Robust Variable Selection with Exponential Squared Loss for the Spatial SingleIndex VaryingCoefficient Model
Abstract
:1. Introduction
 We propose a novel model: the spatial singleindex varyingcoefficient model, which can deal with the spatial correlation and spatial heterogeneity of data at the same time.
 We construct a robust variable selection method for the spatial singleindex varyingcoefficient model, which uses exponential square loss function to resist the influence of strong noise and inaccurate spatial weight matrix. Furthermore, we present the BCD (block coordinate descent) algorithm to solve the optimization problem of the objective function.
 Under reasonable assumptions, we give theoretical properties of this method. In addition, we verify the robustness and effectiveness of the variable selection method through numerical simulation studies. The numerical study shows that the method is more robust than other comparative methods in variable selection and parameter estimation when outliers or noise are presented in the observations.
2. Methodology
2.1. Model Setup
2.2. Basis Function Expansion
2.3. The Penalized Robust Regression Estimator
2.4. Estimation of the Variance of the Noise
3. Theoretical Properties
 (C1)
 The density function $f\left(t\right)$ of $U\alpha $ is uniformly bounded on $T=\{t=U\alpha \}$ and far from 0. Furthermore, $f\left(t\right)$ is assumed to satisfy the Lipschitz condition of order 1 on T.
 (C2)
 The function ${g}_{j}\left(t\right),j=1,\dots ,q$, has bounded and continuous derivatives up to order $r(\ge 2)$ on T, where ${g}_{j}\left(t\right)$ is the jth components of $g\left(t\right)$.
 (C3)
 $E\left(\u2225{U}^{6}\u2225\right)<\infty ,E\left(\u2225{Z}^{6}\u2225\right)<\infty $ and $E\left({\left\epsilon \right}^{6}\right)<\infty $.
 (C4)
 $\left\{\left({y}_{i},{U}_{i},{z}_{i}\right),1\u2a7di\u2a7dn\right\}$ is a strictly stationary and strongly mixing sequence with coefficient $\gamma \left(n\right)=O\left({\xi}^{n}\right)$, where $0<\xi <1$.
 (C5)
 Let ${c}_{1},\dots ,{c}_{K}$ be the interior knots of $[a,b]$, where $a=inf\{t:t\in T\}$, $b=sup\{t:t\in T\}$. Moreover, we set ${c}_{0}=a$, ${c}_{K+1}=b$, ${h}_{i}={c}_{i}{c}_{i1}$, $h=max\left\{{h}_{i}\right\}$. Then, a positive constant ${C}_{0}$ exists such that$$\frac{h}{min\left\{{h}_{i}\right\}}<{C}_{0},\phantom{\rule{1.em}{0ex}}max\left\{{h}_{i+1}{h}_{i}\right\}=o\left({K}^{1}\right).$$
 (C6)
 Let ${b}_{n}={max}_{j}\left\{{\ddot{p}}_{j}\left(\left{\gamma}_{1j0}\right\right)\right:{\gamma}_{1j0}\ne 0\}$ and then ${b}_{n}\to 0$ as $n\to \infty $. Further, let ${lim}_{n\to}{inf}_{\infty}$${lim}_{\left{\gamma}_{1j}\right\to}{inf}_{0}{\lambda}_{j}^{1}\left{\dot{p}}_{j}\left(\left{\gamma}_{1j}\right\right)\right>0$, where $j=d+1,\dots ,q$.
 (C7)
 $H\left(\rho \right)={\left({I}_{n}\rho W\right)}^{1}$ is a nonsingular matrix, invertible for any $\rho \in \Theta $, $\Theta $ is a compact parameter space, and the absolute row and column sums of $H\left(\rho \right)$, $H{\left(\rho \right)}^{1}$ are uniformly bounded on $\rho \in \Theta $;
 (C8)
 Let$$I(\varphi ,{\gamma}_{1};{\gamma}_{2})=\frac{2}{{\gamma}_{2}}\int G\left(\varphi \right){G}^{T}\left(\varphi \right){e}^{{r}^{2}/{\gamma}_{2}}\left(\frac{2{r}^{2}}{{\gamma}_{2}}1\right)dF(G,y)$$
 (C9)
 $\Sigma =E\left(G{G}^{T}\right)$ is positive definite.
 (i)
 $\u2225\alpha {\alpha}_{0}\u2225={O}_{p}\left({n}^{1/(2r+1)}+{a}_{n}\right)$;
 (ii)
 $\u2225{\widehat{g}}_{j}(\xb7){g}_{j0}(\xb7)\u2225={O}_{p}\left({n}^{r/(2r+1)}+{a}_{n}\right)$, for $j=1,\dots ,q,$
 (i)
 ${\widehat{\alpha}}_{l}=0$, $l=s+1,\dots ,p$;
 (ii)
 ${\widehat{g}}_{j}(\xb7)=0$, $j=d+1,\dots ,q$.
