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# robustfit

Robust regression

## Syntax

```b = robustfit(X,y) b = robustfit(X,y,wfun,tune) b = robustfit(X,y,wfun,tune,const) [b,stats] = robustfit(...) ```

## Description

`b = robustfit(X,y)` returns a (p + 1)-by-1 vector `b` of coefficient estimates for a robust multilinear regression of the responses in `y` on the predictors in `X`. `X` is an n-by-p matrix of p predictors at each of n observations. `y` is an n-by-1 vector of observed responses. By default, the algorithm uses iteratively reweighted least squares with a bisquare weighting function.

### Note

By default, `robustfit` adds a first column of 1s to `X`, corresponding to a constant term in the model. Do not enter a column of 1s directly into `X`. You can change the default behavior of `robustfit` using the input `const`, below.

`robustfit` treats `NaN`s in `X` or `y` as missing values, and removes them.

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

The weighting function `wfun` can be any one of the following:

Weight FunctionEquationDefault Tuning Constant
`'andrews'``w = (abs(r)<pi) .* sin(r) ./ r`1.339
`'bisquare'` (default)`w = (abs(r)<1) .* (1 - r.^2).^2`4.685
`'cauchy'``w = 1 ./ (1 + r.^2)`2.385
`'fair'``w = 1 ./ (1 + abs(r))`1.400
`'huber'``w = 1 ./ max(1, abs(r))`1.345
`'logistic'``w = tanh(r) ./ r`1.205
`'ols'`Ordinary least squares (no weighting function)None
`'talwar'``w = 1 * (abs(r)<1)`2.795
`'welsch'``w = exp(-(r.^2))`2.985

If `tune` is 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 value `r` in the weight functions is

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

where `resid` is the vector of residuals from the previous iteration, `h` is the vector of leverage values from a least-squares fit, and `s` is an estimate of the standard deviation of the error term given by

`s = MAD/0.6745`

Here `MAD` is the median absolute deviation of the residuals from their median. The constant 0.6745 makes the estimate unbiased for the normal distribution. If there are p columns in `X`, the smallest p absolute 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, `wfun` is specified using a function handle `@` (as in `@myfun`), and the input `tune` is required.

`b = robustfit(X,y,wfun,tune,const)` controls whether or not the model will include a constant term. `const` is `'on'` to include the constant term (the default), or `'off'` to omit it. When `const` is `'on'`, `robustfit` adds a first column of 1s to `X` and `b` becomes a (p + 1)-by-1 vector . When `const` is `'off'`, `robustfit` does not alter `X`, then `b` is a p-by-1 vector.

`[b,stats] = robustfit(...)` returns the structure `stats`, whose fields contain diagnostic statistics from the regression. The fields of `stats` are:

• `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` — Final estimate of sigma, the larger of `robust_s` and a weighted average of `ols_s` and `robust_s`

• `resid` — Residual

• `rstud` — Studentized residual (see `regress` for more information)

• `se` — Standard error of coefficient estimates

• `covb` — Estimated covariance matrix for coefficient estimates

• `coeffcorr` — Estimated correlation of coefficient estimates

• `t` — Ratio of `b` to `se`

• `p`p-values for `t`

• `w` — Vector of weights for robust fit

• `R`R factor in QR decomposition of `X`

• `dfe` — Degrees of freedom for error

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

The `robustfit` function estimates the variance-covariance matrix of the coefficient estimates using `inv(X'*X)*stats.s^2`. Standard errors and correlations are derived from this estimate.

## Examples

collapse all

Generate data with the trend y = 10 - 2* x , then change one value to simulate an outlier.

```x = (1:10)'; rng default; % For reproducibility y = 10 - 2*x + randn(10,1); y(10) = 0;```

Fit a straight line using ordinary least squares regression.

`bls = regress(y,[ones(10,1) x])`
```bls = 2×1 7.8518 -1.3644 ```

Now use robust regression to estimate a straight-line fit.

`brob = robustfit(x,y)`
```brob = 2×1 8.4504 -1.5278 ```

Create scatter plot of the data together with the fits.

```scatter(x,y,'filled'); grid on; hold on plot(x,bls(1)+bls(2)*x,'r','LineWidth',2); plot(x,brob(1)+brob(2)*x,'g','LineWidth',2) legend('Data','Ordinary Least Squares','Robust Regression')```

The robust fit is less influenced by the outlier than the least-squares fit.

## References

[1] DuMouchel, W. H., and F. L. O'Brien. “Integrating a Robust Option into a Multiple Regression Computing Environment.” Computer Science and Statistics: Proceedings of the 21st Symposium on the Interface. Alexandria, VA: American Statistical Association, 1989.

[2] Holland, P. W., and R. E. Welsch. “Robust Regression Using Iteratively Reweighted Least-Squares.” Communications in Statistics: Theory and Methods, A6, 1977, pp. 813–827.

[3] Huber, P. J. Robust Statistics. Hoboken, NJ: John Wiley & Sons, Inc., 1981.

[4] Street, J. O., R. J. Carroll, and D. Ruppert. “A Note on Computing Robust Regression Estimates via Iteratively Reweighted Least Squares.” The American Statistician. Vol. 42, 1988, pp. 152–154.

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