)[0:780]

#Fig 4.2 (a)
# Set up Figure 
plt.figure(figsize = (15.,7.))

# Add data to plot
plt.plot(xtime,PTA,color='r')

# Add text to plot
plt.text(1950,2.7,'(a)', size = 20, alpha = 0.7)

# Label plot
plt.xlabel('Year')
plt.ylabel('Pressure')
plt.title('Standardized Tahiti SLP Anomalies')
plt.grid()
plt.show()

Download pressure data for Tahiti from http://shen.sdsu.edu/data/PSTANDdarwin.txt into your directory

# Upload and set up data 
Pda = np.loadtxt('PSTANDdarwin.txt',skiprows=0)
PDA = np.reshape(Pda[:,1:],(780))

#Fig 4.2 (b)
plt.figure(figsize = (15.,7.))

plt.plot(xtime,PDA,color='b')
plt.text(1950, 2.7, '(b)', size = 20, alpha = 0.7)

plt.xlabel('Year')
plt.ylabel('Pressure')
plt.title('Standardized Darwin SLP Anomalies')
plt.grid()

plt.show()

Plot the SOI index

#Fig 4.2 (c)
# Set up figure 
fig, soi = plt.subplots(figsize=(15,7))

# Ploting data against time
soi.plot(xtime,PTA-PDA, color = 'k')
soi.text(1950, 4.11, '(c)', size = 20, alpha = 0.7)
soi.axhline(y = 0.0, color = 'k', linestyle = '-')
soi.set_xlabel('Year')
soi.set_ylabel('SOI Index')
soi.grid()

# Making left Y axis 
ax2 = soi.twiny()

# Plotting data against difference of Darwin and Tahiti standardized SLP data
ax2.plot(xtime,PTA-PDA, color='k')
ax2.set_xlabel('Year')

# Mirroring X axis for difference data 
ax3 = soi.twinx()
ax3.plot(xtime,PTA-PDA, color='k')
ax3.set_ylabel('SOI Index')

# Adding some text
ax3.text(1960,4.6,"SOI = Tahiti Standardized SLP - Darwin Standardized SLP", fontsize=18)

plt.show()

CSOI and AMO time series comparison

#Fig 4.2 (d)
# Preparing variables
SOI = PTA-PDA
cnegsoi = -SOI.cumsum()

# Load AMO data
amodat = np.loadtxt('AMO1951-2015.txt',skiprows=0)
amots = amodat[:,1:13].T.flatten("F")

# Set up figures 
fig, ax1 = plt.subplots(figsize=(15,7))
fig.text(0.123, 0.22, '(d)', size = 20, alpha = 0.7)

# Plot the data against CSOI Index 
ax1.set_xlabel('Year')
ax1.set_ylabel('Negative CSOI Index')
ax1.plot(xtime, cnegsoi, color="purple", label="CSOI")
ax1.grid()

# Create a second, vertical axis that shares the same x-axis
ax2 = ax1.twinx()  

# Set labels for new axis 
ax2.set_ylabel('AMO Index')  
ax2.plot(xtime, amots, color="darkgreen", label="AMO Index")

# Set legends 
# ax1.legend(loc = (.169,.89))
# ax2.legend(loc = (.169,.84))
ax1.legend(loc = (.125,.13))
ax2.legend(loc = (.125,.08))

# Yields the full view of the right y-label
fig.tight_layout()
plt.title('CSOI and AMO Index Comparison', pad = 8)
plt.show()

Weighed SOI Computed by the SVD Method

# Yields transpose of the data
ptamonv = PTA.T
pdamonv = PDA.T

# Creates an array of arrays 
ptada = [ptamonv,pdamonv]

# Compute SVD of ptada
svdptd_u, svdptd_s, svdptd_vh = np.linalg.svd(ptada,full_matrices=False)

# Verify SVD 
recontada = np.dot(np.dot(svdptd_u,np.diag(svdptd_s)),svdptd_vh)

# Compute error of calculations
svd_error = ptada - recontada

# Error maximum
svd_error.max()

4.218847493575595e-15

# Error minimum
svd_error.min()

-7.105427357601002e-15

# Gather appropriate variables
Dd = np.diag(svdptd_s)
Uu = svdptd_u
Vv = svdptd_vh

# Set up time array
xtime = np.linspace(1951,2016,(65*12)+1)[0:780]

# Compute WSOI1 for plotting 
wsoi1 = Dd[0,0] * (Vv)[0,:]

Plot WSOI1

#Fig 4.3 (a)
# Prepare figure
fig, pltt = plt.subplots(figsize = (15,7))

# Plot data 
pltt.plot(xtime,wsoi1,color='k')
pltt.set_xlabel('Year')
pltt.set_ylabel('Weighted SOI1')
pltt.text(1980.2, -2.9, "Weighted SOI1 Index", fontsize=20)
pltt.text(1950.2, 3.7, "(a)", size = 16, alpha = 0.7)
pltt.grid()

Plot WSOI2

#Fig 4.3 (b)
# Set up data for WSOI2
wsoi2 = Dd[1,1] * Vv[1,:]

# Prepare figure
fig, wsoi = plt.subplots(figsize = (15,7))

# Plot data 
wsoi.plot(xtime,wsoi2,color='darkblue')
wsoi.set_xlabel('Year')
wsoi.set_ylabel('WSOI2')
wsoi.text(1950.2, 1.8, "(b)", size = 16, alpha = 0.7)
wsoi.grid()

# Create second, horizontal axis
ax2 = wsoi.twiny()
ax2.plot(xtime,wsoi2,color='darkblue')
ax2.text(1970.2,1.76,"Weighted SOI2 Index", fontsize=20)

plt.show()

Plot comparison between CWSOI1 and the smoothed AMO index

#Fig 4.3(c)
# Create needed variables 
cwsoi1 = np.cumsum(wsoi1)
xtime = np.linspace(1951,2016,(65*12)+1)[0:780]

# Prepare figure 
fig, plott = plt.subplots(figsize=(15,7))

# Plot cumulative sums 
plott.plot(xtime,cwsoi1,color='black',label='CWSOI1')
plott.set_xlabel('Year')
plott.set_ylabel('Cumulative Weighted SOI1', size = 20)
plott.grid()
plott.text(1950.2, -172.5, "(c)", size = 20, alpha = 0.7)

# Superimpose CSOI time series on this CWSOI1
cnegsoi = -np.cumsum(ptamonv-pdamonv)
plt.plot(xtime,cnegsoi,color='purple',label='CSOI')
plt.title('Comparison between CWSOI1 and the Smoothed AMO Index', size = 22)

ax2 = plott.twinx()

# Create 24-month ahead moving average of the monthly AMO index
i = 0
window_size = 24
moving_averages = []
while i < len(amots) - window_size + 1:
    this_window = amots[i : i + window_size]
    window_average = sum(this_window) / window_size
    moving_averages.append(window_average)
    i += 1

# Creating time sequence spanning from 1951 to 2016
xtime = np.linspace(1951,2016,(65*12)+1)[0:757]

# Plot the moving averages 
ax2.plot(xtime, moving_averages,color='darkgreen',label='AMO Index')
ax2.set_ylabel('Smoothed AMO Index')

# Set legend 
plott.legend(loc=(.17,.835))
ax2.legend(loc=(.17,.785))

# Ensures we can see all axis labels 
fig.tight_layout() 
plt.show()

Plot the Cumulative WSOI2: CWSOI2

# Create appropriate variables from SVD 
uu, ss, vvh = np.linalg.svd(ptada,full_matrices=False)
PC = vvh
PC1 = PC[0,:]
PC2 = PC[1,:]

#Fig 4.3 (d)
# Prepare figure 
plt.figure(figsize = (15,7))

# Prepare time sequence 
xtime = np.linspace(1951,2016,(65*12)+1)[0:780]

# Plot the cumulative sum
plt.plot(xtime,PC2.cumsum(),color='b',label='Principal Component 2')

# Label appropriately 
plt.xlabel('Year')
plt.ylabel('CWSOI2')
plt.title('CWSOI2 Index: The Cumulative PC2')
plt.text(1950.1, -5.9, '(d)', size = 16, alpha = 0.7)
plt.grid()
plt.show()

Visualization of the ENSO Mode Computed from SVD Method

#Fig 4.4 (a)
# Display the two ENSO modes on a world map


# # Prepare the figure 
# plt.figure(figsize=(15., 7.))

# # Using the basemap package to put the cylindrical projection map into the image
# wmap2 = Basemap(projection='cyl',llcrnrlat=-90.,urcrnrlat=90.,\
#             llcrnrlon=0.,urcrnrlon=360.,resolution='c')

# # Draw coastlines, latitudes, and longitudes on the map
# wmap2.drawcoastlines(color='black', linewidth=1)
# wmap2.drawparallels(np.arange(-90.,91.,30.), labels = [1,0,0,1], fontsize=15)
# wmap2.drawmeridians(np.arange(00.,360.,30.), labels= [1,0,0,1], fontsize=15)

# # Add data to plot
# plt.scatter(131.,-12.,color='b',s=150)
# plt.scatter(220.,-20,color='r',s=150)

# # Add desired text to plot 
# plt.text(100,-24, 'Darwin -0.79', color = 'b', fontsize = 22)
# plt.text(200,-32, 'Tahiti 0.61', color = 'r', fontsize = 22)
# plt.text(60,40, 'El Nino Southern Oscillation Model 1', color = 'm', fontsize = 30)

# plt.show()


# Prepare the Figure
plt.figure(figsize = (15., 7.))

# Using the CartoPy package to put the cylindrical projection map into the image
ax = plt.axes(projection = crs.PlateCarree())
ax.set_extent([-180, 180, -90, 90], crs = crs.PlateCarree())

# Draw coastlines, gridlines, longitudes, and latitudes on the map
ax.coastlines(color='black', linewidth=1)
gl = ax.gridlines(crs=crs.PlateCarree(), draw_labels=True,
                  linewidth=2, color='gray', alpha=0.5, linestyle='--')
gl.xlabels_top = True
gl.ylabels_left = True
gl.xlines = True
gridx = np.linspace(-180, 180, 13)
gl.xlocator = mticker.FixedLocator(gridx)
gl.xformatter = LONGITUDE_FORMATTER
gl.yformatter = LATITUDE_FORMATTER

# Add data to plot
plt.scatter(131.,-12.,color='b',s=150)
plt.scatter(-129.,-30,color='r',s=150)

# Add descriptive text to plot
plt.text(100,-24, 'Darwin -0.79', color = 'b', fontsize = 22)
plt.text(-155,-42, 'Tahiti 0.61', color = 'r', fontsize = 22)
plt.text(-130,40, 'El Nino Southern Oscillation Model 1', color = 'm', fontsize = 30)

Text(-130, 40, 'El Nino Southern Oscillation Model 1')

#Fig 4.4 (b)
# Display the two ENSO modes on a world map
# Prepare figure
plt.figure(figsize=(15, 7))

# Using the basemap package to put the cylindrical projection map into the image
wmap2 = Basemap(projection='cyl',llcrnrlat=-90.,urcrnrlat=90.,\
            llcrnrlon=0.,urcrnrlon=360.,resolution='c')

# Draw coastlines, latitudes, and longitudes on the map
wmap2.drawcoastlines(color='black', linewidth=1)
wmap2.drawparallels(np.arange(-90.,91.,30.), labels = [1,0,0,1], fontsize=15)
wmap2.drawmeridians(np.arange(-180.,181.,30.), labels= [1,0,0,1], fontsize=15)

# Add data to plot 
plt.scatter(131.,-12.,color='r',s=150)
plt.scatter(220.,-20,color='r',s=150)

# Add desired text 
plt.text(105,-24, 'Darwin 0.61', color = 'r', fontsize = 22)
plt.text(200,-32, 'Tahiti 0.79', color = 'r', fontsize = 22)
plt.text(60,40, 'El Nino Southern Oscillation Model 2', color = 'm', fontsize = 30)

plt.show()

4.8: Multivariate Linear Regression Using Matrix Notations

Example 4.9

# Given the coordinates of 3 points 
x1 = [1,2,3]
x2 = [2,1,3]
y = [-1,2,1]

# Put them into a dataframe 
df = pd.DataFrame([x1,x2,y]).T
X_df = pd.DataFrame([x1,x2]).T

# Create the linear regression 
lm = LinearRegression()

# Fit the linear regression 
lm.fit(X_df, df[2])

# Display Coefficients 
lm.coef_

array([ 1.66666667, -1.33333333])

# Verify the 3 points determining a plane 
b1 = [i*1.667 for i in x1]
b2 = [j*1.333 for j in x2]
difference = []

zip_object = zip(b1, b2)
for b1_i, b2_i in zip_object:
    difference.append(b1_i-b2_i)
difference

[-0.9989999999999999, 2.0010000000000003, 1.0020000000000007]

Example 4.10

u = [1,2,3,1]
v = [2,4,3,-1]
w = [1,-2,3,4]

df = pd.DataFrame([u,v,w]).T
X_df = pd.DataFrame([u,v]).T
lm = LinearRegression()
lm.fit(X_df, df[2])
lm.coef_

array([ 2. , -1.5])

lm.intercept_

0.9999999999999982

Example 4.11

# Show a general multivariate linear regression with 3 independent random variables
dat = np.mat([np.random.normal(0, 1, 4),np.random.normal(0, 1, 4),
                np.random.normal(0, 1, 4),np.random.normal(0, 1, 4),
                np.random.normal(0, 1, 4),np.random.normal(0, 1, 4),
                np.random.normal(0, 1, 4),np.random.normal(0, 1, 4),
                np.random.normal(0, 1, 4),np.random.normal(0, 1, 4)])
fdat = pd.DataFrame(dat)
lm = LinearRegression()
lm.fit(fdat.iloc[:,0:3], fdat.iloc[:,3])
lm.coef_

array([0.58955312, 0.22995714, 0.62614479])

lm.intercept_

-0.14682354563605512

---

Chapter 5: Energy Balance Models for Climate

---

import numpy as np 
import matplotlib
from matplotlib import cm
import matplotlib.pyplot as plt
from matplotlib.colors import rgb2hex
from matplotlib.patches import Polygon
import matplotlib.mlab as mlab
import pandas as pd
# from mpl_toolkits.basemap import Basemap
from mpl_toolkits import mplot3d
from netCDF4 import Dataset as ds
from urllib import request
import scipy as sp
from scipy import stats as stt
from sklearn import datasets, linear_model
from sklearn.neighbors import KernelDensity as kd
from sklearn.linear_model import LinearRegression
from random import randint
from scipy.stats import norm
import statsmodels
import sklearn
import math
import statistics
import sympy as sy
from sympy import symbols, diff
import statsmodels.api as sm
from datetime import date
from scipy.stats import norm, kurtosis
from sklearn.linear_model import LinearRegression
import seaborn as sns
import statsmodels.api as sm 
import pylab as py 
from scipy.stats import norm, t
import warnings
import os
warnings.filterwarnings("ignore")

---

5.1: EBM for Modeling the Moon's Surface Temperature

---

# NASA Diviner Data Source:
# http://pds-geosciences.wustl.edu/lro/lro-l-dlre-4-rdr-v1/lrodlr_1001/data/gcp
d19 = pd.read_csv("tbol_snapshot.pbin4d-19.out-180-0.txt", sep='\s+', header=None)
m19 = np.array(d19.iloc[:,2]).reshape(720,360)

# Set up latitude and longitude sequences 
lat1 = np.linspace(-89.75, 89.75, 360)
lon1 = np.linspace(-189.75, 169.75, 720)

# Created desired color sequence
colors = ['skyblue', 'green', 'darkgreen', 'slateblue', 'indigo',
          'darkviolet', 'yellow', 'orange', 'pink', 'red', 'maroon', 'purple', 'black']
myColMap = newColMap(colors)

# Prepare data
mapmat = m19.T 
mapmat[mapmat == mapmat.max()] = 400
mapmat[mapmat == mapmat.min()] = 0

# Create levels for contour map
clev = np.linspace(mapmat.min(), mapmat.max(), 60)

#Fig 5.2 (top panel)
plt.figure(figsize=(14,6))
contf = plt.contourf(lon1, lat1, mapmat, clev, cmap=myColMap)
colbar = plt.colorbar(contf, drawedges=True, ticks=[0, 100, 200, 300, 400], aspect = 10)
plt.title("Moon Surface Temperature Observed by \n NASA Diviner, Snapshot 19")
plt.xlabel("Longitude")
plt.ylabel("Latitude")
plt.text(195, 95, "K", size=20)
plt.grid();

plt.show()

#Fig 5.2 (middle panel)
# Plot the equator temperature for a snapshot 
equ = np.where((lat1 < .5) & (lat1 > -.5))[0]
equatorial = mapmat[equ[0]]
merid = np.where((lon1 < 90.5) & (lon1 > 89.5))[0]
meridian = mapmat[:, merid[0]]

ax = plt.figure()
plt.plot(lon1, equatorial, 'r-')
plt.text(-125, 250, "Nighttime", size=20)
plt.text(50, 250, "Daytime", size=20, color='orange')
plt.xlabel("Longitude")
plt.ylabel("Temperature [K]")
plt.title("Moon's Equatorial Temperature at Snapshot 19");


plt.show()

