PCA Visualization in MATLAB®

How to do PCA Visualization in MATLAB® with Plotly.


Principal Components of a Data Set

Load the sample data set.

load hald

The ingredients data has 13 observations for 4 variables.

Find the principal components for the ingredients data.

 coeff = pca(ingredients)
coeff = 4×4

   -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

The rows of coeff contain the coefficients for the four ingredient variables, and its columns correspond to four principal components.

PCA in the Presence of Missing Data

Find the principal component coefficients when there are missing values in a data set.

Load the sample data set.

load imports-85

Data matrix X has 13 continuous variables in columns 3 to 15: wheel-base, length, width, height, curb-weight, engine-size, bore, stroke, compression-ratio, horsepower, peak-rpm, city-mpg, and highway-mpg. The variables bore and stroke are missing four values in rows 56 to 59, and the variables horsepower and peak-rpm are missing two values in rows 131 and 132.

Perform principal component analysis.

coeff = pca(X(:,3:15));

By default, pca performs the action specified by the 'Rows','complete' name-value pair argument. This option removes the observations with NaN values before calculation. Rows of NaNs are reinserted into score and tsquared at the corresponding locations, namely rows 56 to 59, 131, and 132.

Use 'pairwise' to perform the principal component analysis.

coeff = pca(X(:,3:15),'Rows','pairwise');

In this case, pca computes the (i,j) element of the covariance matrix using the rows with no NaN values in the columns i or j of X. Note that the resulting covariance matrix might not be positive definite. This option applies when the algorithm pca uses is eigenvalue decomposition. When you don’t specify the algorithm, as in this example, pca sets it to 'eig'. If you require 'svd' as the algorithm, with the 'pairwise' option, then pca returns a warning message, sets the algorithm to 'eig' and continues.

If you use the 'Rows','all' name-value pair argument, pca terminates because this option assumes there are no missing values in the data set.

coeff = pca(X(:,3:15),'Rows','all');

Error using pca (line 180)
Raw data contains NaN missing value while 'Rows' option is set to 'all'. Consider using 'complete' or pairwise' option instead.
## Weighted PCA Use the inverse variable variances as weights while performing the principal components analysis. Load the sample data set.
load hald

Perform the principal component analysis using the inverse of variances of the ingredients as variable weights.

 [wcoeff,~,latent,~,explained] = 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

Note that the coefficient matrix, wcoeff, is not orthonormal.

Calculate the orthonormal coefficient matrix.

coefforth = inv(diag(std(ingredients)))* wcoeff
coefforth = 4×4

   -0.4760    0.5090   -0.6755    0.2411
   -0.5639   -0.4139    0.3144    0.6418
    0.3941   -0.6050   -0.6377    0.2685
    0.5479    0.4512    0.1954    0.6767

Check orthonormality of the new coefficient matrix, coefforth.

 coefforth*coefforth'
ans = 4×4

    1.0000    0.0000   -0.0000    0.0000
    0.0000    1.0000   -0.0000   -0.0000
   -0.0000   -0.0000    1.0000    0.0000
    0.0000   -0.0000    0.0000    1.0000

PCA Using ALS for Missing Data

Find the principal components using the alternating least squares (ALS) algorithm when there are missing values in the data.

Load the sample data.

load hald

The ingredients data has 13 observations for 4 variables.

Perform principal component analysis using the ALS algorithm and display the component coefficients.

[coeff,score,latent,tsquared,explained] = pca(ingredients);
coeff
coeff = 4×4

   -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

Introduce missing values randomly.

y = ingredients;
rng('default'); % for reproducibility
ix = random('unif',0,1,size(y))<0.30; 
y(ix) = NaN
y = 13×4

     7    26     6   NaN
     1    29    15    52
   NaN   NaN     8    20
    11    31   NaN    47
     7    52     6    33
   NaN    55   NaN   NaN
   NaN    71   NaN     6
     1    31   NaN    44
     2   NaN   NaN    22
    21    47     4    26
      ⋮

Approximately 30% of the data has missing values now, indicated by NaN.

Perform principal component analysis using the ALS algorithm and display the component coefficients.

