(a) \(-1/+1/pen\) systems with various \(pen\) for \(\lambda_0^{[1]}=0.05\)
\(pen\) 0 1 2 3
Level \(\ell\) \(\tilde{\zeta}(\ell)\) \(\mathbb{P}(L=\ell)\) \(\tilde{\zeta}(\ell)\) \(\mathbb{P}(L=\ell)\) \(\tilde{\zeta}(\ell)\) \(\mathbb{P}(L=\ell)\) \(\tilde{\zeta}(\ell)\) \(\mathbb{P}(L=\ell)\)
9 7.217 0.001 5.749 0.002 4.917 0.004 4.366 0.008
8 6.246 0.000 4.715 0.001 3.889 0.002 3.357 0.002
7 5.573 0.000 4.259 0.001 3.540 0.002 3.073 0.002
6 5.000 0.000 3.834 0.001 3.202 0.002 2.792 0.003
5 4.695 0.001 3.404 0.001 2.854 0.003 2.500 0.004
4 3.805 0.001 2.936 0.002 2.479 0.004 2.184 0.006
3 3.065 0.002 2.407 0.005 2.060 0.008 1.834 0.012
2 2.210 0.007 1.812 0.015 1.591 0.023 1.443 0.030
1 1.369 0.044 1.203 0.073 1.101 0.095 1.026 0.111
0 0.727 0.944 0.692 0.898 0.665 0.858 0.641 0.821
HMSE 14179.63 13189.89 12525.65 12053.38

(b) \(-1/+2/pen\) systems with various \(pen\) for \(\lambda_0^{[1]}=0.05\)
\(pen\) 0 1 2 3
Level \(\ell\) \(\tilde{\zeta}(\ell)\) \(\mathbb{P}(L=\ell)\) \(\tilde{\zeta}(\ell)\) \(\mathbb{P}(L=\ell)\) \(\tilde{\zeta}(\ell)\) \(\mathbb{P}(L=\ell)\) \(\tilde{\zeta}(\ell)\) \(\mathbb{P}(L=\ell)\)
9 4.876 0.002 4.069 0.007 3.582 0.013 3.247 0.020
8 4.093 0.002 3.170 0.004 2.647 0.005 2.307 0.007
7 3.568 0.002 2.882 0.004 2.491 0.005 2.229 0.006
6 3.005 0.003 2.389 0.005 2.045 0.008 1.819 0.011
5 2.672 0.003 2.239 0.005 1.985 0.007 1.809 0.008
4 2.067 0.007 1.706 0.014 1.509 0.020 1.376 0.027
3 1.887 0.009 1.656 0.012 1.510 0.013 1.403 0.014
2 1.259 0.041 1.115 0.066 1.029 0.086 0.966 0.101
1 1.205 0.038 1.117 0.034 1.053 0.030 1.002 0.028
0 0.684 0.892 0.655 0.850 0.631 0.812 0.611 0.778
HMSE 13230.45 12675.38 12303.48 12042.78

(c) \(-1/+1/pen\) systems with various \(pen\) for \(\lambda_0^{[1]}=1\)
\(pen\) 0 1 2 3
Level \(\ell\) \(\tilde{\zeta}(\ell)\) \(\mathbb{P}(L=\ell)\) \(\tilde{\zeta}(\ell)\) \(\mathbb{P}(L=\ell)\) \(\tilde{\zeta}(\ell)\) \(\mathbb{P}(L=\ell)\) \(\tilde{\zeta}(\ell)\) \(\mathbb{P}(L=\ell)\)
14 1.439 0.311 1.230 0.463 1.123 0.562 1.055 0.634
13 0.939 0.089 0.711 0.060 0.590 0.045 0.513 0.035
12 0.796 0.044 0.634 0.036 0.539 0.030 0.475 0.025
11 0.717 0.027 0.581 0.025 0.500 0.022 0.444 0.020
10 0.666 0.020 0.542 0.019 0.468 0.018 0.418 0.016
9 0.627 0.016 0.510 0.016 0.442 0.015 0.395 0.014
8 0.595 0.014 0.483 0.015 0.419 0.014 0.375 0.013
7 0.567 0.014 0.459 0.014 0.397 0.014 0.356 0.013
6 0.540 0.014 0.436 0.015 0.377 0.014 0.337 0.013
5 0.513 0.016 0.413 0.016 0.357 0.016 0.319 0.014
4 0.483 0.020 0.388 0.020 0.335 0.018 0.300 0.017
3 0.450 0.027 0.361 0.026 0.313 0.024 0.280 0.021
2 0.409 0.042 0.330 0.038 0.287 0.033 0.258 0.029
1 0.359 0.078 0.293 0.065 0.256 0.053 0.232 0.044
0 0.291 0.269 0.246 0.172 0.219 0.122 0.201 0.091
HMSE 4649638 5052378 5328804 5532048