#Fig 5.2 (bottom panel)
# Plot the noon time meridional temperature for a snapshot
ax = plt.figure()
plt.plot(lat1, meridian, 'r-')
plt.xlabel("Latitude")
plt.ylabel("Temperature [K]")
plt.title("Moon's Noon Time Meridional Temperature\nat Snapshot 19");

plt.show()

# Compute the bright side average temperature
bt = np.array(d19.iloc[129600:].astype(np.float))
type(bt)
aw = np.cos(bt[:,1]*np.pi/180)
wbt = bt[:,2]*aw
bta = np.sum(wbt)/np.sum(aw)
print("Average temperature of bright side of moon:", round(bta,3), "Kelvin")

Average temperature of bright side of moon: 302.765 Kelvin

# Compute the dark side average temperature
dt = np.array(d19.iloc[:129600].astype(np.float))
aw = np.cos(dt[:,1]*np.pi/180)
wdt = dt[:,2]*aw
dta = np.sum(wdt)/np.sum(aw)
print("Average temperature of dark side of moon:", round(dta,3), "Kelvin")

Average temperature of dark side of moon: 99.557 Kelvin

5.1.3 EBM Prediction for the Moon Surface Temperature

from scipy import optimize as opt

# Making a function to numerically solve an EBM
def ebm_maker(a, s, l, e, si, ka, t0, he):
    """
    This function accepts a set of Energy Balance Model (EBM) parameters
    and returns a lambda function with those parameters as a function
    of t.
    """
    return lambda t: (1-a)*s*np.cos(l) - e*si*(t**4) - ka*(t - t0)/he

# Equator noon temperature of the moon from an EBM
lat = 0*np.pi/180
sigma = 5.670367e-8
alpha = 0.12
S = 1368
ep = 0.98
k = 7.4e-4
h = 0.4
T0 = 260


fEBM = ebm_maker(alpha, S, lat, ep, sigma, k, T0, h)

# Numerically solve the EBM: fEBM = 0 
res = opt.root(fEBM, x0 = 400)
x0 = res['x'][0]
x0

383.629726096338

---

5.3 EBM for the Global Average Surface Temperature of an Earth with a Nonlinear Albedo Feedback

---

# Define a piecewise albedo function 
a1 = 0.7
a2 = 1 - a1
T1 = 250
T2 = 280
a = []

def ab(T):
    try:
        for i in T:
            if i < T1:
                a.append(a1)
            else:
                if i < T2:
                    a.append(((a1-a2)/(T1-T2))*(i-T2) + a2)
                else:
                    a.append(a2)
        return a
    
    except TypeError:
        if T < T1:
                a.append(a1)
        else:
            if T < T2:
                a.append(((a1-a2)/(T1-T2))*(T-T2) + a2)
            else:
                a.append(a2)
        return a
        
# Define the range of temperatures 
t = np.linspace(200, 350, 1001)

#Fig 5.5
# Plot the albedo as a nonlinear function of T
a = []

plt.figure(figsize = (12,9))
plt.plot(t, ab(t), 'k-')
plt.ylim(0,1)
plt.xlabel("Surface Temperature [K]")
plt.ylabel("Albedo");
plt.title("Nonlinear Albedo Feedback")
plt.grid()
plt.show()

#Fig 5.6
# Formulate and solve an EBM
a = []
S = 1368
ep = 0.6
sg = 5.670373e-8
y1 = [(1-i)*(S/4.0) for i in ab(t)]
y2 = ep*sg*(t**4)
y3 = np.zeros(t.size)

plt.figure(figsize=(12,8))
plt.plot(t, y1, 'r-', label="Incoming Energy")
plt.plot(t, y2, 'b-', label="Outgoing Energy")
plt.plot(t, y2 - y1, 'k-', label="Difference between outgoing and incoming energy")
plt.ylim(-10,300)
plt.axhline(y=0, alpha=.7, color='m')
plt.grid()

# Label the plot
plt.title("Simple Nonlinear EBM with Albedo Feedback: $E_{out}$ = $E_{in}$", fontsize = 18, pad = 10)
plt.xlabel("Surface Temperature T[K]")
plt.ylabel("Energy [Wm$^{-2}$]")

plt.text(288,275,"$E_{out}$", fontsize=20, color="b")
plt.text(310,248,"$E_{in}$", fontsize=20, color="r")
plt.text(290,100,"$E_{out} - E_{in}$", fontsize=20, color="k")

plt.text(230,7, "T1", fontsize=16)
plt.text(263,7, "T2", fontsize=16)
plt.text(285,7, "T3", fontsize=16)

plt.scatter(234,0,s=30, c = "black")
plt.scatter(263,0,s=30, c = "black")
plt.scatter(290,0,s=30, c = "black")

plt.show()

# Verify the roots
S = 1368
ep = 0.6
sg = 5.670373e-8
roots = []

# Redefine the albedo smooth function approximation using lambda variable 
ab_smooth = lambda t: 0.5 - 0.2 *np.tanh((t-265)/10)

# Redefine the first albedo function in simpler terms
a1 = 0.7
a2 = 0.3
T1 = 250
T2 = 280
def ab(T):
    if T <= 250:
        a = a1
    elif 250 < T < 280:
        a = ((a1-a2)/(T1-T2))*(T-T2) + a2
    else:
        a = a2
    return a

# Define function f(T)
def f(T):
    try:
        roots = ep*sg*T**4 - (1 - ab(T))*(S/4.0)
        
    except TypeError:
        if len(ab(T)) > 1: # In case there is more than one root
            for i in ab(T):
                r = ep*sg*i**4 - (1 - i)*(S/4.0)
                roots.append(r)
        else: # In case there exists one root
            roots = ep*sg*i**4 - (1 - ab(T)[0])*(S/4.0)
    return roots

# Find roots of Albedo function 
opt.bisect(f, 260, 275)

263.4303178655142

opt.bisect(f, 275, 295)

289.62784771008387

opt.bisect(f, 220, 240)

234.33983543986528

---

Chapter 6: Calculus Applications to Climate Science: Derivatives

---

import numpy as np 
import matplotlib
from matplotlib import cm
import matplotlib.pyplot as plt
import pandas as pd
import warnings
import os
warnings.filterwarnings("ignore")
from scipy import optimize as opt
import sympy as sm
import math as m

---

6.1: Stefan-Boltzmann Law and Budyko's Approximation

---

#Fig 6.1
# Cess' Budyko Parameters 
A = 212
B = 1.6 
ep = 0.6 
sg = 5.670367e-8
T = np.linspace(-50,50,10*100)
S = 1365
alf = 0.3

# Plot the Data 
plt.plot(T, A+B*T, "k-", linewidth=3, label="Budyko Radiation Formula")
plt.plot(T, ep*sg*(273.15+T)**4, "b-", linewidth=3, label="Stefan-Boltzmann Radiation Law")
plt.plot(T, 189.4+2.77*T, "m-", linewidth=3, label="Linear Approximation of Stefan-Boltzmann Law")
plt.plot(T, (1-alf)*S/4*np.repeat(1,len(T)), "r-", linewidth=3, label="Observed Outgoing Radiation")

# Add line Labels 
plt.text(-49,180,"Budyko Formula",size=18, color="k")
plt.text(8,350,"Stefan-Boltzmann Law",size=18, color="b")
plt.text(-32,80,"Linear Approximation of Stefan-Boltzmann Law",size=18, color="m")
plt.text(-40,250,"E$_{out}=239$[Wm$^{-2}$]",size=18, color="r")

# Modify Plot parameters
plt.title("Radiation Law and Its Approximations", pad = 10)
plt.xlabel("Temperature T[$\degree$C]")
plt.ylabel("Outgoing Radiation: E$_{out}$ [Wm$^{-2}$]")
plt.ylim(0,500)
plt.legend(loc='upper left', fontsize="x-large")
# plt.grid()
plt.show()

---

6.2: Linear Approximation

---

Linear approximation of $$f(x)=(1+x)^4$$ by $$L(x)=1+4x$$

# Produce table 6.1 
x = np.linspace(-0.3,0.3,7)
fx = list(range(1,8))
lx = list(range(1,8))
ex = list(range(1,8))
for i in range(7):
    fx[i] = (1+x[i])**4
    lx[i] = 1+4*x[i]
    ex[i] = round(((1+x[i])**4 - (1+4*x[i]))/((1+x[i])**4)*100, 1)

pd.set_option('display.max_columns', None)

my_tbl = pd.DataFrame({"x":x,"f(x)":fx, "L(x)":lx, "Error[%]":ex})

my_tbl

	x	f(x)	L(x)	Error[%]
0	-0.3	0.2401	-0.2	183.3
1	-0.2	0.4096	0.2	51.2
2	-0.1	0.6561	0.6	8.6
3	0.0	1.0000	1.0	0.0
4	0.1	1.4641	1.4	4.4
5	0.2	2.0736	1.8	13.2
6	0.3	2.8561	2.2	23.0

---

6.3: Bisection Method for Solving Nonlinear Equations

---

def func(x):
    return x**3-x-1

opt.bisect(func, 0, 2)

1.324717957244502

# Example 6.2
def func2(x):
    return (1+x)**4 - (2+x)

opt.bisect(func2, 0, 2)

0.22074408460503037

# Example 6.3
S = 1368
ep = 0.6 
sg = 5.670367e-8
def f2(T):
    return ep*sg*T**4 - (1 - (0.5 - 0.2*np.tanh((T-265)/10))) * (S/4)

opt.bisect(f2, 270, 300)

289.30980607804315

---

6.4: Newton's Method

---

# Create Newton Function
def newton(f, tol=1e-12, x0=1,N=20):
    h = 0.001
    i = 1
    x1 = x0
    p = []
    while i <= N:
        df_dx = (f(x0+h)-f(x0))/h
        x1 = (x0 - (f(x0)/df_dx))
        p.append(x1)
        i = i + 1
        if abs(x1-x0) < tol:
            break
        x0 = x1
    return p[:i-1]

# Create test function for input in Newton Function
def f(x):
    return x**3 + 4*x**2 - 10 

root = newton(f, tol=1e-12, x0=1,N=10)
root

[1.454256341032464,
 1.368917103772529,
 1.3652384598365683,
 1.3652300175883723,
 1.3652300134161426,
 1.3652300134140978,
 1.3652300134140969]

#Fig 6.2
# Illustration of Newton's Method
x = np.linspace(0.2,1.7,30)
def f(x): return x**3 + 4*x**2 - 10 
def g(x): return 3*x**2 + 8*x

x0 = 1
x1 = 1.4543
x2 = 1.3689

fig, ax = plt.subplots(frameon=False)

# Plot the data
plt.plot(x,f(x),"k-")
plt.plot(np.linspace(0,1.75,100),np.zeros(100), "k-")
plt.plot(x, f(x0)+g(x0)*(x-x0),"r--")
plt.plot(x, f(x1)+g(x1)*(x-x1),"r--")

plt.scatter(1,0,s=30,c="k")
plt.scatter(x1,0,s=30,c="k")
plt.scatter(x2,0,s=30,c="k")
plt.scatter(x1,f(x1), s=30, c="r")
plt.scatter(x0,f(x0), s=30, c="r")

# Creating Appropriate tick marks on X-Axis
plt.text(0.5,0,"|")
plt.text(0.47,-0.6,"0.5", fontsize=18)
plt.text(0.96,-0.6,"1.0", fontsize=18)
plt.text(1.5,0,"|")
plt.text(1.47,-0.6,"1.5", fontsize=18)
plt.text(1.7,0,"|")
plt.text(1.67,-0.6,"1.7", fontsize=18)

# Add line Labels and text
plt.text(0.25,-7.6,"$y=x^3+4x^2-10$",size=18, color="k")
plt.text(0.97,0.23,"$x_0$",size=18, color="b")
plt.text(0.87,.9,"Initial guess",size=16, color="b")
plt.text(x1-0.05,0.25,"$x_1$",size=18, color="b")
plt.text(x2-0.05,0.25,"$x_2$",size=18, color="b")
plt.text(0.35,5.2,"Follow the blue arrows from the initial guess",size=18, color="b")
plt.text(0.60,4.5,"$x_0 \Rightarrow P_0 \Rightarrow x_1 \Rightarrow P_1 \Rightarrow x_2 \Rightarrow ...$",
         size=18, color="b")
plt.text(x1+0.01,f(x1)-0.3, "$P_1$", color="r",size=18)
plt.text(0.4, -11.2, "Tangent line \n through $P_0$", color="r",size=15)
plt.text(x0+0.01,f(x0)-0.3, "$P_0$", color="r",size=18)
plt.text(1.05, -7, "Tangent line \n through $P_1$", color="r",size=15)

# Add Descriptive arrows
plt.arrow(1,0,0,f(x0)+0.3,head_width=0.02, head_length=0.2,color="b")
plt.arrow(1,f(x0),0.42,4.6,head_width=0.02, head_length=0.2,color="b")
plt.arrow(x1,0,0,f(x1)-0.3,head_width=0.02, head_length=0.2,color="b")
plt.arrow(x1,f(x1),-0.07,-f(x1)+0.3,head_width=0.02, head_length=0.2,color="b")


# Modify Plot parameters
plt.title("Newton's method for finding a root:")
plt.xlim(0,1.75)
plt.ylim(-10,6)
ax.spines['top'].set_visible(False)
ax.spines['bottom'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.get_xaxis().set_ticks([])

plt.show()

def f(x): return (x+1)**4 - (2+x)
root = newton(f, N=10)
root

[0.5809696800109538,
 0.3335992109385367,
 0.2359959176632994,
 0.22108766698749036,
 0.22074474123732507,
 0.22074408554093847,
 0.22074408460709063,
 0.22074408460576125,
 0.22074408460575942]

# Define Parameters 
S = 1365
ep = 0.6
sg = 5.670367e-8

# Define the function for energy
def f3(T): return ep*sg*T**4 - (1 - (0.5 - 0.2*np.tanh((T-265)/10))) * (S/4)

# Compute Solutions 
root1 = newton(f3, x0=220)
root1

[235.69658970294506,
 234.38606171928143,
 234.3817655891268,
 234.38176551811753,
 234.3817655181173]

# Compute Solutions 
root2 = newton(f3, x0=270)
root2

[262.05664244651985,
 264.50704124959947,
 264.33780214259644,
 264.3376470816254,
 264.3376470803433,
 264.3376470803433]

# Compute Solutions 
root3 = newton(f3, x0=300)
root3

[289.9086198038602,
 289.14701243927755,
 289.1401590537761,
 289.1401583849815,
 289.1401583849732,
 289.1401583849732]

# Compute Solutions 
root5 = newton(f3, x0=100)
root5

[827.2552143021114,
 623.5422659266504,
 474.89725763077905,
 372.56214756821834,
 313.36499589662805,
 292.05532787372726,
 289.2231588691877,
 289.1402443882009,
 289.14015838613426,
 289.1401583849732,
 289.1401583849732]

---

6.5: Examples of Higher-Order Derivatives

---

x = symbols("x")
diff(x**2)

$\displaystyle 2 x$

x = symbols("x")
def e(i):
    return math.e**i

diff(e(-x**2),x)

$\displaystyle - 2.0 \cdot 2.71828182845905^{- x^{2}} x$

# Higher order derivatives
x = symbols("x")
g = symbols("g")
t = symbols("t")
v0 = symbols("v0")
h0 = symbols("h0")

# First Derivative
fst_dev = diff(-g*t**2/2+v0*t+h0,t)
fst_dev

$\displaystyle - g t + v_{0}$

# Second Derivative
sec_dev = diff(fst_dev,t)
sec_dev

$\displaystyle - g$

# Third Derivative
thr_dev = diff(sec_dev,t)
thr_dev

$\displaystyle 0$

---

Chapter 9: Python Graphics for Climate Science

---

import numpy as np 
import pandas as pd
# import warnings
# import os
import plotly.express as px 
import sympy as sm
import math as m
import netCDF4 as nc

import matplotlib
from matplotlib import cm as cm1
from matplotlib import pyplot as plt
from matplotlib.colors import LinearSegmentedColormap
from matplotlib.patches import Polygon
from matplotlib import animation as animation
from matplotlib.animation import ArtistAnimation
from mpl_toolkits.basemap import Basemap, cm, shiftgrid, addcyclic

from datetime import datetime

import cartopy 
import cartopy.crs as ccrs
import cartopy.feature as cfeature
from cartopy.mpl.ticker import LongitudeFormatter, LatitudeFormatter

from scipy import optimize as opt
from scipy.ndimage import gaussian_filter
# from tkinter import *
# import time
# warnings.filterwarnings("ignore")

