Robust Variable Selection with Exponential Squared Loss for the Spatial Durbin Model
Abstract
:1. Introduction
- We build a robust variable selection method for SDM, equipped with an exponential squared loss, resistant to the influence of outliers in the observed values and errors estimating the space weight matrix.
- To solve the optimization problem of SDM, we propose a block coordinate descent (BCD) algorithm. Secondly, to solve the subproblems generated by the BCD algorithm, we design the DC decomposition of exponential square loss and construct the CCCP program. Finally, to obtain the BCD algorithm’s convergence, we analyze the algorithm’s convergence rate to the stagnation point under mild conditions.
- We proved the “Oracle” property of the robust variable selection method and conducted numerical experiments to verify the robustness and effectiveness of the model. Numerical studies show that when there are outliers in the observed data, the method proposed in this paper is superior to the comparison method in correctly identifying zero coefficients, nonzero coefficients, and MedSE incorrectly.
2. Variable Selection and Estimation
2.1. Spatial Durbin Model
2.2. Variable Selection Method for SDM
2.3. Oracle Properties and Large Sample Properties
- (1)
- sparsity, that is, with probability 1;
- (2)
- asymptotic normality:where , and ,
2.4. The Selection of Parameter
2.5. The Selection of Parameter and
3. Algorithm for Model Solving
3.1. Block Coordinate Descent Algorithm Frame
Algorithm 1: Block coordinate descent algorithm |
1. Set initial value for ; 2. 3. Solve the subproblem about with initial point : 4. Solve the subproblem with initial value ,
to obtain a solution , ensuring that , and is a stationary point of . 5. convergence. |
3.2. Solving the Subproblem (8)
3.3. Solving the Subproblem (9)
Algorithm 2: The Concave–Convex Procedure (CCCP) |
1. Initialize . Set . 2. 3.
4. convergence of . |
Algorithm 3: FISTA with Backtracking Step for solving (17) |
Require: Ensure: solution 1: Step 0. Select Let 2: Step . 3: Determine the smallest non-negative integer which make satisfy 4:
5: Let according to (19), calculate: 6: 7: 8: 9: Output . |
4. Numerical Examples
4.1. Nonregular Estimation of Normal Data
4.2. Nonregular Estimation for High-Dimensional Data
4.3. Nonregular Estimation of Data with Outliers in Dependent Variable y
4.4. Nonregular Estimation of Data with Noise in Spatial Weight Matrix
4.5. Estimation with Adaptive-l1 Regularizer
5. Application of Practical Examples
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Proof of Theorem 1
Appendix B. Proof of Theorem 2
Appendix B.1. Proof of Theorem 2(i)
Appendix B.2. Proof of Theorem 2(ii)
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Exp | Square | LAD | Exp | Square | LAD | Exp | Square | LAD | |||
---|---|---|---|---|---|---|---|---|---|---|---|
3.0904 | 2.6866 | 3.2503 | 3.1335 | 2.8487 | 3.0801 | 2.8084 | 2.7947 | 2.9486 | |||
2.0303 | 1.9449 | 1.8899 | 1.9594 | 1.9498 | 2.0897 | 2.0949 | 2.1354 | 2.0207 | |||
1.6422 | 1.4689 | 1.8394 | 1.5725 | 1.5409 | 1.3781 | 1.6174 | 1.7394 | 1.6394 | |||
1.242 | 1.3382 | 1.3069 | 1.504 | 1.6117 | 1.3924 | 1.4604 | 1.2156 | 1.3616 | |||
1.5109 | 1.3963 | 1.3582 | 1.1245 | 1.132 | 1.3174 | 1.1786 | 1.1139 | 1.111 | |||
0.8155 | 1.0711 | 1.0625 | 1.1101 | 1.0575 | 0.9693 | 1.0903 | 1.0092 | 1.0871 | |||
0.8001 | 0.8011 | 0.7999 | 0.7999 | 0.8006 | 0.7997 | 0.8002 | 0.7979 | 0.7981 | |||
MedSE | 0.5994 | 0.4158 | 0.4693 | 0.2518 | 0.2827 | 0.3432 | 0.248 | 0.234 | 0.3086 | ||
3.0854 | 3.0349 | 3.0617 | 3.1254 | 2.8039 | 3.0451 | 2.8058 | 3.1899 | 3.2542 | |||
