You can reduce outlier effects in linear regression models by using robust linear
regression. This topic defines robust regression, shows how to use it to fit a linear
model, and compares the results to a standard fit. You can use
fitlm with the
'RobustOpts' name-value pair argument
to fit a robust regression model. Or you can use
robustfit to simply compute the robust regression coefficient
Robust linear regression is less sensitive to outliers than standard linear regression. Standard linear regression uses ordinary least-squares fitting to compute the model parameters that relate the response data to the predictor data with one or more coefficients. (See Estimation of Multivariate Regression Models for more details.) As a result, outliers have a large influence on the fit, because squaring the residuals magnifies the effects of these extreme data points. Models that use standard linear regression, described in What Is a Linear Regression Model?, are based on certain assumptions, such as a normal distribution of errors in the observed responses. If the distribution of errors is asymmetric or prone to outliers, model assumptions are invalidated, and parameter estimates, confidence intervals, and other computed statistics become unreliable.
Robust regression uses a method called iteratively reweighted least squares to assign a weight to each data point. This method is less sensitive to large changes in small parts of the data. As a result, robust linear regression is less sensitive to outliers than standard linear regression.
In weighted least squares, the fitting process includes the weight as an
additional scale factor, which improves the fit. The weights determine how much each
response value influences the final parameter estimates. A low-quality data point
(for example, an outlier) should have less influence on the fit. To compute the
weights wi, you can use predefined weight functions, such as Tukey's bisquare
function (see the name-value pair argument
fitlm for more options).
The iteratively reweighted least-squares algorithm automatically and iteratively calculates the weights. At initialization, the algorithm assigns equal weight to each data point, and estimates the model coefficients using ordinary least squares. At each iteration, the algorithm computes the weights wi, giving lower weight to points farther from model predictions in the previous iteration. The algorithm then computes model coefficients b using weighted least squares. Iteration stops when the values of the coefficient estimates converge within a specified tolerance. This algorithm simultaneously seeks to find the curve that fits the bulk of the data using the least-squares approach, and to minimize the effects of outliers.
For more details, see Steps for Iteratively Reweighted Least Squares.
This example shows how to use robust regression with the
fitlm function, and compares the results of a robust fit to a standard least-squares fit.
moore data. The predictor data is in the first five columns, and the response data is in the sixth.
load moore X = moore(:,1:5); y = moore(:,6);
Fit the least-squares linear model to the data.
mdl = fitlm(X,y)
mdl = Linear regression model: y ~ 1 + x1 + x2 + x3 + x4 + x5 Estimated Coefficients: Estimate SE tStat pValue ___________ __________ _________ ________ (Intercept) -2.1561 0.91349 -2.3603 0.0333 x1 -9.0116e-06 0.00051835 -0.017385 0.98637 x2 0.0013159 0.0012635 1.0415 0.31531 x3 0.0001278 7.6902e-05 1.6618 0.11876 x4 0.0078989 0.014 0.56421 0.58154 x5 0.00014165 7.3749e-05 1.9208 0.075365 Number of observations: 20, Error degrees of freedom: 14 Root Mean Squared Error: 0.262 R-squared: 0.811, Adjusted R-Squared: 0.743 F-statistic vs. constant model: 12, p-value = 0.000118
Fit the robust linear model to the data by using the
'RobustOps' name-value pair argument.
mdlr = fitlm(X,y,'RobustOpts','on')
mdlr = Linear regression model (robust fit): y ~ 1 + x1 + x2 + x3 + x4 + x5 Estimated Coefficients: Estimate SE tStat pValue __________ __________ ________ ________ (Intercept) -1.7516 0.86953 -2.0144 0.063595 x1 1.7006e-05 0.00049341 0.034467 0.97299 x2 0.00088843 0.0012027 0.7387 0.47229 x3 0.00015729 7.3202e-05 2.1487 0.049639 x4 0.0060468 0.013326 0.45375 0.65696 x5 6.8807e-05 7.0201e-05 0.98015 0.34365 Number of observations: 20, Error degrees of freedom: 14 Root Mean Squared Error: 0.249 R-squared: 0.775, Adjusted R-Squared: 0.694 F-statistic vs. constant model: 9.64, p-value = 0.000376
Visually examine the residuals of the two models.
tiledlayout(1,2) nexttile plotResiduals(mdl,'probability') title('Linear Fit') nexttile plotResiduals(mdlr,'probability') title('Robust Fit')
The residuals from the robust fit (right half of the plot) are closer to the straight line, except for the one obvious outlier.
Find the index of the outlier.
outlier = find(isoutlier(mdlr.Residuals.Raw))
outlier = 1
Plot the weights of the observations in the robust fit.
figure b = bar(mdlr.Robust.Weights); b.FaceColor = 'flat'; b.CData(outlier,:) = [.5 0 .5]; xticks(1:length(mdlr.Residuals.Raw)) xlabel('Observations') ylabel('Weights') title('Robust Fit Weights')
The weight of the outlier in the robust fit (purple bar) is much less than the weights of the other observations.
The iteratively reweighted least-squares algorithm follows this procedure:
Start with an initial estimate of the weights and fit the model by weighted least squares.
Compute the adjusted residuals. The adjusted residuals are given by
where ri are the ordinary least-squares residuals, and hi are the least-squares fit leverage values. Leverages adjust the residuals by reducing the weight of high-leverage data points, which have a large effect on the least-squares fit (see Hat Matrix and Leverage).
Standardize the residuals. The standardized adjusted residuals are given by
where K is a tuning constant, and s
is an estimate of the standard deviation of the error term given by
s = MAD/0.6745.
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 the predictor data matrix
p columns, the software excludes the smallest
p absolute deviations when computing the
Compute the robust weights wi as a function of u. For example, the bisquare weights are given by
Estimate the robust regression coefficients b. The weights modify the expression for the parameter estimates b as follows
where W is the diagonal weight matrix, X is the predictor data matrix, and y is the response vector.
Estimate the weighted least-squares error
where wi are the weights, yi are the observed responses, ŷi are the fitted responses, and ri are the residuals.
Iteration stops if the fit converges or the maximum number of iterations is reached. Otherwise, perform the next iteration of the least-squares fitting by returning to the second step.