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Robust regression

`b = robustfit(X,y)`

b = robustfit(X,y,* wfun*,tune)

b = robustfit(X,y,

`wfun`

`const`

[b,stats] = robustfit(...)

`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.

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,`

specifies
a weighting function * wfun*,tune)

`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 Function | Equation | Default 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,`

controls
whether or not the model will include a constant term. * wfun*,tune,

`const`

`const`

`'on'`

to
include the constant term (the default), or `'off'`

to
omit it. When `const`

`'on'`

, `robustfit`

adds
a first column of 1s to `X`

and `b`

becomes
a (`const`

`'off'`

, `robustfit`

does
not alter `X`

, then `b`

is a `[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.

[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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