---

9.1 Two-Dimensional Line Plots and Setups of Margins and Labels

---

Two Different Time Series on the Same Plot

#Set up Variables 
time = np.arange(2001,2011)
Tmean = np.array([12.06, 11.78, 11.81, 11.72, 12.02,
                  12.36, 12.03, 11.27, 11.33, 11.66])
Prec = np.array([737.11, 737.87, 774.95, 844.55, 764.03,
                 757.43, 741.17, 793.50, 820.42, 796.80])

#Fig 9.1
# set up figure
fig, ax = plt.subplots(1,1,figsize=(12, 8))

# plot data 
ax.plot(time, Tmean, 'o-r', label="$T_{mean}$")

# set labels 
ax.set_ylabel("$T_{mean}$ [$\degree$C]", size=18)
ax.set_xlabel("Year", size=18)
ax.set_title("Contiguous U.S. Annual Mean Temperature and Total Precipitation", size=18, fontweight="bold")
ax.grid()
plt.yticks(rotation=90, size=18)
plt.xticks(size=18)

# creates twin axis 
ax1 = ax.twinx()

# plot of twin axis 
ax1.plot(time, Prec, 'o-b', label="Precipitation")
ax1.set_ylabel("Precipitation [mm]")

plt.yticks(rotation=90, size=18)

# create legend 
hand1, lab1 = ax.get_legend_handles_labels()
hand2, lab2 = ax1.get_legend_handles_labels()

ax.legend(handles=hand1+hand2, labels=lab1+lab2,
          loc='upper left', prop={'size': 20});

ax1.tick_params(direction='out')
ax.tick_params(direction='out')

# show plot
plt.show()

Figure Setups: Margins, Fonts, Mathematical Symbols, and More

#Fig 9.2
x = 0.25*np.arange(-30, 31)

# find constraint on variables
yPositive = np.where(np.sin(x) > 0)[0]
yNegative = np.where(np.sin(x) < 0)[0]

# set up figure
fig, ax = plt.subplots(figsize=(14,8))
plt.rcParams['hatch.color'] = 'blue'

# create bar plot 
ax.bar(x[yPositive], np.sin(x[yPositive]), width=.1, color='red')
ax.bar(x[yNegative], np.sin(x[yNegative]), hatch='.', fc='white', color="blue")

# add annotations
ax.text(-5, .7, "$Sine\ Waves$", size=75, color='green')
ax.set_xlabel(r"Angle in Radians: $\theta - \phi_0$", color='red',
              labelpad=-75)
ax.set_xticks([2*i for i in range(-3, 4)])
ax.set_xlim(x.min(), x.max())
ax.set_ylabel(r"$y = \sin(\theta - \hat{\phi})$", color='blue')
ax.set_yticks([-1., 0., .5, 1.])
ax.spines['bottom'].set_position(('data',-1.4))
ax.spines['bottom'].set_bounds(-6,6)

# twin axis 
ax1 = ax.twinx()

# plot on twin axis 
ax1.plot(x, np.sin(x), alpha=0)

# set labels
ax1.set_yticks([-1., -.5, 0., .5, 1.])
ax1.set_yticklabels(['A', 'B', 'C', 'D', 'E'])
ax1.tick_params(width=1, length=9)

fig.text(0.28,.95,"Text outside of the figure on side 3", size=25, fontweight="bold")

# show plot 
fig.show()

#Fig 9.3
#Illustrating how to adjust font size, axis labels space, and margins.

# create Random Variable
randVals = np.random.standard_normal(200)
x = np.linspace(0, 10, randVals.size)

# set up Figure 
fig, ax = plt.subplots(1, figsize=(12,8))
fig.suptitle("Normal Random Values", fontsize=25, y=.93)

# plot data 
ax.scatter(x, randVals, alpha=1, facecolors='none', edgecolors='k')

# add plot labels 
ax.set_title("Subtitle: 200 Random Values", y=-.2, fontsize=20)
ax.set_ylabel("Random Values")
plt.yticks(rotation=90)

ax.set_yticks([i for i in range(-2, 3)])
ax.set_xlabel("Time")

# show plot and grid
plt.grid()
fig.show()

# read in data 
NOAATemp = np.loadtxt("aravg.ann.land_ocean.90S.90N.v4.0.1.2016.txt",skiprows=0)

# set up variables
x=NOAATemp[:,0]
y=NOAATemp[:,1]
z = [-99] * x.size

def forz(z,y):
    for i in range(2,134):
        
        rslt = [(y[i-2],y[i-1],y[i],y[i+1],y[i+2])]
        z[i] = np.mean(rslt)
    return z

zz = forz(z,y)

def n1func():
    n1 = []
    for i in range(0,137):
        if y[i] >= 0:
            n1.append(i)
    return n1

def n2func():
    n2 = []
    for i in range(0,137):
        if y[i] < 0:
            n2.append(i)
    return n2

n1 = n1func()
n2 = n2func()

x1= x[n1]
y1= y[n1]
x2 = x[n2]
y2 = y[n2]
x3 = x[2:134]
y3 = z[2:134]

#Fig 9.4
# fancy plot of the NOAAGlobalTemp time series
plt.bar(x1,y1,color='r', width=0.5)
plt.bar(x2,y2,color='b', width=0.5)
plt.plot(x3,y3,color='k')
plt.xlim(1877,2020)

plt.xlabel('Time')
plt.ylabel('Temperature [$\degree$C]')
plt.title('NOAA Global Average Annual Mean Temperature Anomalies', fontweight="bold", size=20)
plt.grid()
plt.show()

#Fig 9.5
# Contiguous United States (a) annual mean temperature and (b) annual total precipitation 

# plot the US temp and prec time series on the same figure 
fig, ax = plt.subplots(2,1, figsize=(10,8))

# add additional axis 
ax[0].axes.get_xaxis().set_visible(False)

# plot first subplot data
ax[0].plot(time, Tmean, 'ro-', alpha=.6, label="(a)")

ax[0].set_yticks([11.4, np.mean([11.4, 11.8]), 11.8
                  , np.mean([11.8, 12.2]), 12.2])
ax[0].set_yticklabels([11.4, None, 11.8, None, 12.2])
ax[0].set_ylabel("$T_{mean}$ [$\degree$C]")
ax[0].text(2004,12.0,"US Annual Mean Temperature"
           , fontsize = "xx-large", fontweight="bold")

ax[0].text(2001,12.3,"(a)", size=18)

# plot second subplot data 
ax[1].plot(time, Prec, 'bo-', alpha=.6, label="(b)")

ax[1].text(2004,810,"US Annual Total Precipitation"
           , fontsize = "xx-large", fontweight="bold")
plt.ylabel("Precipitation [mm]")
ax[1].set_yticks([740, 760, 780, 800, 820, 840])
ax[1].set_yticklabels([740, None, 780, None, 820, None])

#set labels 
plt.text(2001,840,"(b)", size=18)

# ensures no overlap in plots 
fig.tight_layout(pad=-1.5)

# show plot
fig.show()

---

9.2 Color Contour Maps

---

Basic Principals for a Python Contour Plot

#9.2.1
# set up variables
x, y = np.linspace(-1, 1, 25), np.linspace(-1, 1, 25)
z = np.random.standard_normal(size=(25,25))

# set up figure 
fig, ax = plt.subplots(1,figsize=(12,12))

# plot contour plot
mymap = ax.contourf(x,y,z, cmap=cm1.get_cmap('jet'))

# create and set labels
ticklabs = [-1., None, -.5, None, 0., None, .5, None, 1.]
ax.set_yticklabels(ticklabs)
ax.set_xticklabels(ticklabs)

ax.set_ylabel("$y$")
ax.set_xlabel("$x$")

ax.set_title("Filled Contour Plot of Normal Random Values", size = 22)
fig.colorbar(mymap)

# show plot
fig.show()

Plot Contour Color Maps for Random Values on a Map

#Fig 9.6
# plot a 5-by-5 grid of global map of standard normal random values
# set up variables
lat = np.linspace(-90, 90, 37)
lon = np.linspace(0, 360, 72)

# create data to be plotted 
mapmat = np.random.standard_normal(size=(lat.size, lon.size))

levels = np.linspace(-3, 3, 81)

# set up figure 
fig, ax = plt.subplots(figsize=(14, 7))

ax = plt.subplot(1,1,1, projection=cartopy.crs.PlateCarree(central_longitude=180))
ax.coastlines()
ax.add_feature(cartopy.feature.BORDERS)
ax.set_global()
ax.set_title("Standard Normal Random Values")

# create desired Colormap 
colors = ["darkred","firebrick","firebrick","indianred","lightcoral","lightcoral","lightpink"
          ,"bisque","moccasin","yellow","yellow","greenyellow","yellowgreen","limegreen","limegreen"
          ,"mediumspringgreen","palegreen","honeydew","lavender","mediumpurple",
         "blueviolet","blueviolet","mediumorchid","mediumvioletred","darkmagenta","indigo","black"]
colors.reverse()
myColMap = newColMap(colors)

# plot data
contf = ax.contourf(lon-180, lat, mapmat, levels, cmap=myColMap)

# add labels and colorbar
colbar = plt.colorbar(contf, drawedges=True, aspect=12)
colbar.set_ticks([i for i in range(-3, 4)])
ax.set_aspect('auto')

ax.set_xticks([0, 50, 100, 150, 200, 250, 300, 350]
              ,crs=cartopy.crs.PlateCarree())
ax.set_xticklabels([0, 50, 100, 150, 200, 250, 300, 350])
ax.set_xlabel("Longitude")

ax.set_yticks([-50, 0, 50], crs= cartopy.crs.PlateCarree())
ax.set_ylabel("Latitude")

# show plot 
fig.show()

#Fig 9.7
# plot a 5-by-5 grid regional map to cover USA and Canada 
# set up varibles 
lat3 = np.linspace(10, 70, 13)
lon3 = np.linspace(230, 295, 14)
mapdat = np.random.standard_normal(size=(lat3.size,lon3.size))

# set up figure
fig, ax = plt.subplots(figsize=(14,10))

ax = plt.subplot(111, projection=cartopy.crs.PlateCarree())
ax.set_extent([230, 295, 10, 65])
ax.coastlines()
ax.add_feature(cartopy.feature.BORDERS)
ax.set_title("Standard Normal Random Values")

# plot data 
contf = ax.contourf(lon3, lat3, mapdat, levels, cmap=myColMap)

# add colorbar and labels 
colbar = plt.colorbar(contf, drawedges=True, aspect=12)
colbar.set_ticks([i for i in range(-3, 4)])
ax.set_aspect('auto')

ax.set_yticks([10*i for i in range(1, 8)],crs=cartopy.crs.PlateCarree())
ax.set_ylabel("Latitude")

ax.set_xticks([10*i for i in range(-13, -6)], crs=cartopy.crs.PlateCarree())
ax.set_xticklabels([10*i for i in range(23,30)])
ax.set_xlabel("Longitude")

ax.tick_params(length=9)

# show plot 
fig.show()

9.2.3 Plot Contour Maps from Climate Model Data in NetCDF Files

# Import Data
datamat = nc.Dataset("air.mon.mean.nc")

# Preparing data for plotting 
Lon = datamat.variables['lon'][:]
Lat = datamat.variables['lat'][:]
Time = datamat.variables['time']
precnc = datamat.variables['air']

#Fig 9.8
# set up figure
fig, ax = plt.subplots(1, figsize=(12,8))

# plot data 
ax.plot(np.linspace(90, -90, precnc.shape[1]), precnc[0, :, 64], '-k',
       linewidth=4)

# set labels and ticks
ax.set_ylabel("Temperature [$\degree$C]")
ax.set_yticks([10*i for i in range(-3, 3)])
ax.set_xlabel("Latitude")
ax.set_xticks([-50, 0, 50])
ax.set_title("90S-90N Temperature [$\degree$C] \nalong a Meridional Line at 160$\degree$E: January 1948",
             size = 22, pad = 10)
ax.tick_params(length=9, width=1)
ax.grid()

# show plot 
fig.show()

#Fig 9.9(a)
#Compute and plot climatology and standard deviation from Jan 1984 to Dec 2015

# set up variables for map
# number of colors that correspond to data values
contour_levels = np.linspace(-50,50,81)

# create color palette
myColMap = LinearSegmentedColormap.from_list(name='my_list', colors=['black','blue','darkgreen',
                                            'green','white','yellow','pink','red','maroon'], N=256)

JMon = precnc[12*np.arange(67)]
sdmat = np.std(precnc, axis=0)
climmat = np.mean(JMon, axis=0)
levels1 = np.linspace(-50, 50, 81)
levels2 = np.linspace(0, 20, 81)

# set up figure
fig, ax = plt.subplots(figsize=(14,7))

ax = plt.subplot(111, projection=cartopy.crs.PlateCarree(central_longitude=180))
ax.coastlines()
ax.add_feature(cartopy.feature.BORDERS, edgecolor='black')
ax.set_title("NCEP RA 1948-2015 January Climatology [$\degree$C]", size = 22, pad = 10)

# plot data
contf = ax.contourf(Lon-180, Lat, climmat, levels1, cmap=myColMap)

# add colorbar and labels
colbar = plt.colorbar(contf, drawedges=True, aspect=12, ticks=[])
colbar.set_ticks([20*i for i in range(-2, 3)])
ax.set_aspect('auto')

ax.set_yticks([50*i for i in range(-1, 2)],crs=cartopy.crs.PlateCarree())
ax.set_ylabel("Latitude")

ax.set_xticks([50*i for i in range(8)], crs=cartopy.crs.PlateCarree())
ax.set_xticklabels([50*i for i in range(8)])
ax.set_xlabel("Longitude")

ax.tick_params(length=9, width=1)
plt.text(198, 93, "[$\degree$C]", size=20)

# show plot 
fig.show()

#Fig 9.9(b)
# plot the standard deviation
# set up figure 
fig, ax = plt.subplots(figsize=(14,7))

ax = plt.subplot(111, projection=cartopy.crs.PlateCarree(central_longitude=180))
ax.coastlines()
ax.add_feature(cartopy.feature.BORDERS, edgecolor='black')
ax.set_title("NCEP 1948-2015 Jan. SAT RA Standard Deviation [$\degree$C]",size = 22, pad = 10)

# plot data
contf = ax.contourf(Lon-180, Lat, sdmat, levels2, cmap=myColMap)

# add colorbar and labels
colbar = plt.colorbar(contf, drawedges=True, aspect=12)
colbar.set_ticks([5*i for i in range(5)])
ax.set_aspect('auto')

ax.set_yticks([50*i for i in range(-1, 2)],crs=cartopy.crs.PlateCarree())
ax.set_ylabel("Latitude")

ax.set_xticks([50*i for i in range(8)], crs=cartopy.crs.PlateCarree())
ax.set_xticklabels([50*i for i in range(8)])
ax.set_xlabel("Longitude")

ax.tick_params(length=9, width=1)
plt.text(198, 93, "[$\degree$C]", size=20)

# show and save figure 
fig.savefig("CH9;JanuarySTD.jpg", bbox_inches='tight');

Plot Data for Displaying Climate Features

#Fit 9.10
# plot the January 1983 temperature using the above setup
mapmat = climmat

# set up figure 
fig, ax = plt.subplots(figsize=(14,7))

ax = plt.subplot(111, projection=cartopy.crs.PlateCarree(central_longitude=180))
ax.coastlines()
ax.add_feature(cartopy.feature.BORDERS, edgecolor='black')
ax.set_title("January 1983 Surface Air Temperature [$\degree$C]", size = 22, pad = 10)

# plot data 
contf = ax.contourf(Lon-180, Lat, mapmat, levels1, cmap=myColMap)

# add colorbar and labels
colbar = plt.colorbar(contf, drawedges=True, aspect=12)
colbar.set_ticks([20*i for i in range(-2, 3)])
ax.set_aspect('auto')

ax.set_yticks([50*i for i in range(-1, 2)],crs=cartopy.crs.PlateCarree())
ax.set_ylabel("Latitude")

ax.set_xticks([50*i for i in range(8)], crs=cartopy.crs.PlateCarree())
ax.set_xticklabels([50*i for i in range(8)])
ax.set_xlabel("Longitude")

plt.text(198,93,"[$\degree$C]", size=20)

ax.tick_params(length=9, width=1)

# show figure 
fig.show()

#Fig 9.11
# NOTE: the color distribution compared to the figures in the textbook look slightly different
# due to differences between R and Python ColorMapping characteristics

# plot the January 1983 anomaly from NCEP data
# create variables
levels1 = np.linspace(-6, 6, 81)
anomat = precnc[420,:,:] - climmat

## suppress values into [-6,6] range
# for i in range(73):
#     for j in range(144):
#         if anomat[i,j] > 6:
#             anomat[i,j] = 6
#         if anomat[i,j] < -6:
#             anomat[i,j] = -6

## another method to suppress anomat values into [-6,6]
anomat = np.maximum(np.minimum(anomat, 6), -6) 

# set up figure            
fig, ax = plt.subplots(figsize=(14,7))

ax = plt.subplot(111, projection=cartopy.crs.PlateCarree(central_longitude=180))
ax.coastlines()
ax.add_feature(cartopy.feature.BORDERS, edgecolor='black')
ax.set_title("January 1983 Surface Air Temperature Anomaly [$\degree$C]", size = 22, pad = 10)

