fitrqlinear
Syntax
Description
Mdl = fitrqlinear(Tbl,ResponseVarName)Mdl. The function
trains the model using the predictors in the table Tbl and the response
values in the ResponseVarName table variable.
By default, the function uses the median (0.5 quantile).
Mdl = fitrqlinear(___,Name=Value)Quantiles name-value argument.
[
also returns Mdl,AggregateOptimizationResults] = fitrqlinear(___)AggregateOptimizationResults, which contains
hyperparameter optimization results when you specify the
OptimizeHyperparameters and
HyperparameterOptimizationOptions name-value arguments. You must also
specify the ConstraintType and ConstraintBounds
options of HyperparameterOptimizationOptions. You can use this syntax
to optimize on the compact model size instead of the cross-validation loss, and to solve a
set of multiple optimization problems that have the same options but different constraint
bounds. (since R2025a)
Note
Hyperparameter optimization is supported only for models with one quantile.
Examples
Fit a quantile linear regression model using the 0.25, 0.50, and 0.75 quantiles.
Load the carbig data set, which contains measurements of cars made in the 1970s and early 1980s. Create a matrix X containing the predictor variables Acceleration, Displacement, Horsepower, and Weight. Store the response variable MPG in the variable Y.
load carbig
X = [Acceleration,Displacement,Horsepower,Weight];
Y = MPG;Delete rows of X and Y where either array has missing values.
R = rmmissing([X Y]); X = R(:,1:end-1); Y = R(:,end);
Partition the data into training data (XTrain and YTrain) and test data (XTest and YTest). Reserve approximately 20% of the observations for testing, and use the rest of the observations for training.
rng(0,"twister") % For reproducibility of the partition c = cvpartition(length(Y),"Holdout",0.20); trainingIdx = training(c); XTrain = X(trainingIdx,:); YTrain = Y(trainingIdx); testIdx = test(c); XTest = X(testIdx,:); YTest = Y(testIdx);
Train a quantile linear regression model. Specify to use the 0.25, 0.50, and 0.75 quantiles (that is, the lower quartile, median, and upper quartile). To improve the model fit, change the beta tolerance to 1e-6 instead of the default value 1e-4. Use a ridge (L2) regularization term of 1. Adjusting the regularization term can help prevent quantile crossing.
Mdl = fitrqlinear(XTrain,YTrain,Quantiles=[0.25,0.50,0.75], ...
BetaTolerance=1e-6,Lambda=1)Mdl =
RegressionQuantileLinear
ResponseName: 'Y'
CategoricalPredictors: []
ResponseTransform: 'none'
Beta: [4×3 double]
Bias: [17.0004 23.0029 29.5243]
Quantiles: [0.2500 0.5000 0.7500]
Properties, Methods
Mdl is a RegressionQuantileLinear model object. You can use dot notation to access the properties of Mdl. For example, Mdl.Beta and Mdl.Bias contain the linear coefficient estimates and estimated bias terms, respectively. Each column of Mdl.Beta corresponds to one quantile, as does each element of Mdl.Bias.
In this example, you can use the linear coefficient estimates and estimated bias terms directly to predict the test set responses for each of the three quantiles in Mdl.Quantiles. In general, you can use the predict object function to make quantile predictions.
predictedY = XTest*Mdl.Beta + Mdl.Bias
predictedY = 78×3
12.3963 16.2569 19.5263
5.8328 10.1568 12.6058
17.1726 20.6398 24.9748
23.3790 28.1122 31.3617
17.0036 22.5314 23.0539
16.6120 17.0713 20.1062
10.9274 12.3302 13.2707
14.9130 14.6659 12.7100
16.3103 17.7497 20.8477
19.6229 25.7109 30.5389
19.5583 24.6621 30.4345
12.9525 14.4508 16.0004
14.8525 16.1338 16.4112
24.1648 31.1758 33.9310
15.1039 17.8497 19.2013
⋮
isequal(predictedY,predict(Mdl,XTest))
ans = logical
1
Each column of predictedY corresponds to a separate quantile (0.25, 0.5, or 0.75).
Visualize the predictions of the quantile linear regression model. First, create a grid of predictor values.
minX = floor(min(X))
minX = 1×4
8 68 46 1613
maxX = ceil(max(X))
maxX = 1×4
25 455 230 5140
gridX = zeros(100,size(X,2)); for p = 1:size(X,2) gridp = linspace(minX(p),maxX(p))'; gridX(:,p) = gridp; end
Next, use the trained model Mdl to predict the response values for the grid of predictor values.
gridY = predict(Mdl,gridX)
gridY = 100×3
20.8073 25.4104 29.1436
20.6991 25.2907 29.0251
20.5909 25.1711 28.9066
20.4828 25.0514 28.7881
20.3746 24.9318 28.6696
20.2664 24.8121 28.5512
20.1583 24.6924 28.4327
20.0501 24.5728 28.3142
19.9419 24.4531 28.1957
19.8337 24.3335 28.0772
19.7256 24.2138 27.9587
19.6174 24.0941 27.8402
19.5092 23.9745 27.7217
19.4011 23.8548 27.6032
19.2929 23.7351 27.4848
⋮
For each observation in gridX, the predict object function returns predictions for the quantiles in Mdl.Quantiles.