2.0058 | 2.1532 | 1.927 | 1.9556 | 2.1823 | 2.2277 | 2.0986 | 1.9975 | 2.0256 | |||
1.6799 | 1.3788 | 1.6744 | 1.5702 | 1.4268 | 1.7227 | 1.6208 | 1.6813 | 1.303 | |||
1.2338 | 1.219 | 1.7939 | 1.4848 | 1.1814 | 1.3734 | 1.458 | 1.4145 | 1.6612 | |||
1.4943 | 1.4233 | 1.5202 | 1.1322 | 1.3266 | 1.3411 | 1.186 | 0.9884 | 1.3373 | |||
0.8849 | 1.0766 | 0.5961 | 1.1036 | 0.9614 | 0.8644 | 1.0966 | 0.9671 | 0.9961 | |||
0.5021 | 0.4999 | 0.5 | 0.5003 | 0.5 | 0.5 | 0.4998 | 0.4999 | 0.4999 | |||
MedSE | 0.6007 | 0.388 | 0.4564 | 0.2808 | 0.2829 | 0.3287 | 0.2452 | 0.2262 | 0.2939 | ||
3.0072 | 2.7008 | 2.7579 | 3.0283 | 2.9572 | 2.9077 | 2.635 | 2.8836 | 3.0321 | |||
1.8903 | 1.7081 | 1.93 | 1.8152 | 2.081 | 1.9718 | 1.8603 | 2.0794 | 1.4453 | |||
1.5386 | 1.4297 | 1.571 | 1.4646 | 1.2788 | 1.3697 | 1.4512 | 1.5486 | 1.5858 | |||
0.8622 | 1.2241 | 0.9667 | 1.2297 | 1.2184 | 1.015 | 1.0963 | 1.184 | 1.1689 | |||
1.4427 | 1.0584 | 0.7845 | 0.8279 | 0.8104 | 1.2721 | 0.7684 | 0.889 | 1.0247 | |||
0.6609 | 0.5715 | 1.1202 | 1.0265 | 0.9351 | 0.4027 | 0.7551 | 0.8819 | 0.9224 | |||
0.2417 | 0.2419 | 0.2519 | 0.2271 | 0.2365 | 0.2437 | 0.25 | 0.2216 | 0.2317 | |||
MedSE | 0.9037 | 0.8134 | 0.9757 | 0.5492 | 0.6286 | 0.7235 | 0.9921 | 0.5287 | 0.6407 | ||
3.0727 | 2.8882 | 3.0548 | 3.2634 | 2.7795 | 3.423 | 2.6808 | 3.1902 | 3.0531 | |||
2.1164 | 1.8141 | 2.0596 | 2.0387 | 1.6757 | 2.1636 | 2.1004 | 1.9817 | 1.9528 | |||
1.6747 | 1.3996 | 1.4515 | 1.7457 | 1.7597 | 1.7635 | 1.5395 | 1.3549 | 1.2795 | |||
0.9807 | 1.5773 | 1.6723 | 1.4112 | 1.3493 | 1.5103 | 1.4831 | 1.6088 | 1.5378 | |||
1.7814 | 0.8604 | 0.9949 | 1.0979 | 1.1257 | 1.0441 | 1.1439 | 1.0058 | 1.3651 | |||
0.7301 | 0.9358 | 0.8807 | 1.0707 | 0.6032 | 1.0806 | 1.0423 | 1.0172 | 0.8286 | |||
0.801 | 0.8045 | 0.7943 | 0.7996 | 0.8042 | 0.7978 | 0.7989 | 0.7989 | 0.7897 | |||
MedSE | 1.2058 | 0.7731 | 0.9502 | 0.5131 | 0.5493 | 0.6914 | 0.5536 | 0.4719 | 0.5597 | ||
3.0762 | 3.1916 | 3.1159 | 3.2325 | 3.0528 | 2.8093 | 2.6731 | 3.0178 | 3.0432 | |||
2.0839 | 2.2408 | 1.5295 | 1.8826 | 1.9422 | 2.1974 | 2.1207 | 2.1175 | 1.9125 | |||
1.7169 | 1.384 | 1.8689 | 1.6075 | 1.7206 | 1.5534 | 1.5565 | 1.355 | 1.2292 | |||
0.9706 | 1.3618 | 1.483 | 1.437 | 1.5269 | 1.7395 | 1.4862 | 1.3611 | 1.1453 | |||
1.7802 | 1.1337 | 1.2179 | 1.0509 | 1.0885 | 1.4445 | 1.1825 | 1.2947 | 1.3262 | |||
0.8067 | 1.2863 | 1.0691 | 1.2173 | 0.8635 | 0.901 | 1.0777 | 0.9428 | 0.9601 | |||
0.5044 | 0.5035 | 0.5 | 0.5007 | 0.4996 | 0.4986 | 0.4994 | 0.4973 | 0.4998 | |||
MedSE | 1.2459 | 0.8065 | 0.9201 | 0.6033 | 0.5434 | 0.6822 | 0.5387 | 0.4699 | 0.5622 | ||
2.9838 | 3.0811 | 3.0512 | 3.1253 | 2.965 | 2.7197 | 2.5382 | 2.8219 | 2.7438 | |||
1.9525 | 1.8371 | 2.3438 | 1.7215 | 1.9963 | 2.2379 | 1.8893 | 1.886 | 1.8351 | |||
1.5477 | 1.5198 | 1.2998 | 1.4741 | 1.6448 | 1.0015 | 1.4075 | 1.7982 | 1.7257 | |||
0.5511 | 1.5501 | 1.1878 | 1.1662 | 1.1816 | 0.8059 | 1.1623 | 1.0545 | 1.1654 | |||
1.6614 | 0.7911 | 1.9069 | 0.6863 | 0.9868 | 1.2406 | 0.8082 | 1.0211 | 1.3427 | |||
0.5739 | 0.3887 | 0.1885 | 1.1021 | 0.8204 | 0.8706 | 0.7875 | 0.5343 | 0.8185 | |||
0.2624 | 0.2341 | 0.2342 | 0.235 | 0.2307 | 0.2313 | 0.2472 | 0.2349 | 0.238 | |||
MedSE | 1.5138 | 1.1417 | 1.5123 | 0.8143 | 0.816 | 0.907 | 1.0921 | 0.7422 | 0.8705 |
Exp | Square | LAD | Exp | Square | LAD | Exp | Square | LAD | |||
---|---|---|---|---|---|---|---|---|---|---|---|
2.9991 | 2.9355 | 2.8898 | 3 | 3.1579 | 3.273 | 2.1076 | 2.8197 | 3.0519 | |||
1.87 | 1.8534 | 2.148 | 2.2471 | 2.097 | 1.7728 | 1.4959 | 2.1326 | 2.4031 | |||
1.6641 | 1.7286 | 1.3997 | 1.7933 | 1.4782 | 1.3566 | 1.2695 | 1.5688 | 1.1838 | |||