# plot data
contf = ax.contourf(Lon-180, Lat, anomat, levels1, cmap=myColMap)

# add colorbar and labels 
colbar = plt.colorbar(contf, drawedges=True, aspect=12)
colbar.set_ticks([2*i for i in range(-4, 4)])
ax.set_aspect('auto')

ax.set_yticks([50*i for i in range(-1, 2)],crs=cartopy.crs.PlateCarree())
ax.set_ylabel("Latitude")

ax.set_xticks([50*i for i in range(8)], crs=cartopy.crs.PlateCarree())
ax.set_xticklabels([50*i for i in range(8)])
ax.set_xlabel("Longitude")

plt.text(198,93,"[$\degree$C]", size=20)
ax.tick_params(length=9, width=1)

# show figure 
fig.show()

Plot the January 1983 Temperature Anomaly from NCEP data

#Fig 9.12(a)
# create and prepare variables 
myslice = range(144)
anomat = anomat[:,myslice]
int1 = np.linspace(-6,6,81)

# set up figure
fig, ax = plt.subplots(figsize=(14,6))

ax = plt.subplot(111, projection=cartopy.crs.PlateCarree(central_longitude=200))
ax.coastlines()
ax.add_feature(cartopy.feature.BORDERS, edgecolor='black')
ax.set_title("January 1983 Surface Air Temperature Anomaly [$\degree$C]", size = 22, pad = 10)

# plot contour 
contf = ax.contourf(Lon-200, Lat, anomat, int1, cmap=myColMap) 
# subtract longitude by 200 to adjust for perspective change when zooming in

# add colorbar and labels
colbar = plt.colorbar(contf, drawedges=True, aspect=12)
colbar.set_ticks([2*i for i in range(-3, 4)])
ax.set_aspect('auto')

ax.set_yticks([20*i for i in range(-3, 3)],crs=cartopy.crs.PlateCarree())
ax.set_ylabel("Latitude")

ax.set_xticks([50*i for i in range(8)], crs=cartopy.crs.PlateCarree())
ax.set_xticklabels([50*i for i in range(8)])
ax.set_xlabel("Longitude")

ax.set_extent([100,300,50,-50],crs=cartopy.crs.PlateCarree())
ax.tick_params(length=9, width=1)

plt.text(111.5,52,"[$\degree$C]", size=20)

# show figure
fig.show()

#Zoom in to a specific lat-lon region: Pacific

# prepare data 
JMon = list(12*np.arange(68))
climmat = np.zeros((144,73))
for i in range(144):
    for j in range(73):
        climmat[i,j] = np.mean(precnc[JMon,j,i])

anomat = precnc[420,:,:] - climmat.T

# suppress values into the (-5,5) range 
for i in range(73):
    for j in range(144):
        if anomat[i,j] > 5:
            anomat[i,j] = 5
        if anomat[i,j] < -5:
            anomat[i,j] = -5
            
matdiff = anomat
int1 = np.linspace(-5,5,81)

#Fig 9.12(b)
# set up figure 
fig, ax = plt.subplots(figsize=(14,8))

ax = plt.subplot(111, projection=cartopy.crs.PlateCarree(central_longitude=250))
ax.coastlines()
ax.set_title("January 1983 Surface Air Temperature Anomaly [$\degree$C]", size = 22, pad = 10)

# plot data 
contf = ax.contourf(Lon-250, Lat, matdiff, int1, cmap=myColMap)
# subtract longitude by 250 to adjust for perspective change when zooming in

# add colorbar and labels 
colbar = plt.colorbar(contf, drawedges=True, aspect=12)
colbar.set_ticks([2*i for i in range(-3, 4)])
ax.set_aspect('auto')

ax.set_yticks([10*i for i in range(2, 7)],crs=cartopy.crs.PlateCarree())
ax.set_ylabel("Latitude")

ax.set_xticks([200+i*(10) for i in range(3,11)], crs=cartopy.crs.PlateCarree())
ax.set_xticklabels([200+i*(10) for i in range(3,11)])
ax.set_xlabel("Longitude")

ax.set_extent([230,300,20,60],crs=cartopy.crs.PlateCarree())
ax.add_feature(cartopy.feature.BORDERS, edgecolor='black')
ax.tick_params(length=9, width=1)

plt.text(54,61,"[$\degree$C]", size=20)

# show figure 
fig.show()

datamat.close()

---

9.3 Plot Wind Velocity Field on a Map

---

#Fig 9.13
# prepare variables 
lat = np.repeat(np.linspace(-75,75,num = 6),12)
lon = np.repeat(np.linspace(-165,165,num=12),6)

# create wind vector variables
u = np.repeat([-1,1,-1,-1,1,-1],864)
v = np.repeat([1,-1,1,-1,1,-1],864)

X, Y = np.meshgrid(lon, lat)

# set up figure
fig, ax = plt.subplots(figsize=(14,8))

ax = plt.axes(projection=cartopy.crs.PlateCarree())
ax.coastlines()
ax.add_feature(cartopy.feature.BORDERS, edgecolor='black')

# plot every wind vector in a for loop
for i in lon:
    for j in lat:
        plt.plot(i,j, markersize=3, marker='o', color='black')

# create the quiver plot
ax.quiver(X,Y,u,v, scale=65, color="blue")

# add labels and text
# plt.title("Polar High", color = 'green', size = 25)
ax.text(-165,80,"Polar High", color="green", size = 20)
ax.text(-165,30,"Subtropical High", color="green", size = 20)
ax.text(-165,0,"Intertropical Convergence Zone (ITCZ)", color="green", size = 20)
ax.text(-165,-30,"Subtropical High", color="green", size = 20)

ax.axis([-180,180,-90,90])
ax.set_xticks([-165,-105,-45,15,75,135], crs=cartopy.crs.PlateCarree())
ax.set_yticks([-75,-45,-15,15,45,75], crs=cartopy.crs.PlateCarree())

# format lon and lat 
lon_formatter = LongitudeFormatter(degree_symbol='', dateline_direction_label=True)
lat_formatter = LatitudeFormatter(degree_symbol='')
ax.xaxis.set_major_formatter(lon_formatter)
ax.yaxis.set_major_formatter(lat_formatter)

# show figure 
plt.show()

#-- open netcdf file
ncd = nc.Dataset('uvclm95to05.nc')

#-- read variable
u10 = ncd.variables['u'][0,:,:]
v10 = ncd.variables['v'][0,:,:]
lat = ncd.variables['lat'][::-1]
lon = ncd.variables['lon'][:]

#Fig 9.14 
#The NOAA sea wind field of January 1, 1995. 

## THIS FIGURE IS BEING ADJUSTED SEPARATELY AND WILL BE UPDATED SHORTLY ##

ncd.close()

---

9.4 GG-Plot for Data

# create dict of state names and labels
state_names = {"AL":"Alabama","AK":"Alaska","AZ":"Arizona","AR":"Arkansas","CA":"California","CO":"Colorado"
               ,"CT":"Connecticut","DE":"Delaware","FL":"Florida","GA":"Georgia","HI":"Hawaii","ID":"Idaho"
               ,"IL":"Illinois","IN":"Indiana","IA":"Iowa","KS":"Kansas","KY":"Kentucky","LA":"Louisiana"
               ,"ME":"Maine","MD":"Maryland","MA":"Massachusetts","MI":"Michigan","MN":"Minnesota","MS":"Mississippi"
               ,"MO":"Missouri","MT":"Montana","NE":"Nebraska","NV":"Nevada","NH":"New Hampshire"
               ,"NJ":"New Jersey","NM":"New Mexico","NY":"New York","NC":"North Carolina","ND":"North Dakota"
               ,"OH":"Ohio","OK":"Oklahoma","OR":"Oregon","PA":"Pennsylvania","RI":"Rhode Island","SC":"South Carolina"
               ,"SD":"South Dakota","TN":"Tennessee","TX":"Texas","UT":"Utah","VT":"Vermont","VA":"Virginia"
               ,"WA":"Washington","WV":"West Virginia","WI":"Wisconsin","WY":"Wyoming"}

# create state Abbreviations
state_abr = ["AL", "AK", "AZ", "AR", "CA", "CO", "CT", "DC", "DE", "FL", "GA", 
          "HI", "ID", "IL", "IN", "IA", "KS", "KY", "LA", "ME", "MD", 
          "MA", "MI", "MN", "MS", "MO", "MT", "NE", "NV", "NH", "NJ", 
          "NM", "NY", "NC", "ND", "OH", "OK", "OR", "PA", "RI", "SC", 
          "SD", "TN", "TX", "UT", "VT", "VA", "WA", "WV", "WI", "WY"]
state_df = pd.DataFrame({"State":list(state_names.values()),"Abbreviation":list(state_names.keys())})

#Fig 9.15
# set up and create figure 
fig = px.choropleth(state_df,  # Input Pandas DataFrame
                    locations="Abbreviation",  # DataFrame column with locations
                    color="State",  # DataFrame column with color values
                    locationmode = 'USA-states',
                   color_continuous_scale = 'rainbow') # Set to plot as US States

# update layout of interactive plot 
fig.update_layout(
    title_text = 'Color Map of the United States', # Create a Title
    geo_scope='usa',  # Plot only the USA instead of globe
)

# show figure
fig.show()

---

9.5 Animation

---

import turtle
import numpy as np
# animation will open in another window

# window setup
width = 400
height = 500
window = turtle.Screen()
window.setup(width, height)
# window.setup()
window.title('Free Fall Ball')

# ball characteristics
ball = turtle.Turtle()
ball.penup()
ball.color("green")
ball.shape("circle")
# ball.speed('slowest')

g = -9.81 # meters/second^2 = gravitational acceleration
v = 0 # meters/second = initial velocity
y = 490 # meters = initial height
time = np.linspace(0,20.5,100) # 0 to 10 seconds

try:
    while y > 0.0:
        for t in time:
    #         print(t,y)
            v0 = v
            y0 = y
            y = y0 + (v0 * t) + (0.5 * g * t**2)
            v = np.sqrt(v0**2 + 2*g*(y-y0))
            ball.sety(y)
            ball.speed(1 - y/490)
            window.update()

    window.bye()
    # exit window
    
    turtle.done()
except Exception: 
    # so the animation does not end in an error
    pass

# turtle.done()
# uncomment above if program does not stop alone

---

Chapter 10: Advanced Python Analysis and Plotting: EOFs, Trends, and Global Data

---

import numpy as np 
import matplotlib
from matplotlib import cm as cm1
import matplotlib.pyplot as plt
import pandas as pd
import warnings
import os

from scipy import optimize as opt
import sympy as sm
import math as m
import cartopy 
import netCDF4 as nc
from matplotlib.colors import LinearSegmentedColormap
from cartopy.mpl.ticker import LongitudeFormatter, LatitudeFormatter
from mpl_toolkits.basemap import Basemap, cm, shiftgrid, addcyclic
from matplotlib.patches import Polygon

from datetime import datetime

import cartopy.crs as ccrs
import cartopy.feature as cfeature

from scipy.ndimage import gaussian_filter
from sklearn import datasets, linear_model
from sklearn.neighbors import KernelDensity as kd
from sklearn.linear_model import LinearRegression
from pylab import *
warnings.filterwarnings("ignore")

---

10.2: 2Dim Spatial Domain EOFs and 1Dim Temporal PCs

---

# create two arrays, x and y, ranging from 0 to 2*pi 
# create a 3D array full of zeros for use later
x = y = np.linspace(0.,2*np.pi,100)
mydat = np.zeros((100,100,10))

# define a function that iterates term by term for the two variables into a specific function
def zfunc(x,y):
    # the triple nested for loop accomplishes term by term order we are looking for
    for t in range(0,10):
        for i in range(0,100):
            for j in range(0,100):
                mydat[i,j,t] = (np.sin(t+1)*(1/np.pi)*np.sin(x[i])*np.sin(y[j]) 
                                + np.exp(-0.3*(t+1))*(1/np.pi)*np.sin(8*x[i])*np.sin(8*y[j]))
    # print the shape and maximum of the result
    print(np.shape(mydat))
    print(np.max(mydat))
    return mydat


# plug in our x and y arrays 
mydata = zfunc(x,y)

(100, 100, 10)
0.4909421672431401

#Fig 10.1
# set a range for the contour plot
maxi = np.linspace(-.5,.5,100)

# define the contour plot using previous x, y, and mydata arrays
cmap = cm.get_cmap('hsv', 20)
CS = plt.contourf(x,y,mydata[:,:,0],maxi,cmap=cmap)

# add titles for the graph and axes
plt.title('Original field at t = 1', pad = 10)
plt.xlabel('x')
plt.ylabel('y')

# add a color bar with a label
cbar = plt.colorbar(CS, ticks=[0.4,0.2,0,-0.2,-0.4])
plt.text(6.55, 6.45, "Scale", size=20)

# show figure
plt.show()

# set a range for the contour plot
maxi = np.linspace(-.2,.2,100)

# define the contour plot using previous x, y, and mydata arrays
cmap = cm.get_cmap('hsv', 20)
# CS = plt.contourf(x,y,np.flip(mydata[:,:,10], axis=1),maxi,cmap=cmap)
CS = plt.contourf(x,y,mydata[:,:,9],maxi,cmap=cmap)

# add titles for the graph and axes
plt.title('Original field at t = 10', pad = 10)
plt.xlabel('x')
plt.ylabel('y')

# add a color bar with a label
cbar = plt.colorbar(CS, ticks=[0.2,0.1,0,-0.1,-0.2])
plt.text(6.5, 6.5, "Scale", size=20)

# save figure as a .png file
plt.show()

# define variables
x = y = np.linspace(0.,2.*np.pi,100)

# create a matrix of zeros of size (10000,10)
da = np.matrix(np.zeros((np.size(x)*np.size(y),10)))

# define a function that reshapes mydata into a new array of size
def twod_func(x,y,a):
    for i in range(0,10):
        a[:,i] = np.reshape(mydata[:,:,i],(10000,1))
    return a
    
da1 = twod_func(x,y,da)
# SVD for EOFs
u, s, vh = np.linalg.svd(da1,full_matrices=False) 
v = np.transpose(vh)

# to show the accuracy of the svd
x_a = np.dot(np.dot(u,np.diag(s)),vh)

# create an array of zeros of size (100,100)
u2 = np.zeros((100,100))

# define a function that takes all the entries in the first column
# of the u matrix, transposes it and reshapes it into size (100,100)
def dig():
    y1 = np.transpose(u[:,0])
    u2 = np.reshape(-y1,(100,100))
    return u2

def dig2():
    y1 = np.transpose(u[:,1])
    u2 = np.reshape(-y1,(100,100))
    return u2

# create zero matrix, as well as u-mat, from the dig fuction and u-mat2 from the dig2 fuction
zeros = np.zeros((100,100))
u_mat = dig()
u_mat2 = dig2()

# define two more functions that call upon the above z functions to generate spatial data
def fcn1(x,y):
    for i in range(0,100):
        for j in range(0,100):
            zeros[i,j] = z1(x[i],y[j])
    return zeros

def fcn2(x,y):
    for i in range(0,100):
        for j in range(0,100):
            zeros[i,j] = z2(x[i],y[j])
    return zeros

# define two different functions to be used below
def z1(x,y):
    return (1./np.pi)*np.sin(x)*np.sin(y)

def z2(x,y):
    return (1./np.pi)*np.sin(8.*x)*np.sin(8.*y)

# create desired Colormap
colors = ["crimson","red","red","red","tomato","lightpink",
          "pink","peachpuff","yellow","yellow", "yellowgreen","limegreen","limegreen","white","lavenderblush",
          "darkorchid","indigo","indigo","rebeccapurple","darkmagenta","mediumvioletred","mediumvioletred","black"]
colors.reverse()
cmap = newColMap(colors)

colors1 = ["crimson","red","tomato","lightpink",
          "pink","peachpuff","yellow", "yellowgreen","limegreen","white","lavenderblush",
          "darkorchid","indigo","rebeccapurple","darkmagenta","mediumvioletred","black"]
colors1.reverse()
cmap1 = newColMap(colors1)

#Fig 10.2
# graph the EOF
size = np.linspace(-2.5e-2,2.5e-2,25)
bounds = np.array([-0.02,-0.01,0.0,.01,0.02])
bounds_d = np.array([-0.3,-0.2,-0.1,0,0.1,0.2,0.3])

# set up the figure
plt.subplot(221)

# plot the data
dig = plt.contourf(x,y,u_mat,size,cmap=cmap)

# set labels and title
plt.title('SVD Mode 1: EOF1')
plt.ylabel('y')
plt.xticks([0,1,2,3,4,5,6], size=14)
plt.yticks([0,1,2,3,4,5,6],rotation=90, size=14)

