InterpolationOrder (1) 

The option InterpolationOrder specifies the polynomial order of the B-spline basis. Its values are expected to be non-negative integers:

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Method (1) 

QuantileRegression uses LinearProgramming. Additional parameters can be passed to LinearProgramming with the Method option:

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Fit for heteroscedastic data (3) 

Here is heteroscedastic data (the variance is not constant with respect to the predictor variable):

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Find quantile regression fits:

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Plot the data and the regression quantiles:

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Note that the regression quantiles clearly outline the heteroscedastic nature of the data.

Find variance anomalies (4) 

A certain contextual type of anomaly is a subset of points that have variance very different than other subsets. Using quantile regression we can (1) evaluate the regressor-dependent variance for each point using the regression quantiles 0.25 and 0.75; and (2) find the points that have outlier variances.

Here we compute and plot the variance estimates for a signal:

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Find the lower and upper thresholds for the variance outliers:

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Find the outlier positions:

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Plot the data and the outliers found:

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Fit and anomalies for financial time series (5) 

Here is a financial time series:

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Do a quantile regression fit and plot it:

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Here are the errors of the fit found:

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Find anomalies' positions in the list of fit errors:

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Plot the data, fit and anomalies:

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Analyze temperature time series (6) 

Get temperature data:

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Convert the time series into a list of numeric pairs:

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Compute quantile regression fits:

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Plot the data and the regression quantiles:

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Find an estimate of the conditional cumulative distribution function (CDF) at the date 2017-10-01:

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Find outliers in the temperature data—outliers are defined as points below or above the 0.02 and 0.98 regression quantiles respectively:

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Slow computations

Because of the linear programming formulation for some data and knots specifications, the computations can be slow.

Fitting for probabilities 0 and 1 (3) 

For most data, the quantile regression fitting for probabilities 0 and 1 produces regression quantiles that are "too far away from the data."

Find regression quantiles for probabilities 0 and 0.5 and plot them:

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Find regression quantiles for probabilities 0.5 and 1 and plot them:

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One way to fix this is to use probabilities that are close to 0 and 1 from above and below, respectively:

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Overfitting (4) 

Consider the following nonlinear data:

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Make a quantile regression fit with 20 knots:

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Make a quantile regression fit with 40 knots:

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Plot the regression quantiles and the data:

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You can see that the regression quantile computed with 40 knots is "overfitted" between 0 and 8—the B-spline basis knots are too densely placed between 0 and 8.

Intersecting regression quantiles (3) 

When regression quantiles are overfitted, then the estimate of the conditional cumulative distribution function (CDF) can be problematic—the estimated CDF is not a monotonically increasing function.

Compute regression quantiles using "too many" knots:

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Plot the regression quantiles:

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Here is the estimated conditional CDF:

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Rescaling (1) 

For certain data, it is beneficial to rescale the predictor values, predicted values, or both before doing the quantile regression computations:

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