How to related PCA output to the original data?

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Yaser Khojah 2019-4-18

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评论： Paul Gershon 2023-11-20

Hello,

I'm new to PCA and I would like to learn the outcome of pca function. I have read the document and checked others works but I'm a bit confused on how to related the results (wcoeff, latent, explained) to the original data. For example, I'm using the example from the document as below. I understand the (wcoeff) presents the eigenvector vectors. the (latent) presents the eigenvalues. the (explained) is the percentage of the total variance explained by each principal component. NOW, how are all these information are related to the main data which is the ingredients here? how do I know from looking at the results in (explained) that the 55 % is related to which variables or columns in the ingredients matrix?

load hald
[wcoeff,~,latent,~,exp
lained] = pca(ingredients,'VariableWeights','variance')

wcoeff = 4×4

-2.7998 2.9940 -3.9736 1.4180

-8.7743 -6.4411 4.8927 9.9863

2.5240 -3.8749 -4.0845 1.7196

9.1714 7.5529 3.2710 11.3273

latent = 4×1

2.2357

1.5761

0.1866

0.0016

explained = 4×1

55.8926

39.4017

4.6652

0.0406

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Yaser Khojah 2019-4-18

编辑：Yaser Khojah 2019-4-18

在 MATLAB Online 中打开

I guess you did not understand my qeustion? Another way to ask.

Which variables from my dataset are related to PC1, PC2, PC3 and PC4?

For example:

explained = 4×1

55.8926 Which variable?

39.4017 Which variable?

4.6652 Which variable?

0.0406 Which variable?

Here is the explinaiton of each componete which relates to PC and nothing is related to original data in terms of which variables are related to the PC.

coeff: contains coefficients for one principal component, and the columns are in descending order of component variance
score: Rows of score correspond to observations, and columns correspond to components.
explained: the percentage of the total variance explained by each principal component
latent: Principal component variances, that is the eigenvalues of the covariance matrix of X, returned as a column vector.

I tried the following:

load hald
[coeff,score,latent,tsquared,explained] = pca(ingredients)
coeff =
   -0.0678   -0.6460    0.5673    0.5062
   -0.6785   -0.0200   -0.5440    0.4933
    0.0290    0.7553    0.4036    0.5156
    0.7309   -0.1085   -0.4684    0.4844
    score =
   36.8218   -6.8709   -4.5909    0.3967
   29.6073    4.6109   -2.2476   -0.3958
  -12.9818   -4.2049    0.9022   -1.1261
   23.7147   -6.6341    1.8547   -0.3786
   -0.5532   -4.4617   -6.0874    0.1424
  -10.8125   -3.6466    0.9130   -0.1350
  -32.5882    8.9798   -1.6063    0.0818
   22.6064   10.7259    3.2365    0.3243
   -9.2626    8.9854   -0.0169   -0.5437
   -3.2840  -14.1573    7.0465    0.3405
    9.2200   12.3861    3.4283    0.4352
  -25.5849   -2.7817   -0.3867    0.4468
  -26.9032   -2.9310   -2.4455    0.4116
latent =
  517.7969
   67.4964
   12.4054
    0.2372
tsquared =
    5.6803
    3.0758
    6.0002
    2.6198
    3.3681
    0.5668
    3.4818
    3.9794
    2.6086
    7.4818
    4.1830
    2.2327
    2.7216
explained =
   86.5974
   11.2882
    2.0747
    0.0397
    
% compared against:
covarianceMatrix = cov(ingredients);
[V,D] = eig(covarianceMatrix);
V =
    0.5062    0.5673    0.6460   -0.0678
    0.4933   -0.5440    0.0200   -0.6785
    0.5156    0.4036   -0.7553    0.0290
    0.4844   -0.4684    0.1085    0.7309
D =
    0.2372         0         0         0
         0   12.4054         0         0
         0         0   67.4964         0
         0         0         0  517.7969

So D here I believe is matching the covarianceMatrix which its is column meets the orignal data in (ingredients). In addtion, the first row in D represent the first column in covarianceMatrix and so on. However, latent order is not matching D. I know it matches the PC only and how someone like me can relate the laten output to the original data (ingredients)

Paul Gershon 2023-11-20

Is the answer to square all numbers in the coeff matrix and then do the division for relative contribution? For example:

PC1 PC2

var1: -2 0.3

var2: 5 3.8

Square it:

PC1 PC2

var1: 4 0.09

var2: 25 14.4

Now relative contribution of var2 to PC1 = 25/4? or is it 25/29? (the denominator is the sum for the column)?

Thanks!

Paul Gershon 2023-11-20

..Or is the answer not to square the coefficients, but simply to take absolute values because sign simply shows whether the point is pushed negative on the PC axis by the sign of the coefficient, but the sign is irrelevant to magnitude of contribution?

Thanks!

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