ols

Syntax

ols(Y, X, [intercept=true], [mode=0])

Arguments

Y is the dependent variable; X is the independent variable(s).

Y is a vector; X is a matrix/table/tuple. When X is a matrix, if the number of rows equals the length of Y, each column of X is a factor; if the number of rows is not the same as the length of Y, and if the number of columns equals the length of Y, each row of X is a factor.

intercept is a Boolean variable indicating whether the regression includes the intercept. If it is true, the system automatically adds a column of 1's to X to generate the intercept. The default value is true.

mode is an integer and could be 0, 1 or 2. It indicates the contents in the output. The default value is 0.

0: a vector of the coefficient estimates.
1: a table with coefficient estimates, standard error, t-statistics, and p-values.
2: a dictionary with the following keys: ANOVA, RegressionStat, Coefficient and Residual

ANOVA (one-way analysis of variance)


Source of Variance	DF (degree of freedom)	SS (sum of square)	MS (mean of square)	F (F-score)	Significance
Regression	p	sum of squares regression, SSR	regression mean square, MSR=SSR/R	MSR/MSE	p-value
Residual	n-p-1	sum of squares error, SSE	mean square error, MSE=MSE/E
Total	n-1	sum of squares total, SST

RegressionStat (Regression statistics)


Item	Description
R2	R-squared
AdjustedR2	The adjusted R-squared corrected based on the degrees of freedom by comparing the sample size to the number of terms in the regression model.
StdError	The residual standard error/deviation corrected based on the degrees of freedom.
Observations	The sample size.

Coefficient


Item	Description
factor	Independent variables
beta	Estimated regression coefficients
StdError	Standard error of the regression coefficients
tstat	t statistic, indicating the significance of the regression coefficients

Residual: the difference between each predicted value and the actual value.

Details

Return the result of an ordinary-least-squares regression of Y on X.

Note that NULL values in X and Y are treated as 0 in calculations.

Examples

x1=1 3 5 7 11 16 23
x2=2 8 11 34 56 54 100
y=0.1 4.2 5.6 8.8 22.1 35.6 77.2;

ols(y, x1);
// output
[-9.912821,3.378632]

ols(y, (x1,x2));
// output
[-9.494813,2.806426,0.13147]

ols(y, (x1,x2), 1, 1);


factor	beta	stdError	tstat	pvalue
intercept	-9.494813	5.233168	-1.814353	0.143818
x1	2.806426	1.830782	1.532911	0.20007
x2	0.13147	0.409081	0.321379	0.764015

ols(y, (x1,x2), 1, 2);
// output
ANOVA->
Breakdown  DF SS          MS          F         Significance
---------- -- ----------- ----------- --------- ------------
Regression 2  4204.416396 2102.208198 31.467739 0.003571
Residual   4  267.220747  66.805187
Total      6  4471.637143

RegressionStat->
item         statistics
------------ ----------
R2           0.940241
AdjustedR2   0.910361
StdError     8.173444
Observations 7

Coefficient->
factor    beta      stdError tstat     pvalue
--------- --------- -------- --------- --------
intercept -9.494813 5.233168 -1.814353 0.143818
x1        2.806426  1.830782 1.532911  0.20007
x2        0.13147   0.409081 0.321379  0.764015

x=matrix(1 4 8 2 3, 1 4 2 3 8, 1 5 1 1 5);
x;


#0	#1	#2
1	1	1
4	4	5
8	2	1
2	3	1
3	8	5

ols(1..5, x);
// output
[1.156537,0.105505,0.91055,-0.697821]

ols(1..5, x.transpose());
// output
[1.156537,0.105505,0.91055,-0.697821]
// the system adjusts the dimensions of the dependent variable and the independent variables for the regression to go through.