# Theory and Applications

### SSP Shen and RCJ Somerville

Version 1.0 released in July 2019 and coded by Dr. Samuel Shen, Distinguished Professor
San Diego State University, California, USA
Email:

Version 2.0 compiled and updated by Momtaza Sayd
San Diego State University May 2021
Version 3.0 compiled and updated by Joaquin Stawsky
San Diego State University June 2022

## Chapter 4: Climate Data Matrices and Linear Algebra

### Matrix algebra

``````#Matrix subtraction
matrix(c(1,1,1,-1), nrow = 2) - matrix(c(1,3,2,0), nrow = 2)``````
``````##      [,1] [,2]
## [1,]    0   -1
## [2,]   -2   -1``````
``````#Matrix multiplication
matrix(c(1,1,1,-1), nrow = 2) %*% matrix(c(1,2,3,4), nrow = 2)``````
``````##      [,1] [,2]
## [1,]    3    7
## [2,]   -1   -1``````
``matrix(c(1,2,3,4), nrow = 2) %*% matrix(c(1,1,1,-1), nrow = 2) ``
``````##      [,1] [,2]
## [1,]    4   -2
## [2,]    6   -2``````
``````#Find the inverse of a matrix
solve(matrix(c(1,1,1,-1), nrow = 2))``````
``````##      [,1] [,2]
## [1,]  0.5  0.5
## [2,]  0.5 -0.5``````
``````#Verify the inverse matrix
matrix(c(0.5,0.5,0.5,-0.5), nrow = 2) %*% matrix(c(1,1,1,-1), nrow = 2) ``````
``````##      [,1] [,2]
## [1,]    1    0
## [2,]    0    1``````
``````#Solve a group of linear equations
solve(matrix(c(1,1,1,-1),nrow = 2),c(20,4))``````
``## [1] 12  8``
``````#This is the result x1 = 12, and x2 = 8.

#Compute a covariance matrix
A <- matrix(c(1,-1,2,0,3,1),nrow = 2)
A``````
``````##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]   -1    0    1``````
``````covm <- (1/(dim(A)[2]))*A%*%t(A)
covm #is the covariance matrix.``````
``````##           [,1]      [,2]
## [1,] 4.6666667 0.6666667
## [2,] 0.6666667 0.6666667``````
``````#Matrix multiplication of a vector changes the vector direction
u <- c(1,-1)
v <- covm%*%u
v``````
``````##      [,1]
## [1,]    4
## [2,]    0``````
``````#u and v are in different directions.

#Matrix mutiplication of an eigenvector does change direction
ew <- eigen(covm)
eigen(covm)\$values``````
``## [1] 4.7748518 0.5584816``
``eigen(covm)\$vectors``
``````##            [,1]       [,2]
## [1,] -0.9870875  0.1601822
## [2,] -0.1601822 -0.9870875``````
``````#Verify the eigenvectors and eigenvalues
covm%*%ew\$vectors[,1]/ew\$values[1]``````
``````##            [,1]
## [1,] -0.9870875
## [2,] -0.1601822``````
``````#This is the first eigenvector

#SVD: Decomposition and recovery of a space-time data matrix
#Develop a 2-by-3 space-time data matrix for SVD
A <- matrix(c(1,-1,2,0,3,1),nrow = 2)
A``````
``````##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]   -1    0    1``````
``````#Perform SVD calculation
msvd <- svd(A)
msvd``````
``````## \$d
## [1] 3.784779 1.294390
##
## \$u
##            [,1]       [,2]
## [1,] -0.9870875 -0.1601822
## [2,] -0.1601822  0.9870875
##
## \$v
##            [,1]       [,2]
## [1,] -0.2184817 -0.8863403
## [2,] -0.5216090 -0.2475023
## [3,] -0.8247362  0.3913356``````
``msvd\$d``
``## [1] 3.784779 1.294390``
``msvd\$u``
``````##            [,1]       [,2]
## [1,] -0.9870875 -0.1601822
## [2,] -0.1601822  0.9870875``````
``msvd\$v``
``````##            [,1]       [,2]
## [1,] -0.2184817 -0.8863403
## [2,] -0.5216090 -0.2475023
## [3,] -0.8247362  0.3913356``````
``````#One can verify that A = UDV', where V' is transpose of V.
verim <- msvd\$u%*%diag(msvd\$d)%*%t(msvd\$v)
verim``````
``````##      [,1]         [,2] [,3]
## [1,]    1 2.000000e+00    3
## [2,]   -1 1.665335e-16    1``````
``round(verim)``
``````##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]   -1    0    1``````
``````#This is the original data matrix A

#Covariance matrix and its eigen-problem
covm <- (1/(dim(A)[2]))*A%*%t(A)
eigcov <- eigen(covm)
eigcov\$values``````
``## [1] 4.7748518 0.5584816``
``eigcov\$vectors``
``````##            [,1]       [,2]
## [1,] -0.9870875  0.1601822
## [2,] -0.1601822 -0.9870875``````
``````#Eigenvalues from SVD and from a covariance matrix
((msvd\$d)^2)/(dim(A)[2])``````
``## [1] 4.7748518 0.5584816``
``eigcov\$values``
``## [1] 4.7748518 0.5584816``