1.5449 | 1.4123 | 1.2135 | 1.1743 | 0.9747 | 1.6914 | 0.8435 | 1.4414 | 1.6309 | |||
1.1133 | 1.2998 | 1.4747 | 1.193 | 1.006 | 0.6772 | 1.5524 | 1.0102 | 1.3473 | |||
1.104 | 1.0165 | 0.8516 | 1.5647 | 0.6918 | 0.8134 | 0.9871 | 0.8072 | 0.3043 | |||
0.7841 | 0.7976 | 0.7812 | 0.7814 | 0.7626 | 0.7698 | 0.7785 | 0.7921 | 0.7578 | |||
MedSE | 1.0389 | 1.1975 | 2.4913 | 3.9024 | 1.3519 | 3.216 | 3.5719 | 1.4983 | 3.4753 | ||
2.9674 | 3.0461 | 3.0203 | 2.8981 | 3.3433 | 2.8355 | 3.0614 | 3.0561 | 2.7106 | |||
1.9594 | 1.956 | 2.2012 | 2.2222 | 1.8716 | 2.041 | 2.1271 | 2.0035 | 2.1597 | |||
1.7014 | 1.5528 | 1.4707 | 1.698 | 1.5486 | 1.533 | 1.4004 | 1.8345 | 1.5845 | |||
1.401 | 1.4171 | 1.8907 | 1.4537 | 1.3244 | 1.4937 | 1.4151 | 1.6572 | 1.516 | |||
1.3144 | 1.3097 | 1.1209 | 1.3471 | 1.0562 | 1.4014 | 1.2555 | 1.1235 | 0.9392 | |||
1.1186 | 1.0907 | 0.9799 | 0.9439 | 0.977 | 1.1517 | 1.3021 | 1.2148 | 1.0304 | |||
0.4984 | 0.499 | 0.5 | 0.5008 | 0.5004 | 0.5 | 0.4997 | 0.5007 | 0.5 | |||
MedSE | 0.7268 | 0.8361 | 1.0131 | 0.846 | 0.8735 | 1.0286 | 1.152 | 0.9011 | 1.1143 | ||
2.6774 | 2.7337 | 2.2269 | 2.7222 | 2.6372 | 2.5639 | 2.7679 | 2.625 | 2.3669 | |||
1.8758 | 1.5391 | 2.0274 | 1.9726 | 1.4435 | 1.4924 | 1.7517 | 1.8328 | 1.7334 | |||
1.6073 | 1.1691 | 1.5513 | 1.5515 | 1.4517 | 1.9718 | 1.3588 | 1.4107 | 1.5919 | |||
0.6648 | 0.1578 | −0.522 | 1.3422 | 0.5994 | 0.0136 | 0.4778 | 0.5541 | 0.5164 | |||
1.3401 | 0.5349 | 0.2749 | 1.0115 | 0.1372 | 0.3757 | −0.186 | −0.095 | −0.533 | |||
0.8527 | 0.2299 | 1.1821 | 0.4795 | 0.6847 | −0.254 | 0.4057 | 0.9201 | −0.205 | |||
0.2731 | 0.368 | 0.424 | 0.309 | 0.3721 | 0.4449 | 0.429 | 0.4203 | 0.4601 | |||
MedSE | 1.6545 | 3.3194 | 4.2853 | 2.3481 | 3.2567 | 4.9894 | 5.0342 | 3.9322 | 4.9488 | ||
3.0412 | 2.9279 | 3.2539 | 3.0001 | 2.9425 | 3.3778 | 3.0661 | 2.9154 | 2.9285 | |||
1.8198 | 1.731 | 1.3838 | 2.2004 | 2.009 | 1.6173 | 2.0961 | 1.7923 | 1.9155 | |||
1.5695 | 1.6793 | 1.9783 | 1.818 | 1.7579 | 2.1066 | 1.3409 | 1.8222 | 1.9951 | |||
1.3762 | 1.8133 | 0.6013 | 1.0067 | 1.2925 | 1.0236 | 1.2358 | 1.3486 | 1.1922 | |||
1.4344 | 1.0001 | 1.841 | 1.3286 | 1.3243 | 1.0053 | 1.0631 | 1.0136 | 1.148 | |||
0.9648 | 1.2588 | 1.7202 | 1.4715 | 1.123 | 1.021 | 1.629 | 0.9606 | 0.9799 | |||
0.7775 | 0.7886 | 0.7847 | 0.7808 | 0.7977 | 0.7196 | 0.787 | 0.7941 | 0.7652 | |||
MedSE | 1.9747 | 1.9176 | 3.2383 | 4.2531 | 2.1049 | 3.9547 | 2.641 | 2.139 | 4.1654 | ||
2.9924 | 3.127 | 3.3996 | 2.9071 | 2.9556 | 3.2707 | 3.1224 | 2.9192 | 2.7059 | |||
1.937 | 1.7325 | 2.2317 | 2.1674 | 2.1939 | 1.9265 | 2.1034 | 2.0763 | 2.0959 | |||
1.6366 | 1.9284 | 1.3111 | 1.7516 | 1.5356 | 1.6939 | 1.329 | 1.2335 | 1.5695 | |||
1.1814 | 1.3155 | 1.5799 | 1.3022 | 1.3435 | 1.781 | 1.3229 | 1.2476 | 1.5013 | |||
1.6792 | 1.4335 | 1.1793 | 1.5096 | 1.4914 | 0.8587 | 1.1117 | 1.3711 | 0.9883 | |||
1.0417 | 0.9891 | 1.1182 | 0.8649 | 0.8226 | 1.0424 | 1.6105 | 0.9617 | 1.3937 | |||
0.4982 | 0.5005 | 0.5 | 0.5013 | 0.5015 | 0.5 | 0.4992 | 0.4996 | 0.5 | |||
MedSE | 1.7761 | 1.6992 | 2.0438 | 1.9405 | 1.7439 | 2.1404 | 2.3691 | 1.8227 | 2.2753 | ||
2.7126 | 2.463 | 3.0218 | 2.7365 | 2.6766 | 2.3112 | 2.819 | 2.7498 | 2.8593 | |||
1.8389 | 1.7703 | 1.1025 | 1.9165 | 1.625 | 1.8649 | 1.7335 | 1.7638 | 1.7436 | |||
1.5575 | 1.302 | 1.5166 | 1.5931 | 1.1868 | 1.3592 | 1.299 | 1.4037 | 1.2125 | |||
0.5004 | 0.375 | 1.4855 | 1.1763 | 0.9432 | −0.186 | 0.4655 | 0.406 | 0.846 | |||
1.6485 | 0.5337 | −0.864 | 1.0869 | 0.1892 | −0.044 | −0.227 | 0.1407 | −1.145 | |||
0.8129 | 0.676 | −0.697 | 0.4247 | −0.641 | 0.3875 | 0.6011 | −0.289 | 0.6946 | |||
