The use of the term " n " " 1 is called Bessel's correction, and it is also used in sample covariance and the sample standard deviation ( the square root of variance ).
12.
However, when \ alpha > 0, i . e ., when ridge regression is used, the addition of \ alpha I to the sample covariance matrix ensures that all of its eigenvalues will be strictly greater than 0.
13.
When \ alpha = 0, i . e ., in the case of ordinary least squares, the condition that d > n causes the sample covariance matrix X ^ T X to not have full rank and so it cannot be inverted to yield a unique solution.
14.
The reason the sample covariance matrix has \ textstyle N-1 in the denominator rather than \ textstyle N is essentially that the population mean \ operatorname { E } ( X ) is not known and is replaced by the sample mean \ mathbf { \ bar { X } }.
15.
Then " D " is the data transformed so every random variable has zero mean, and " T " is the data transformed so all variables have zero mean and zero correlation with all other variables the sample covariance matrix of " T " will be the identity matrix.
16.
If only one variable has had values observed, then the sample mean is a single number ( the arithmetic average of the observed values of that variable ) and the sample covariance matrix is also simply a single value ( a 1x1 matrix containing a single number, the sample variance of the observed values of that variable ).
17.
The sample mean is a matrix whose " i, j " element is the sample covariance ( an estimate of the population covariance ) between the sets of observed values of two of the variables and whose " i, i " element is the sample variance of the observed values of one of the variables.
18.
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector \ textstyle \ mathbf { X }, a row vector whose " j " th element ( " j " = 1, . . ., " K " ) is one of the random variables.
19.
Specifically, if \ mathbf { X } : = ( x _ { ij } ) \ in \ mathbb { R } ^ { n \ times m } is a prewhitened data matrix, that is, the sample mean of each column is zero and the sample covariance of the rows is the m \ times m dimensional identity matrix, Kernel ICA estimates a m \ times m dimensional orthogonal matrix \ mathbf { A } so as to minimize finite-sample \ mathcal { F }-correlations between the columns of \ mathbf { S } : = \ mathbf { X } \ mathbf { A } ^ { \ prime }.
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