### Plot Fig. 4.2

``````#SOI and the Standardized SLP data at Darwin and Tahiti
setwd("~/sshen/climmath")
#setEPS()
#postscript("fig0402a.eps", height = 4.9, width = 7)
#par(mar = c(4.0,4.2,1.5,0.5))
# Remove the first column that is the year
ptamon <- Pta[,seq(2,13)]
#Convert the matrix into a vector according to mon: Jan 1951, Feb 1951, ..., Dec 2015
ptamonv <- c(t(ptamon))
#Generate time ticks from Jan 1951 to Dec 2015
xtime <- seq(1951, 2016-1/12, 1/12)
# Plot the Tahiti standardized SLP anomalies
plot(xtime, ptamonv,type = "l",xlab = "Year",
ylab = "Pressure Anomalies", cex.lab = 1.2, cex.axis = 1.2,
main = "Standardized Tahiti SLP Anomalies", col = "red",
xlim = range(xtime), ylim = c(-4,4))
text(1952,3.5, "(a)", cex = 1.2)``````

``# dev.off()``

### Do the same for Darwin SLP

``````setwd("~/sshen/climmath/")
#setEPS()
#postscript("fig0402b.eps", height = 4.9, width = 7)
#par(mar = c(4.0,4.2,1.5,0.5))
pdamon <- Pda[,seq(2,13)]
pdamonv <- c(t(pdamon))
plot(xtime, pdamonv,type = "l",cex.lab = 1.2, cex.axis = 1.2,
xlab = "Year",ylab = "Pressure",
main = "Standardized Darwin SLP Anomalies", col = "blue",
xlim = range(xtime), ylim = c(-4,4))
text(1952,3.5, "(b)", cex = 1.2)``````

``# dev.off()``

### Plot the SOI index

``````#setEPS()
#postscript("fig0402c.eps", height = 4.9, width = 7)
#par(mar = c(4.0,4.2,2.0,0.5))
SOI <- ptamonv-pdamonv
plot(xtime, SOI ,type = "l", cex.lab = 1.2, cex.axis = 1.2,
xlab = "Year",ylab = "SOI index",
col = "black",xlim = range(xtime), ylim = c(-6,6), lwd = 1)
#Add ticks on top edge of the plot box
axis (3, at = seq(1951,2015,4), labels = seq(1951,2015,4))
#Add ticks on the right edge of the plot box
axis (4, at = seq(-4,4,2), labels = seq(-4,4,2))
lines(xtime, rep(0,length(xtime)))
text(1985, 5, "SOI = Tahiti Pressure - Darwin Pressure", cex = 1.2)
text(1952,5.6, "(c)", cex = 1.2)``````

``````#abline(lm(SOI ~ xtime), col = "red", lwd = 2)
# dev.off()``````

### CSOI and AMO time series comparison

``````setwd("~/sshen/climmath/")
# setEPS()
#postscript("fig0402d.eps", height = 4.9, width = 7)
par(mar = c(4,4.2,1.6,4))
cnegsoi <- -cumsum(ptamonv-pdamonv)
#par(mgp = c(2,2,4,0))
plot(xtime, cnegsoi,type = "l", cex.lab = 1.2, cex.axis = 1.2,
xlab = "Year",ylab = "Negative CSOI index",
main = "CSOI and AMO Index Comparison",
col = "purple",xlim = range(xtime), ylim = range(cnegsoi), lwd = 1.5)
legend(1960,15, col = c("purple"),lty = 1,lwd = 2.0,
legend = c("CSOI"),bty = "n",text.font = 2,cex = 1.2)
text(1951,12, "(d)", cex = 1.2)
#AMO data and plot
amots <- as.vector(t(amodat[,2:13]))
par(new = TRUE)
plot(xtime, amots,type = "l",col = "darkgreen",
cex.lab = 1.2, cex.axis = 1.2,
lwd = 1.5,axes = FALSE,xlab = "",ylab = "")
legend(1960,0.45, col = c("darkgreen"),lty = 1,lwd = 2.0,
legend = c("AMO index"),bty = "n",text.font = 2,cex = 1.2)
#Suppress the axes and assign the y-axis to side 4
axis(4, cex.axis = 1.3)
mtext("AMO index",side = 4,line = 3, cex = 1.2)``````

``# dev.off()``

### Compute the weighted SOI by SVD

