Freedom degrees, chemometrics

Recall, the standard deviation of the added noise in Y was lxlO-3. It is reached approximately after the removal of 3 sets of eigenvectors (at t=4). Note that, from a strictly statistical point of view, it is not quite appropriate to use Matlab s std function for the determination of the residual standard deviation since it doesn t properly take into account the gradual reduction in the degrees of freedom in the calculation of R. But it is not our intention to go into the depths of statistics here. For more rigorous statistical procedures to determine the number of significant factors, we refer to the relevant chemometrics literature on this topic. 

The t value is the number of standard deviations that the single value differs from the mean value. This t value is then compared to the critical t value obtained from a t-table, given a desired statistical confidence (i.e., 90%, 95%, or 99% confidence) and the number of degrees of freedom (typically iV-1), to assess whether the value is statistically different from the other values in the series. In chemometrics, the t test can be useful for evaluating outliers in data sets. 

M.R. Valhm and V.F. Juhano analyzed the data of Table 6.9 as an assignment in a chemometrics course, and soon perceived that, since there are 27 different runs, it is possible to fit a fimction with up to 27 parameters to the data. The linear and quadratic functions for three variables are defined by only 4 and 10 parameters, respectively. Using them to model the data in the table, we will have many degrees of freedom left over to assess lack of fit. 

Van der Voet, H. (1999). Pseudo-degrees of freedom for complex predictive models The example of partial least squares. Journal of Chemometrics, Vol. 13, pp. 195-208. 

It can be calculated by means of a tabulated value for Student s t distribution with v = Ni, 1 degrees of freedom and order = 1 - a (one-sided values) (- Chemometrics). 

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