4. Algorithm
4.1. Choice of the Tuning Parameter ${\gamma}_{2}$
4.2. Choice of the Regularization Parameter $\lambda $ and ${\eta}_{j}$
4.3. Block Coordinate Descent (BCD) Algorithm
Algorithm 1 The block coordinate descent (BCD) algorithm 

4.4. DC Decomposition and CCCP Algorithm
Algorithm 2 The Concave–Convex Procedure 

5. Simulation Studied
5.1. Simulation Sampling
5.2. Simulation Results
6. Summary
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
SAR model:  Spatial autoregressive model; 
BCD algorithm:  Blockcoordinate descent algorithm; 
DC function:  Difference between two convex functions; 
CCCP:  concave–convex procedure; 
ISTA:  Iterative shrinkagethresholding algorithm; 
FISTA:  Fast iterative shrinkagethresholding algorithm; 
MedSE:  Median of squared error; 
MAISE:  Square root of mean deviation. 
Appendix A. Proofs
Appendix A.1. The Related Lemmas
Appendix A.2. Poof of Main Theorems
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n = 25, q = 5  n = 144, q = 5  n = 324, q = 5  

E+null  S+null  L+null  E+null  S+null  L+null  E+null  S+null  L+null  
${\rho}_{1}=0.8$, ${\sigma}_{1}=1$  
${\alpha}_{1}$  0.80  0.61  0.83  0.50  0.66  0.28  0.53  0.62  0.67 
${\alpha}_{2}$  0.88  0.61  0.75  0.78  0.77  0.66  0.74  0.79  0.96 
$\widehat{\rho}$  0.80  0.94  0.75  0.90  0.80  0.91  0.88  0.89  0.84 
${\widehat{\sigma}}^{2}$  0.45  0.78  0.82  0.79  0.72  0.70  0.86  0.71  0.80 
MedSE  0.28  0.44  0.70  0.22  0.23  0.42  0.17  0.16  0.19 
${\rho}_{1}=0.5$, ${\sigma}_{1}=1$  
${\alpha}_{1}$  0.69  0.65  0.64  0.49  0.66  0.28  0.53  0.63  0.67 
${\alpha}_{2}$  0.82  0.61  0.88  0.78  0.76  0.72  0.74  0.80  0.96 
$\widehat{\rho}$  0.52  0.61  0.50  0.68  0.40  0.74  0.62  0.65  0.55 
${\widehat{\sigma}}^{2}$  0.44  0.82  0.71  0.83  0.71  0.74  0.89  0.73  0.82 
MedSE  0.22  0.45  0.77  0.22  0.23  0.41  0.17  0.16  0.19 
${\rho}_{1}=0.2$, ${\sigma}_{1}=1$  
${\alpha}_{1}$  0.67  0.66  0.57  0.51  0.65  0.32  0.53  0.65  0.65 
${\alpha}_{2}$  0.81  0.59  0.96  0.80  0.75  0.72  0.76  0.81  0.98 
$\widehat{\rho}$  0.14  0.00  0.27  0.33  0.00  0.50  0.19  0.26  0.16 
${\widehat{\sigma}}^{2}$  0.44  0.82  0.68  0.87  0.70  0.78  0.91  0.76  0.84 
MedSE  0.23  0.49  0.72  0.21  0.23  0.41  0.17  0.16  0.22 
${\rho}_{1}=0$, ${\sigma}_{1}=1$  
${\alpha}_{1}$  0.68  0.66  0.61  0.51  0.66  0.32  0.53  0.65  0.65 
${\alpha}_{2}$  0.81  0.60  0.95  0.81  0.73  0.70  0.76  0.81  0.98 
$\widehat{\rho}$  0.00  0.00  0.22  0.19  0.00  0.36  0.03  0.11  0.04 
${\widehat{\sigma}}^{2}$  0.44  0.83  0.69  0.88  0.72  0.79  0.91  0.76  0.84 
MedSE  0.22  0.47  0.70  0.21  0.25  0.39  0.16  0.16  0.22 
${\rho}_{1}=0.8$, ${\sigma}_{1}=2$  
${\alpha}_{1}$  0.73  0.77  1.37  0.41  0.79  0.34  0.58  0.63  0.42 
${\alpha}_{2}$  0.74  0.27  0.64  0.66  0.65  0.55  0.69  0.68  0.96 
$\widehat{\rho}$  0.86  0.98  0.62  0.95  0.83  0.97  0.93  0.94  0.90 
${\widehat{\sigma}}^{2}$  1.83  3.26  5.78  3.20  3.08  2.68  3.53  2.89  3.25 
MedSE  0.46  1.00  1.95  0.49  0.52  0.90  0.38  0.33  0.48 
${\rho}_{1}=0.5$, ${\sigma}_{1}=2$  
${\alpha}_{1}$  0.72  0.79  0.66  0.43  0.78  0.23  0.59  0.65  0.42 
${\alpha}_{2}$  0.74  0.32  0.90  0.68  0.68  0.54  0.72  0.70  1.01 
$\widehat{\rho}$  0.58  0.68  0.50  0.77  0.29  0.86  0.70  0.74  0.62 