``````#Time-space data format
#Space-time data format
#Implement SVD for the sapce-time data
recontd <- svdptd\$u%*%diag(svdptd\$d[1:2])%*%t(svdptd\$v)
#Two EOFs from the space-time data
U <- svdptd\$u
U``````
``````##            [,1]      [,2]
## [1,] -0.6146784 0.7887779
## [2,]  0.7887779 0.6146784``````
``````#These are the weights for the WSOI
svdptd\$d``````
``## [1] 31.34582 22.25421``
``````D <- diag(svdptd\$d)
D``````
``````##          [,1]     [,2]
## [1,] 31.34582  0.00000
## [2,]  0.00000 22.25421``````
``````#PCs: The principal components
V <- svdptd\$v
V``````
``````##                 [,1]          [,2]
##   [1,] -5.820531e-02  1.017018e-02
##   [2,] -4.026198e-02 -4.419324e-02
##   [3,] -2.743069e-03 -8.276652e-02
##   [4,]  7.583129e-03 -4.294790e-02
##   [5,]  2.379220e-02 -1.376248e-02
##   [6,] -1.476952e-02 -9.027043e-02
##   [7,]  3.278087e-02 -6.656126e-02
##   [8,]  4.772043e-03 -2.324615e-02
##   [9,]  3.444712e-02 -4.764183e-02
##  [10,]  3.470782e-02 -1.887153e-02
##  [11,]  2.630857e-02 -1.100041e-02
##  [12,]  3.326827e-02  4.221348e-02
##  [13,]  2.941437e-02 -5.316599e-02
##  [14,]  2.072041e-02 -2.283104e-02
##  [15,] -1.568766e-02  2.835519e-02
##  [16,]  1.006549e-02  1.104831e-02
##  [17,] -3.160202e-02 -2.329405e-02
##  [18,] -2.575316e-02  1.730688e-02
##  [19,] -1.790933e-02  3.129287e-03
##  [20,] -5.848866e-03 -4.060094e-02
##  [21,]  7.288417e-03 -2.048407e-02
##  [22,] -1.817003e-02 -2.564102e-02
##  [23,] -6.109569e-03 -6.937124e-02
##  [24,]  4.421790e-02 -1.412969e-02
##  [25,] -8.693959e-03  3.033495e-02
##  [26,]  1.790933e-02 -3.129287e-03
##  [27,]  4.216628e-03 -2.955263e-02
##  [28,] -1.088161e-02 -4.612509e-02
##  [29,]  6.323806e-02 -4.646029e-03
##  [30,] -6.143578e-03 -1.813711e-02
##  [31,]  2.221662e-03  2.522591e-02
##  [32,]  3.755291e-02 -8.980741e-02
##  [33,]  4.617886e-02 -1.767409e-02
##  [34,] -1.960958e-03  3.544399e-03
##  [35,]  1.258187e-02  1.381039e-02
##  [36,]  1.705920e-02  1.302806e-02
##  [37,] -3.045719e-02 -6.191523e-02
##  [38,]  1.006549e-02  1.104831e-02
##  [39,] -1.313728e-02 -2.011686e-02
##  [40,] -3.267885e-02 -8.714114e-02
##  [41,] -2.290806e-02 -5.362900e-02
##  [42,] -7.254409e-03 -3.075006e-02
##  [43,] -1.676449e-02 -3.549189e-02
##  [44,] -4.307306e-02 -2.449149e-02
##  [45,] -7.549120e-03 -8.286232e-03
##  [46,] -1.036021e-02  1.141552e-02
##  [47,] -2.811085e-03  1.970175e-02
##  [48,] -4.954536e-02  3.106937e-02
##  [49,]  2.597985e-02  6.269755e-02
##  [50,] -7.634133e-02 -6.670497e-02
##  [51,] -2.990177e-02 -5.560876e-02
##  [52,] -1.076823e-03 -6.384709e-02
##  [53,] -4.388918e-02 -5.956827e-02
##  [54,] -5.117760e-02 -3.908419e-02
##  [55,] -5.987156e-02 -8.749246e-03
##  [56,] -5.706047e-02 -2.845100e-02
##  [57,] -4.784510e-02 -1.245336e-03
##  [58,] -5.624435e-02  6.625785e-03
##  [59,] -4.784510e-02 -1.245336e-03
##  [60,] -3.026450e-02  6.932334e-02
##  [61,] -5.258314e-02 -2.923332e-02
##  [62,] -6.153780e-02 -2.766868e-02
##  [63,] -5.036148e-02 -4.007413e-03
##  [64,] -3.719019e-02 -3.512468e-02
##  [65,] -5.928213e-02 -5.367690e-02
##  [66,] -4.362848e-02 -3.079796e-02
##  [67,] -4.640555e-02 -6.233034e-02
##  [68,] -4.866122e-02 -3.632212e-02
##  [69,] -6.404281e-03 -4.690741e-02
##  [70,] -6.438290e-02  4.326721e-02
##  [71,] -6.993705e-03 -1.979755e-03
##  [72,] -4.222293e-02 -4.064884e-02
##  [73,] -2.072041e-02  2.283104e-02
##  [74,] -2.607036e-04 -2.877031e-02
##  [75,] -1.062091e-02 -1.735479e-02
##  [76,] -1.202645e-02 -7.503910e-03
##  [77,]  2.545845e-02  5.156945e-03
##  [78,] -8.104536e-03 -1.459271e-02
##  [79,] -1.062091e-02 -1.735479e-02
##  [80,]  1.513225e-02 -3.466167e-02
##  [81,]  3.385769e-02 -2.714175e-03
##  [82,]  3.887908e-03  4.414534e-02
##  [83,]  3.581865e-02 -6.258575e-03
##  [84,]  1.147104e-02  1.197434e-03
##  [85,]  7.219272e-02 -6.210673e-03
##  [86,]  1.679850e-02 -1.574224e-02
##  [87,] -8.954663e-03  1.564644e-03
##  [88,] -1.676449e-02 -3.549189e-02
##  [89,]  1.291059e-02 -5.988758e-02
##  [90,] -1.450882e-02 -6.150012e-02
##  [91,] -1.676449e-02 -3.549189e-02
##  [92,] -3.830102e-02 -4.773764e-02
##  [93,]  5.916882e-03 -6.186733e-02
##  [94,] -1.003149e-02 -6.228244e-02
##  [95,]  1.679850e-02 -1.574224e-02
##  [96,]  2.405290e-02  1.500782e-02
##  [97,]  2.663729e-02 -8.469837e-02
##  [98,]  5.454410e-02  2.568892e-02
##  [99,] -4.840052e-02 -7.551812e-03
## [100,] -1.872545e-02 -3.194749e-02
## [101,] -2.124182e-02 -3.470957e-02
## [102,]  3.366501e-03 -1.339527e-02
## [103,]  1.036021e-02 -1.141552e-02
## [104,]  2.947118e-04 -2.246383e-02
## [105,] -1.666246e-03 -1.891943e-02
## [106,] -1.679850e-02  1.574224e-02
## [107,] -3.552394e-02 -1.620525e-02
## [108,] -3.607936e-02 -2.251173e-02
## [109,] -6.993705e-03 -1.979755e-03
## [110,] -2.516373e-03 -2.762077e-03
## [111,] -3.944586e-02 -9.116456e-03
## [112,] -3.467381e-02 -3.236261e-02
## [113,] -2.542444e-02 -5.639108e-02
## [114,] -6.143578e-03 -1.813711e-02
## [115,] -2.013099e-02 -2.209662e-02
## [116,] -2.993578e-02 -4.374623e-03
## [117,] -3.045719e-02 -6.191523e-02
## [118,] -4.477332e-03  7.823218e-04
## [119,] -1.875945e-02  1.928664e-02
## [120,] -2.660329e-02  3.346424e-02
## [121,]  1.062091e-02  1.735479e-02
## [122,] -3.163603e-02  2.794008e-02
## [123,]  6.297735e-02 -3.341633e-02
## [124,] -2.882495e-02  8.238330e-03
## [125,] -1.398741e-02 -3.959511e-03
## [126,] -2.516373e-03 -2.762077e-03
## [127,] -9.510078e-03 -4.741833e-03
## [128,] -6.993705e-03 -1.979755e-03
## [129,] -2.516373e-03 -2.762077e-03
## [130,]  1.009950e-02 -4.018582e-02
## [131,] -1.402142e-02  4.727462e-02
## [132,] -5.039549e-02  4.722672e-02
## [133,] -7.052647e-02  2.513010e-02
## [134,]  1.258187e-02  1.381039e-02
## [135,] -3.366501e-03  1.339527e-02
## [136,] -6.438290e-03  4.326721e-03
## [137,] -4.251765e-02 -1.818501e-02
## [138,] -3.075190e-02 -3.945140e-02
## [139,] -7.515112e-03 -5.952037e-02
## [140,] -2.212595e-02  3.268191e-02
## [141,] -1.928086e-02 -3.825397e-02
## [142,] -3.889044e-02 -2.809979e-03
## [143,] -1.369270e-02 -2.642334e-02
## [144,] -6.438290e-03  4.326721e-03
## [145,] -3.722419e-02  1.610945e-02
## [146,] -1.153905e-02  1.012708e-01
## [147,] -4.558943e-02 -2.725356e-02
## [148,] -3.467381e-02 -3.236261e-02
## [149,] -1.787532e-02 -4.810485e-02
## [150,]  1.735391e-02 -9.435764e-03
## [151,]  4.477332e-03 -7.823218e-04
## [152,] -4.998739e-03 -5.675829e-02
## [153,]  2.183124e-02 -1.021809e-02
## [154,]  4.755039e-02  2.370916e-02
## [155,]  3.467381e-02  3.236261e-02
## [156,]  4.951135e-02  2.016476e-02
## [157,]  1.509824e-02  1.657246e-02
## [158,] -8.501272e-04  1.615735e-02
## [159,] -4.392319e-02 -8.334134e-03
## [160,] -3.918515e-02  1.965385e-02
## [161,] -1.284257e-02 -4.258069e-02
## [162,] -2.908565e-02 -2.053197e-02
## [163,] -2.486902e-02 -5.008460e-02
## [164,] -5.568894e-02  1.293226e-02
## [165,] -5.062218e-02 -3.277772e-02
## [166,] -4.728969e-02  5.061141e-03
## [167,] -1.143703e-02 -5.243157e-02
## [168,]  1.091562e-02 -5.109043e-03
