Table 9:
Fitting results for 31 Zhou and Ren’s circular tunnel dataset. As explained in Section 4, we use Steering law (Equation 1) to predict the mean (or location) parameter, and one of the six variance models (Table 1) to predict the variance (or scale) parameter, as explained in Table 2. a and b are parameters of Steering law, and c, d and e are parameters of variance models. Parameter estimations are shown in mean and 95% credible interval of posterior distributions. Fitness results are reported in AIC and WAIC metrics. As shown, the quadratic variance model (Model #4 in Table 1) has the best fitting results across all distribution types, measured by AIC, and WAIC.
| Model | Model Parameters (Mean and 95% Credible Interval) | Information Criteria | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Variance Model | Distribution Type | a | b | c | d | e | k | AIC | WAIC |
| #1. σ2 = c | Gaussian | 0.65 [0.55, 0.75] | 0.084 [0.08, 0.089] | 0.617 [0.559, 0.681] | N/A | N/A | N/A | 13103.7 | 13102.0 |
| Truncated Gaussian | 0.538 [0.421, 0.65] | 0.088 [0.083, 0.092] | 0.69 [0.617, 0.771] | N/A | N/A | N/A | 13066.4 | 13064.3 | |
| Lognormal | 0.88 [0.80, 0.97] | 0.075 [0.071, 0.079] | 0.825 [0.722, 0.941] | N/A | N/A | N/A | 13022.3 | 13022.8 | |
| Gamma | 0.84 [0.76, 0.93] | 0.076 [0.072, 0.080] | 0.665 [0.595, 0.742] | N/A | N/A | N/A | 13025.3 | 13024.3 | |
| Extreme value | 1.024 [0.928, 1.123] | 0.68 [0.063, 0.072] | 0.901 [0.815, 0.996] | N/A | N/A | N/A | 13154.4 | 13157.9 | |
| #2. σ2 = (c · ID)2 | Gaussian | 0.56 [0.46, 0.65] | 0.090 [0.082, 0.099] | 0.058 [0.056, 0.061] | N/A | N/A | N/A | 13356.1 | 13354.9 |
| Truncated Gaussian | 0.612 [0.507, 0.716] | 0.083 [0.073, 0.092] | 0.062 [0.058, 0.066] | N/A | N/A | N/A | 13322.9 | 13321.5 | |
| Lognormal | 0.24 [0.17, 0.31] | 0.12 [0.11, 0.12] | 0.056 [0.053, 0.060] | N/A | N/A | N/A | 13077.3 | 13074.8 | |
| Gamma | 0.27 [0.20, 0.35] | 0.11 [0.11, 0.12] | 0.054 [0.052, 0.058] | N/A | N/A | N/A | 13140.9 | 13138.5 | |
| Extreme value | 0.22 [0.159, 0.282] | 0.114 [0.109, 0.12] | 0.052 [0.049, 0.054] | N/A | N/A | N/A | 13081.6 | 12078.9 | |
| #3. σ2 = c + d · ID | Gaussian | 0.64 [0.54, 0.73] | 0.085 [0.080, 0.90] | 0.269 [0.179, 0.362] | 0.02 [0.01, 0.03] | N/A | N/A | 13057.2 | 13053.7 |
| Truncated Gaussian | 0.583 [0.48, 0.68] | 0.087 [0.082, 0.092] | 0.345 [0.233, 0.469] | 0.016 [0.01, 0.023] | N/A | N/A | 13040.2 | 13037.0 | |
| Lognormal | 0.62 [0.53, 0.72] | 0.086 [0.081, 0.092] | 0.166 [0.0445, 0.306] | 0.03 [0.02, 0.04] | N/A | N/A | 12965.4 | 12963.3 | |