${\widehat{\sigma}}^{2}$  1.87  3.49  3.05  3.44  3.03  3.00  3.74  3.08  3.45 
MedSE  0.46  0.98  1.59  0.48  0.48  0.95  0.38  0.32  0.48 
${\rho}_{1}=0.2$, ${\sigma}_{1}=2$  
${\alpha}_{1}$  0.76  0.77  0.53  0.46  0.78  0.19  0.60  0.68  0.46 
${\alpha}_{2}$  0.76  0.35  1.06  0.72  0.65  0.65  0.75  0.72  1.01 
$\widehat{\rho}$  0.14  0.00  0.39  0.39  0.00  0.61  0.21  0.32  0.23 
${\widehat{\sigma}}^{2}$  1.88  3.52  3.00  3.68  2.99  3.19  3.88  3.24  3.57 
MedSE  0.45  0.97  1.51  0.46  0.50  0.84  0.36  0.32  0.48 
${\rho}_{1}=0$, ${\sigma}_{1}=2$  
${\alpha}_{1}$  0.77  0.78  0.57  0.47  0.79  0.23  0.59  0.68  0.47 
${\alpha}_{2}$  0.77  0.34  1.07  0.72  0.64  0.55  0.76  0.72  1.04 
$\widehat{\rho}$  0.00  0.00  0.31  0.23  0.00  0.52  0.01  0.14  0.07 
${\widehat{\sigma}}^{2}$  1.88  3.54  3.04  3.74  3.09  3.32  3.90  3.26  3.60 
MedSE  0.45  0.97  1.50  0.46  0.52  0.89  0.35  0.33  0.48 
$\mathit{n}=25$, $\mathit{q}=20$  $\mathit{n}=144$, $\mathit{q}=80$  $\mathit{n}=324$, $\mathit{q}=200$  

E+null  S+null  L+null  E+null  S+null  L+null  E+null  S+null  L+null  
${\rho}_{1}=0.8$, ${\sigma}_{1}=1$  
${\alpha}_{1}$  0.74  0.39  0.24  0.04  0.49  −0.21  0.78  0.72  0.62 
${\alpha}_{2}$  0.67  0.18  2.81  1.05  0.88  2.15  0.86  0.86  0.71 
$\widehat{\rho}$  0.84  0.93  0.50  0.54  0.80  0.50  0.80  0.80  0.50 
${\widehat{\sigma}}^{2}$  0.18  0.53  3.54  0.30  0.26  1.00  0.37  0.41  1.68 
MedSE  2.79  2.20  7.92  2.78  1.72  4.56  1.44  1.66  2.21 
${\rho}_{1}=0.5$, ${\sigma}_{1}=1$  
${\alpha}_{1}$  0.72  0.39  0.10  0.19  0.49  0.06  0.75  0.68  0.56 
${\alpha}_{2}$  0.65  0.27  1.76  0.94  0.83  1.68  0.84  0.84  0.65 
$\widehat{\rho}$  0.61  0.66  0.50  0.46  0.54  0.50  0.52  0.53  0.50 
${\widehat{\sigma}}^{2}$  0.17  0.56  0.44  0.23  0.24  0.47  0.36  0.40  0.72 
MedSE  2.87  2.04  3.30  1.57  1.71  2.43  1.40  1.60  1.62 
${\rho}_{1}=0.2$, ${\sigma}_{1}=1$  
${\alpha}_{1}$  0.70  0.37  0.08  0.17  0.50  0.24  0.75  0.67  0.55 
${\alpha}_{2}$  0.64  0.40  1.45  0.93  0.79  1.60  0.84  0.85  0.60 
$\widehat{\rho}$  0.45  0.04  0.50  0.00  0.31  0.50  0.13  0.19  0.50 
${\widehat{\sigma}}^{2}$  0.18  0.57  0.22  0.22  0.24  0.50  0.36  0.40  0.73 
MedSE  3.16  1.71  2.40  1.59  1.82  2.50  1.41  1.59  1.64 
${\rho}_{1}=0$, ${\sigma}_{1}=1$  
${\alpha}_{1}$  0.71  0.37  0.06  0.20  0.50  0.26  0.75  0.68  0.61 
${\alpha}_{2}$  0.64  0.38  1.47  0.94  0.80  1.60  0.84  0.85  0.62 
$\widehat{\rho}$  0.35  0.00  0.50  0.00  0.16  0.50  0.00  0.04  0.50 
${\widehat{\sigma}}^{2}$  0.18  0.57  0.21  0.23  0.24  0.52  0.36  0.40  0.77 
MedSE  3.19  1.76  2.42  1.57  1.81  2.62  1.41  1.60  1.84 
${\rho}_{1}=0.8$, ${\sigma}_{1}=2$  
${\alpha}_{1}$  0.58  0.08  −0.47  −1.05  0.29  −1.22  0.78  0.68  0.54 
${\alpha}_{2}$  0.45  −0.57  2.66  1.31  0.87  3.60  0.91  0.71  0.72 
$\widehat{\rho}$  0.77  0.97  0.50  0.61  0.81  0.50  0.84  0.87  0.50 
${\widehat{\sigma}}^{2}$  4.18  2.27  12.43  2.20  1.06  4.13  1.63  1.63  6.24 
MedSE  8.37  4.41  9.85  5.68  3.59  9.27  2.97  3.24  4.38 
${\rho}_{1}=0.5$, ${\sigma}_{1}=2$  
${\alpha}_{1}$  0.63  0.11  −0.38  −0.43  0.33  −0.45  0.75  0.68  0.57 
${\alpha}_{2}$  1.23  −0.23  2.89  1.28  0.78  2.71  0.93  0.71  0.62 
$\widehat{\rho}$  0.64  0.60  0.50  0.47  0.59  0.50  0.60  0.61  0.50 