## [169,]  1.206046e-02 -4.373022e-02
## [170,] -1.928086e-02 -3.825397e-02
## [171,] -3.075190e-02 -3.945140e-02
## [172,]  2.431361e-02  4.377813e-02
## [173,] -7.549120e-03 -8.286232e-03
## [174,]  2.516373e-02  2.762077e-02
## [175,]  6.938163e-02  1.349108e-02
## [176,]  2.323678e-02 -2.006896e-02
## [177,]  4.869523e-02 -1.491202e-02
## [178,]  3.774560e-02  4.143116e-02
## [179,]  5.679977e-02 -3.193082e-04
## [180,] -9.770782e-03 -3.351214e-02
## [181,]  4.702899e-02 -3.383145e-02
## [182,]  7.288417e-03 -2.048407e-02
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## [520,]  4.562344e-02 -2.398057e-02
## [521,]  2.434762e-02 -7.456008e-03
## [522,]  1.568766e-02 -2.835519e-02
## [523,]  4.647357e-02 -4.013792e-02
## [524,]  4.699498e-02  1.740269e-02
## [525,]  5.709448e-02 -2.278314e-02
## [526,]  4.281236e-02 -4.278819e-03
## [527,]  2.209195e-02  1.855222e-02
## [528,]  4.866122e-02  3.632212e-02
## [529,]  1.539295e-02 -5.891365e-03
## [530,]  8.920655e-03  4.966949e-02
## [531,] -2.908565e-02 -2.053197e-02
## [532,]  1.990429e-02 -5.790782e-02
## [533,]  1.232116e-02 -1.495992e-02
## [534,] -3.366501e-03  1.339527e-02
## [535,] -1.957557e-02 -1.579014e-02
## [536,] -1.287658e-02  8.653442e-03
## [537,] -8.954663e-03  1.564644e-03
## [538,] -2.777077e-03 -3.153238e-02
## [539,] -2.516373e-03 -2.762077e-03
## [540,]  1.624308e-02 -2.204872e-02
## [541,] -3.833503e-02  3.496497e-03
## [542,] -1.313728e-02 -2.011686e-02
## [543,] -3.974057e-02  1.334737e-02
## [544,] -2.993578e-02 -4.374623e-03
## [545,] -6.472298e-03  5.556085e-02
## [546,] -4.199624e-02  3.935560e-02
## [547,] -2.405290e-02 -1.500782e-02
## [548,] -3.049119e-02 -1.068110e-02
## [549,] -3.156802e-02 -7.452819e-02
## [550,] -2.516373e-02 -2.762077e-02
## [551,] -8.161190e-04 -3.507678e-02
## [552,] -3.830102e-02 -4.773764e-02
## [553,] -1.235517e-02  6.619405e-02
## [554,] -6.490430e-02 -1.427340e-02
## [555,]  1.483754e-02 -1.219784e-02
## [556,]  2.960706e-02  7.807259e-02
## [557,]  4.536274e-02 -5.275088e-02
## [558,]  5.398868e-02  1.938244e-02
## [559,]  2.993578e-02  4.374623e-03
## [560,]  5.594964e-02  1.583804e-02
## [561,]  5.732117e-02  5.722130e-02
## [562,]  6.235392e-02  6.274546e-02
## [563,]  5.395467e-02  7.061658e-02
## [564,]  3.748490e-02  1.266085e-02
## [565,]  9.183631e-02 -9.288880e-02
## [566,]  8.389045e-02  7.499120e-02
## [567,]  8.676955e-02 -4.717882e-02
## [568,]  5.317256e-02 -1.569434e-02
## [569,] -7.843832e-03  1.417760e-02
## [570,] -3.581865e-02  6.258575e-03
## [571,] -4.366248e-02  2.043617e-02
## [572,] -4.059070e-02  2.950473e-02
## [573,] -3.666878e-02  2.241593e-02
## [574,] -4.140681e-02 -5.572056e-03
## [575,] -4.166752e-02 -3.434236e-02
## [576,] -5.287785e-02 -6.769490e-03
## [577,] -6.771539e-02  5.428351e-03
## [578,] -3.470782e-02  1.887153e-02
## [579,] -5.117760e-02 -3.908419e-02
## [580,] -5.121160e-02  1.214994e-02
## [581,] -4.511340e-03  5.201646e-02
## [582,] -6.733001e-03  2.679055e-02
## [583,] -1.875945e-02  1.928664e-02
## [584,] -6.472298e-03  5.556085e-02
## [585,]  5.554154e-04  6.306476e-03
## [586,] -3.359699e-02  3.148448e-02
## [587,] -3.777961e-02  9.802974e-03
## [588,] -4.898994e-02  3.737585e-02
## [589,] -2.157054e-02  3.898839e-02
## [590,] -5.405670e-02  8.308582e-02
## [591,] -4.758440e-02  2.752497e-02
## [592,] -4.588415e-02 -4.789735e-03
## [593,] -1.065492e-02  3.387935e-02
## [594,]  8.659951e-03  2.089918e-02
## [595,]  4.216628e-03 -2.955263e-02
## [596,] -2.964107e-02 -2.683845e-02
## [597,] -2.856424e-02  3.700864e-02
## [598,] -4.362848e-02 -3.079796e-02
## [599,] -7.300884e-02 -2.886611e-02
## [600,] -4.022797e-02 -9.542737e-02
## [601,] -2.859825e-02  8.824277e-02
## [602,] -7.156929e-02 -8.995112e-02
## [603,] -3.163603e-02  2.794008e-02
## [604,] -8.104536e-03 -1.459271e-02
## [605,]  1.902016e-02  9.483666e-03
## [606,] -1.565366e-02 -2.287894e-02
## [607,]  7.843832e-03 -1.417760e-02
## [608,]  1.317129e-02 -3.111727e-02
## [609,] -8.104536e-03 -1.459271e-02
## [610,] -2.482365e-03 -5.399621e-02
## [611,] -2.630857e-02  1.100041e-02
## [612,]  2.911966e-02 -3.070216e-02
## [613,] -1.094963e-02  5.634318e-02
## [614,] -3.444712e-02  4.764183e-02
## [615,]  1.173174e-02  2.996774e-02
## [616,]  2.516373e-03  2.762077e-03
## [617,]  3.078591e-02 -1.178273e-02
## [618,]  8.954663e-03 -1.564644e-03
## [619,]  2.375819e-02  3.747165e-02
## [620,]  3.304158e-02 -3.779096e-02
## [621,]  2.349749e-02  8.701344e-03
## [622,]  1.261588e-02 -3.742375e-02
## [623,]  1.817003e-02  2.564102e-02
## [624,]  4.362848e-02  3.079796e-02
## [625,]  6.993705e-03  1.979755e-03
## [626,]  2.045971e-02 -5.160134e-02
## [627,]  8.399248e-03 -7.871120e-03
## [628,]  9.215367e-03  2.720566e-02
## [629,]  9.804790e-03 -1.772200e-02
## [630,]  2.016500e-02 -2.913752e-02
## [631,] -9.804790e-03  1.772200e-02
## [632,] -3.627204e-03 -1.537503e-02
## [633,]  3.366501e-03 -1.339527e-02
## [634,] -2.221662e-03 -2.522591e-02
## [635,]  1.202645e-02  7.503910e-03
## [636,] -4.336777e-02 -2.027657e-03
## [637,]  4.758440e-02 -2.752497e-02
## [638,] -4.451261e-02  3.659352e-02
## [639,] -2.150252e-02 -6.347988e-02
## [640,]  3.078591e-02 -1.178273e-02
## [641,] -3.777961e-02  9.802974e-03
## [642,]  3.522923e-02  3.866908e-02
## [643,]  1.790933e-02 -3.129287e-03
## [644,]  1.036021e-02 -1.141552e-02
## [645,]  1.620907e-02  2.918542e-02
## [646,]  3.627204e-03  1.537503e-02
## [647,]  2.323678e-02 -2.006896e-02
## [648,]  2.493704e-02 -5.238367e-02
## [649,] -1.147104e-02 -1.197434e-03
## [650,]  1.130781e-01 -5.817923e-02
## [651,] -1.121033e-02  2.757287e-02
## [652,]  2.571915e-02  3.392725e-02
## [653,]  2.741940e-02  1.612546e-03
## [654,] -2.013099e-02 -2.209662e-02
## [655,] -6.993705e-03 -1.979755e-03
## [656,]  1.036021e-02 -1.141552e-02
## [657,] -1.343199e-02  2.346965e-03
## [658,] -4.366248e-02  2.043617e-02
## [659,]  3.105797e-03 -4.216558e-02
## [660,]  3.627204e-03  1.537503e-02
## [661,] -6.294334e-02 -1.781780e-02
## [662,] -1.144839e-03  3.862118e-02
## [663,] -6.742068e-02 -1.703548e-02
## [664,] -4.640555e-02 -6.233034e-02
## [665,]  2.098111e-02  5.939266e-03
## [666,]  1.062091e-02  1.735479e-02
## [667,]  2.460832e-02  2.131430e-02
## [668,]  3.944586e-02  9.116456e-03
## [669,]  2.516373e-02  2.762077e-02
## [670,]  4.977205e-02  4.893507e-02
## [671,]  5.214071e-04  5.754061e-02
## [672,]  1.646978e-02  5.795572e-02
## [673,]  2.549245e-02 -4.607719e-02
## [674,]  3.071789e-03  9.068554e-03
## [675,] -1.117632e-02 -2.366126e-02
## [676,]  7.254409e-03  3.075006e-02
## [677,]  5.588162e-03  1.183063e-02
## [678,] -2.486902e-02 -5.008460e-02
## [679,]  1.369270e-02  2.642334e-02
## [680,] -1.372671e-02  2.481079e-02
## [681,] -9.770782e-03 -3.351214e-02
## [682,] -2.741940e-02 -1.612546e-03
## [683,] -3.411840e-02 -2.605613e-02
## [684,] -6.212723e-02  1.725898e-02
## [685,] -6.575443e-02  1.883952e-03
## [686,] -9.487409e-02  3.258611e-02
## [687,] -4.454662e-02  8.782766e-02
## [688,] -2.157054e-02  3.898839e-02
