Statistics: pcares

### Function Reference: `pcares`

statistics: residuals = pcares (x, ndim)
statistics: [residuals, reconstructed] = pcares (x, ndim)

Calculate residuals from principal component analysis.

`residuals = pcares (x, ndim)` returns the residuals obtained by retaining ndim principal components of the N×D matrix x. Rows of x correspond to observations, columns of x correspond to variables. ndim is a scalar and must be less than or equal to D. residuals is a matrix of the same size as x. Use the data matrix, not the covariance matrix, with this function.

`[residuals, reconstructed] = pcares (x, ndim)` returns the reconstructed observations, i.e. the approximation to x obtained by retaining its first ndim principal components.

`pcares` does not normalize the columns of x. Use `pcares (zscore (x), ndim)` in order to perform the principal components analysis based on standardized variables, i.e. based on correlations. Use `pcacov` in order to perform principal components analysis directly on a covariance or correlation matrix without constructing residuals.

### References

1. Jolliffe, I. T., Principal Component Analysis, 2nd Edition, Springer, 2002

See also: factoran, pcacov, pca

Source Code: pcares

### Example: 1

     x = [ 7    26     6    60;
           1    29    15    52;
          11    56     8    20;
          11    31     8    47;
           7    52     6    33;
          11    55     9    22;
           3    71    17     6;
           1    31    22    44;
           2    54    18    22;
          21    47     4    26;
           1    40    23    34;
          11    66     9    12;
          10    68     8    12];

     ## As we increase the number of principal components, the norm
     ## of the residuals matrix will decrease
     r1 = pcares (x,1);
     n1 = norm (r1)
     r2 = pcares (x,2);
     n2 = norm (r2)
     r3 = pcares (x,3);
     n3 = norm (r3)
     r4 = pcares (x,4);
     n4 = norm (r4)

    n1 = 28.460
    n2 = 12.201
    n3 = 1.6870
    n4 = 4.2168e-14