Formula Engine Reference Guide

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• the linear least squares solution, X, to the overdetermined system of equations AX=Y
• the standard errors of the least square estimates of each element of the parameter vector X, the T-statistics that compare each parameter to zero, and the significance level of each.
• the variance of the model, the R-squared correlation for the model, and its corresponding T-statistic and significance level. (Note: These values are only valid if A contains a column of 1's, which corresponds to the y-intercept. Omitting the column of 1's is equivalent to forcing the y-intercept of the model to be equal to zero.)

If any columns of A are linearly dependent, the function produces an error.

The output of LLS is a a table with four rows and n+1 columns, where n is the number of independent variables in the model (i.e., the number of columns in the A matrix). The table is presented in the following format:
where

= the least squares estimate of the ith coefficient, corresponding to the independent variable in the ith column of A.

= the standard error of xi.

= the t-statistic for testing whether xi is significantly different from zero.

= the probability of error in rejecting the null hypothesis that xi=0 , based on two-sided t-test.

= the mean squared error of the model.

MSE = the model coefficient of determination (the square of the model correlation coefficient, R).

= the t-statistic for testing whether R is significantly different from zero.

p(tr) = the probability of error in rejecting the null hypothesis that R=0, based on a two-sided t-test.

Parameters
A
A range

Y
A range

Examples

Matrix D3..E5 =
D E
1 3
2 4
7 4

Matrix G3..G5 =
9
5
4

LLS(D3..E5, G3..G5) =

-0.76479076   2.2640693   15.308802

0.9516909   1.0921967   0.54349619

-0.80361256   2.0729501   0.49703082

0.56904671   0.2861429   0.60906673

LLS(A1..A2, B1..C2) = Error - LLS, improper dimensions