Functional data analysis of generalized regression quantiles

Abstract

Generalized regression quantiles, including the conditional quantiles and expectiles as special cases, are useful alternatives to the conditional means for characterizing a conditional distribution, especially when the interest lies in the tails. We develop a functional data analysis approach to jointly estimate a family of generalized regression quantiles. Our approach assumes that the generalized regression quantiles share some common features that can be summarized by a small number of principal component functions. The principal component functions are modeled as splines and are estimated by minimizing a penalized asymmetric loss measure. An iterative least asymmetrically weighted squares algorithm is developed for computation. While separate estimation of individual generalized regression quantiles usually suffers from large variability due to lack of sufficient data, by borrowing strength across data sets, our joint estimation approach significantly improves the estimation efficiency, which is demonstrated in a simulation study. The proposed method is applied to data from 159 weather stations in China to obtain the generalized quantile curves of the volatility of the temperature at these stations.

Appendix

1.1 A.1 The complete PLAWS algorithm

We give the complete algorithm here. The parameters that appear on the right hand side of the equations are all fixed at the values from the previous iteration.

• 1. Initialization the algorithm using the procedure described in Sect. A.2.

• 2. Update $\widehat{\boldsymbol{\theta}}_{\mu}$ using

  $$ \begin{aligned}[c] \widehat{\boldsymbol{\theta}}_{\mu} &= \Biggl\{ \sum _{i=1}^{N}\mathbf {B}_{i}^{\top}\widehat{ \mathbf {W}}_{i}\mathbf {B}_{i}+\lambda_{\mu} {\boldsymbol {\varOmega }}\Biggr\} ^{-1} \\ &\quad {}\times\Biggl\{ \sum_{i=1}^{N} \mathbf {B}_{i}^{\top}\widehat{\mathbf {W}}_{i}(\mathbf {Y}_{i}- \mathbf {B}_{i}\widehat{{\boldsymbol {\varTheta }}}_{\psi}\widehat{{\boldsymbol {\alpha }}}_{i}) \Biggr\} . \end{aligned} $$

• 3. For l=1,…,K, update the l-th column of $\widehat{{\boldsymbol {\varTheta }}}_{\psi}$ using

  $$ \begin{aligned}[c] \widehat{\boldsymbol{\theta}}_{\psi,l} &= \Biggl\{ \sum _{i=1}^{N} \widehat{{\boldsymbol {\alpha }}}_{il}^{2} \mathbf {B}_{i}^{\top}\widehat{\mathbf {W}}_{i}\mathbf {B}_{i}+ \lambda_{\psi} {\boldsymbol {\varOmega }}\Biggr\} ^{-1} \\ &\quad {}\times\Biggl\{ \sum _{i=1}^{N} \widehat{\alpha}_{il} \mathbf {B}_{i}^{\top}\widehat{\mathbf {W}}_{i}(\mathbf {Y}_{i}- \mathbf {B}_{i}\widehat{\boldsymbol {\theta}}_{\mu}-\mathbf {B}_{i} {\mathbf {q}}_{il}) \Biggr\} , \end{aligned} $$

  where $\widehat{\boldsymbol{\theta}}_{\psi,k}$ is the k-th column of $\widehat{{\boldsymbol {\varTheta }}}_{\psi}$, and

  $$\begin{aligned} {\mathbf {q}}_{il} =\sum_{k \neq l} \widehat{\boldsymbol{ \theta}}_{\psi ,k}\widehat{{\boldsymbol {\alpha }}}_{ik},\quad i=1, \ldots, N. \end{aligned}$$

• 4. Use the QR decomposition to orthonormalize the columns of $\widehat{{\boldsymbol {\varTheta }}}_{\psi}$.

