The qrnn package for R implements the quantile regression neural network (QRNN) (Taylor, 2000; Cannon, 2011; Cannon, 2018), which is a flexible nonlinear form of quantile regression. While low level modelling functions are available, it is recommended that the mcqrnn.fit and mcqrnn.predict wrappers be used for most applications. More information is provided below.
The goal of quantile regression is to estimate conditional quantiles of a response variable that depend on covariates in some form of regression equation. The QRNN adopts the multi-layer perceptron neural network architecture. The implementation follows from previous work on the estimation of censored regression quantiles, thus allowing predictions for mixed discrete-continuous variables like precipitation (Friederichs and Hense, 2007). A differentiable approximation to the quantile regression cost function is adopted so that a simplified form of the finite smoothing algorithm (Chen, 2007) can be used to estimate model parameters. This approximation can also be used to force the model to solve a standard least squares regression problem or an expectile regression problem (Cannon, 2018). Weight penalty regularization can be added to help avoid overfitting, and ensemble models with bootstrap aggregation are also provided.
An optional monotone constraint can be invoked, which guarantees monotonic non-decreasing behaviour of model outputs with respect to specified covariates (Zhang, 1999). The input-hidden layer weight matrix can also be constrained so that model relationships are strictly additive (see gam.style; Cannon, 2018). Borrowing strength by using a composite model for multiple regression quantiles (Zou et al., 2008; Xu et al., 2017) is also possible (see composite.stack). Weights can be applied to individual cases (Jiang et al., 2012).
Applying the monotone constraint in combination with the composite model allows one to simultaneously estimate multiple non-crossing quantiles (Cannon, 2018); the resulting monotone composite QRNN (MCQRNN) is provided by the mcqrnn.fit and mcqrnn.predict wrapper functions. Examples for qrnn.fit and qrnn2.fit show how the same functionality can be achieved using the low level composite.stack and fitting functions.
QRNN models with a single layer of hidden nodes can be fitted using the qrnn.fit function. Predictions from a fitted model are made using the qrnn.predict function. The function gam.style can be used to visualize and investigate fitted covariate/response relationships from qrnn.fit (Plate et al., 2000). Note: a single hidden layer is usually sufficient for most modelling tasks. With added monotonicity constraints, a second hidden layer may sometimes be beneficial (Lang, 2005; Minin et al., 2010). QRNN models with two hidden layers are available using the qrnn2.fit and qrnn2.predict functions. For non-crossing quantiles, the mcqrnn.fit and mcqrnn.predict wrappers also allow models with one or two hidden layers to be fitted and predictions to be made from the fitted models.
In general, mcqrnn.fit offers a convenient, single function for fitting multiple quantiles simultaneously. Note, however, that default settings in mcqrnn.fit and other model fitting functions are not optimized for general speed, memory efficiency, or accuracy and should be adjusted for a particular regression problem as needed. In particular, the approximation to the quantile regression cost function eps.seq, the number of trials n.trials, and number of iterations iter.max can all influence fitting speed (and accuracy), as can changing the optimization algorithm via method. Non-crossing quantiles are implemented by stacking multiple copies of the x and y data, one copy per value of tau. Depending on the dataset size, this can lead to large matrices being passed to the optimization routine. In the adam adaptive stochastic gradient descent method, the minibatch size can be adjusted to help offset this cost. Model complexity is determined via the number of hidden nodes, n.hidden and n.hidden2, as well as the optional weight penalty penalty; values of these hyperparameters are crucial to obtaining a well performing model.
When using mcqrnn.fit, it is also possible to estimate the full quantile regression process by specifying a single integer value for tau. In this case, tau is the number of random samples used in the stochastic estimation. For more information, see Tagasovska and Lopez-Paz (2019). It may be necessary to restart the optimization multiple times from the previous weights and biases, in which case init.range can be set to the weights values from the previously completed optimization run. For large datasets, it is recommended that the adam method with an integer tau and an appropriate minibatch size be used for optimization.
If models for multiple quantiles have been fitted, for example by mcqrnn.fit or multiple calls to either qrnn.fit or qrnn2.fit, the (experimental) dquantile function and its companion functions are available to create proper probability density, distribution, and quantile functions (Quiñonero-Candela et al., 2006; Cannon, 2011). Alternative distribution, quantile, and random variate functions based on the Nadaraya-Watson estimator (Passow and Donner, 2020) are also available in [p,q,r]quantile.nw. These can be useful for assessing probabilistic calibration and evaluating model performance.
Note: the user cannot easily change the output layer transfer function to be different than hramp, which provides either the identity function or a ramp function to accommodate optional left censoring. Some applications, for example fitting smoothed binary quantile regression models for a binary target variable (Kordas, 2006), require an alternative like the logistic sigmoid. While not straightforward, it is possible to change the output layer transfer function by switching off scale.y in the call to the fitting function and reassigning hramp and hramp.prime as follows:
library(qrnn)
# Use the logistic sigmoid as the output layer transfer function
To.logistic <- function(x, lower, eps) 0.5 + 0.5*tanh(x/2)
environment(To.logistic) <- asNamespace("qrnn")
assignInNamespace("hramp", To.logistic, ns="qrnn")
# Change the derivative of the output layer transfer function
To.logistic.prime <- function(x, lower, eps) 0.25/(cosh(x/2)^2)
environment(To.logistic.prime) <- asNamespace("qrnn")
assignInNamespace("hramp.prime", To.logistic.prime, ns="qrnn")
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