View the gridY predictions for the second predictor (Displacement). Compare the quantile predictions to the true test data values.
predictorIdx = 2; plot(XTest(:,predictorIdx),YTest,".") hold on plot(gridX(:,predictorIdx),gridY(:,1)) plot(gridX(:,predictorIdx),gridY(:,2)) plot(gridX(:,predictorIdx),gridY(:,3)) hold off xlabel("Predictor (Displacement)") ylabel("Response (MPG)") legend(["True values","0.25 predicted values", ... "0.50 predicted values","0.75 predicted values"]) title("Test Data")
The red line shows the predictions for the 0.25 quantile, the yellow line shows the predictions for the 0.50 quantile, and the purple line shows the predictions for the 0.75 quantile. The blue points indicate the true test data values.
Notice that the quantile prediction lines do not cross each other.
When training a quantile linear regression model, you can use a ridge (L2) regularization term to prevent quantile crossing.
Load the carbig data set, which contains measurements of cars made in the 1970s and early 1980s. Create a table containing the predictor variables Acceleration, Cylinders, Displacement, and so on, as well as the response variable MPG.
load carbig cars = table(Acceleration,Cylinders,Displacement, ... Horsepower,Model_Year,Weight,MPG);
Remove rows of cars where the table has missing values.
cars = rmmissing(cars);
Partition the data into training and test sets using cvpartition. Use approximately 80% of the observations as training data, and 20% of the observations as test data.
rng(0,"twister") % For reproducibility of the data partition c = cvpartition(height(cars),"Holdout",0.20); trainingIdx = training(c); carsTrain = cars(trainingIdx,:); testIdx = test(c); carsTest = cars(testIdx,:);
Train a quantile linear regression model. Use the 0.25, 0.50, and 0.75 quantiles (that is, the lower quartile, median, and upper quartile). To improve the model fit, change the beta tolerance to 1e-6 instead of the default value 1e-4.
Mdl = fitrqlinear(carsTrain,"MPG",Quantiles=[0.25 0.5 0.75], ... BetaTolerance=1e-6);
Mdl is a RegressionQuantileLinear model object.
Determine if the test data predictions for the quantiles in Mdl.Quantiles cross each other by using the predict object function of Mdl. The crossingIndicator output argument contains a value of 1 (true) for any observation with quantile predictions that cross.
[~,crossingIndicator] = predict(Mdl,carsTest); sum(crossingIndicator)
ans = 2
In this example, two of the observations in carsTest have quantile predictions that cross each other.
To prevent quantile crossing, specify the Lambda name-value argument in the call to fitrqlinear. Use a 0.1 ridge (L2) penalty term.
newMdl = fitrqlinear(carsTrain,"MPG",Quantiles=[0.25 0.5 0.75], ... BetaTolerance=1e-6,Lambda=0.1); [predictedY,newCrossingIndicator] = predict(newMdl,carsTest); sum(newCrossingIndicator)
ans = 0
With regularization, the predictions for the test data set do not cross for any observations.
Visualize the predictions returned by newMdl by using a scatter plot with a reference line. Plot the predicted values along the vertical axis and the true response values along the horizontal axis. Points on the reference line indicate correct predictions.
plot(carsTest.MPG,predictedY(:,1),".") hold on plot(carsTest.MPG,predictedY(:,2),".") plot(carsTest.MPG,predictedY(:,3),".") plot(carsTest.MPG,carsTest.MPG) hold off xlabel("True MPG") ylabel("Predicted MPG") legend(["0.25 quantile values","0.50 quantile values", ... "0.75 quantile values","Reference line"], ... Location="southeast") title("Test Data")
Blue points correspond to the 0.25 quantile, red points correspond to the 0.50 quantile, and yellow points correspond to the 0.75 quantile.
For a more in-depth example, see Regularize Quantile Regression Model to Prevent Quantile Crossing.
Input Arguments
Name-Value Arguments
Output Arguments
Tips
You can use the α/2 and 1 – α/2 quantiles to create a prediction interval that captures an estimated 100*(1 – α) percent of the variation in the response. For an example, see Create Prediction Interval Using Quantiles.
You can use quantile regression models to fit models that are robust to outliers. For an example, see Fit Regression Models to Data with Outliers.