${\widehat{\sigma}}^{2}$  2.33  2.44  1.88  1.76  1.00  2.03  1.73  1.68  3.08 
MedSE  5.36  3.80  6.83  4.34  3.55  5.02  2.97  3.26  3.35 
${\rho}_{1}=0.2$, ${\sigma}_{1}=2$  
${\alpha}_{1}$  0.65  0.13  −0.81  −0.15  0.36  −0.17  0.75  0.69  0.65 
${\alpha}_{2}$  1.14  −0.04  2.37  0.88  0.74  2.48  0.96  0.72  0.51 
$\widehat{\rho}$  0.37  0.00  0.50  0.00  0.31  0.50  0.26  0.22  0.50 
${\widehat{\sigma}}^{2}$  1.89  2.43  0.95  1.95  1.03  2.01  1.91  1.72  3.17 
MedSE  5.12  3.43  5.01  3.42  3.66  4.81  3.07  3.30  3.32 
${\rho}_{1}=0$, ${\sigma}_{1}=2$  
${\alpha}_{1}$  0.64  0.12  −0.80  −0.10  0.35  −0.14  0.76  0.70  0.64 
${\alpha}_{2}$  0.58  −0.18  2.23  0.88  0.76  2.47  0.93  0.72  0.53 
$\widehat{\rho}$  0.21  0.00  0.50  0.00  0.15  0.50  0.01  0.01  0.50 
${\widehat{\sigma}}^{2}$  0.78  2.46  0.97  2.09  1.03  2.07  1.73  1.71  3.24 
MedSE  5.91  3.72  4.90  3.42  3.64  4.90  2.98  3.30  3.40 
$\mathit{n}=25$, $\mathit{q}=5$  $\mathit{n}=144$, $\mathit{q}=5$  $\mathit{n}=324$, $\mathit{q}=5$  

E+null  S+null  L+null  E+null  S+null  L+null  E+null  S+null  L+null  
${\rho}_{1}=0.5$, ${\sigma}_{1}=$1, ${\delta}_{1}=0.01$  
${\alpha}_{1}$  0.71  0.70  0.59  0.52  0.53  0.54  0.52  0.60  0.42 
${\alpha}_{2}$  0.49  0.76  1.20  0.73  0.94  0.86  0.72  0.66  1.14 
$\widehat{\rho}$  0.45  0.64  0.50  0.67  0.54  0.51  0.59  0.49  0.55 
${\widehat{\sigma}}^{2}$  0.80  0.82  0.73  0.87  0.86  0.98  1.02  1.09  0.91 
MedSE  0.46  0.34  0.51  0.30  0.25  0.21  0.18  0.22  0.35 
${\rho}_{1}=0.5$, ${\sigma}_{1}=2$, ${\delta}_{1}=0.01$  
${\alpha}_{1}$  1.06  0.69  0.64  0.45  0.40  0.45  0.59  0.61  0.35 
${\alpha}_{2}$  0.14  0.86  1.58  0.62  1.28  0.80  0.71  0.54  1.38 
$\widehat{\rho}$  0.48  0.69  0.66  0.76  0.59  0.70  0.67  0.52  0.63 
${\widehat{\sigma}}^{2}$  3.84  3.24  2.82  3.72  3.48  3.98  4.21  4.46  3.79 
MedSE  1.52  0.70  1.07  0.55  0.63  0.56  0.39  0.40  0.67 
${\rho}_{1}=0.5$, ${\sigma}_{1}=1$, ${\delta}_{1}=0.05$  
${\alpha}_{1}$  0.88  1.01  0.67  0.57  0.74  0.39  0.42  0.58  0.58 
${\alpha}_{2}$  0.43  0.72  1.01  0.56  0.78  1.00  0.65  0.51  1.05 
$\widehat{\rho}$  0.60  0.75  0.50  0.75  0.85  0.60  0.67  0.74  0.65 
${\widehat{\sigma}}^{2}$  2.64  3.72  4.32  3.75  3.02  4.44  4.04  4.64  3.83 
MedSE  0.67  0.59  1.07  0.73  0.45  0.47  0.35  0.48  0.28 
${\rho}_{1}=0.5$, ${\sigma}_{1}=2$, ${\delta}_{1}=0.05$  
${\alpha}_{1}$  1.09  1.00  0.74  0.56  0.61  0.61  0.48  0.61  0.41 
${\alpha}_{2}$  0.23  0.83  1.58  0.43  1.13  0.73  0.64  0.40  1.42 
$\widehat{\rho}$  0.39  0.76  0.50  0.77  0.82  0.50  0.69  0.69  0.64 
${\widehat{\sigma}}^{2}$  5.02  6.17  6.23  6.13  5.67  7.62  7.08  7.94  6.33 
MedSE  0.98  0.83  1.61  0.87  0.59  0.72  0.46  0.64  0.67 
$\mathit{n}=25$, $\mathit{q}=5$  $\mathit{n}=144$, $\mathit{q}=5$  $\mathit{n}=324$, $\mathit{q}=5$  

E+null  S+null  L+null  E+null  S+null  L+null  E+null  S+null  L+null  
Remove 30% nonzero weights  
${\alpha}_{1}$  0.59  0.58  0.17  0.43  0.32  0.47  0.55  0.65  0.46 
${\alpha}_{2}$  0.59  0.97  1.63  0.82  0.80  0.97  0.73  0.78  1.12 
$\widehat{\rho}$  0.61  0.55  0.52  0.70  0.57  0.50  0.65  0.54  0.53 
${\widehat{\sigma}}^{2}$  1.05  1.13  0.99  1.08  1.08  1.03  1.14  1.09  1.22 
MedSE  0.48  0.45  1.10  0.35  0.33  0.49  0.20  0.25  0.25 
Remove 50% nonzero weights  