0.2716 | 0.3598 | 0.5 | 0.311 | 0.355 | 0.4821 | 0.4261 | 0.4147 | 0.4469 | |||
MedSE | 2.1955 | 3.5415 | 5.0709 | 2.8997 | 3.8662 | 5.2612 | 5.3588 | 4.256 | 5.583 |
Exp | Square | LAD | Exp | Square | LAD | Exp | Square | LAD | |||
---|---|---|---|---|---|---|---|---|---|---|---|
3.053 | 3.333 | 2.873 | 3.02 | 3.237 | 2.754 | 2.882 | 3.102 | 2.827 | |||
2.213 | 1.48 | 1.958 | 2.125 | 2.048 | 1.836 | 2.126 | 1.948 | 1.754 | |||
1.577 | 1.579 | 2.046 | 1.57 | 1.857 | 1.908 | 1.604 | 1.677 | 1.757 | |||
1.341 | 1.983 | 0.811 | 1.444 | 0.966 | 1.7 | 1.464 | 1.165 | 1.344 | |||
1.311 | 0.959 | 1.736 | 1.127 | 1.113 | 1.199 | 0.962 | 0.906 | 1.722 | |||
0.876 | 1.224 | 0.644 | 1.182 | 1.284 | 0.752 | 1.119 | 0.382 | 0.954 | |||
0.801 | 0.791 | 0.798 | 0.8 | 0.8 | 0.786 | 0.794 | 0.756 | 0.799 | |||
MedSE | 0.609 | 1.866 | 1.009 | 0.398 | 1.392 | 0.755 | 0.405 | 1.268 | 0.623 | ||
3.035 | 3.371 | 3.049 | 3.036 | 3.123 | 3.179 | 2.874 | 3.031 | 2.972 | |||
2.217 | 2.19 | 2.259 | 2.094 | 1.672 | 1.981 | 2.14 | 1.687 | 2.249 | |||
1.561 | 1.875 | 1.646 | 1.555 | 2.008 | 1.582 | 1.699 | 1.474 | 1.436 | |||
1.143 | 1.37 | 1.623 | 1.448 | 1.432 | 1.227 | 1.492 | 1.464 | 1.264 | |||
1.395 | 0.621 | 1.193 | 1.136 | 1.642 | 0.954 | 1.075 | 1.472 | 1.079 | |||
0.929 | 0.918 | 0.768 | 1.119 | 0.742 | 1.213 | 1.186 | 1.161 | 1.138 | |||
0.5 | 0.497 | 0.5 | 0.5 | 0.499 | 0.499 | 0.501 | 0.5 | 0.5 | |||
MedSE | 0.738 | 1.341 | 0.84 | 0.347 | 1.097 | 0.627 | 0.341 | 0.959 | 0.511 | ||
3.032 | 2.817 | 2.936 | 2.961 | 3.077 | 3.17 | 2.7 | 3.333 | 2.752 | |||
2.085 | 2.497 | 1.757 | 1.946 | 1.646 | 1.802 | 1.869 | 1.9 | 1.853 | |||
1.52 | 1.224 | 1.654 | 1.441 | 1.817 | 1.329 | 1.506 | 1.614 | 1.479 | |||
0.882 | 1.381 | 1.323 | 1.216 | 0.883 | 1.478 | 1.151 | 1.654 | 1.018 | |||
1.667 | 1.117 | 0.84 | 0.879 | 0.785 | 0.688 | 0.631 | 1.323 | 1.034 | |||
0.723 | 0.49 | 1.022 | 0.996 | 1.207 | 0.72 | 0.809 | 0.618 | 1.096 | |||
0.213 | 0.207 | 0.222 | 0.225 | 0.23 | 0.223 | 0.256 | 0.188 | 0.232 | |||
MedSE | 1.091 | 1.509 | 1.065 | 0.556 | 1.076 | 0.807 | 1.061 | 1.052 | 0.625 | ||
2.845 | 2.064 | 3.28 | 2.914 | 3.493 | 3.116 | 2.935 | 3.857 | 3.369 | |||
2.228 | 3.473 | 1.405 | 2.148 | 2.501 | 1.688 | 2.109 | 2.336 | 1.564 | |||
1.85 | 1.076 | 2.271 | 1.703 | 0.18 | 1.86 | 1.675 | 2.19 | 1.487 | |||
1.41 | 2.957 | 0.344 | 1.361 | 0.522 | 0.856 | 1.791 | 1.181 | 1.062 | |||
0.863 | 0.258 | 2.166 | 1.05 | -0.3 | 2.015 | 0.899 | 0.782 | 2.052 | |||
0.869 | 3.323 | 0.661 | 1.255 | 2.546 | 0.566 | 0.95 | 1.568 | 1.107 | |||
0.796 | 0.788 | 0.782 | 0.799 | 0.794 | 0.793 | 0.788 | 0.77 | 0.789 | |||
MedSE | 0.984 | 4.778 | 2.45 | 0.716 | 3.707 | 1.366 | 0.835 | 3.077 | 1.048 | ||
2.894 | 3.636 | 2.978 | 2.8 | 3.655 | 2.922 | 2.882 | 3.027 | 3.152 | |||
2.169 | 1.293 | 2.131 | 2.177 | 2.57 | 2.54 | 2.149 | 1.971 | 2.333 | |||
1.766 | 0.053 | 1.419 | 1.572 | 1.919 | 1.607 | 1.705 | 2.184 | 1.673 | |||
1.215 | -0.05 | 0.866 | 1.436 | 1.79 | 1.238 | 1.729 | 0.58 | 0.811 | |||
1.283 | 1.593 | 2.262 | 1.02 | 0.449 | 1.381 | 1.01 | 1.381 | 1.316 | |||
0.833 | 0.489 | 0.595 | 1.068 | -0.27 | 0.837 | 1.042 | -0.14 | 1.032 | |||
0.497 | 0.473 | 0.5 | 0.499 | 0.494 | 0.501 | 0.499 | 0.499 | 0.5 | |||
MedSE | 0.627 | 3.878 | 1.536 | 0.506 | 2.989 | 0.996 | 0.799 | 2.467 | 0.803 | ||
3.111 | 2.373 | 3.24 | 2.597 | 3.516 | 2.797 | 2.784 | 3.617 | 2.885 | |||
2.294 | 3.955 | 2.314 | 2.195 | 2.032 | 1.871 | 2.013 | 1.844 | 2.128 | |||
1.791 | 0.154 | 1.293 | 1.335 | 1.097 | 1.344 | 1.589 | 1.296 | 1.467 | |||
1.687 | 2.849 | 1.613 | 1.542 | 2.57 | 1.161 | 1.418 | 1.035 | 1.627 | |||
