<p> PCA Utilities </p>

93a67b18-0993-4e9f-88f1-41b1c963eb79 35 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

un-Nest Matrix ... Row/col

data UnestCol v data 1 el : j 2 data cols range v v data j el augment : for v 31 line :

data UnestRow v data 1 el : j 2 data rows range v v data j el stack : for v 31 line :

data Explode data UnestRow v data 1 el : j 2 data rows range v v data j el stack : for v 31 line : j 1 data cols range U 1 j el data j col UnestRow : for v U 1 el : j 2 U cols range v v U j el augment : for v 41 line 21 line :

D Covar j 1 D cols range vx 1 j el D j col D j col Mean - : for U vx transpose vx * D rows 1 - / : 21 line :

X 14.1 0.86 0.985 13.6 16 1.06 1.094 16.6 29.8 2.03 1.165 23.5 8 0.67 0.928 10.2 4.1 0.4 0.946 5.4 15 1.04 0.888 15 8.8 0.76 1.027 9 12.4 0.95 0.922 12.3 16.6 1.12 0.937 16.3 14.9 1.02 0.886 15.4 13.7 1.01 0.964 13 15.1 0.9 0.932 14.4 7.8 0.57 0.971 10 11.4 0.78 1.124 10.2 9 0.74 0.852 9.5 1 0.13 0.785 1.5 17 1.26 0.919 18.5 12.8 1.08 1.04 12.6 15.8 0.96 0.957 17.5 4.5 0.42 0.911 4.9 14.5 1.01 1.007 15.9 7.3 0.61 0.981 8.5 8.6 0.69 0.969 10.6 15.2 1.02 0.95 13.9 12 0.82 1.118 14.9 254 mat :

X data

1. Each row represents one observation

2. Each col, a measured characteristic.

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25 brands of cigarettes

<1> Tar content [mg]

<2> Nicotine content [mg]

<3> Weight [g]

<4> Carbone monoxide content [mg]

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X Covar 32.1014 11 mat 1.9591 11 mat 0.2438 11 mat 25.7124 11 mat 1.9591 11 mat 0.1254 11 mat 0.0155 11 mat 1.5539 11 mat 0.2438 11 mat 0.0155 11 mat 0.0077 11 mat 0.1927 11 mat 25.7124 11 mat 1.5539 11 mat 0.1927 11 mat 22.4646 11 mat 44 mat

Covar X Covar Explode :

Covar 32.101 1.959 0.244 25.712 1.959 0.125 0.016 1.554 0.244 0.016 0.008 0.193 25.712 1.554 0.193 22.465 44 mat

Covar Covar :

variance of the PC cols = VarPC below λ Covar dn_LinAlgEigenvalues sort reverse :

λ 53.5618 1.1258 0.0064 0.0051 41 mat

Evec p 1 X cols range Evec 11 el Covar dn_LinAlgEigenvectors : for Evec 1 el 21 line :

Evec Evec :

M New j 1 M cols range new 1 j el M j col M j col Mean - : for v new 1 el : j 2 new cols range v v new j el augment : for v 41 line :

Evec 0.7687 0.636 - 0.0506 - 0.0457 0.0467 0.0496 - 0.6486 0.758 - 0.0058 0.0066 - 0.7594 0.6506 0.6379 0.77 0.0065 0.0055 - 44 mat

New X New :

principal component PC of 'New' pc New Evec * :

idx 1 X rows range : pc1 idx pc 1 col augment : pc2 idx pc 2 col augment : pc3 idx pc 4 col augment : pc4 idx pc 4 col augment : 51 line

pc1 pc2 pc3 pc4 4 1 sys

<= Tar Nicotine Weight Co2 41 sys

Examine the PCA [Principal Component Analysis]

New 1.884 0.016 - 0.015 1.072 3.784 0.184 0.124 4.072 17.584 1.154 0.195 10.972 4.216 - 0.206 - 0.042 - 2.328 - 8.116 - 0.476 - 0.024 - 7.128 - 2.784 0.164 0.082 - 2.472 3.416 - 0.116 - 0.057 3.528 - 0.184 0.074 0.048 - 0.228 - 4.384 0.244 0.033 - 3.772 2.684 0.144 0.084 - 2.872 1.484 0.134 0.006 - 0.472 2.884 0.024 0.038 - 1.872 4.416 - 0.306 - 0.001 2.528 - 0.816 - 0.096 - 0.154 2.328 - 3.216 - 0.136 - 0.118 - 3.028 - 11.216 - 0.746 - 0.185 - 11.028 - 4.784 0.384 0.051 - 5.972 0.584 0.204 0.07 0.072 3.584 0.084 0.013 - 4.972 7.716 - 0.456 - 0.059 - 7.628 - 2.284 0.134 0.037 3.372 4.916 - 0.266 - 0.011 4.028 - 3.616 - 0.186 - 0.001 - 1.928 - 2.984 0.144 0.02 - 1.372 0.216 - 0.056 - 0.148 2.372 254 mat