# print colorbar
cbar = plt.colorbar(dig, ticks = bounds)
cbar.ax.tick_params(labelsize='large')

# create second plot 
plt.subplot(223)
lvl = np.linspace(-.35,.35,35)

# plot data and set labels
contour = plt.contourf(x,y,fcn1(x,y),lvl,cmap=cmap1)

plt.title('Accurate Mode 1')
plt.xlabel('x')
plt.ylabel('y')
plt.yticks([0,1,2,3,4,5,6], rotation=90, size=14)
plt.xticks([0,1,2,3,4,5,6], size=14)

cbar = plt.colorbar(contour, ticks = bounds_d)
cbar.ax.tick_params(labelsize='large')

# plot third plot
plt.subplot(222)
dig = plt.contourf(x,y,np.flip(u_mat2,axis=1),size,cmap=cmap)

plt.title('SVD Mode 2: EOF2')
plt.ylabel('y')
plt.yticks([0,1,2,3,4,5,6],rotation=90, size=14)
plt.xticks([0,1,2,3,4,5,6], size=14)

cbar = plt.colorbar(dig, ticks = bounds)
cbar.ax.tick_params(labelsize='large')

# plot fourth plot
plt.subplot(224)
lvl = np.linspace(-.35,.35,35)
contour = plt.contourf(x,y,fcn2(x,y),lvl,cmap=cmap1)

plt.title('Accurate Mode 2')
plt.xlabel('x')
plt.ylabel('y')
plt.yticks([0,1,2,3,4,5,6],rotation=90, size=14)
plt.xticks([0,1,2,3,4,5,6], size=14)

plt.tight_layout()

cbar = plt.colorbar(contour, ticks = bounds_d)
cbar.ax.tick_params(labelsize='large')

plt.show()

#Fig 10.3
# create variable
t = np.array(range(1,11))

# plot separate lines based on different formulas
plt.plot(t,v[:,0],color='k',marker='o',label='PC1: 83% Variance')
plt.plot(t,v[:,1],color='r',marker='o',label='PC2: 17% Variance')
plt.plot(t,-np.sin(t),color='b',marker='o',label='Original Mode 1 Coefficient: 91% variance')
plt.plot(t,-np.exp(-.3*t),color='m',marker='o',label='Original Mode 2 Coefficient: 9% variance')

# set labels
plt.ylim(-1.1,1.1)

plt.xlabel('Time')
plt.ylabel('PC or Coefficient')
plt.yticks([1,0.5,0,-0.5,-1])
plt.title('SVD PCs vs. Accurate Temporal Coefficients')

# show legend and plot 
plt.legend(loc = 'upper left', fontsize=20)
plt.show()

# create arrays to compute a dot product
t = np.array(range(1,11))

# make sine function
def sinarr(t):
    rslt = np.zeros(t)
    for ii in range(1,t+1):
        rslt[ii-1] = -np.sin(ii)
    return rslt

# make exponential function
def exparr(t):
    rslt = np.zeros(t)
    for jj in range(1,t+1):
        rslt[jj-1] = -np.exp(-.3*jj)
    return rslt

# create variables using above functions
sin = sinarr(10)
exp = exparr(10)
print('Dot Product:',np.dot(np.transpose(sin),exp))

# find the variances and compare them
v1 = np.var(sin)
v2 = np.var(exp)
print("Variance:",v1/(v1+v2))

# find the product of the PC's
v_1 = np.transpose(v[:,0])
v_2 = v[:,1]
print("Dot Product of PCs:" , np.dot(v_1,v_2))

Dot Product: 0.8625048359266784
Variance: 0.9098719287510537
Dot Product of PCs: [[2.08166817e-17]]

# create arrays that will be used below 
# and check the sizes of the arrays for matrix multiplication
sdiag = np.diag(s)
vtran = np.transpose(v[:,0:1])

# create two new arrays, B and B1 multiplying the previous
# matrices together, being careful to match their sizes
B = np.transpose(np.matmul(np.matmul(u[:,0:1],sdiag[0:1,0:1]),vtran))
B1 = np.matmul(np.matmul(u,sdiag),np.transpose(v))

#Fig 10.4
# reshaping B to plot it
BB = np.reshape(B[4,:],(100,100))
scale = np.linspace(-.4,.4,25)
bounds = [0.4,0.2,0,-0.2,-0.4]

cmap = cm.get_cmap('hsv', 15)

# set up figure
fig = plt.figure(figsize=(14,6))

# create subplot
plt.subplot(121)
BBplot = plt.contourf(x,y,BB,scale,cmap = cmap)

# set labels 
plt.title('(a) 2-mode SVD reconstructed field t = 5', size=18)
plt.xlabel('x')
plt.ylabel('y')
plt.xticks([0,1,2,3,4,5,6])

# print colorbar
cbar = plt.colorbar(BBplot, ticks=bounds)
cbar.ax.tick_params(labelsize='large')


# reshaping B to plot it
new_B = B1.T
BB = np.reshape(new_B[4,:],(100,100))

# create subplot
plt.subplot(122)
BBplot = plt.contourf(x,y,BB,scale,cmap = cmap)

# set labels
plt.title('(b) All-mode SVD reconstructed field t = 5', size=18)
plt.xlabel('x')
plt.ylabel('y')
plt.xticks([0,1,2,3,4,5,6])

# create colorbar
cbar = plt.colorbar(BBplot, ticks=bounds)
cbar.ax.tick_params(labelsize='large')

plt.tight_layout()

---

10.3 From Climate Data Download to EOFs and PC Visualization: An NCEP/NCAR Reanalysis Example

---

#Download the following data set from NOAA into your active folder
# https://www.esrl.noaa.gov/psd/repository/entry/show?
# entryid=synth%3Ae570c8f9-ec09-4e89-93b4-babd5651e7a9%
# 3AL25jZXAucmVhbmFseXNpcy5kZXJpdmVkL3N1cmZhY2UvYWlyLm1vbi5tZWFuLm5j

ncd = nc.Dataset("air.mon.mean.nc","r+")

#Define variables
lon_vals = ncd.variables['lon']
lat_vals = ncd.variables['lat']
time = ncd.variables['time']
air = ncd.variables['air']
time_unit = time.units
precnc = ncd.variables['air']

#Fig 10.5
# plot a 90S-90N temp along a meridional line at 180E

# create variables
nc_x = np.linspace(-90,90,73)
y = list(precnc[0,:,71])
y.reverse()


# plot data
plt.plot(nc_x,y,marker="o",color='k' )

# set labels and show figure
plt.xlabel('Latitude')
plt.ylabel('Temperature [$\degree$C]')
plt.yticks(rotation=90)
plt.title('NCEP/NCAR Reanalysis Surface Air Temperature [$\degree$C] \n along a Meridional Line at $180\degree$E: Jan. 1948'
         , pad = 10)
plt.grid()
plt.show()

list(precnc[0,:,71]).reverse()

precnc.shape

(826, 73, 144)

# define variables 
precst = np.zeros((10512,826))

temp = np.reshape(precnc[0,:,:],(144*73))
print(np.shape(temp))
print(temp[0:6])

# create reshaping function
def spmat(x,y):
    for i in range(0,826):
        y[:,i] = np.reshape(x[i,:,:],(144*73))
    return y

# use function and save result
precst2 = spmat(precnc,precst)

print(np.shape(precst2))

# build lat and lon for 10512 spatial positions using rep
def rep(x,y,n):
    for j in range(0,n):
        x = np.append(x,y)
    return x

arra = np.zeros((0))
LAT = rep(arra,lat_vals,144)
LON = rep(arra,lon_vals[0],73)
print(LON)

# create reconstruction function
def recon():
    rslt = LON
    for jj in range(0,143):
        rslt = np.append(rslt,rep(arra,lon_vals[jj],73))
    return rslt
LON2 = recon()
print("The last 6 entries are:",LON2[-6:])

# creating an array with first two columns of lat and lon
# with 826 months spanning from 1948-2016
gpcpst_ = np.column_stack((LAT,(LON2)))
gpcpst = np.column_stack((gpcpst_,precst2))

(10512,)
[-34.926773 -34.926773 -34.926773 -34.926773 -34.926773 -34.926773]
(10512, 826)
[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0.]
The last 6 entries are: [355. 355. 355. 355. 355. 355.]

# create an array of all ones for the days of the data
ones = np.ones(826)

# define a function to create a list of months repeating to match the size of our data
def monthfunc():
    montharr = np.zeros(0)
    for i in range(1,71):
        for j in range(1,13):
             montharr = np.append(montharr,int(j))
    for k in range(1,6):
        montharr = np.append(montharr,int(k))
    return montharr

months = monthfunc()

# define a function to create a list of years to match the size of our data
def yearfunc():
    yeararr = np.zeros(0)
    for i in range(1948,2018):
        for j in range(1,13):
            yeararr = np.append(yeararr,i)
    for k in range(1,6):
        yeararr = np.append(yeararr,2018)
    return yeararr

years = yearfunc()

# define a function that pairs a string version of our year and month arrays with a dash
# in between these will serve as our column names for our data
def tmfunc():
    empty = np.zeros(0)
    for i in range(0,826):
        empty = np.append(empty,str(years[i])  + '-'  + str(months[i]))
    return empty

tm1 = tmfunc()

# add the 'Lat' and 'Lon' columns to the beginning
# of our column names array
tm2 = np.append(['Lat','Lon'],tm1)

# use the package pandas.DataFrame to match the column names with the data
GPCPST = pd.DataFrame(gpcpst,columns=tm2)

monJ = np.array(range(0,826,12))
gpcpdat = GPCPST.iloc[:,2:826]
gpcpJ = gpcpdat.iloc[:,monJ]
climJ = gpcpJ.mean(axis=1)

# create desired functions
def rowSDS2():
    rslt = []
    for jj in range(0,10512):
        rslt = np.append(rslt,np.std(gpcpJ.iloc[jj,:]))
    return rslt

sdJ = gpcpJ.std(axis=1)

def anomJ_func2():
    rslt = pd.DataFrame(np.zeros((10512, 69)),columns=tm1[monJ])
    for i in range(0,69):
        rslt.iloc[:,i] = (gpcpdat.iloc[:,monJ[i]]-climJ)/sdJ
    return rslt

anomJ = anomJ_func2()

def anomJ_func3():
    rslt = pd.DataFrame(np.zeros((10512, 69)),columns=tm1[monJ])
    for i in range(0,69):
        rslt.iloc[:,i] = np.sqrt(np.cos(GPCPST.iloc[:,0]*np.pi/180.))*anomJ.iloc[:,i]
    return rslt
        
anomAW = anomJ_func3()

# computing SVD from the area-weighted anomaly matrix anomAW
U, S, Vh = np.linalg.svd(anomAW,full_matrices=False)

print(U.shape,S.shape,Vh.shape)

(10512, 69) (69,) (69, 69)

#Fig 10.6
# create variables
xvals = np.linspace(0.,70.,69)
yvals = 100.*(S**2.)/np.sum(S**2)

# create two subplots
fig, ax1 = plt.subplots()

# plot of first plot
ax1.scatter(xvals,yvals,marker = "o",facecolors='none',edgecolors='k')
ax1.plot(xvals,yvals,color='k',label = "Percentage Variance ")
ax1.set_ylabel('Percentage Variance [%]',color='k')
ax1.set_xlabel('Mode number',color='k')
ax1.tick_params(axis='y',labelcolor='k')
ax1.grid()

# twin the axis to prepare second plot
ax2 = ax1.twinx()

# plot data on second axis 
ax2.scatter(xvals,yvals.cumsum(),marker = "o",facecolors='none',edgecolors='b')
ax2.plot(xvals,yvals.cumsum(),color='b',label = "Cumulative Variance")

# set labels
ax2.set_ylabel('Cumulative Variance [%]',color='b')
ax2.tick_params(axis='y',labelcolor='b')

ax1.legend(loc=(.5,.5))
ax2.legend(loc=(.5,.45))
plt.title("Eigenvalues of Covariance Matrix")

# show figure
plt.show()

# define function
def mapmatr():
    q = np.zeros((144,73))
    for ii in range(0,144):
        q[ii,:] = U[:,0]/(np.sqrt(np.cos(gpcpst[:,0]
          *np.pi/180.)))
    return q

# reshape data using result of SVD 
mapmatr = (np.reshape(U[:,0]/(np.sqrt(np.cos(gpcpst[:,0]
          *np.pi/180.))),(73,144)))

mapmatr2 = mapmatr[:,range(0,mapmatr[0,:].size)]

# make Colormap and create desired Colormap
colors = ["darkred","red","darkorange","orange","yellow","lightyellow","white","lightgreen"
          ,"green","blue","darkblue", "black"]
colors.reverse()
myColMap = newColMap(colors)

#Fig 10.7
# prepare figure
plt.figure(figsize=(12,12))

# plot EOF1
plt.subplot(211)

# using the basemap package to put the cylindrical projection map into the image
eof = Basemap(projection='cyl',llcrnrlat=-90.,urcrnrlat=90.,\
            llcrnrlon=0.,urcrnrlon=360.,resolution='c')

# shifting the data because 
lons2, data2 = eof.shiftdata(lon_vals, datain = -mapmatr2, lon_0=180)


# draw coastlines, latitudes, and longitudes on the map
eof.drawcoastlines(color='black', linewidth=1)
eof.drawparallels(np.arange(-90.,91.,30.), labels = [1,0,0,1], fontsize=15)
eof.drawmeridians(np.arange(-180.,181.,60.), labels= [1,0,0,1], fontsize=15)
eof.drawcountries()

limits = np.linspace(-.04,.04,51)

# plot data on contour map
eof_plt = plt.contourf(np.array(lons2),np.array(lat_vals),data2,limits,cmap = "jet")

# set labels and colorbar bounds
bounds = np.array([-0.04,-0.02,0,0.02,0.04])
efo = plt.colorbar(eof_plt, ticks = bounds, shrink = 0.85)
efo.ax.tick_params(labelsize='large')
plt.text(375, 93, 'Scale', size=16)
plt.xlabel('Longitude',labelpad = 30)
plt.ylabel('Latitude',labelpad =40)
plt.title('January EOF1 from 1948-2018 NCEP Temp. Data', pad = 10)

# retrieve PC1 from SVD 
pcdat = Vh[0,:]

# open bottom plot
plt.subplot(212)

#create time variable
TIME = np.array(range(1948,2017))

# plot data
plt.scatter(TIME,-pcdat,marker = "o",facecolors='none',edgecolors='k')
plt.plot(TIME,-pcdat,color='k')
plt.title("PC1 of NCEP RA Jan. SAT: 1948-2018", pad = 10)
plt.xlabel("Year")
plt.ylabel("PC Values")
plt.grid()
plt.tight_layout()
plt.show()

#Fig 10.8
# prepare figure
plt.figure(figsize=(12,12))

# plot EOF2 by reshaping data from SVD
mapmatr = (np.reshape(U[:,1]/(np.sqrt(np.cos(gpcpst[:,0]
          *np.pi/180.))),(73,144)))

mapmatr2 = mapmatr[:,range(0,mapmatr[0,:].size)]

# open first subplot
plt.subplot(211)

# using the basemap package to put the cylindrical projection map into the image
eof = Basemap(projection='cyl',llcrnrlat=-90.,urcrnrlat=90.,\
            llcrnrlon=0.,urcrnrlon=360.,resolution='c')

# shifting the data
lons2, data2 = eof.shiftdata(lon_vals, datain = -mapmatr2, lon_0=180)

# draw coastlines, latitudes, and longitudes on the map
eof.drawcoastlines(color='black', linewidth=1)
eof.drawparallels(np.arange(-90.,91.,30.), labels = [1,0,0,1], fontsize=15)
eof.drawmeridians(np.arange(-180.,181.,60.), labels= [1,0,0,1], fontsize=15)
eof.drawcountries()

# create contour limits
limits = np.linspace(-.04,.04,51)

# plot data
eof_plt = plt.contourf(np.array(lons2),np.array(lat_vals),data2,limits,cmap = "jet")

# set labels and colorbar bounds
bounds = np.array([-0.04,-0.02,0,0.02,0.04])
efo = plt.colorbar(eof_plt, ticks=bounds, shrink = 0.85)
efo.ax.tick_params(labelsize='large')
plt.text(375, 93, 'Scale', size=16)
plt.xlabel('Longitude',labelpad = 30)
plt.ylabel('Latitude',labelpad =40)
plt.title('January EOF2 from 1948-2018 NCEP Temp. Data', pad = 10)

pcdat = Vh[1,:]

# open bottom plot
plt.subplot(212)

# create time variable
TIME = np.array(range(1948,2017))

# plot PC2
plt.scatter(TIME,-pcdat,marker = "o",facecolors='none',edgecolors='k')
plt.plot(TIME,-pcdat,color='k')
plt.title("PC2 of NCEP RA Jan. SAT: 1948-2018", pad = 10)
plt.xlabel("Year")
plt.ylabel("PC Values")
plt.grid()
plt.tight_layout()
plt.show()

#Fig 10.9
# set figure up
plt.figure(figsize=(12,12))