1.536 | 1.503 | 0.8 | 0.98 | 1.2 | 1.125 | 0.8 | 0.872 | 1.222 | |||
0.805 | 1.732 | 1.143 | 0.884 | 0.344 | 0.76 | 0.934 | 1.147 | 1.145 | |||
0.133 | 0.034 | 0.183 | 0.189 | 0.088 | 0.226 | 0.221 | 0.174 | 0.189 | |||
MedSE | 0.975 | 3.461 | 1.326 | 1.074 | 2.428 | 0.822 | 0.81 | 2.047 | 0.67 |
Exp | Square | LAD | Exp | Square | LAD | Exp | Square | LAD | |||
---|---|---|---|---|---|---|---|---|---|---|---|
3.125 | 3.143 | 2.909 | 3.286 | 2.614 | 3.142 | 2.82 | 2.138 | 2.895 | |||
1.692 | 1.89 | 1.934 | 1.3 | 2.39 | 2.025 | 2.07 | 2.367 | 1.826 | |||
1.919 | 1.633 | 1.597 | 1.68 | 0.716 | 1.622 | 1.651 | 2.761 | 1.473 | |||
1.167 | 0.737 | 1.612 | 0.997 | 1.4 | 1.418 | 1.365 | 0.86 | 1.452 | |||
1.422 | 1.235 | 1.059 | −0.28 | 0.584 | 1.318 | 1.247 | 0.485 | 1.191 | |||
0.898 | 0.038 | 1.076 | 2.083 | 1.16 | 1.273 | 0.978 | 0.799 | 1.164 | |||
0.501 | 0.492 | 0.496 | 0.501 | 0.477 | 0.5 | 0.5 | 0.486 | 0.5 | |||
MedSE | 0.636 | 2.596 | 0.562 | 2.657 | 2.623 | 0.411 | 0.275 | 2.591 | 0.341 | ||
2.941 | 1.728 | 2.955 | 1.38 | 2.193 | 3.15 | 2.83 | 2.295 | 2.963 | |||
1.143 | 2.575 | 2.278 | 0.298 | 2.322 | 1.952 | 1.88 | 2.292 | 2.002 | |||
1.771 | 2.299 | 1.386 | 1.195 | 0.383 | 1.267 | 1.867 | 1.809 | 1.619 | |||
1.008 | −0.63 | 1.607 | 0.019 | 0.211 | 1.156 | 0.396 | 2.348 | 1.218 | |||
0.605 | 2.475 | 1.326 | 0.255 | 0.283 | 0.961 | 0.974 | 1.317 | 0.994 | |||
0.579 | 0.146 | 0.922 | 0.58 | 0.826 | 0.921 | 0.881 | 0.699 | 1.136 | |||
0.503 | 0.468 | 0.495 | 0.503 | 0.449 | 0.499 | 0.494 | 0.45 | 0.498 | |||
MedSE | 1.561 | 3.925 | 0.819 | 3.645 | 3.922 | 0.547 | 1.227 | 4.972 | 0.454 | ||
3.02 | 1.981 | 3.046 | 3.183 | 2.054 | 2.849 | 2.893 | 2.507 | 3.187 | |||
1.479 | 0.857 | 2.072 | 1.259 | 0.636 | 2.253 | 1.911 | 0.865 | 2.265 | |||
1.978 | 0.753 | 0.897 | 1.721 | 1.649 | 1.299 | 1.827 | 2.039 | 1.362 | |||
0.837 | 0.645 | 1.349 | 0.785 | 0.67 | 1.362 | 0.61 | 0.934 | 0.962 | |||
1.557 | −0.56 | 1.26 | 0.551 | −0.68 | 1.275 | 0.813 | 1.384 | 1.135 | |||
−0.23 | −0.13 | 0.379 | 1.105 | 0.509 | 0.536 | 1.067 | −1.56 | 1.24 | |||
0.502 | 0.431 | 0.489 | 0.504 | 0.431 | 0.493 | 0.493 | 0.459 | 0.491 | |||
MedSE | 1.922 | 5.079 | 1.207 | 2.191 | 4.588 | 0.805 | 1.034 | 5.232 | 0.69 |
Exp | Square | LAD | Exp | Square | LAD | Exp | Square | LAD | |||
---|---|---|---|---|---|---|---|---|---|---|---|
Correct | 10 | 5.23 | 5.78 | 10 | 5.53 | 5.61 | 10 | 5.61 | 5.64 | ||
Incorrect | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0.8008 | 0.8035 | 0.8011 | 0.7999 | 0.8014 | 0.7982 | 0.7997 | 0.7995 | 0.801 | |||
MedSE | 0.3747 | 0.3887 | 0.4697 | 0.1468 | 0.2843 | 0.3259 | 0.1316 | 0.2374 | 0.2944 | ||
Correct | 10 | 5.27 | 5.61 | 10 | 5.47 | 5.74 | 10 | 5.69 | 5.75 | ||
Incorrect | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0.5013 | 0.5008 | 0.502 | 0.5001 | 0.5003 | 0.5005 | 0.4999 | 0.4997 | 0.5005 | |||
MedSE | 0.3575 | 0.3514 | 0.4354 | 0.1342 | 0.2751 | 0.3161 | 0.1207 | 0.2217 | 0.2699 | ||
Correct | 10 | 5.42 | 5.52 | 9.98 | 5.42 | 5.36 | 9 | 5.3 | 5.46 | ||
Incorrect | 0 | 0.05 | 0.14 | 0 | 0.01 | 0.03 | 0 | 0 | 0 | ||
0.2351 | 0.2375 | 0.2508 | 0.2257 | 0.231 | 0.2426 | 0.245 | 0.2265 | 0.2407 | |||
MedSE | 0.7905 | 0.8637 | 1.0335 | 0.4758 | 0.6328 | 0.7898 | 0.8372 | 0.5565 | 0.6443 | ||
Correct | 10 | 5.34 | 5.12 | 10 | 5.1 | 5.42 | 9 | 5.17 | 5.18 | ||
Incorrect | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | ||
0.8017 | 0.7992 | 0.8087 | 0.5033 | 0.7988 | 0.8018 | 0.8002 | 0.8036 | 0.7986 | |||
MedSE | 0.8202 | 0.7826 | 0.9687 | 4.5219 | 0.5524 | 0.6503 | 0.2753 | 0.4729 | 0.5452 | ||
Correct | 10 | 5.34 | 5.21 | 10 | 5.4 | 5.27 | 9 | 5.27 | 5.23 | ||