[coeff1,score1,latent,tsquared,explained,mu1] = pca(y,...
'algorithm','als');
coeff1
coeff1 = 4×4

   -0.0362    0.8215   -0.5252    0.2190
   -0.6831   -0.0998    0.1828    0.6999
    0.0169    0.5575    0.8215   -0.1185
    0.7292   -0.0657    0.1261    0.6694

Display the estimated mean.

mu1
mu1 = 1×4

    8.9956   47.9088    9.0451   28.5515

Reconstruct the observed data.

t = score1*coeff1' + repmat(mu1,13,1)
t = 13×4

    7.0000   26.0000    6.0000   51.5250
    1.0000   29.0000   15.0000   52.0000
   10.7819   53.0230    8.0000   20.0000
   11.0000   31.0000   13.5500   47.0000
    7.0000   52.0000    6.0000   33.0000
   10.4818   55.0000    7.8328   17.9362
    3.0982   71.0000   11.9491    6.0000
    1.0000   31.0000   -0.5161   44.0000
    2.0000   53.7914    5.7710   22.0000
   21.0000   47.0000    4.0000   26.0000
      ⋮

The ALS algorithm estimates the missing values in the data.

Another way to compare the results is to find the angle between the two spaces spanned by the coefficient vectors. Find the angle between the coefficients found for complete data and data with missing values using ALS.

subspace(coeff,coeff1)
ans = 9.1336e-16

This is a small value. It indicates that the results if you use pca with 'Rows','complete' name-value pair argument when there is no missing data and if you use pca with 'algorithm','als' name-value pair argument when there is missing data are close to each other.

Perform the principal component analysis using 'Rows','complete' name-value pair argument and display the component coefficients.

[coeff2,score2,latent,tsquared,explained,mu2] = pca(y,...
'Rows','complete');
coeff2
coeff2 = 4×3

   -0.2054    0.8587    0.0492
   -0.6694   -0.3720    0.5510
    0.1474   -0.3513   -0.5187
    0.6986   -0.0298    0.6518

In this case, pca removes the rows with missing values, and y has only four rows with no missing values. pca returns only three principal components. You cannot use the 'Rows','pairwise' option because the covariance matrix is not positive semidefinite and pca returns an error message.

Find the angle between the coefficients found for complete data and data with missing values using listwise deletion (when 'Rows','complete').

subspace(coeff(:,1:3),coeff2)
ans = 0.3576

The angle between the two spaces is substantially larger. This indicates that these two results are different.

Display the estimated mean.

mu2
mu2 = 1×4

    7.8889   46.9091    9.8750   29.6000

In this case, the mean is just the sample mean of y.

Reconstruct the observed data.

score2*coeff2'
ans = 13×4

       NaN       NaN       NaN       NaN
   -7.5162  -18.3545    4.0968   22.0056
       NaN       NaN       NaN       NaN
       NaN       NaN       NaN       NaN
   -0.5644    5.3213   -3.3432    3.6040
       NaN       NaN       NaN       NaN
       NaN       NaN       NaN       NaN
       NaN       NaN       NaN       NaN
       NaN       NaN       NaN       NaN
   12.8315   -0.1076   -6.3333   -3.7758
      ⋮

This shows that deleting rows containing NaN values does not work as well as the ALS algorithm. Using ALS is better when the data has too many missing values.

Principal Component Coefficients, Scores, and Variances

Find the coefficients, scores, and variances of the principal components.

Load the sample data set.

load hald

The ingredients data has 13 observations for 4 variables.

Find the principal component coefficients, scores, and variances of the components for the ingredients data.

[coeff,score,latent] = pca(ingredients)
coeff = 4×4

   -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 = 13×4

   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
      ⋮

latent = 4×1

  517.7969
   67.4964
   12.4054
    0.2372

Each column of score corresponds to one principal component. The vector, latent, stores the variances of the four principal components.

Reconstruct the centered ingredients data.

Xcentered = score*coeff'
Xcentered = 13×4

   -0.4615  -22.1538   -5.7692   30.0000
   -6.4615  -19.1538    3.2308   22.0000
    3.5385    7.8462   -3.7692  -10.0000
    3.5385  -17.1538   -3.7692   17.0000
   -0.4615    3.8462   -5.7692    3.0000
    3.5385    6.8462   -2.7692   -8.0000
   -4.4615   22.8462    5.2308  -24.0000
   -6.4615  -17.1538   10.2308   14.0000
   -5.4615    5.8462    6.2308   -8.0000
   13.5385   -1.1538   -7.7692   -4.0000
      ⋮

The new data in Xcentered is the original ingredients data centered by subtracting the column means from corresponding columns.