# create data from SVD 
mapmatr = (np.reshape(U[:,2]/(np.sqrt(np.cos(gpcpst[:,0]
          *np.pi/180.))),(73,144)))
mapmatr2 = mapmatr[:,range(0,mapmatr[0,:].size)]

# open first plot
plt.subplot(211)

# using the basemap package to put the cylindrical projection map into the image
eof = Basemap(projection='cyl',llcrnrlat=-90.,urcrnrlat=90.,\
            llcrnrlon=0.,urcrnrlon=360.,resolution='c')

# shifting the data because 
lons2, data2 = eof.shiftdata(lon_vals, datain = -mapmatr2, lon_0=180)

# draw coastlines, latitudes, and longitudes on the map
eof.drawcoastlines(color='black', linewidth=1)
eof.drawparallels(np.arange(-90.,91.,30.), labels = [1,0,0,1], fontsize=15)
eof.drawmeridians(np.arange(-180.,181.,60.), labels= [1,0,0,1], fontsize=15)
eof.drawcountries()

limits = np.linspace(-.04,.04,51)

# plot data 
eof_plt = plt.contourf(np.array(lons2),np.array(lat_vals),data2,limits,cmap = "jet")

# set labels and colorbar bounds
bounds = np.array([-0.04,-0.02,0,0.02,0.04])
efo = plt.colorbar(eof_plt, ticks=bounds, shrink = 0.85)
efo.ax.tick_params(labelsize='large')
plt.text(375, 93, 'Scale', size=16)
plt.xlabel('Longitude',labelpad = 30)
plt.ylabel('Latitude',labelpad =40)
plt.title('January EOF3 from 1948-2018 NCEP Temp. Data', pad = 10)

pcdat = Vh[2,:]

# open bottom plot
plt.subplot(212)

# create time variable
TIME = np.array(range(1948,2017))

# plot data 
plt.scatter(TIME,-pcdat,marker = "o",facecolors='none',edgecolors='k')
plt.plot(TIME,-pcdat,color='k')

# set labels 
plt.title("PC3 of NCEP RA Jan. SAT: 1948-2018", pad = 10)
plt.xlabel("Year")
plt.ylabel("PC Values")
plt.grid()
plt.tight_layout()
plt.show()

GPCPST.shape

(10512, 828)

# EOF from de-trended data
monJ = np.array(range(0,815,12))
gpcpdat = GPCPST.iloc[:,2:817]
gpcpJ = gpcpdat.iloc[:,monJ]
climJ = gpcpJ.mean(axis=1)
sdJ = gpcpJ.std(axis=1)

anomJ = ((gpcpdat.iloc[:,monJ].T-climJ)/sdJ).T

# create linear regression object
regr = linear_model.LinearRegression()
print(anomJ.iloc[0,:].shape)

(68,)

# create time variable 
TIME = np.array(range(1948,2016))

# create trend V
def trendV():
    trendV = np.zeros(10512)
    for i in range(0,10512):
        regr.fit(np.reshape(TIME,(68,1)),
                 np.reshape(np.array(anomJ.iloc[i,:]),(68,1)))
        trendV[i] = regr.coef_
    return trendV
 
# create trend M
def trendM():
    rslt = np.zeros((10512,68))
    trendM = pd.DataFrame(np.zeros((10512, 68)),columns=tm1[monJ])
    for i in range(0,10512):
        regr.fit(np.reshape(TIME,(68,1)),
                 np.reshape(np.array(anomJ.iloc[i,:]),(68,1)))
        rslt[i,:] = regr.intercept_
    return rslt

# make trend V instance 
trendV = trendV()
print(trendV)

[ 0.02303803  0.02303803  0.02303803 ... -0.03220623 -0.03220623
 -0.03220623]

# make trend M instance
trendM = trendM()
print(trendM)

[[-45.64986127 -45.64986127 -45.64986127 ... -45.64986127 -45.64986127
  -45.64986127]
 [-45.64986127 -45.64986127 -45.64986127 ... -45.64986127 -45.64986127
  -45.64986127]
 [-45.64986127 -45.64986127 -45.64986127 ... -45.64986127 -45.64986127
  -45.64986127]
 ...
 [ 63.81665437  63.81665437  63.81665437 ...  63.81665437  63.81665437
   63.81665437]
 [ 63.81665437  63.81665437  63.81665437 ...  63.81665437  63.81665437
   63.81665437]
 [ 63.81665437  63.81665437  63.81665437 ...  63.81665437  63.81665437
   63.81665437]]

print(trendV.shape)
print(trendM.shape)

(10512,)
(10512, 68)

dtanomJ = anomJ - trendM
dtanomJ.shape

(10512, 68)

vArea = np.cos(gpcpst[:,0]*np.pi/180.)
dtanomAW = np.sqrt(vArea)*dtanomJ.T

# computing SVD from the area-weighted anomaly matrix anomAW
U, S, Vh = np.linalg.svd(dtanomAW.T,full_matrices=False)

print(U.shape,S.shape,Vh.shape)

(10512, 68) (68,) (68, 68)

# define function
def mapmatr():
    q = np.zeros((144,73))
    for ii in range(0,144):
        q[ii,:] = U[:,0]/(np.sqrt(np.cos(gpcpst[:,0]
          *np.pi/180.)))
    return q

#Fig 10.10
# plot EOF1
# prepare figure
plt.figure(figsize=(12,12))
plt.subplot(211)

# create data using SVD 
mapmatr = (np.reshape(U[:,1]/(np.sqrt(np.cos(gpcpst[:,0]
          *np.pi/180.))),(73,144)))

mapmatr2 = mapmatr[:,range(0,mapmatr[0,:].size)]

# using the basemap package to put the cylindrical projection map into the image
eof = Basemap(projection='cyl',llcrnrlat=-90.,urcrnrlat=90.,\
            llcrnrlon=0.,urcrnrlon=360.,resolution='c')

# shifting the data because 
lons2, data2 = eof.shiftdata(lon_vals, datain = mapmatr2)

# draw coastlines, latitudes, and longitudes on the map
eof.drawcoastlines(color='black', linewidth=1)
eof.drawparallels(np.arange(-90.,91.,30.), labels = [1,0,0,1], fontsize=15)
eof.drawmeridians(np.arange(-180.,181.,60.), labels= [1,0,0,1], fontsize=15)
eof.drawcountries()

limits = np.linspace(-.04,.04,51)

# plot EOF1 on contour map
eof_plt = plt.contourf(np.array(lons2),np.array(lat_vals),data2,limits,cmap = "jet")

# set labels and colorbar
bounds = np.array([-0.04,-0.02,0,0.02,0.04])
efo = plt.colorbar(eof_plt, ticks=bounds, shrink = 0.85)
efo.ax.tick_params(labelsize='large')
plt.text(375, 93, 'Scale', size=16)
plt.xlabel('Longitude',labelpad = 30)
plt.ylabel('Latitude',labelpad =40)
plt.title('January EOF1 from 1948-2018 NCEP \n De-Trended Standardized Temp. Anomaly Data', pad = 10)

# retrieve PC1 from SVD 
pcdat = Vh[1,:]

# open bottom plot
plt.subplot(212)

# plot PC1
TIME = np.array(range(1948,2016))
plt.scatter(TIME,pcdat,marker = "o",facecolors='none',edgecolors='k')
plt.plot(TIME,pcdat,color='k')

# set labels 
plt.title("PC1 of NCEP RA Jan De-Trended Standardized SAT: 1948-2018", pad = 10)
plt.xlabel("Year")
plt.ylabel("PC Values")
plt.grid()
plt.tight_layout()
plt.show()

#Fig 10.11
# prepare figure
plt.figure(figsize=(12,12))

# create data from SVD 
mapmatr = (np.reshape(U[:,2]/(np.sqrt(np.cos(gpcpst[:,0]
          *np.pi/180.))),(73,144)))

mapmatr2 = mapmatr[:,range(0,mapmatr[0,:].size)]

plt.subplot(211)

# using the basemap package to put the cylindrical projection map into the image
eof = Basemap(projection='cyl',llcrnrlat=-90.,urcrnrlat=90.,\
            llcrnrlon=0.,urcrnrlon=360.,resolution='c')

# shifting the data 
lons2, data2 = eof.shiftdata(lon_vals, datain = mapmatr2)

# draw coastlines, latitudes, and longitudes on the map
eof.drawcoastlines(color='black', linewidth=1)
eof.drawparallels(np.arange(-90.,91.,30.), labels = [1,0,0,1], fontsize=15)
eof.drawmeridians(np.arange(-180.,181.,60.), labels= [1,0,0,1], fontsize=15)
eof.drawcountries()

limits = np.linspace(-.04,.04,51)

# plot EOF2 
eof_plt = plt.contourf(np.array(lons2),np.array(lat_vals),data2,limits,cmap = "jet")

# set labels and colorbar
bounds = np.array([-0.04,-0.02,0,0.02,0.04])
efo = plt.colorbar(eof_plt, ticks=bounds, shrink = 0.85)
efo.ax.tick_params(labelsize='large')
plt.text(375, 93, 'Scale', size=16)
plt.xlabel('Longitude',labelpad = 30)
plt.ylabel('Latitude',labelpad =40)
plt.title('January EOF2 from 1948-2018 NCEP \n De-Trended Standardized Temp. Anomaly Data', pad = 10)

# retrieve PC2 data
pcdat = Vh[2,:]

# open bottom plot
plt.subplot(212)

# plot PC1
TIME = np.array(range(1948,2016))
plt.scatter(TIME,-pcdat,marker = "o",facecolors='none',edgecolors='k')
plt.plot(TIME,-pcdat,color='k')

# set labels 
plt.title("PC2 of NCEP RA Jan. Standardized SAT: 1948-2018", pad = 10)
plt.xlabel("Year")
plt.ylabel("PC Values")
plt.grid()
plt.tight_layout()
plt.show()

---

10.4 Area-Weighted Average and Spatial Distribution of Trend

---

# EOF from de-trended data
monJ = np.array(range(0,816,12))
gpcpdat = GPCPST.iloc[:,2:817]
gpcpJ = gpcpdat.iloc[:,monJ]
climJ = gpcpJ.mean(axis=1)
sdJ = gpcpJ.std(axis=1)

# create desired variables for de-trended data
anomJ = ((gpcpdat.iloc[:,monJ].T-climJ)/sdJ).T
vArea = np.cos(gpcpst[:,0]*np.pi/180.)
anomA = vArea*anomJ.T
anomA.shape

(68, 10512)

# compute the de-trended data
JanSAT = anomA.T.sum(axis=0)/vArea.sum()
JanSAT.shape

(68,)

#Fig 10.12
# set up figure 
plt.figure(figsize=(12,7))

# create time variable
TIME = np.linspace(1948,2017, 68)

# create linear regression object
lm = linear_model.LinearRegression()
lm.fit(TIME.reshape(-1,1), JanSAT)
pred = lm.predict(TIME.reshape(-1,1))

# plot linear regression
plt.plot(TIME, pred, "-",color="red", linewidth=2)

# plot data 
plt.plot(TIME, JanSAT, "-ok")

# set labels 
plt.title("Global Average Jan. SAT Anomalies from NCEP RA", pad = 10)
plt.text(1950,0.35,"Linear Trend ${}\degree$C per Century".format(np.round(lm.coef_,3)[0]*100), color="r", fontsize=18)
plt.ylabel("Temperature [$\degree$C]")
plt.xlabel("Year")
plt.grid()
plt.show()

# recreate desired variables 
monJ = np.array(range(0,826,12))
gpcpdat = GPCPST.iloc[:,2:826]
gpcpJ = gpcpdat.iloc[:,monJ]
climJ = gpcpJ.mean(axis=1)

# define function and make instance of function 
def anomJ_func3():
    rslt = pd.DataFrame(np.zeros((10512, 69)),columns=tm1[monJ])
    for i in range(0,69):
        rslt.iloc[:,i] = gpcpdat.iloc[:,monJ[i]]-climJ
    return rslt

anomJ = anomJ_func3()

# create time variable
TIME = np.linspace(1948,2017, 69)

# create trend V function and instance of trend V
def trendV():
    trendV = np.zeros(10512)
    for i in range(0,10512):
        regr.fit(np.reshape(TIME,(69,1)),
                 np.reshape(np.array(anomJ.iloc[i,:]),(69,1)))
        trendV[i] = regr.coef_
    return trendV

trendV = trendV()

# reshape data into desired form 
mapmat1 = np.reshape([10*trendV],(73,144))

# create and use function
def mapv1_func():
    zeros = np.zeros((73,144))
    for i in range(0,73):
        for j in range(0,144):
            if mapmat1[i,j] >= 1.:
                zeros[i,j] = 1.
            else:
                zeros[i,j] = mapmat1[i,j]
    return zeros
mapv1 = mapv1_func()

# create and use function
def mapmat_func():
    zeros = np.zeros((73,144))
    for i in range(0,73):
        for j in range(0,144):
            if mapv1[i,j] <= -1.:
                zeros[i,j] = -1.
            else:
                zeros[i,j] = mapv1[i,j]
    return zeros

mapmat3 = mapmat_func()

#Fig 10.13
# prepare figure
plt.figure(figsize=(14., 7.))

# using the basemap package to put the cylindrical projection map into the image
trend = Basemap(projection='cyl',llcrnrlat=-90.,urcrnrlat=90.,\
            llcrnrlon=0.,urcrnrlon=360.,resolution='c')

# shifting the data because 
lons2, data3 = trend.shiftdata(lon_vals, datain = mapmat3, lon_0=180)

# draw coastlines, latitudes, and longitudes on the map
trend.drawcoastlines(color='black', linewidth=1)
trend.drawparallels(np.arange(-90.,91.,30.), labels = [1,0,0,1], fontsize=17)
trend.drawmeridians(np.arange(-180.,181.,60.), labels= [1,0,0,1], fontsize=17)
trend.drawcountries()

limits = np.linspace(-1.1,1.1,50)

# plot data 
eof_plt = plt.contourf(np.array(lons2),np.array(lat_vals),data3,limits,cmap = "jet")

# set labels and colorbar
efo = plt.colorbar(eof_plt, ticks=[1,0.5,0,-0.5,-1], shrink = 0.80)
plt.text(375, 95, 'Scale', size=16)
plt.xlabel('Longitude',labelpad = 40)
plt.ylabel('Latitude',labelpad = 65)
plt.title('Trend of the NCEP RA1 Jan. 1948-2018 \n Anomaly Temp. [$\degree$C/Century]', pad = 8)

plt.show()

---

10.5 GPCP Precipitation Data: Analysis and Visualization by Python

---

#Download the following data set from NOAA into your active folder
#https://www.esrl.noaa.gov/psd/repository/entry/show?
# entryid=synth%3Ae570c8f9-ec09-4e89-93b4-babd5651e7a9%
# 3AL25jZXAucmVhbmFseXNpcy5kZXJpdmVkL3N1cmZhY2UvYWlyLm1vbi5tZWFuLm5j

ncd = nc.Dataset("precip.mon.mean.nc","r+")

# define variables
lon_vals = ncd.variables['lon'][:]
lat_vals = ncd.variables['lat'][:]
time = ncd.variables['time']
time_unit = time.units
precnc = ncd.variables['precip']

#Fig 10.14
# plot a 90S-90N temp along a meridional line at 180E
nc_x = np.linspace(-90,90,72)
plt.plot(nc_x,precnc[0,:,63],color='k' )

# set variables
plt.xlabel('Latitude')
plt.ylabel('Precipitation [mm/day]')
plt.title('90S-90N Precipitation along a Meridional Line at 160E: Jan. 1979', size=20)
plt.grid()
plt.show()

# write the data as space-time matrix with a header
precst = np.zeros((10368,451))
temp = precnc[0,:,:]

# fill precst with data from precnc 
for i in range(451):
    precst[:,i] = precnc[i,:,:].flatten()

# create and reshape Lat and Lon variable for DataFrame
LAT = list(lat_vals.data) * 144
LON1 = [np.repeat(i,72) for i in lon_vals.data]
LON = [i for j in LON1 for i in j]

gpcpst_ = pd.DataFrame({"Lat":LAT,"Lon":LON})
gpcpst = pd.concat([gpcpst_, pd.DataFrame(precst)], axis=1)

gpcpst.iloc[889:892,:5]

	Lat	Lon	0	1	2
889	-26.25	31.25	0.057521	0.180531	0.210445
890	-23.75	31.25	0.108556	0.256279	0.288389
891	-21.25	31.25	0.087190	0.273805	0.256987

# create desired Colormap
colors = ["black", "black","purple","crimson","red","red","red","tomato","lightpink",
          "pink","peachpuff","yellow","yellow", "yellowgreen","limegreen","limegreen","mediumaquamarine",
          "skyblue","lightskyblue","wheat","wheat"]
colors.reverse()
myColMap = newColMap(colors)