Incorrect | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0.5034 | 0.5014 | 0.4998 | 0.5005 | 0.5001 | 0.5001 | 0.4997 | 0.4988 | 0.4998 | |||
MedSE | 0.813 | 0.75 | 0.9107 | 0.3039 | 0.5554 | 0.6583 | 0.2723 | 0.4426 | 0.5261 | ||
Correct | 8 | 5.17 | 5.11 | 9 | 5.51 | 5.29 | 9 | 5.31 | 5.27 | ||
Incorrect | 0 | 0.03 | 0.23 | 0 | 0.01 | 0.02 | 0 | 0 | 0 | ||
0.2601 | 0.2382 | 0.2301 | 0.2359 | 0.2241 | 0.2535 | 0.2432 | 0.246 | 0.246 | |||
MedSE | 1.3903 | 1.0942 | 1.3318 | 0.6905 | 0.7508 | 1.0301 | 0.8508 | 0.7031 | 0.8261 |
Exp | Square | LAD | Exp | Square | LAD | Exp | Square | LAD | |||
---|---|---|---|---|---|---|---|---|---|---|---|
Correct | 40 | 21.32 | 21.19 | 80 | 42.42 | 42.63 | 119.01 | 65.11 | 64.21 | ||
Incorrect | 0 | 0.02 | 0.04 | 0 | 0 | 0.03 | 0 | 0.02 | 0.07 | ||
0.7991 | 0.8011 | 0.769 | 0.8 | 0.788 | 0.775 | 0.7995 | 0.773 | 0.773 | |||
MedSE | 0.1818 | 1.091 | 1.746 | 0.1553 | 1.348 | 1.969 | 0.2672 | 1.478 | 1.992 | ||
Correct | 40 | 21.61 | 22.4 | 80 | 43.52 | 45.49 | 119.99 | 66.78 | 69.73 | ||
Incorrect | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0.4984 | 0.5018 | 0.5 | 0.5003 | 0.5 | 0.5 | 0.5005 | 0.5 | 0.5 | |||
MedSE | 0.1826 | 0.8458 | 0.767 | 0.1489 | 0.867 | 0.788 | 0.2289 | 0.915 | 0.809 | ||
Correct | 39.99 | 20.74 | 21.06 | 73.99 | 41.98 | 42.53 | 109.99 | 62.69 | 63.91 | ||
Incorrect | 0 | 0.65 | 0.89 | 0 | 0.72 | 1.15 | 0.99 | 0.72 | 1.14 | ||
0.2206 | 0.3554 | 0.375 | 0.2476 | 0.3644 | 0.437 | 0.3424 | 0.371 | 0.431 | |||
MedSE | 0.4032 | 2.9396 | 3.381 | 0.8237 | 3.4479 | 3.853 | 2.6921 | 3.691 | 3.975 | ||
Correct | 38 | 20.31 | 20.9 | 77.98 | 41.06 | 42.05 | 117.99 | 62.58 | 62.78 | ||
Incorrect | 0 | 0.02 | 0.05 | 0 | 0.01 | 0.01 | 0 | 0 | 0.16 | ||
0.7962 | 0.7944 | 0.785 | 0.8002 | 0.7937 | 0.778 | 0.7982 | 0.793 | 0.753 | |||
MedSE | 0.4685 | 1.7795 | 1.959 | 0.3963 | 2.0106 | 2.263 | 0.7239 | 2.123 | 2.766 | ||
Correct | 38.02 | 20.51 | 21.13 | 76.99 | 41.76 | 43.31 | 118.01 | 63.37 | 65.18 | ||
Incorrect | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0.4963 | 0.5009 | 0.5 | 0.5006 | 0.4987 | 0.5 | 0.5008 | 0.499 | 0.5 | |||
MedSE | 0.4416 | 1.6128 | 1.464 | 0.3987 | 1.7193 | 1.522 | 0.6623 | 1.776 | 1.571 | ||
Correct | 38.99 | 20.73 | 20.87 | 75 | 41.22 | 42.04 | 115 | 62.43 | 63.03 | ||
Incorrect | 0 | 0.81 | 1.22 | 0 | 0.59 | 1.05 | 0 | 0.8 | 1.1 | ||
0.219 | 0.3583 | 0.461 | 0.2383 | 0.3523 | 0.434 | 0.2962 | 0.413 | 0.462 | |||
MedSE | 0.5759 | 3.656 | 3.804 | 0.7593 | 3.6017 | 3.887 | 1.8711 | 4.392 | 4.039 |
Exp | Square | LAD | Exp | Square | LAD | Exp | Square | LAD | |||
---|---|---|---|---|---|---|---|---|---|---|---|
Correct | 10 | 5.3 | 5.47 | 10 | 5.05 | 5.45 | 9.8 | 5 | 5.5 | ||
Incorrect | 0 | 0.23 | 0.01 | 0 | 0.07 | 0 | 0 | 0.09 | 0 | ||
0.8016 | 0.7759 | 0.781 | 0.7997 | 0.7978 | 0.7949 | 0.7957 | 0.7606 | 0.7991 | |||
MedSE | 0.4001 | 1.8977 | 0.5016 | 0.2229 | 1.6261 | 0.3415 | 0.2547 | 1.4235 | 0.287 | ||
Correct | 10 | 5.23 | 5.53 | 10 | 5.11 | 5.68 | 10 | 5.03 | 5.74 | ||
Incorrect | 0 | 0.1 | 0 | 0 | 0.02 | 0 | 0 | 0.01 | 0 | ||
0.4999 | 0.5003 | 0.5024 | 0.4997 | 0.4967 | 0.4987 | 0.5004 | 0.4981 | 0.4999 | |||
MedSE | 0.5384 | 1.4093 | 0.4443 | 0.1554 | 1.2247 | 0.3282 | 0.1962 | 1.1521 | 0.2811 | ||
Correct | 9.9 | 5.15 | 5.43 | 10 | 5.11 | 5.44 | 9.1 | 5.33 | 5.62 | ||
Incorrect | 0 | 0.17 | 0.14 | 0 | 0.02 | 0.03 | 0 | 0.01 | 0 | ||
0.2096 | 0.2569 | 0.2543 | 0.2236 | 0.2091 | 0.2362 | 0.2513 | 0.2344 | 0.2358 | |||
MedSE | 0.7234 | 1.5711 | 1.0847 | 0.4201 | 1.1447 | 0.8001 | 0.857 | 0.9537 | 0.6616 | ||
Correct | 9 | 5.56 | 0.7973 | 10 | 5.33 | 0.7993 | 8.2 | 5.23 | 0.7991 | ||