• 5. Update $(\widehat{{\boldsymbol {\alpha }}}_{1}, \dots, \widehat{{\boldsymbol {\alpha }}}_{N})$ using

  $$\widehat{{\boldsymbol {\alpha }}}_{i} = \bigl(\widehat{{\boldsymbol {\varTheta }}}_{\psi}^{\top} \mathbf {B}_{i}^{\top} \mathbf {W}_{i}\mathbf {B}_{i}\widehat{ {\boldsymbol {\varTheta }}}_{\psi} \bigr)^{-1} \ \bigl\{ \widehat{ {\boldsymbol {\varTheta }}}_{\psi}^{\top} \mathbf {B}_{i}^{\top} \mathbf {W}_{i}(\mathbf {Y}_{i}-\mathbf {B}_{i}\widehat{\boldsymbol{ \theta}}_{\mu}) \bigr\} , $$

  and then center $\widehat{{\boldsymbol {\alpha }}}_{i}$ such that $\sum_{i}^{N} \widehat{{\boldsymbol {\alpha }}}_{i}=0$.

• 6. Update the weights, defined in (21) for expectiles and (22) for quantiles.

• 7. Iterate Steps 2–6 until convergence is reached.

1.2 A.2 Initial values for the PLAWS algorithm

The following procedure is used for initialization of the PLAWS algorithm:

• 1. Estimate N expectile/quantile curves $\widehat{l}_{i}(t)$ separately by applying the single curve estimation algorithm described in Sect. 2.

• 2. Set $\widehat{\mathbf {l}}_{i} = \{\widehat{l}(t_{i1}), \dots, \widehat {l}(t_{iT_{i}})\}^{\top}$. Run the linear regression

  $$ \widehat{\mathbf {l}}_{i}= \mathbf {B}_{i}\boldsymbol{ \theta}_{\mu}+\boldsymbol {\varepsilon }_{i}, \quad i=1, \dots, N, $$

  (28)

  to get the initials of $\widehat{\boldsymbol{\theta}}_{\mu}$ as follows

  $$\widehat{\boldsymbol{\theta}}_{\mu}^{(0)}= \Biggl(\sum _{i=1}^{N}\mathbf {B}_{i}^{\top} \mathbf {B}_{i} \Biggr)^{-1} \Biggl(\sum_{i=1}^{N} \mathbf {B}_{i}^{\top}\widehat{\mathbf {l}}_{i} \Biggr). $$

  Add a small ridge penalty if needed to overcome the singularity problem

• 3. Calculate the residuals of the regression (28), denoted as $\widetilde{\mathbf {l}}_{i}= \widehat{\mathbf {l}}_{i}-\mathbf {B}_{i}\widehat {\boldsymbol{\theta}}_{\mu}^{(0)}$. For each i, run the following linear regression

  $$\begin{aligned} \widetilde{\mathbf {l}}_{i}= \mathbf {B}_{i}\boldsymbol {\varGamma }_{i}+ \boldsymbol {\varepsilon }_{i}. \end{aligned}$$

  The solution, denoted as $\widehat{\boldsymbol {\varGamma }}_{i}^{(0)}$, is used in later steps for finding initials of Θ _ψ and α _i . Set $\widehat{\boldsymbol {\varGamma }}_{0}=(\widehat{\boldsymbol {\varGamma }}_{1}^{(0)}, \ldots, \widehat{\boldsymbol {\varGamma }}_{N}^{(0)})$.

• 4. Calculate the singular value decomposition of $\widehat{\boldsymbol {\varGamma }}^{(0)\top}$:

  $$\begin{aligned} \widehat{\boldsymbol {\varGamma }}^{(0)\top}= \mathbf {U}\mathbf {D}\mathbf {V}^{\top}. \end{aligned}$$

  The initial of Θ _ψ is chosen as $\widehat{{\boldsymbol {\varTheta }}}_{\psi }^{(0)} = \mathbf {V}_{k}\mathbf {D}_{k}$ where V _k consists of the first K columns of V and D _k is the corresponding K×K block of D.

• 5. Run the following regression

  $$ \widehat{\boldsymbol {\varGamma }}_{i}^{(0)}= \widehat{{\boldsymbol {\varTheta }}}_{\psi} \widehat {{\boldsymbol {\alpha }}}_{i} + \boldsymbol {\varepsilon }_{i} $$

  (29)

  to get the initials of $\widehat{{\boldsymbol {\alpha }}}_{i}$; use a ridge penalty if the regression is singular. Center $\widehat{{\boldsymbol {\alpha }}}_{i}$’s such that $\sum_{i=1}^{N} \widehat{{\boldsymbol {\alpha }}}_{i} = 0$.