${\alpha}_{1}$  0.67  0.57  0.16  0.44  0.30  0.31  0.52  0.66  0.54 
${\alpha}_{2}$  0.61  0.90  1.48  0.73  0.81  1.15  0.74  0.79  1.10 
$\widehat{\rho}$  0.54  0.48  0.50  0.64  0.49  0.41  0.57  0.49  0.49 
${\widehat{\sigma}}^{2}$  1.07  1.17  0.97  1.11  1.11  1.09  1.18  1.12  1.25 
MedSE  0.41  0.26  1.12  0.34  0.37  0.64  0.19  0.27  0.31 
Remove 80% nonzero weights  
${\alpha}_{1}$  0.68  0.63  0.26  0.46  0.33  0.24  0.58  0.70  0.48 
${\alpha}_{2}$  0.59  0.94  1.29  0.71  0.82  1.31  0.77  0.78  1.19 
$\widehat{\rho}$  0.40  0.34  0.51  0.52  0.34  0.37  0.42  0.33  0.36 
${\widehat{\sigma}}^{2}$  1.15  1.30  0.96  1.26  1.24  1.14  1.34  1.26  1.40 
MedSE  0.50  0.40  0.89  0.40  0.40  0.78  0.20  0.29  0.33 
$\mathit{n}=25$, $\mathit{q}=5$  $\mathit{n}=324$, $\mathit{q}=5$  

$\mathbf{E}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{E}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{S}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{S}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{L}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{L}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{E}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{E}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{S}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{S}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{L}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{L}+{\tilde{\mathit{\ell}}}_{1}$  
${\rho}_{1}=0.8$, ${\sigma}_{1}=1$  
Correct  4.00  5.00  4.00  5.00  0.00  3.00  5.00  5.00  5.00  5.00  5.00  5.00 
Incorrect  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  1.00 
$\widehat{\rho}$  0.99  0.86  0.86  0.97  0.73  0.82  0.89  0.88  0.88  0.89  0.89  0.92 
MedSE  0.42  0.37  0.44  0.36  1.43  0.54  0.14  0.14  0.20  0.16  0.21  0.24 
${\rho}_{1}=0.5$, ${\sigma}_{1}=1$  
Correct  4.00  4.00  3.00  5.00  5.00  4.00  5.00  5.00  5.00  5.00  5.00  5.00 
Incorrect  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  1.00 
$\widehat{\rho}$  0.52  0.57  0.58  0.81  0.51  0.46  0.50  0.58  0.58  0.57  0.56  0.68 
MedSE  0.24  0.31  0.45  0.40  0.49  0.43  0.17  0.11  0.15  0.16  0.23  0.22 
${\rho}_{1}=0.2$, ${\sigma}_{1}=1$  
Correct  4.00  4.00  3.00  5.00  5.00  3.00  5.00  5.00  5.00  5.00  5.00  5.00 
Incorrect  0.00  0.00  0.00  1.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  1.00 
$\widehat{\rho}$  0.26  0.38  0.40  0.61  0.37  0.20  0.33  0.35  0.35  0.31  0.35  0.49 
MedSE  0.24  0.32  0.47  0.42  0.50  0.52  0.14  0.12  0.16  0.16  0.22  0.21 
${\rho}_{1}=0$, ${\sigma}_{1}=1$  
Correct  4.00  4.00  3.00  5.00  5.00  3.00  5.00  5.00  5.00  5.00  5.00  5.00 
Incorrect  0.00  0.00  0.00  1.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  1.00 
$\widehat{\rho}$  0.00  0.06  0.09  0.30  0.01  0.00  0.00  0.00  0.00  0.00  0.00  0.11 
MedSE  0.24  0.31  0.46  0.44  0.56  0.55  0.14  0.13  0.17  0.16  0.17  0.21 
${\rho}_{1}=0.8$, ${\sigma}_{1}=2$  
Correct  4.00  2.00  1.00  3.00  0.00  1.00  5.00  5.00  5.00  4.00  2.00  4.00 
Incorrect  0.00  0.00  0.00  1.00  0.00  0.00  1.00  0.00  0.00  0.00  0.00  1.00 
$\widehat{\rho}$  0.88  0.90  0.92  0.99  0.86  0.88  0.94  0.92  0.92  0.94  0.92  0.96 
MedSE  0.53  0.67  0.94  0.65  2.09  1.05  0.32  0.29  0.32  0.36  0.45  0.44 