## [689,]  5.293450e-03  3.429446e-02
## [690,] -1.653779e-02  4.451255e-02
## [691,] -7.843832e-03  1.417760e-02
## [692,] -3.278087e-02  6.656126e-02
## [693,] -4.395720e-02  4.290000e-02
## [694,] -4.006929e-02  8.704534e-02
## [695,] -4.702899e-02  3.383145e-02
## [696,] -5.150632e-02  3.461377e-02
## [697,] -3.862974e-02  2.596033e-02
## [698,] -6.912093e-02  1.527923e-02
## [699,] -1.317129e-02  3.111727e-02
## [700,] -2.826953e-02  1.454481e-02
## [701,] -5.214071e-04 -5.754061e-02
## [702,] -5.032747e-03 -5.524155e-03
## [703,] -9.510078e-03 -4.741833e-03
## [704,]  3.661213e-03 -3.585910e-02
## [705,] -1.424811e-02 -3.272982e-02
## [706,]  4.255165e-02 -3.304912e-02
## [707,]  1.960958e-02 -3.544399e-02
## [708,]  2.630857e-02 -1.100041e-02
## [709,]  3.140934e-02 -1.079445e-01
## [710,]  5.402269e-02 -3.185169e-02
## [711,]  2.434762e-02 -7.456008e-03
## [712,] -4.088541e-02  5.196855e-02
## [713,] -3.607936e-02 -2.251173e-02
## [714,] -1.065492e-02  3.387935e-02
## [715,] -6.186652e-02  4.602929e-02
## [716,] -6.493831e-02  3.696073e-02
## [717,] -7.977585e-02  4.915857e-02
## [718,] -6.212723e-02  1.725898e-02
## [719,] -4.536274e-02  5.275088e-02
## [720,] -1.069005e-01  2.508220e-02
## [721,] -8.229222e-02  4.639650e-02
## [722,] -9.964613e-02  5.583226e-02
## [723,] -9.039676e-02  3.180379e-02
## [724,] -6.578844e-02  5.311809e-02
## [725,] -8.988671e-03  5.279878e-02
## [726,] -5.066755e-03  4.570998e-02
## [727,] -3.248616e-02  4.409743e-02
## [728,] -1.261588e-02  3.742375e-02
## [729,] -3.167004e-02  7.917422e-02
## [730,] -2.575316e-02  1.730688e-02
## [731,] -3.696349e-02  4.487976e-02
## [732,] -9.261842e-02  6.577883e-03
## [733,] -3.862974e-02  2.596033e-02
## [734,] -1.402142e-02  4.727462e-02
## [735,] -3.156802e-02 -7.452819e-02
## [736,]  1.173174e-02  2.996774e-02
## [737,] -3.071789e-03 -9.068554e-03
## [738,]  1.539295e-02 -5.891365e-03
## [739,] -3.332492e-03 -3.783886e-02
## [740,]  9.215367e-03  2.720566e-02
## [741,] -5.882874e-03  1.063320e-02
## [742,] -1.176575e-02  2.126639e-02
## [743,] -5.622171e-03  3.940350e-02
## [744,]  2.127583e-02 -1.652456e-02
## [745,] -2.482365e-03 -5.399621e-02
## [746,]  7.549120e-03  8.286232e-03
## [747,] -5.261715e-02  2.200081e-02
## [748,] -8.954663e-03  1.564644e-03
## [749,] -2.882495e-02  8.238330e-03
## [750,] -4.810581e-02 -3.001564e-02
## [751,] -2.797482e-02 -7.919022e-03
## [752,] -9.510078e-03 -4.741833e-03
## [753,] -1.509824e-02 -1.657246e-02
## [754,] -5.554154e-04 -6.306476e-03
## [755,] -2.964107e-02 -2.683845e-02
## [756,] -2.947118e-04  2.246383e-02
## [757,] -5.343327e-02 -1.307597e-02
## [758,] -6.404281e-03 -4.690741e-02
## [759,]  3.526324e-02 -1.256505e-02
## [760,] -3.049119e-02 -1.068110e-02
## [761,] -1.653779e-02  4.451255e-02
## [762,] -8.104536e-03 -1.459271e-02
## [763,]  6.143578e-03  1.813711e-02
## [764,]  2.908565e-02  2.053197e-02
## [765,]  2.797482e-02  7.919022e-03
## [766,]  2.405290e-02  1.500782e-02
## [767,]  3.441311e-02  3.592301e-03
## [768,]  1.820404e-02 -2.559312e-02
## [769,]  2.438162e-02 -5.869014e-02
## [770,] -2.845094e-03  7.093588e-02
## [771,]  2.630857e-02 -1.100041e-02
## [772,]  5.293450e-03  3.429446e-02
## [773,]  3.186273e-02  5.206436e-02
## [774,]  1.931487e-02 -1.298016e-02
## [775,]  4.503402e-02  2.094709e-02
## [776,]  5.428339e-02 -3.081385e-03
## [777,]  6.127710e-02 -1.101630e-03
## [778,]  7.241942e-02  7.379377e-02
## [779,]  1.846474e-02  3.177189e-03
## [780,]  1.542696e-02 -5.712550e-02``````