Incorrect | 0.2 | 0.73 | 5.34 | 0 | 0.5 | 5.43 | 0 | 0.42 | 5.73 | ||
0.797 | 0.7892 | 0 | 0.7998 | 0.7954 | 0 | 0.7857 | 0.7892 | 0 | |||
MedSE | 0.9265 | 4.6649 | 0.4994 | 0.3901 | 3.7479 | 0.3404 | 0.5873 | 2.9429 | 0.2881 | ||
Correct | 10 | 5.31 | 0.5 | 9.8 | 5.24 | 0.4999 | 8,2 | 5.13 | 0.5002 | ||
Incorrect | 0.1 | 0.45 | 5.34 | 0 | 0.23 | 5.87 | 0 | 0.18 | 5.76 | ||
0.4969 | 0.4999 | 0 | 0.4991 | 0.4974 | 0 | 0.4994 | 0.4961 | 0 | |||
MedSE | 0.475 | 3.8028 | 0.4602 | 0.2339 | 2.9176 | 0.332 | 0.4473 | 2.4682 | 0.2825 | ||
Correct | 10 | 5.15 | 0.2815 | 8.2 | 5.06 | 0.2357 | 9.1 | 5.04 | 0.2364 | ||
Incorrect | 0 | 0.25 | 5.34 | 0 | 0.09 | 5.47 | 0 | 0.03 | 5.34 | ||
0.1366 | 0.1648 | 0.25 | 0.1762 | 0.1159 | 0.02 | 0.2152 | 0.1538 | 0 | |||
MedSE | 0.4858 | 3.1858 | 1.0803 | 0.2833 | 2.4497 | 0.7324 | 0.5134 | 2.1076 | 0.6375 |
Exp | Square | LAD | Exp | Square | LAD | Exp | Square | LAD | |||
---|---|---|---|---|---|---|---|---|---|---|---|
Correct | 8.1 | 5.24 | 5.4 | 10 | 5.04 | 5.45 | 10 | 5.11 | 5.54 | ||
Incorrect | 0 | 0.34 | 0 | 0 | 0.41 | 0 | 0 | 0.24 | 0 | ||
0.4991 | 0.4974 | 0.5 | 0.5005 | 0.489 | 0.4989 | 0.5 | 0.4835 | 0.5001 | |||
MedSE | 1.1925 | 2.4503 | 0.54 | 0.1897 | 2.5552 | 0.3621 | 0.1557 | 2.2579 | 0.2916 | ||
Correct | 9.8 | 5.54 | 5.4 | 6.3 | 5.61 | 5.3 | 6.1 | 5.14 | 5.76 | ||
Incorrect | 0 | 0.87 | 0 | 1.1 | 0.84 | 0 | 1.9 | 0.72 | 0 | ||
0.5012 | 0.4644 | 0.4996 | 0.4982 | 0.4665 | 0.4998 | 0.4974 | 0.476 | 0.4976 | |||
MedSE | 0.7048 | 4.6858 | 0.6433 | 1.724 | 3.5277 | 0.4495 | 3.0656 | 3.7178 | 0.3789 | ||
Correct | 9.96 | 5.2 | 5.43 | 7.02 | 5.37 | 5.38 | 6.02 | 5.29 | 5.77 | ||
Incorrect | 0.04 | 1.21 | 0 | 0 | 1.2 | 0 | 1 | 1.04 | 0 | ||
0.5019 | 0.4653 | 0.4993 | 0.4993 | 0.4758 | 0.4953 | 0.4974 | 0.4491 | 0.495 | |||
MedSE | 0.832 | 4.9396 | 0.806 | 1.3822 | 4.3962 | 0.5848 | 2.2125 | 4.0998 | 0.4734 |
Variable | Description |
---|---|
STATION | ID variable |
PRICE | sales price of house iin $1000 (MLS) |
NROOM | the number of rooms |
DWELL | 1 if detached unit, 0 otherwise |
NBATH | the number of bathrooms |
PATIO | 1 if patio, 0 otherwise |
FIREPL | 1 if fireplace, 0 otherwise |
AC | 1 if air conditioning, 0 otherwise |
BMENT | 1 if basement, 0 otherwise |
NSTOR | number of stories |
GAR | number of car spaces in garage (0 = no garage) |
AGE | age of dwelling in years |
CITCOU | 1 if dwelling is in Baltimore County, 0 otherwise |
LOTSZ | lot size in hundreds of square feet |
SQFT | interior living space in hundreds of square feet |
X | x coordinate on the Maryland grid |
Y | y coordinate on the Maryland grid |
EXP | Square | LAD | ||||||
---|---|---|---|---|---|---|---|---|
Adaptive-l1 | Null | Adaptive-l1 | Null | Adaptive-l1 | Null | |||
NROOM | 0.49674002 | 0.20409727 | 0.01051881 | 0.1929546 | 0.0037123 | 0.2362159 | ||
DWELL | −1.3922 × 10−17 | 0.45980677 | 0.00075831 | 0.4703206 | 0.00029162 | 0.5097926 | ||
NBATH | 0.030063578 | 0.36030577 | 0.00385469 | 0.3514846 | 0.0012552 | 0.4254525 | ||
PATIO | 4.91478 × 10−18 | 0.01072777 | 0.00097357 | 0.017285 | 0.00014135 | −0.092123 | ||
FIREPL | −9.5477 × 10−18 | −0.01059 | 0.0002972 | 0.0013726 | 0.00012271 | −0.077913 | ||
AC | −1.2919 × 10−17 | 0.3021609 | 0.001 | 0.311554 | 0.00020267 | 0.3138447 | ||
BMENT | −2.2645 × 10−17 | 0.1187834 | 0.0044111 | 0.1235361 | 0.00116474 | 0.1317025 | ||
NSTOR | −9.9947 × 10−17 | 0.47809045 | 0.00359286 | 0.503778 | 0.00129079 | 0.4164672 | ||
GAR | −2.2383 × 10−17 | −0.1040092 | 0.00038606 | −0.099652 | 0.00039553 | −0.0844 | ||
AGE | 4.72988 × 10−17 | 0.01105389 | 0.03319136 | 0.0113282 | 0.02091404 | 0.0100436 | ||
