pcaFS
pcaFS performs Principal Component Analysis (PCA) on raw data.
Description
The main differences with respect to MATLAB function pca are: 1) accepts an input X also as table;
2) produces in table format the percentage of the variance explained single and cumulative of the various components and the associated scree plot, in order to decide about the number of components to retain.
3) returns the loadings in table format and shows them graphically.
4) provides guidelines about the automatic choice of the number of components;
5) returns the communalities for each variable with respect to the first k principal components in table format;
6) retuns the orthogonal distance ($OD_i$) of each observation to the PCA subspace. For example, if the subspace is defined by the first two principal components, $OD_i$ is computed as:
\[ OD_i=|| z_i- V_{(2)} V_{(2)}' z_i || \]where z_i is the i-th row of the original centered data matrix $Z$ of dimension $n \times v$ and $V_{(2)}=(v_1 v_2)$ is the matrix of size $p \times 2$ containing the first two eigenvectors of $Z'Z/(n-1)$. The observations with large $OD_i$ are not well represented in the space of the principal components.
7) returns the score distance $SD_i$ of each observation. For example, if the subspace is defined by the first two principal components, $SD_i$ is computed as:
\[ SD_i=\sqrt{(z_i'v_1)^2/l_1+ (z_i'v_2)^2/l_2 } \]where $l_1$ and $l_2$ are the first two eigenvalues of $Z'Z/(n-1)$.
8) calls app biplotFS which enables to obtain an interactive biplot in which points, rowslabels or arrows can be shown or hidden. This app also gives the possibility of controlling the length of the arrows and the position of the row points through two interactive slider bars. In the app it is also possible to color row points depending on the orthogonal distance ($OD_i$) of each observation to the PCA subspace. If optional input argument bsb or bdp is specified, it is possible to have in the app two tabs which enable the user to select the breakdown point of the analysis or the subset size to use in the svd. The units which are declared as outliers or the units outside the subset are shown in the biplot with filled circles.
use of pcaFS on the ingredients dataset.out
=pcaFS(Y,
Name, Value)
Examples
use of pcaFS on the ingredients dataset.
use of pcaFS on the ingredients dataset.
load hald % Operate on the covariance matrix. out=pcaFS(ingredients,'standardize',false,'biplot',0);
The first PC already explains more than 0.95^v variability
In what follows we still extract the first 2 PCs
Initial covariance matrix
Y1 Y2 Y3 Y4
______ _______ ______ _______
Y1 34.60 20.92 -31.05 -24.17
Y2 20.92 242.14 -13.88 -253.42
Y3 -31.05 -13.88 41.03 3.17
Y4 -24.17 -253.42 3.17 280.17
Explained variance by PCs
Eigenvalues Explained_Variance Explained_Variance_cum
___________ __________________ ______________________
PC1 517.80 86.60 86.60
PC2 67.50 11.29 97.89
PC3 12.41 2.07 99.96
PC4 0.24 0.04 100.00
Loadings = correlations between variables and PCs
PC1 PC2
_____ _____
Y1 0.26 0.90
Y2 0.99 0.01
Y3 -0.10 -0.97
Y4 -0.99 0.05
Communalities
PC1 PC2 PC1-PC2
____ ____ _______
Y1 0.07 0.81 0.88
Y2 0.98 0.00 0.98
Y3 0.01 0.94 0.95
Y4 0.99 0.00 0.99
Units with the 5 largest values of (combined) score and orthogonal distance
5 10 6 7 1
ans =
1×3 Axes array:
Axes Axes Axes