### Plot Fig. 4.3

``````U <- svdptd\$u #The matrix of the EOF spatial patterns
V <- svdptd\$v #The matrix of the PC temporal patterns
D <- diag(svdptd\$d) #The diagonal matrix of standard deviations
recontada <- U%*%D%*%t(V) #Verify the SVD decomposition
max(svd_error)``````
``## [1] 3.996803e-15``
``min(svd_error)``
``## [1] -4.218847e-15``

### Plot WSOI1

``````#setEPS()
#postscript("fig0403a.eps", height = 4.9, width = 7)
par(mar = c(4,4.2,2.0,0.5))
xtime <- seq(1951, 2016-1/12, 1/12)
wsoi1 <- D[1,1]*t(V)[1,]
plot(xtime, wsoi1,type = "l", cex.lab = 1.2, cex.axis = 1.2,
xlab = "Year",ylab = "WSOI1",
col = "black",xlim = range(xtime), ylim = c(-4,4))
axis (3, at = seq(1951,2015,4), labels = seq(1951,2015,4))
text(1985,-3.7,"Weighted SOI1 Index", cex = 1.2)
text(1952,3.5, "(a)", cex = 1.2)``````

``# dev.off()``

### Plot WSOI2

``````#setEPS()
#postscript("fig0403b.eps", height = 4.9, width = 7)
par(mar = c(4,4.2,2.0,0.5))
xtime <- seq(1951, 2016-1/12, 1/12)
wsoi2 <- D[2,2]*t(V)[2,]
plot(xtime, wsoi2,type = "l", cex.lab = 1.3, cex.axis = 1.3,
xlab = "Year",ylab = "WSOI2",
col = "blue",xlim = range(xtime), ylim = c(-4,4))
axis (3, at = seq(1951,2015,4), labels = seq(1951,2015,4))
text(1985,-3.7,"Weighted SOI2 Index", cex = 1.5)
text(1952,3.5, "(b)", cex = 1.3)``````

``# dev.off()``

### Cumulative WSOIs

``````setwd("~/sshen/climmath/")
#setEPS()
#postscript("fig0403c.eps", height = 4.9, width = 7)
par(mar = c(4,4.2,2.0,4.2))
cwsoi1 <- cumsum(wsoi1)
plot(xtime, cwsoi1,type = "l",cex.lab = 1.2, cex.axis = 1.2,
xlab = "Year",ylab = "Weighted SOI1",
col = "black",lwd = 3, ylim = c(-200,50),
main = "Comparison between CWSOI1 and the Smoothed AMO Index")
text(1951,40, "(c)", cex = 1.3)
#axis (3, at = seq(1951,2015,4), labels = seq(1951,2015,4))
legend(1970,20, col = c("black"),lty = 1,lwd = 3.0,
legend = c("CWSOI1"),bty = "n",text.font = 2,cex = 1.2)
#Superimpose CSOI time series on this CWSOI1
cnegsoi <- -cumsum(ptamonv-pdamonv)
lines(xtime, cnegsoi,type = "l",col = "purple", lwd = 3.0)
legend(1970,2, col = c("purple"),lty = 1,lwd = 3.0,
legend = c("CSOI"),bty = "n",text.font = 2,cex = 1.2)
#24-month ahead moving average of the monthly AMO index
amots <- as.vector(t(amodat[,2:13]))
#install.packages("TTR")
library("TTR")``````
``## Warning: package 'TTR' was built under R version 4.1.3``
``````amomv <- SMA(amots,n = 24, fill = NA)
#Average of the previous n points
par(new = TRUE)
xtime <- seq(1951,2015,len = 780)
plot(xtime, amomv[37:816],type = "l",col = "darkgreen",
lwd = 3,axes = FALSE,xlab = "",ylab = "",
cex.lab = 1.2, cex.axis = 1.2,
ylim = c(-0.5, 0.5))
legend(1970,0.23, col = c("darkgreen"),lty = 1,lwd = 3.0,
legend = c("AMO index"),bty = "n",text.font = 2,cex = 1.2)
#Suppress the axes and assign the y-axis to side 4
axis(4, cex.axis = 1.2, col.axis = "darkgreen")
mtext("Smoothed AMO index",
side = 4,line = 3, cex = 1.2, col = "darkgreen")``````

``# dev.off()``

### Plot cumulative WSOI2: CWSOI2

``````#setEPS()
#postscript("fig0403d.eps", height = 4.9, width = 7)
par(mar = c(4,4.2,2.0,0.5))
cwsoi1 <- cumsum(wsoi1)
wsoi2 <- D[2,2]*t(V)[2,]
cwsoi2 <- cumsum(wsoi2)
plot.new()
# par(mar = c(4,4,3,4))
plot(xtime, cwsoi2,type = "l",xlab = "Year",ylab = "CWSOI2",
cex.lab = 1.2, cex.axis = 1.2,
col = "blue",lwd = 3, ylim = c(-200,50),
main = "CWSOI2 Index: The Cumulative PC2")
text(1951,40, "(d)", cex = 1.3)``````