CITCOU | −6.0274 × 10−17 | 0.68202393 | 0.00093599 | 0.6868509 | 0.00025428 | 0.4701451 | ||
LOTSZ | 1.04997 × 10−17 | 0.0011463 | 0.002195 | 0.0011849 | 0.00443912 | 4.606×10−5 | ||
SQFT | −7.0764 × 10−17 | −0.0362982 | 0.0316769 | −0.037256 | 0.01075005 | −0.034887 | ||
NROOM_W | −0.28507527 | −0.1092328 | −0.012775 | −0.098775 | −0.0059391 | −0.17384 | ||
DWELL_W | 1.61355 × 10−33 | −0.1051101 | −0.0010685 | −0.102124 | −0.0005411 | −0.107508 | ||
NBATH_W | −7.3257 × 10−18 | −0.1575366 | −0.0033218 | −0.159341 | −0.0014154 | −0.095605 | ||
PATIO_W | −8.7577 × 10−18 | 0.0642994 | −8.43 × 10−5 | 0.0601125 | 1.1236 × 10−5 | 0.1341514 | ||
FIREPL_W | −4.6005 × 10−34 | −0.0387668 | 0.00068287 | −0.036572 | −2.587 × 10−5 | −0.042569 | ||
AC_W | −3.3933 × 10−17 | −0.2098427 | −0.001 | −0.223255 | −0.0004486 | −0.187274 | ||
BMENT_W | −0.02780421 | −0.111448 | −0.0074127 | −0.117315 | −0.0032221 | −0.1317 | ||
NSTOR_W | −2.6009 × 10−32 | −0.1648658 | −0.0047893 | −0.159493 | −0.0023109 | −0.178134 | ||
GAR_W | 1.94949 × 10−17 | 0.15495116 | 0.001 | 0.1566788 | 0.00017173 | 0.1186948 | ||
AGE_W | 5.87444 × 10−33 | −0.0069188 | −0.0228056 | −0.007876 | −0.0204497 | −0.001951 | ||
CITCOU_W | −0.06178084 | −0.4116914 | −0.0030172 | −0.426973 | −0.001 | −0.240381 | ||
LOTSZ_W | −2.2196 × 10−32 | −0.0005826 | −0.0034412 | −0.000541 | −0.0057558 | −0.000449 | ||
SQFT_W | 2.75575 × 10−17 | 0.01072435 | −0.0207167 | 0.0098465 | −0.0112853 | 0.0143746 | ||
0.498613719 | 0.49970041 | 0.49571237 | 0.4997043 | 0.49780529 | 0.499992 | |||
MSE | 0.121911727 | 0.11475312 | 0.13792606 | 0.1149259 | 0.14467343 | 0.114829 | ||
BIC | −304.892336 | −317.66083 | −278.85061 | −317.3434 | −268.77299 | −317.5214 |
EXP | Square | LAD | ||||||
---|---|---|---|---|---|---|---|---|
Adaptive-l1 | Null | Adaptive-l1 | Null | Adaptive-l1 | Null | |||
NROOM | + | + | + | + | + | + | ||
DWELL | + | + | + | |||||
NBATH | + | + | + | + | + | + | ||
PATIO | + | + | − | |||||
FIREPL | − | + | − | |||||
AC | + | + | + | |||||
BMENT | + | + | + | + | + | |||
NSTOR | + | + | + | + | + | |||
GAR | − | − | − | |||||
AGE | + | + | + | + | + | |||
CITCOU | + | + | + | |||||
LOTSZ | + | + | + | + | ||||
SQFT | − | + | − | + | − | |||
NROOM_W | − | − | − | − | − | − | ||
DWELL_W | − | − | − | − | ||||
NBATH_W | − | − | − | − | − | |||
PATIO_W | + | + | + | |||||
FIREPL_W | − | − | − | |||||
AC_W | − | − | − | |||||
BMENT_W | − | − | − | − | − | − | ||
NSTOR_W | − | − | − | − | − | |||
GAR_W | + | + | + | |||||
AGE_W | − | − | − | − | − | |||
CITCOU_W | − | − | − | − | − | |||
LOTSZ_W | − | − | ||||||
SQFT_W | + | − | + | − | + | |||
count “+” | 2 | 13 | 7 | 14 | 7 | 11 | ||
count “−” | 3 | 12 | 9 | 11 | 7 | 13 | ||
count | 5 | 25 | 16 | 25 | 14 | 24 | ||
BIC | −304.892336 | −317.66083 | −278.85061 | −317.3434 | −268.77299 | −317.5214 |
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Liu, Z.; Song, Y.; Cheng, Y. Robust Variable Selection with Exponential Squared Loss for the Spatial Durbin Model. Entropy 2023, 25, 249. https://doi.org/10.3390/e25020249
Liu Z, Song Y, Cheng Y. Robust Variable Selection with Exponential Squared Loss for the Spatial Durbin Model. Entropy. 2023; 25(2):249. https://doi.org/10.3390/e25020249
Chicago/Turabian StyleLiu, Zhongyang, Yunquan Song, and Yi Cheng. 2023. "Robust Variable Selection with Exponential Squared Loss for the Spatial Durbin Model" Entropy 25, no. 2: 249. https://doi.org/10.3390/e25020249
APA StyleLiu, Z., Song, Y., & Cheng, Y. (2023). Robust Variable Selection with Exponential Squared Loss for the Spatial Durbin Model. Entropy, 25(2), 249. https://doi.org/10.3390/e25020249