Syntax

Description

b = robustfit(X,y)returns a (p+ 1)-by-1 vectorbof coefficient estimates for a robust multilinear regression of the responses iny在预测X.Xis ann-by-pmatrix ofppredictors at each ofnobservations.yis ann-by-1 vector of observed responses. By default, the algorithm uses iteratively reweighted least squares with a bisquare weighting function.

Note:By default,robustfitadds a first column of 1s toX, corresponding to a constant term in the model. Do not enter a column of 1s directly intoX. You can change the default behavior ofrobustfitusing the inputconst, below.

robustfittreatsNaNs inXoryas missing values, and removes them.

b = robustfit(X,y,wfun,调)specifies a weighting functionwfun.tuneis a tuning constant that is divided into the residual vector before computing weights.

The weighting functionwfuncan be any one of the following:

Iftuneis unspecified, the default value in the table is used. Default tuning constants give coefficient estimates that are approximately 95% as statistically efficient as the ordinary least-squares estimates, provided the response has a normal distribution with no outliers. Decreasing the tuning constant increases the downweight assigned to large residuals; increasing the tuning constant decreases the downweight assigned to large residuals.

The valuerin the weight functions is

r = resid/(tune*s*sqrt(1-h))

whereresidis the vector of residuals from the previous iteration,his the vector of leverage values from a least-squares fit, andsis an estimate of the standard deviation of the error term given by

s = MAD/0.6745

HereMADis the median absolute deviation of the residuals from their median. The constant 0.6745 makes the estimate unbiased for the normal distribution. If there arepcolumns inX, the smallestpabsolute deviations are excluded when computing the median.

You can write your own weight function. The function must take a vector of scaled residuals as input and produce a vector of weights as output. In this case,wfunis specified using a function handle@(as in@myfun), and the inputtuneis required.

b = robustfit(X,y,wfun,调,const)controls whether or not the model will include a constant term.constis'on'to include the constant term (the default), or'off'to omit it. Whenconstis'on',robustfitadds a first column of 1s toXandbbecomes a (p+ 1)-by-1 vector . Whenconstis'off',robustfitdoes not alterX, thenbis ap-by-1 vector.

[b,stats] = robustfit(...)returns the structurestats, whose fields contain diagnostic statistics from the regression. The fields ofstatsare:

• ols_s— Sigma estimate (RMSE) from ordinary least squares

• robust_s— Robust estimate of sigma

• mad_s— Estimate of sigma computed using the median absolute deviation of the residuals from their median; used for scaling residuals during iterative fitting

• s——最终估计的σ,较大的robust_sand a weighted average ofols_sandrobust_s

• resid— Residual

• rstud— Studentized residual (seeregressfor more information)

• se— Standard error of coefficient estimates

• covb— Estimated covariance matrix for coefficient estimates

• coeffcorr— Estimated correlation of coefficient estimates

• t— Ratio ofbtose

• p—p-values fort

• w— Vector of weights for robust fit

• R—Rfactor inQRdecomposition ofX

• dfe— Degrees of freedom for error

• h— Vector of leverage values for least-squares fit

Therobustfitfunction estimates the variance-covariance matrix of the coefficient estimates usinginv(X'*X)*stats.s^2. Standard errors and correlations are derived from this estimate.

References

See Also

regress|robustdemo