pc 2.131 0.372 - 0.088 - 0.102 5.516 0.719 0.048 0.092 20.57 2.793 - 0.078 0.004 - 4.736 - 0.899 0.032 0.051 - 10.808 - 0.303 - 0.037 0.013 3.724 0.125 0.081 - 0.064 - 4.881 - 0.539 - 0.117 0.012 - 0.001 - 0.296 - 0 0.078 - 5.787 0.104 0.064 - 0.027 - 3.901 0.498 0.088 - 0.057 - 1.448 0.587 - 0.01 0.04 - 3.412 0.394 - 0.147 - 0.079 5.021 - 0.877 0.009 0.045 2.116 - 1.27 - 0.08 0.148 4.411 - 0.279 - 0.035 - 0.104 - 15.692 - 1.32 - 0.129 - 0.007 - 7.505 1.537 0.007 0.138 - 0.505 0.327 - 0.156 0.083 - 5.93 1.545 0.105 - 0.065 10.819 - 0.943 - 0 0.004 - 3.913 1.137 0.021 0.009 6.361 - 0.038 0.058 0.006 4.018 - 0.824 0.048 0.014 - 3.175 0.848 - 0.064 - 0.007 1.345 1.966 0.102 0.116 254 mat

Variance of the columns  of the Principal Components VarPC j 1 Evec cols range data j el Evec j col transpose Covar * Evec j col * : for v data 1 el : j 2 data rows range v v data j el stack : for v 41 line :

VarPC λ 12 mat 53.5618 1.1258 0.0064 0.0051 41 mat 53.5618 1.1258 0.0064 0.0051 41 mat 12 mat

NIPALS Nonlinear Iterative Partial Least Squares ≡

Nipals(data,NumPC,maxiter,"scale/noscale",acc)

Takes a matrix of multivariate data, and returns numPC principal components calculated with

a maximum of maxiter iterations. The 4th argument may be set "scale" if you wish to scale

the data to the standard deviation, or "noscale" if you don't. The convergence accuracy is

given by acc, typically of the ordere-10 or smaller . . .

Returns numPC scores, loading, and accumulated variance explained by each PC.

and principal components [eigenvalues], from multivariate data using a maximum number

of iterations ... acc specifies the termination accuracyused for eigenvalue generation.

Data may be optionally scaled to the standard deviation.

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Mahalanobis distance for each parameter

The Mahalanobis distance is a measure of the distance between a point P and a distribution D, introduced by P. C. Mahalanobis in 1936. It is a multi-dimensional generalization of the idea of measuring how many standard deviations away P is from the mean of D

x New :

N X rows : 25

MD c 1 : p 1 X cols range data c el X transpose p col transpose 1 N 1 - / X transpose * X * (* X transpose p col * : c c 1 + : 21 line for v data 1 el : j 2 data rows range v v data j el stack : for v abs sqrt vectorize 51 line :

MD 379.476 446.554 730.037 249.628 41 mat

md c 1 : p 1 x cols range data c el x transpose p col transpose 1 N 1 - / x transpose * x * (* x transpose p col * : c c 1 + : 21 line for v data 1 el : j 2 data rows range v v data j el stack : for v abs sqrt vectorize 51 line :

md 15.603 40.373 150.576 34.671 41 mat

MD_pc c 1 : p 1 pc cols range data c el pc transpose p col transpose 1 N 1 - / pc transpose * pc * (* x transpose p col * : c c 1 + : 21 line for v data 1 el : j 2 data rows range v v data j el stack : for v abs sqrt vectorize 51 line :

MD_pc 14.666 33.437 139.177 32.698 41 mat

Y i 1 X cols range module i el MD i el (2^vectorize N / : for 10 module log10 * vectorize 21 line :

y i 1 X cols range module i el md i el (2^vectorize N / : for 10 module log10 * vectorize 21 line :

Y 37.604 39.018 43.287 33.966 41 mat

y 9.885 18.142 29.576 16.82 41 mat