| Gamma | 0.63 [0.54, 0.72] | 0.085 [0.080, 0.091] | 0.171 [0.0781, 0.273] | 0.02 [0.175, 0.031] | N/A | N/A | 12956.8 | 12953.8 | |
| Extreme value | 0.578 [0.487, 0.67] | 0.089 [0.084, 0.095] | 0.003 [−0.081, 0.093] | 0.042 [0.035, 0.049] | N/A | N/A | 12985.3 | 12985.1 | |
| #4. σ2 = c + d · ID2 | Gaussian | 0.65 [0.55, 0.74] | 0.085 [0.080, 0.090] | 0.419 [0.356, 0.489] | 0.0004 [0.0003, 0.0006] | N/A | N/A | 13058.7 | 13055.4 |
| Truncated Gaussian | 0.587 [0.48, 0.69] | 0.086 [0.081, 0.092] | 0.483 [0.406, 0.574] | 0.0003 [0.0002, 0.0005] | N/A | N/A | 13040.1 | 13036.9 | |
| Lognormal | 0.61 [0.51, 0.71] | 0.087 [0.081, 0.093] | 0.360 [0.274, 0.463] | 0.0008 [0.0006, 0.001] | N/A | N/A | 12959.3 | 12957.5 | |
| Gamma | 0.64 [0.54, 0.73] | 0.085 [0.080, 0.091] | 0.344 [0.276, 0.417] | 0.0006 [0.0004, 0.0008] | N/A | N/A | 12954.4 | 12951.5 | |
| Extreme value | 0.55 [0.46, 0.65] | 0.091 [0.085, 0.097] | 0.258 [0.196, 0.327] | 0.0011 [0.0009, 0.0014] | N/A | N/A | 12974.1 | 12971.1 | |
| exGaussian | 0.342 [0.131, 0.652] | 0.107 [0.085, 0.124] | 0.119 [0.001, 0.369] | 0.0002 [0.00006, 0.0004] | N/A | 0.03 [0.02, 0.05] | 13039.6 | 13074.5 | |
| #5. = (c + d · ID)2 | Gaussian | 0.64 [0.55, 0.73] | 0.085 [0.080, 0.090] | 0.56 [0.50, 0.63] | 0.011 [0.008, 0.015] | N/A | N/A | 13057.2 | 13053.7 |
| Truncated Gaussian | 0.585 [0.477, 0.683] | 0.087 [0.082, 0.092] | 0.615 [0.538, 0.697] | 0.01 [0.006, 0.013] | N/A | N/A | 13039.9 | 13036.6 | |
| Lognormal | 0.60 [0.50, 0.70] | 0.087 [0.082, 0.094] | 0.48 [0.39, 0.57] | 0.019 [0.014, 0.024] | N/A | N/A | 12961.8 | 12960.2 | |
| Gamma | 0.63 [0.53, 0.72] | 0.086 [0.080, 0.091] | 0.48 [0.41, 0.56] | 0.015 [0.012, 0.020] | N/A | N/A | 12954.9 | 12952.1 | |
| Extreme value | 0.546 [0.453, 0.642] | 0.091 [0.085, 0.097] | 0.356 [0.28, 0.434] | 0.026 [0.022, 0.031] | N/A | N/A | 12976.9 | 12974.9 | |
| #6. σ2 = c + d · ID + e · ID2 | Gaussian | 0.64 [0.55, 0.74] | 0.085 [0.080, 0.090] | 0.358 [0.221, 0.521] | 0.007 [−0.102, 0.021] | 0.0003 [0.00001, 0.0006] | N/A | 13060.2 | 13055.4 |
| Truncated Gaussian | 0.587 [0.469, 0.7] | 0.087 [0.081, 0.092] | 0.477 [0.0.299, 0.7] | 0.0004 [−0.022, 0.017] | 0.0003 [0.00002, 0.0008] | N/A | 13042.7 | 13038.1 | |
| Lognormal | 0.61 [0.52, 0.72] | 0.087 [0.081, 0.093] | 0.454 [0.174, 0.766] | −0.012 [−0.047, 0.022] | 0.001 [0.0002, 0.002] | N/A | 12962.2 | 12960.2 | |
| Gamma | 0.63 [0.54, 0.73] | 0.086 [0.080, 0.091] | 0.326 [0.147, 0.539] | 0.002 [−0.02, 0.02] | 0.0005 [0.00005, 0.001] | N/A | 12957.2 | 12953.3 | |
| Extreme value | 0.551 [0.456, 0.65] | 0.091 [0.085, 0.098] | 0.3 [0.118, 0.499] | −0.006 [−0.032, 0.02] | 0.001 [0.0006, 0.002] | N/A | 12976.9 | 12973.7 | |