#Fig 10.15(a)
# prepare variables for plotting 
climmat = precnc[0,:,:]
for i in range(451):
    climmat = climmat + precnc[i,:,:]

climmat = climmat/451
mapmat = climmat

# suppress values into the (0,10) range 
for i in range(72):
    for j in range(144):
        if mapmat[i,j] > 10:
            mapmat[i,j] = 10
        if mapmat[i,j] < 0:
            mapmat[i,j] = 0

myint = np.linspace(0, 10,11)

# prepare figure
plt.figure(figsize=(15., 7.))

# using the basemap package to put the cylindrical projection map into the image
trend = Basemap(projection='cyl',llcrnrlat=-90.,urcrnrlat=90.,\
            llcrnrlon=0.,urcrnrlon=360.,resolution='c')

# shifting the data because 
lons2, data3 = trend.shiftdata(lon_vals, datain = mapmat, lon_0=180)

# draw coastlines, latitudes, and longitudes on the map
trend.drawcoastlines(color='black', linewidth=1)
trend.drawparallels(np.arange(-90.,91.,30.), labels = [1,0,0,1], fontsize=15)
trend.drawmeridians(np.arange(-180.,181.,60.), labels= [1,0,0,1], fontsize=15)
trend.drawcountries()

# plot data 
eof_plt = plt.contourf(np.array(lons2),np.array(lat_vals),mapmat,myint,cmap = myColMap)

# set labels and colorbar
efo = plt.colorbar(eof_plt, ticks=[10,9,8,7,6,5,4,3,2,1])
plt.xlabel('Longitude',labelpad = 30)
plt.ylabel('Latitude',labelpad =40)
plt.title('GPCP 1979-2016 Precipitation Climatology [mm/day]', size=18, fontweight="bold")

plt.show()

mapmat.shape

(72, 144)

# create and prepare data 
sdmat = (precnc[0,:,:]-climmat)**2
for i in range(1,451):
    sdmat = sdmat + (precnc[i,:,:]-climmat)**2

sdmat = np.sqrt(sdmat/451)
mapmat = sdmat

# suppress values into the (0,5) range 
for i in range(72):
    for j in range(144):
        if mapmat[i,j] > 5:
            mapmat[i,j] = 5
        if mapmat[i,j] < 0:
            mapmat[i,j] = 0

#Fig 10.15(b)
myint = np.linspace(0, 5,25)

# prepare figure
plt.figure(figsize=(15., 7.))

# using the basemap package to put the cylindrical projection map into the image
trend = Basemap(projection='cyl',llcrnrlat=-90.,urcrnrlat=90.,\
            llcrnrlon=0.,urcrnrlon=360.,resolution='c')

# shifting the data because 
lons2, data3 = trend.shiftdata(lon_vals, datain = mapmat, lon_0=180)

# draw coastlines, latitudes, and longitudes on the map
trend.drawcoastlines(color='black', linewidth=1)
trend.drawparallels(np.arange(-90.,91.,30.), labels = [1,0,0,1], fontsize=15)
trend.drawmeridians(np.arange(-180.,181.,60.), labels= [1,0,0,1], fontsize=15)
trend.drawcountries()

# plot data 
eof_plt = plt.contourf(np.array(lons2),np.array(lat_vals),mapmat,myint,cmap = myColMap)

# set labels and colorbar
efo = plt.colorbar(eof_plt, ticks=[5,4.5,4,3.5,3,2.5,2,1.5,1,0.5,0])
plt.xlabel('Longitude',labelpad = 30)
plt.ylabel('Latitude',labelpad =40)
plt.title('GPCP 1979-2016 Standard Deviation of Precipitation [mm/day]', size=18, fontweight="bold", pad = 10)
plt.show()

---

Chapter 11: Python Analysis of Incomplete Climate Data

---

import numpy as np 
import matplotlib
from matplotlib import cm as cm1
import matplotlib.pyplot as plt
import pandas as pd
import warnings
import os
warnings.filterwarnings("ignore")

from scipy import optimize as opt
import sympy as sm
import math as m
import os
import cartopy 
import netCDF4 as nc
from matplotlib.colors import LinearSegmentedColormap
from cartopy.mpl.ticker import LongitudeFormatter, LatitudeFormatter
from mpl_toolkits.basemap import Basemap, cm, shiftgrid, addcyclic
from matplotlib.patches import Polygon

from datetime import datetime

import cartopy.crs as ccrs
import cartopy.feature as cfeature

from scipy.ndimage import gaussian_filter
from sklearn.linear_model import LinearRegression

Starting with Figure 11.4, due to the fact that Figure 11.1-11.3 are snapshots of the NCDC NOAA data access tool. Please follow link below for more information regarding these plots. www.ncdc.noaa.gov/data-access/marineocean-data/noaa-global-surface-temperature-noaaglobaltemp

---

11.2 Read NOAAGlobalTemp and From the Space-Time Data Matrix

---

# open .asc file 
file = open('NOAAGlobalTemp.gridded.v4.0.1.201701.asc', "r")

# read float numbers from the file
da1 = []
for line in file:
    x = line.strip().split()
    for f in x:
        da1.append(float(f))

len(da1)

4267130

print(da1[0 : 3])

# create time variables
tm1 = np.arange(0, 4267129, 2594)
tm2 = np.arange(1, 4267130, 2594)
print(len(tm1))
print(len(tm2))

# extract months
mm1 = []   
for i in range(len(tm1)):
    mm1.append(da1[tm1[i]])
    
# extract years
yy1 = []
for i in range(len(tm2)):
    yy1.append(da1[tm2[i]])
    
# combine YYYY with MM
rw1 = []
for i in range(len(mm1)):
    rw1.append(str(int(yy1[i])) + "-" + str(int(mm1[i])))    
print(mm1[0 : 6])
print(yy1[0 : 6])
print(len(mm1))
print(len(yy1))

print(tm1[0 : 6])
print(tm2[0 : 6])

tm3 = pd.concat([pd.DataFrame(tm1), pd.DataFrame(tm2)], axis = 1)

# remove the months and years data from the scanned data
tm4 = [] 
for r in range(len(tm1)):
    tm4.append(tm1[r])
    tm4.append(tm2[r])

tm4.reverse()
da2 = da1.copy() 
for f in tm4:
    del da2[f]

print(len(da2)/(36*72)) # months, 137 yrs 1 mon: Jan 1880-Jan 2017

# generates the space-time data
# 2592 (=36*72) rows and 1645 months (=137 yrs 1 mon)
var = [[0 for x in range(1645)] for y in range(2592)]
i = 0
for c in range(1645):
    for r in range(2592):
        var[r][c] = da2[i]
        i += 1

da3 = pd.DataFrame(var, columns = rw1)

lat1=np.linspace(-87.5, 87.5, 36).tolist()
lon1=np.linspace(2.5, 357.5, 72).tolist()

Lat = sorted(lat1 * 72)
Lon = lon1 * 36

rw1.insert(0, "LAT")
rw1.insert(1, "LON")
gpcpst = pd.concat([pd.DataFrame(Lat), pd.DataFrame(Lon), 
                    pd.DataFrame(da3)], axis = 1)
gpcpst.columns = rw1

[1.0, 1880.0, -999.9]
1645
1645
[1.0, 2.0, 3.0, 4.0, 5.0, 6.0]
[1880.0, 1880.0, 1880.0, 1880.0, 1880.0, 1880.0]
1645
1645
[    0  2594  5188  7782 10376 12970]
[    1  2595  5189  7783 10377 12971]
1645.0

# the first two columns are Lat and Lon
# -87.5 to 87.5 and then 2.5 to 375.5
# the first row for time is header, not counted as data. 
gpcpst.to_csv("NOAAGlobalT1.csv") 
# Output the data as a csv file

#Fig 11.4
# column '2015-12' corresponding to Dec 2015
# convert the vector into a lon-lat matrix for contour map plotting
# compresses numbers to [-6, 6] range
mapmat = np.reshape(gpcpst['2015-12'].values, (-1, 72)) #36 X 72
# this command compresses numbers to the range of -6 to 6
mapmat = np.maximum(np.minimum(mapmat, 6), -6)  

dpi = 100
fig = plt.figure(figsize = (1100/dpi, 1100/dpi), dpi = dpi)
ax  = fig.add_axes([0.1, 0.1, 0.8, 0.9])

# create map
dmap = Basemap(projection = 'cyl', llcrnrlat = -87.5, urcrnrlat = 87.5,
              resolution = 'c',  llcrnrlon = 2.5, urcrnrlon = 357.5)

# draw coastlines, state and country boundaries, edge of map
dmap.drawcoastlines()
# dmap.drawstates()
dmap.drawcountries()

# create and draw meridians and parallels grid lines
dmap.drawparallels(np.arange( -50, 100, 50.), labels = [1, 0, 0, 0], 
                  fontsize = 15)
dmap.drawmeridians(np.arange(50, 350, 50.), labels = [0, 0, 0, 1], 
                  fontsize = 15)

# convert latitude/longitude values to plot x/y values
# need to reverse sequence of lat1
# lat1_rev = lat1[::-1] 
lat1_rev = lat1
x, y = dmap(*np.meshgrid(lon1, lat1_rev))

# contour levels
clevs = np.arange(-7, 7, 0.5)
print(x.shape,y.shape, mapmat.shape)

# draw filled contours
cnplot = dmap.contourf(x, y, mapmat, clevs, cmap = plt.cm.jet)

# add colorbar
# pad: distance between map and colorbar
cbar = dmap.colorbar(cnplot, location = 'right', pad = "2%", ticks=[-6,-4,-2,0,2,4,6])      
cbar.ax.set_yticklabels(['-6','-4','-2','0','2','4','6'])
plt.text(362, 90, '[$\degree$C]', size=20) # add colorbar title string

# add plot title
plt.title('NOAAGlobalTemp Anomalies [$\degree$C]: Dec. 2015')

# label x and y
plt.xlabel('Longitude', labelpad = 30)
plt.ylabel('Latitude', labelpad = 40)

# display on screen
plt.show()

(36, 72) (36, 72) (36, 72)

# keep only the data for the Pacific region for years from 1951-2000 

# get data for region with -20 < Latitude < 20 and 160 < Longitude < 260
n2 = gpcpst[(gpcpst['LAT'] > -20) & (gpcpst['LAT'] < 20) & 
            (gpcpst['LON'] > 160) & (gpcpst['LON'] < 260)]
print(gpcpst.shape)
print(n2.size)

# from 1951-2000 
# note Jan 1951 data starts from column 854 (note python is zero based)
pacificdat = n2.iloc[:, 854 : 1454]

(2592, 1647)
263520

#Fig 11.5
Lat = np.arange(-17.5, 22.5, 5)
Lon = np.arange(162.5, 262.5, 5)

plt.close()

# get Dec 1997 data from pacificdat (tropical region)
mapmat = np.reshape(pacificdat['1997-12'].values, (8, -1)) # 8 * 20

fig = plt.figure(figsize=(1100/dpi, 1100/dpi), dpi = dpi)
ax  = fig.add_axes([0.1, 0.1, 0.8, 0.9])

# create map
smap = Basemap(projection = 'cyl', llcrnrlat = -40, urcrnrlat = 40,
              resolution = 'c', llcrnrlon = 120, urcrnrlon = 300)

# draw coastlines, state and country boundaries, edge of map
smap.drawcoastlines()
smap.drawcountries()

# create and draw meridians and parallels grid lines
smap.drawparallels(np.arange( -40, 60, 20.),labels=[1, 0, 0, 0],fontsize = 15)
smap.drawmeridians(np.arange(50, 350, 50.),labels=[0, 0, 0, 1],fontsize = 15)

# convert latitude/longitude values to plot x/y values
x, y = smap(*np.meshgrid(Lon, Lat))

# contour levels
clevs = np.arange(-4, 4.2, 0.1)

# draw filled contours
cnplot = smap.contourf(x, y, mapmat, clevs, cmap = plt.cm.jet)

# add colorbar
# pad: distance between map and colorbar
cbar = smap.colorbar(cnplot, location = 'right', pad = "2%", ticks=[-4,-2,0,2,4]) 
# add colorbar title string
plt.text(302, 42, '[$\degree$C]', size=20)

# add plot title
plt.title('Tropic Pacific SAT Anomalies [$\degree$C]: Dec. 1997')

# label x and y
plt.xlabel('Longitude', labelpad = 30)
plt.ylabel('Latitude', labelpad = 40)

# display on screen
plt.show()

---

11.3 Spatial Averages and Their Trends

---

# prepare the data
temp = gpcpst.copy()
areaw = np.zeros((2592, 1647))
areaw.shape 
areaw[:, 0] = temp['LAT'].values
areaw[:, 1] = temp['LON'].values
# convert degrees to radians 
veca=np.cos(temp['LAT'].values * np.pi / 180).tolist()
tempvalues = temp.values # get the underlying ndarray from the temp DataFrame
# area-weight matrix equal to cosine lat of the boxes with data 
# and to zero for the boxes of missing data -999.9
for j in range(2, 1647):
    for i in range(0, 2592):
        if tempvalues[i, j] > -290.0:
            areaw[i][j] = veca[i]

# area-weight data matrix first two columns as lat-lon
# create monthly global average vector for 1645 months (Jan 1880 - Jan 2017)
tempw = np.multiply(areaw, tempvalues)
tempw[:,0] = temp['LAT'].values
tempw[:,1] = temp['LON'].values
avev = np.divide(tempw.sum(axis=0)[2:1647], areaw.sum(axis = 0)[2:1647])

#Fig 11.6
plt.close()

# create time variable 
timemo = np.linspace(1880, 2017, 1645)

# create plot and plot labels 
plt.plot(timemo, avev, '-', color="k")
plt.title('Area-Weighted Global Average of Monthly \n SAT Anomalies: Jan. 1880 - Jan. 2017')
plt.xlabel('Year')
plt.ylabel('Temperature Anomaly [$\degree$C]')

plt.yticks([-1,-0.5,0,0.5,1])

# create linear regression model
lm = LinearRegression()
lm.fit(timemo.reshape(-1, 1), avev)
predictions = lm.predict(timemo.reshape(-1, 1))

# plot linear regression model
plt.plot(timemo, predictions, '-', color="blue",linewidth=2)


# add text indicating the trend 
plt.text(1885, .55, 'Linear Trend: 0.69 [$\degree$C] per Century', fontsize=16, color="blue")
plt.grid()
plt.show()  

gpcpst

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2592 rows × 1647 columns

# Edmonton, Canada
n2_0 = set(np.where(gpcpst.LAT == 52.5)[0])
n2_1 = set(np.where(gpcpst.LON == 247.5)[0])
print(n2_0.intersection(n2_1))

# San Diego, USA
n2_3 = set(np.where(gpcpst.LAT == 32.5)[0])
n2_4 = set(np.where(gpcpst.LON == 242.5)[0])
print(n2_3.intersection(n2_4))

{2065}
{1776}

#Fig 11.7
# extract data for a specified box with given lat and lon
n2 = 2065
dedm = gpcpst.iloc[n2,854:1453]
t = np.linspace(1880,2017, len(dedm))

# set up figure and plot Edmonton data
fig = plt.figure(figsize=(12,10))
plt.plot(t, dedm, "red", label="Edmonton, Canada: Trend 1.18 $\degree$C/Century, SD 3.01 $\degree$C")

# set plot labels
plt.title("Monthly Temperature Anomalies History of \n Edmonton, Canada, and San Diego, USA")
plt.ylabel("Temperature Anomalies [$\degree$C]")
plt.xlabel("Year")
plt.ylim((-15,15))
plt.yticks(rotation=90)

# plot linear regression
lm = LinearRegression()
lm.fit(t.reshape(-1, 1), dedm)
predictions = lm.predict(t.reshape(-1, 1))
plt.plot(t, predictions, '-', color="red", linewidth = 2)

# create San Diego Data 
n2 = 1776
dsan = gpcpst.iloc[n2,854:1453]
t = np.linspace(1880,2017, len(dsan))

# plot san diego data 
plt.plot(t, dsan, "blue", label="San Diego, USA: Trend 0.76 $\degree$C/Century, SD 0.87 $\degree$C")

# plot linear regression
lmsan = LinearRegression()
lmsan.fit(t.reshape(-1, 1), dsan)
predictions = lmsan.predict(t.reshape(-1, 1))
plt.plot(t, predictions, '-', color="blue", linewidth = 2)

plt.legend()
plt.grid()
plt.show()

# print linear model coefficients and standard deviation of each data set
print(lm.coef_)
print(lmsan.coef_)
print(np.std(dedm))
print(np.std(dsan))

[0.01200846]
[0.00760766]
3.0049052677575125
0.8650167715422953

# download the NCEI spatial average time series of monthly data
# https://www1.ncdc.noaa.gov/pub/data/noaaglobaltemp/operational/
# timeseries/aravg.mon.land_ocean.90S.90N.v4.0.1.201702.asc
aveNCEI = pd.read_table('http://shen.sdsu.edu/data/' + 
                        'aravg.mon.land_ocean.90S.90N.v4.0.1.201703.asc.txt', 
                        header = None, delim_whitespace = True)
aveNCEI.shape 

aveNCEI = aveNCEI.values

print(aveNCEI.shape) #(1647,10)