${\rho}_{1}=0.5$, ${\sigma}_{1}=2$  
Correct  4.00  2.00  1.00  2.00  3.00  3.00  5.00  5.00  5.00  4.00  2.00  4.00 
Incorrect  0.00  0.00  0.00  1.00  1.00  0.00  1.00  0.00  0.00  0.00  0.00  1.00 
$\widehat{\rho}$  0.45  0.64  0.69  0.89  0.51  0.52  0.66  0.63  0.65  0.64  0.62  0.81 
MedSE  0.52  0.67  0.96  0.70  0.99  0.95  0.31  0.29  0.31  0.35  0.46  0.48 
${\rho}_{1}=0.2$, ${\sigma}_{1}=2$  
Correct  4.00  2.00  1.00  1.00  2.00  3.00  5.00  5.00  5.00  4.00  2.00  3.00 
Incorrect  0.00  0.00  0.00  1.00  1.00  0.00  1.00  0.00  0.00  0.00  0.00  1.00 
$\widehat{\rho}$  0.03  0.45  0.51  0.76  0.50  0.50  0.38  0.37  0.39  0.34  0.40  0.57 
MedSE  0.55  0.69  0.97  0.76  1.13  1.02  0.29  0.30  0.33  0.34  0.47  0.48 
${\rho}_{1}=0$, ${\sigma}_{1}=2$  
Correct  4.00  2.00  1.00  1.00  2.00  2.00  5.00  5.00  5.00  4.00  3.00  3.00 
Incorrect  0.00  0.00  0.00  1.00  1.00  0.00  1.00  0.00  0.00  0.00  0.00  1.00 
$\widehat{\rho}$  0.00  0.10  0.18  0.49  0.03  0.41  0.00  0.00  0.00  0.00  0.00  0.21 
MedSE  0.51  0.68  0.96  0.82  1.09  1.21  0.28  0.31  0.36  0.34  0.38  0.50 
$\mathit{n}=25$, $\mathit{q}=20$  $\mathit{n}=324$, $\mathit{q}=200$  

$\mathbf{E}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{E}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{S}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{S}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{L}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{L}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{E}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{E}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{S}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{S}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{L}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{L}+{\tilde{\mathit{\ell}}}_{1}$  
${\rho}_{1}=0.8$, ${\sigma}_{1}=1$  
Correct  7.00  9.00  5.00  6.00  8.00  16.00  195.00  200.00  187.00  195.00  180.00  187.00 
Incorrect  1.00  1.00  0.00  0.00  1.00  0.00  1.00  0.00  0.00  0.00  0.00  1.00 
$\widehat{\rho}$  0.81  0.82  0.89  0.88  0.58  0.69  0.84  0.87  0.85  0.88  0.65  0.73 
MedSE  2.89  1.38  3.38  2.06  1.39  0.56  1.23  0.53  1.52  1.36  1.69  1.53 
${\rho}_{1}=0.5$, ${\sigma}_{1}=1$  
Correct  6.00  10.00  5.00  4.00  17.00  13.00  197.00  200.00  192.00  196.00  199.00  197.00 
Incorrect  0.00  1.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  1.00 
$\widehat{\rho}$  0.50  0.25  0.55  0.61  0.51  0.42  0.62  0.55  0.54  0.61  0.50  0.50 
MedSE  2.02  1.41  3.38  2.20  0.64  0.76  1.05  0.52  1.43  1.33  0.88  1.00 
${\rho}_{1}=0.2$, ${\sigma}_{1}=1$  
Correct  5.00  10.00  5.00  5.00  13.00  14.00  197.00  200.00  191.00  193.00  199.00  198.00 
Incorrect  1.00  1.00  0.00  0.00  0.00  0.00  0.00  0.00  1.00  0.00  0.00  1.00 
$\widehat{\rho}$  0.55  0.00  0.40  0.47  0.50  0.23  0.48  0.34  0.40  0.48  0.50  0.50 
MedSE  2.98  1.39  3.47  2.45  0.90  0.74  1.11  0.52  1.41  1.36  0.91  1.02 
${\rho}_{1}=0$, ${\sigma}_{1}=1$  
Correct  9.00  11.00  5.00  4.00  13.00  13.00  197.00  200.00  192.00  193.00  200.00  196.00 
Incorrect  1.00  1.00  0.00  0.00  0.00  0.00  0.00  0.00  1.00  0.00  0.00  1.00 
$\widehat{\rho}$  0.55  0.00  0.00  0.12  0.38  0.00  0.13  0.00  0.03  0.16  0.50  0.49 