Visualize both the orthonormal principal component coefficients for each variable and the principal component scores for each observation in a single plot.

biplot(coeff(:,1:2),'scores',score(:,1:2),'varlabels',{'v_1','v_2','v_3','v_4'});

fig2plotly()

All four variables are represented in this biplot by a vector, and the direction and length of the vector indicate how each variable contributes to the two principal components in the plot. For example, the first principal component, which is on the horizontal axis, has positive coefficients for the third and fourth variables. Therefore, vectors v3 and v4 are directed into the right half of the plot. The largest coefficient in the first principal component is the fourth, corresponding to the variable v4.

The second principal component, which is on the vertical axis, has negative coefficients for the variables v1, v2, and v4, and a positive coefficient for the variable v3.

This 2-D biplot also includes a point for each of the 13 observations, with coordinates indicating the score of each observation for the two principal components in the plot. For example, points near the left edge of the plot have the lowest scores for the first principal component. The points are scaled with respect to the maximum score value and maximum coefficient length, so only their relative locations can be determined from the plot.

T-Squared Statistic

Find the Hotelling’s T-squared statistic values.

Load the sample data set.

load hald

The ingredients data has 13 observations for 4 variables.

Perform the principal component analysis and request the T-squared values.

[coeff,score,latent,tsquared] = pca(ingredients);
tsquared
tsquared = 13×1

    5.6803
    3.0758
    6.0002
    2.6198
    3.3681
    0.5668
    3.4818
    3.9794
    2.6086
    7.4818
      ⋮

Request only the first two principal components and compute the T-squared values in the reduced space of requested principal components.

[coeff,score,latent,tsquared] = pca(ingredients,'NumComponents',2);
tsquared
tsquared = 13×1

    5.6803
    3.0758
    6.0002
    2.6198
    3.3681
    0.5668
    3.4818
    3.9794
    2.6086
    7.4818
      ⋮

Note that even when you specify a reduced component space, pca computes the T-squared values in the full space, using all four components.

The T-squared value in the reduced space corresponds to the Mahalanobis distance in the reduced space.

tsqreduced = mahal(score,score)
tsqreduced = 13×1

    3.3179
    2.0079
    0.5874
    1.7382
    0.2955
    0.4228
    3.2457
    2.6914
    1.3619
    2.9903
      ⋮

Calculate the T-squared values in the discarded space by taking the difference of the T-squared values in the full space and Mahalanobis distance in the reduced space.

tsqdiscarded = tsquared - tsqreduced
tsqdiscarded = 13×1

    2.3624
    1.0679
    5.4128
    0.8816
    3.0726
    0.1440
    0.2362
    1.2880
    1.2467
    4.4915
      ⋮

Percent Variability Explained by Principal Components

Find the percent variability explained by the principal components. Show the data representation in the principal components space.

Load the sample data set.

load imports-85

Data matrix X has 13 continuous variables in columns 3 to 15: wheel-base, length, width, height, curb-weight, engine-size, bore, stroke, compression-ratio, horsepower, peak-rpm, city-mpg, and highway-mpg.

Find the percent variability explained by principal components of these variables.

[coeff,score,latent,tsquared,explained] = pca(X(:,3:15));

explained
explained = 13×1

   64.3429
   35.4484
    0.1550
    0.0379
    0.0078
    0.0048
    0.0013
    0.0011
    0.0005
    0.0002
      ⋮

The first three components explain 99.95% of all variability.

Visualize the data representation in the space of the first three principal components.

scatter3(score(:,1),score(:,2),score(:,3))
axis equal
xlabel('1st Principal Component')
ylabel('2nd Principal Component')
zlabel('3rd Principal Component')

fig2plotly()

The data shows the largest variability along the first principal component axis. This is the largest possible variance among all possible choices of the first axis. The variability along the second principal component axis is the largest among all possible remaining choices of the second axis. The third principal component axis has the third largest variability, which is significantly smaller than the variability along the second principal component axis. The fourth through thirteenth principal component axes are not worth inspecting, because they explain only 0.05% of all variability in the data.

To skip any of the outputs, you can use ~ instead in the corresponding element. For example, if you don’t want to get the T-squared values, specify

[coeff,score,latent,~,explained] = pca(X(:,3:15));