When performing data analysis, it is sometimes desirable to assign weights to selected data according to their perceived values. For example, data that are more reliable are assigned a higher weight, or weight value that is inversely proportional to variance of that data value. This technique can be applied to multiple linear regression as well. In the more common regression method by ordinary least squares (OLS), all observed data are of the same weight. In weighted least squares (WLS), an arbitrary weight value is assigned to each of the observations. WLS is a special case of generalized least squares (GLS) method.
The regression analysis in Nspire supports only the OLS method. Programming is required to adopt the WLS. Fortunately, the built-in programming by the Nspire supports accessing data stored in the spreadsheet application in the form of lists and matrices, which are heavily relied upon on the calculation of WLS statistics. Needless to say Nspire is good at performing matrix operations.
Similar to OLS, the WLS approach is based on the minimization of the sum of squares between sets of data, from which the parameters for the regression equation are obtained. In WLS, the equation is given by
β = (XT Λ-1 X)-1XTΛ-1Y
where Λ is the covariance matrix used to determine the weights, and can be represented by the piece-wise equation
The total sum of squares in WLS is given by
and the sum of squares error by
For visualization, the response plane plot of the regression equations obtained from a sample data set by OLS with the Nspire built-in multiple linear regression and the WLS program respectively are generated using the 3D function plot.