# create a time variable that spans the needed years
timeo = np.linspace(aveNCEI[0,0], aveNCEI[-1,0],len(aveNCEI[:,0]))

# create a matrix of 136 years of data matrix
# row = year from 1880 to 2016 column = month 1 to 12
ave = aveNCEI[:,2]
myear = len(ave)/12
nyear = int(np.floor(myear))
nmon = nyear*12

# reshape the monthly averages into a matrix
avem = ave[0:nmon].reshape((137,12))

# compute annual average 
annv = np.zeros(137)
for i in range(0,nyear):
    annv[i] = np.mean(avem[i,:])

# put the monthly averages and annual ave in a matrix
avemy = pd.DataFrame(avem)

# add a column that indicated the averages 
avemy["Average"] = annv

(1647, 10)

avemy

	0	1	2	3	4	5	6	7	8	9	10	11	Average
0	-0.254882	-0.390030	-0.414509	-0.314104	-0.327618	-0.421961	-0.386787	-0.303434	-0.307535	-0.395433	-0.490574	-0.357690	-0.363713
1	-0.281037	-0.298598	-0.256906	-0.201682	-0.233895	-0.348272	-0.274636	-0.282329	-0.395464	-0.437131	-0.444607	-0.331158	-0.315476
2	-0.174060	-0.240485	-0.210161	-0.390328	-0.408842	-0.392581	-0.300641	-0.236459	-0.217395	-0.425229	-0.352755	-0.452639	-0.316798
3	-0.524041	-0.563601	-0.419921	-0.443575	-0.402770	-0.259521	-0.315984	-0.278074	-0.370997	-0.376471	-0.392778	-0.358323	-0.392171
4	-0.445016	-0.421579	-0.493839	-0.453975	-0.390322	-0.447676	-0.534206	-0.499795	-0.449471	-0.386881	-0.494061	-0.428869	-0.453807
...	...	...	...	...	...	...	...	...	...	...	...	...	...
132	0.165989	0.153761	0.240487	0.463265	0.463908	0.455566	0.453870	0.445553	0.501094	0.490923	0.488859	0.185138	0.375701
133	0.331214	0.365247	0.333766	0.292351	0.456119	0.433921	0.435057	0.435280	0.463543	0.466267	0.605887	0.420357	0.419917
134	0.435523	0.207236	0.486705	0.530473	0.535486	0.511096	0.465122	0.572558	0.567807	0.571151	0.466506	0.553309	0.491914
135	0.558792	0.613820	0.617789	0.507067	0.599994	0.633337	0.574553	0.645725	0.705505	0.774835	0.738430	0.836829	0.650556
136	0.791078	0.922814	0.949248	0.804055	0.623138	0.654642	0.636169	0.675929	0.654386	0.519169	0.519705	0.519317	0.689137

137 rows × 13 columns

#Fig 11.8
# plot 12 panels on the same figure: Trend for each month
fig = plt.figure(figsize=(15,15))

# create time variable
timeyr = np.linspace(aveNCEI[0,0],aveNCEI[0,0]+nyear-1, 137)

# locate and plot each month on seperate panels
for i in range(0,12):
    plt.subplot(4,3,i+1)
    plt.plot(timeyr, avemy.iloc[:,i], color="black")
    plt.title("Month = {}".format(i+1), size=14)
    plt.yticks(rotation=90)
    plt.ylabel("Temp. [$\degree$C]", size=14)
    plt.xlabel("Year", size=14)
    
    #plot linear regression
    lm = LinearRegression()
    lm.fit(timeyr.reshape(-1, 1), avemy.iloc[:,i])
    predictions = lm.predict(timeyr.reshape(-1, 1))
    plt.plot(timeyr, predictions, '-', color="red")
    plt.text(1890,0.75, "Trend $\degree$C/Century = {}".format(np.round(lm.coef_,5)[0]*100), size=14, color="red")
    plt.grid()
    plt.ylim((-1,1))
    plt.xticks([1880,1920,1960,2000], size =14)
    plt.yticks([-1,-0.5,0,0.5,1], size =14)

# allows no overlap betweeen monthly plots
plt.tight_layout()    

plt.show()

# create the differenced data
avediff = np.subtract(avev, aveNCEI[0:1645, 2]) 

#Fig 11.9
# compute annual average, then plot the annual mean global average temp
avem = np.reshape(avev[0 : 1644], (-1, 12)) 
avem.shape #size 137 * 12, with row for the year, column for the month

# average over columns (annual average over 12 months), 
# there are 137 years, 1880/01 - 2016/12
annv = np.mean(avem, axis = 1)  

# plot the annual mean global average temp (1880-1996)
timeyr = np.arange(1880, 2017)

plt.close()
plt.step(timeyr, annv, '-', color="black")
plt.title('Area-Weighted Global Average of \n Annual SAT Anomalies: 1880-2016')
plt.xlabel('Year')
plt.ylabel('Temperature Anomaly [$\degree$C]')

# plot linear regression
lm = LinearRegression()
lm.fit(timeyr.reshape(-1, 1), annv)
predictions = lm.predict(timeyr.reshape(-1 , 1))

plt.plot(timeyr, predictions, '-', color="purple")


# create text on plot
plt.text(1900, .45, 'Linear Trend: 0.69 [$\degree$C]/Century', color="purple", fontsize=16)
plt.text(1900, 0.05, 'Base Line', color="red", fontsize=16)

# plot standard abline
plt.plot(timeyr, np.zeros(137), 'r-')
plt.grid()
plt.show()

# one can compare with the NOAAGlobalTemp annual time series
# there are some small differences
aveannNCEI = pd.read_table('http://shen.sdsu.edu/data/' + 
                          'aravg.ann.land_ocean.90S.90N.v4.0.1.201703.asc.txt', 
                          header = None, delim_whitespace = True)
aveannNCEI.shape 

aveannNCEI = aveannNCEI.values 
diff2 = np.subtract(annv, aveannNCEI[0 : 137, 1])

# find the value range (minimum and maximum values in diff2 array)
[np.amin(diff2, axis = 0), np.amax(diff2, axis = 0)] 

[-0.01609496875429861, 0.007508731446406958]

#Fig 11.10
# polynomial fitting to the global average annual mean
# the following polynomial fitting from numpy.polyfit is NOT orthogonal, 
# we know orthogonal polynomial fitting is usually better as we saw in the R code implemention
polyor9 = np.polyfit(timeyr, annv, 9)
polyor20 = np.polyfit(timeyr, annv, 20)

plt.close()
plt.step(timeyr, annv, '-', color="black")
plt.title('Annual SAT Time Series and Its Orthogonal \n Polynomial Fits: 1880-2016', pad = 10)
plt.xlabel('Year')
plt.ylabel('Temperature Anomaly [$\degree$C]')

# plot the first poly fit 
plt.plot(timeyr, np.polyval(polyor9, timeyr), "-", color="purple", 
         label = "9th Order Orthogonal Polynomial Fit")

# plot the second poly fit 
plt.plot(timeyr, np.polyval(polyor20, timeyr), "-r",  
         label = "20th Order Orthogonal Polynomial Fit")

plt.legend(loc = 'upper left')
plt.grid()
plt.show()

---

11.4 Spatial Characteristics of the Temperature Change Trends

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# Compute the trend for each box for the 20th century
timemo1 = np.linspace(1900, 2000, 1200)
temp1 = temp.copy()

# replace missing values (any value less than -490.00) with NaN in temp1 dataframe
temp1 = temp1.where(temp1 >= -490.0, np.NaN)
trendgl = np.zeros(2592)

# for grids in (Lon: 2.5-257.5, Lat: -87.5-87.5), perform linear regression 
# fitting on each grid temperature over months between Jan 1900 and Dec 1999
# linear slope cofficient will be saved in trendgl list to be used for 
# subsequent plotting 
for i in range(0, 2592):
    if not(np.isnan(temp1.iloc[i, 242]) or np.isnan(temp1.iloc[i, 1441])): 
        # column 242 corresponds to month 1990-01, 
        # and column 1441 corresponds to month 1999-12
        # temperature data for a grid from 1990-01 to 1999-12
        gridtemp = temp1.iloc[i, 242 : 1442].values 
        # filter out any month where grid temperature is missing 
        # (i.e., having a NaN value)
        idx = np.isfinite(timemo1) & np.isfinite(gridtemp) 
        # check if there is at least one month that has 
        # non-NaN number for linear regression
        if np.any(idx[:]): 
            ab = np.polyfit(timemo1[idx], gridtemp[idx], 1)
            # ab[0] has the linear regression slope coefficient
            trendgl[i] = ab[0] 
        else:
            trendgl[i] = np.NaN
    else:
        trendgl[i] = np.NaN

#Fig 11.11
mapmat = np.reshape([n * 100 for n in trendgl], (-1, 72)) #36 X 72 

# this command compresses numbers to the range of -2 to 2 
mapmat = np.maximum(np.minimum(mapmat, 2), -2) 

# set up figure
dpi = 100
fig = plt.figure(figsize = (1100 / dpi, 1100 / dpi), dpi = dpi)
ax  = fig.add_axes([0.1, 0.1, 0.8, 0.9])

# create map
amap = Basemap(projection = 'cyl', llcrnrlat = -87.5, urcrnrlat = 87.5,
              resolution = 'c',  llcrnrlon = 2.5, urcrnrlon = 357.5)

# draw coastlines, state and country boundaries, edge of map
amap.drawcoastlines()
# amap.drawstates()
amap.drawcountries()

# create and draw meridians and parallels grid lines
amap.drawparallels(np.arange( -50, 100, 50.),
                  labels = [1, 0, 0, 0], fontsize = 15)
amap.drawmeridians(np.arange(50, 350, 50.), 
                  labels = [0, 0, 0, 1], fontsize = 15)

# convert latitude/longitude values to plot x/y values
x, y = amap(*np.meshgrid(lon1, lat1))

# contour levels
clevs = np.linspace(-2, 2, 21)

# draw filled contours
cnplot = amap.contourf(x, y, mapmat, clevs, cmap = plt.cm.jet)

# add colorbar
# pad: distance between map and colorbar
cbar = amap.colorbar(cnplot, location = 'right', pad = "2%", ticks=[-2,-1,0,1,2])      
              
# add plot title
plt.title('Jan. 1900 - Dec. 1999 Temperature Trends: [$\degree$C/century]', fontsize=18)

# label x and y
plt.xlabel('Longitude', labelpad = 30)
plt.ylabel('Latitude', labelpad = 40)

plt.show()

# Compute trend for each box for the 20th century - Version 2: Allow 2/3 of data, i.e., 1/3 missing
# compute the trend and plot the 20C V2 trend map 
timemo1 = np.linspace(1900, 2000, 1200)
temp1 = temp.iloc[:, 242 : 1442]

# replace missing values (any value less than -490.00) 
# with NaN in temp1 dataframe
temp1 = temp1.where(temp1 >= -490.0, np.NaN)
temptf = np.isnan(temp1)

countOfNotMissing = np.zeros(2592)

# count the number of missing data points
for i in range(0, 2592):
    countOfNotMissing[i] = temptf.iloc[i, :].values.tolist().count(False)

trend20c = np.zeros(2592)
for i in range(0, 2592):
    if countOfNotMissing[i] > 800: # allow 2/3 of data, i.e., 1/3 missing
        # column 242 corresponds to month 1990-01, 
        # and column 1441 corresponds to month 1999-12
        gridtemp = temp1.iloc[i, :].values # temperature for a grid 
                                           # from 1990-01 to 1999-12
        # filter out any month where grid temperature is missing 
        # (i.e., having a NaN value)
        idx = np.isfinite(timemo1) & np.isfinite(gridtemp) 
        ab = np.polyfit(timemo1[idx], gridtemp[idx], 1)
        # ab[0] has the linear regression slope coefficient
        trend20c[i] = ab[0] 
    else:
        trend20c[i] = np.NaN
    

temp1.shape

(2592, 1200)

#Fig 11.12
mapmat = np.reshape([n * 120 for n in trend20c], (-1, 72)) #36 X 72 
mapmat = mapmat/10

# this command compresses numbers to the range of -.2 to .2
mapmat = np.maximum(np.minimum(mapmat, .2), -.2)  

dpi = 100
fig = plt.figure(figsize = (1100 / dpi, 1100 / dpi), dpi = dpi)
ax  = fig.add_axes([0.1, 0.1, 0.8, 0.9])

# create map
fmap = Basemap(projection = 'cyl', llcrnrlat = -87.5, urcrnrlat = 87.5,
              resolution = 'c',  llcrnrlon = 2.5, urcrnrlon = 357.5)

# draw coastlines and country boundaries, edge of map
fmap.drawcoastlines()
fmap.drawcountries()

# create and draw meridians and parallels grid lines
fmap.drawparallels(np.arange( -50, 100, 50.), 
                  labels = [1, 0, 0, 0], fontsize = 15)
fmap.drawmeridians(np.arange(50, 350, 50.), 
                  labels = [0, 0, 0, 1], fontsize = 15)

# convert latitude/longitude values to plot x/y values
x, y = fmap(*np.meshgrid(lon1, lat1))

# contour levels
clevs = np.linspace(-.2, .2, 41)

# draw filled contours
cnplot = fmap.contourf(x, y, mapmat, clevs, cmap = plt.cm.jet)

# add colorbar
# pad: distance between map and colorbar
cbar = fmap.colorbar(cnplot, location = 'right', pad = "2%", ticks=[0.2,0.1,0,-0.1,-0.2])      

              
# add plot title
plt.title('Jan. 1900 - Dec. 1999 Temperature Trends: [$\degree$C/Decade]', fontsize=18)

# label x and y
plt.xlabel('Longitude', labelpad = 30)
plt.ylabel('Latitude', labelpad = 40)

plt.show()

# compute trend for each box for the 20th century - Version 2: Allow 2/3 of data, i.e., 1/3 missing
# compute the trend and plot the 20C V2 trend map 
timemo1 = np.linspace(1976, 2016, 492)
temp1 = temp.iloc[:, 1154 : 1646]

# replace missing values (any value less than -490.00) 
# with NaN in temp1 dataframe
temp1 = temp1.where(temp1 >= -490.0, np.NaN)
temptf = np.isnan(temp1)

countOfNotMissing = np.zeros(2592)

for i in range(0, 2592):
    countOfNotMissing[i] = temptf.iloc[i, :].values.tolist().count(False)

trend20c = np.zeros(2592)
for i in range(0, 2592):
    if countOfNotMissing[i] > 320: # allow 2/3 of data, i.e., 1/3 missing
        # column 1154 corresponds to month 1976-01, 
        # and column 1646 corresponds to month 2016-12
        gridtemp = temp1.iloc[i, :].values # temperature for a grid 
                                           # from 2003-06 to 2016-12
        # filter out any month where grid temperature is missing 
        # (i.e., having a NaN value)
        idx = np.isfinite(timemo1) & np.isfinite(gridtemp) 
        ab = np.polyfit(timemo1[idx], gridtemp[idx], 1)
        # ab[0] has the linear regression slope coefficient
        trend20c[i] = ab[0] 
    else:
        trend20c[i] = np.NaN

temp1.shape

(2592, 492)

#Fig 11.13
mapmat = np.reshape([n * 120 for n in trend20c], (-1, 72)) #36 X 72 
# this command compresses numbers to the range of -4 to 4
mapmat = np.maximum(np.minimum(mapmat, 4), -4)  

dpi = 100
fig = plt.figure(figsize = (1100 / dpi, 1100 / dpi), dpi = dpi)
ax  = fig.add_axes([0.1, 0.1, 0.8, 0.9])

# create map
fmap = Basemap(projection = 'cyl', llcrnrlat = -87.5, urcrnrlat = 87.5,
              resolution = 'c',  llcrnrlon = 2.5, urcrnrlon = 357.5)

# draw coastlines, state and country boundaries, edge of map
fmap.drawcoastlines()
# fmap.drawstates()
fmap.drawcountries()

# create and draw meridians and parallels grid lines
fmap.drawparallels(np.arange( -50, 100, 50.), 
                  labels = [1, 0, 0, 0], fontsize = 15)
fmap.drawmeridians(np.arange(50, 350, 50.), 
                  labels = [0, 0, 0, 1], fontsize = 15)

# convert latitude/longitude values to plot x/y values
x, y = fmap(*np.meshgrid(lon1, lat1))

# contour levels
clevs = np.linspace(-4, 4, 41)

# draw filled contours
cnplot = fmap.contourf(x, y, mapmat, clevs, cmap = plt.cm.jet)

# add colorbar
# pad: distance between map and colorbar
cbar = fmap.colorbar(cnplot, location = 'right', pad = "2%", ticks=[-4,-2,0,2,4])      

              
# add plot title
plt.title('Jan. 1976 - Dec. 2016 Temperature Trends: [$\degree$C/Decade]', fontsize=18)

# label x and y
plt.xlabel('Longitude', labelpad = 30)
plt.ylabel('Latitude', labelpad = 40)

plt.show()