MedSE  1.78  1.24  3.41  2.28  1.04  0.82  1.07  0.52  1.42  1.36  0.92  1.13 
${\rho}_{1}=0.8$, ${\sigma}_{1}=2$  
Correct  6.00  6.00  5.00  1.00  5.00  13.00  162.00  172.00  133.00  137.00  156.00  138.00 
Incorrect  1.00  0.00  0.00  0.00  1.00  0.00  0.00  1.00  1.00  0.00  0.00  1.00 
$\widehat{\rho}$  0.81  0.92  1.00  0.98  0.76  0.73  0.94  0.89  0.89  0.94  0.73  0.73 
MedSE  3.33  3.65  7.30  5.30  2.30  0.92  2.28  1.84  2.99  2.71  2.30  2.78 
${\rho}_{1}=0.5$, ${\sigma}_{1}=2$  
Correct  8.00  6.00  4.00  1.00  9.00  9.00  160.00  173.00  134.00  135.00  185.00  173.00 
Incorrect  1.00  1.00  0.00  0.00  0.00  0.00  0.00  1.00  1.00  0.00  0.00  1.00 
$\widehat{\rho}$  0.77  0.53  0.91  0.90  0.60  0.47  0.74  0.63  0.61  0.75  0.50  0.50 
MedSE  2.52  3.39  7.63  5.95  1.87  1.44  2.34  1.81  2.96  2.77  1.64  1.94 
${\rho}_{1}=0.2$, ${\sigma}_{1}=2$  
Correct  7.00  7.00  4.00  1.00  8.00  9.00  162.00  167.00  129.00  130.00  181.00  173.00 
Incorrect  1.00  1.00  0.00  0.00  0.00  0.00  0.00  1.00  1.00  0.00  1.00  1.00 
$\widehat{\rho}$  0.24  0.10  0.68  0.79  0.63  0.15  0.53  0.46  0.46  0.55  0.50  0.50 
MedSE  2.87  3.42  7.62  5.98  1.84  1.47  2.33  1.83  2.97  2.81  1.72  1.99 
${\rho}_{1}=0$, ${\sigma}_{1}=2$  
Correct  8.00  6.00  4.00  1.00  9.00  9.00  144.00  172.00  131.00  131.00  178.00  171.00 
Incorrect  1.00  0.00  0.00  0.00  0.00  0.00  0.00  1.00  1.00  0.00  1.00  1.00 
$\widehat{\rho}$  0.00  0.00  0.32  0.54  0.50  0.00  0.35  0.00  0.00  0.26  0.50  0.50 
MedSE  2.98  3.60  7.65  5.70  1.95  1.58  2.73  1.83  2.99  2.86  1.87  2.11 
$\mathit{n}=25$, $\mathit{q}=5$  $\mathit{n}=324$, $\mathit{q}=5$  

$\mathbf{E}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{E}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{S}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{S}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{L}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{L}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{E}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{E}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{S}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{S}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{L}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{L}+{\tilde{\mathit{\ell}}}_{1}$  
${\rho}_{1}=0.5$, ${\sigma}_{1}=1$, ${\delta}_{1}=0.01$  
Correct  4.00  4.00  4.00  5.00  4.00  3.00  5.00  5.00  5.00  5.00  5.00  5.00 
Incorrect  0.00  0.00  0.00  1.00  0.00  0.00  0.00  0.00  0.00  0.00  1.00  0.00 
$\widehat{\rho}$  0.77  0.64  0.63  0.77  0.70  0.79  0.74  0.66  0.66  0.70  0.64  0.61 
MedSE  0.48  0.32  0.48  0.36  0.62  0.53  0.14  0.14  0.17  0.18  0.31  0.30 
${\widehat{\sigma}}^{2}$  
${\rho}_{1}=0.5$, ${\sigma}_{1}=2$, ${\delta}_{1}=0.01$  
Correct  3.00  1.00  2.00  3.00  1.00  1.00  5.00  5.00  5.00  5.00  3.00  3.00 
Incorrect  0.00  0.00  0.00  0.00  2.00  0.00  0.00  0.00  0.00  0.00  1.00  0.00 
$\widehat{\rho}$  0.58  0.47  0.52  0.71  0.51  0.76  0.57  0.56  0.58  0.64  0.50  0.50 
MedSE  0.61  0.97  0.93  0.79  1.19  1.17  0.18  0.35  0.35  0.34  0.66  0.63 
${\rho}_{1}=0.5$, ${\sigma}_{1}=1$, ${\delta}_{1}=0.05$  
Correct  1.00  4.00  3.00  3.00  0.00  3.00  3.00  4.00  4.00  4.00  4.00  5.00 
Incorrect  0.00  1.00  1.00  1.00  1.00  1.00  0.00  1.00  0.00  0.00  0.00  0.00 
