<< Models Models scidoe_regress >>

Scidoe >> Models > scidoe_multilinreg

scidoe_multilinreg

Multivariate linear regression

Calling Sequence

B = scidoe_multilinreg(Y,X)
B = scidoe_multilinreg(Y,X,level)
[B,bint] = scidoe_multilinreg(...)
[B,bint,r] = scidoe_multilinreg(...)
[B,bint,r,rint] = scidoe_multilinreg(...)
[B,bint,r,rint,stats] = scidoe_multilinreg(...)
[B,bint,r,rint,stats,fullstats] = scidoe_multilinreg(...)

Parameters

Y :

a m-by-1 matrix of doubles, the responses.

X :

a m-by-n matrix of doubles, the inputs, where m is the number of observations and n is the number of variables.

level :

a 1-by-1 matrix of doubles, the confidence level (default level=0.05, meaning that the default level is 95%). level is expected to be in the range [0.,0.5]

B :

a n-by-1 matrix of doubles, the coefficients of the linear model.

bint :

a 2-by-2 matrix of doubles, intervals with confidence level. The column bin(:,1) are the lower bounds and bin(:,2) are the upper bounds.

r :

a m-by-1 matrix of doubles, the residuals Y-B(1)-B(2)*x

rint :

a m-by-2 matrix of doubles, the confidence intervals of the residuals. The column rint(:,1) are the lower bounds and rint(:,2) are the upper bounds.

stats :

a 4-by-1 matrix of doubles, the statistics. stats(1) is the R2 statistics, stats(2) is the F statistics, stats(3) is the p-value of the F statistics, stats(4) is an estimate of the error variance.

fullstats :

a struct, the statistics, see below for details.

Description

This function computes a linear model with n independent variables x1, x2, ..., xn which best fit the data in the least squares sense.

The linear model we compute is

Y = B(1)*x1 + B(2)*x2 +...+ B(n)*xn.

The fields in fullstats are :

fullstats.RegressionSS: the sum of squares of the regression

fullstats.RegressionDof: the number of degrees of freedom of the regression

fullstats.RegressionMean: the mean of the sum of squares of the regression

fullstats.ResidualSS: the sum of squares of the residuals

fullstats.ResidualDof: the number of degrees of freedom of the residuals

fullstats.ResidualMean: the mean of the sum of squares of the residuals

fullstats.F: the F statistics

fullstats.pval : the p-value corresponding to the F statistics

fullstats.Bstddev: a (n+1)-by-1 matrix of doubles, the standard deviation of B

fullstats.R2: the R-squared statistics

The residual r is :

r == Y-X*B

Examples

// We use dataset provided by NIST in
// http://www.itl.nist.gov/div898/strd/lls/data/LINKS/DATA/Longley.dat
// Longley.dat contains 1 Response Variable y, 6 Predictor Variables x
// and 16 Observations.
X = [
83.0 234289 2356 1590 107608 1947
88.5 259426 2325 1456 108632 1948
88.2 258054 3682 1616 109773 1949
89.5 284599 3351 1650 110929 1950
96.2 328975 2099 3099 112075 1951
98.1 346999 1932 3594 113270 1952
99.0 365385 1870 3547 115094 1953
100.0 363112 3578 3350 116219 1954
101.2 397469 2904 3048 117388 1955
104.6 419180 2822 2857 118734 1956
108.4 442769 2936 2798 120445 1957
110.8 444546 4681 2637 121950 1958
112.6 482704 3813 2552 123366 1959
114.2 502601 3931 2514 125368 1960
115.7 518173 4806 2572 127852 1961
116.9 554894 4007 2827 130081 1962
];
Y = [
60323
61122
60171
61187
63221
63639
64989
63761
66019
67857
68169
66513
68655
69564
69331
70551
];
[B,bint,r,rint,stats,fullstats] = scidoe_multilinreg(Y,X)
// Print an analysis of variance table
scidoe_regressprint(fullstats)

// Create an example with known coefficients
grand("setsd",0);
// The number of observations
N = 100;
// The exact coefficients
Bexact = [2;3;4]
// The input 1
X1min = 50;
X1max = 150;
X1=grand(N,1,"uin",X1min,X1max);
X1=gsort(X1,"g","i");
// The input 2
X2min = -500;
X2max = -200;
X2=grand(N,1,"uin",X2min,X2max);
X2=gsort(X2,"g","i");
// Make the observations
X = [X1,X2];
// Compute the ouput from the input
Y = Bexact(1)+Bexact(2)*X1+Bexact(3)*X2;
// Add a normal noise to the output
sigma = 50;
e = grand(N,1,"nor",0,sigma);
Y = int(Y + e);
// Linear regression
B = scidoe_multilinreg(Y,X)
// Compare B (exact) with B (estimate)
[B,Bexact]
L = B(1)+B(2)*X(:,1)+B(3)*X(:,2);
// Check visually
scf();
plot(Y,L,"bo");
plot(Y,Y,"r-");
xtitle("Linear Regression","Data","Linear Fit");

Authors

Bibliography

"Introduction to probability and statistics for engineers and scientists.", Third Edition, Sheldon Ross, Elsevier Academic Press, 2004

http://en.wikipedia.org/wiki/Linear_regression

Octave's regress, William Poetra Yoga Hadisoeseno


Report an issue
<< Models Models scidoe_regress >>