$\widehat{\rho}$  0.78  0.73  0.73  0.90  0.77  0.88  0.83  0.75  0.79  0.79  0.83  0.81 
MedSE  0.69  0.87  0.92  0.69  1.75  0.66  0.38  0.30  0.35  0.32  0.52  0.22 
${\rho}_{1}=0.5$, ${\sigma}_{1}=2$, ${\delta}_{1}=0.05$  
Correct  1.00  3.00  1.00  3.00  0.00  3.00  5.00  4.00  4.00  4.00  4.00  3.00 
Incorrect  0.00  0.00  1.00  1.00  1.00  0.00  0.00  1.00  0.00  0.00  1.00  0.00 
$\widehat{\rho}$  0.65  0.42  0.41  0.83  0.51  0.79  0.75  0.59  0.64  0.67  0.63  0.57 
MedSE  0.96  1.28  1.26  0.74  1.93  0.54  0.36  0.46  0.46  0.46  0.75  0.54 
$\mathit{n}=25$, $\mathit{q}=5$  $\mathit{n}=324$, $\mathit{q}=5$  

$\mathbf{E}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{E}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{S}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{S}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{L}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{L}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{E}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{E}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{S}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{S}+{\tilde{\mathit{\ell}}}_{1}$  $\mathbf{L}+\phantom{\rule{4pt}{0ex}}{\mathit{l}}_{1}$  $\mathbf{L}+{\tilde{\mathit{\ell}}}_{1}$  
Remove 30% nonzero weights  
Correct  4.00  5.00  5.00  5.00  0.00  5.00  5.00  5.00  5.00  5.00  5.00  5.00 
Incorrect  0.00  0.00  0.00  0.00  1.00  0.00  1.00  0.00  0.00  0.00  0.00  0.00 
$\widehat{\rho}$  0.54  0.53  0.50  0.64  0.28  0.48  0.57  0.55  0.55  0.50  0.55  0.56 
MedSE  0.50  0.22  0.41  0.32  0.90  0.28  0.16  0.18  0.26  0.16  0.17  0.28 
Remove 50% nonzero weights  
Correct  4.00  5.00  2.00  4.00  2.00  4.00  5.00  5.00  5.00  5.00  5.00  5.00 
Incorrect  0.00  0.00  0.00  0.00  1.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00 
$\widehat{\rho}$  0.55  0.36  0.37  0.54  0.16  0.35  0.38  0.41  0.40  0.35  0.42  0.46 
MedSE  0.57  0.43  0.63  0.36  0.86  0.40  0.09  0.22  0.29  0.17  0.12  0.28 
Remove 80% nonzero weights  
Correct  4.00  5.00  5.00  3.00  0.00  5.00  5.00  4.00  4.00  5.00  4.00  4.00 
Incorrect  0.00  0.00  0.00  0.00  1.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00 
$\widehat{\rho}$  0.37  0.47  0.43  0.89  0.50  0.72  0.52  0.48  0.46  0.37  0.58  0.49 
MedSE  0.63  0.35  0.50  0.77  0.91  0.42  0.20  0.48  0.50  0.29  0.36  0.41 
$\mathit{h}=16$  $\mathit{h}=18$  

$\widehat{{g}_{1}}$  0.0437  0.0388 
$\widehat{{g}_{2}}$  0.0542  0.0539 
$\widehat{{g}_{3}}$  0.0515  0.0496 
$\widehat{y}$  0.0566  0.0548 
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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Wang, Y.; Wang, Z.; Song, Y. Robust Variable Selection with Exponential Squared Loss for the Spatial SingleIndex VaryingCoefficient Model. Entropy 2023, 25, 230. https://doi.org/10.3390/e25020230
Wang Y, Wang Z, Song Y. Robust Variable Selection with Exponential Squared Loss for the Spatial SingleIndex VaryingCoefficient Model. Entropy. 2023; 25(2):230. https://doi.org/10.3390/e25020230
Chicago/Turabian StyleWang, Yezi, Zhijian Wang, and Yunquan Song. 2023. "Robust Variable Selection with Exponential Squared Loss for the Spatial SingleIndex VaryingCoefficient Model" Entropy 25, no. 2: 230. https://doi.org/10.3390/e25020230