Model run using Apollo for R, version 0.2.3 on Darwin by stephane.hess www.ApolloChoiceModelling.com Model name : Apollo_example_16 Model description : Mixed logit model on Swiss route choice data, WTP space with correlated and flexible distributions, inter and intra-individual heterogeneity Model run at : 2021-02-06 16:31:17 Estimation method : bfgs Model diagnosis : successful convergence Number of individuals : 388 Number of rows in database : 3492 Number of modelled outcomes : 3492 Number of cores used : 3 Number of inter-individual draws : 100 (halton) Number of intra-individual draws : 100 (mlhs) LL(start) : -2406.92 LL(0) : -2420.47 LL(final) : -1440.254 Rho-square (0) : 0.405 Adj.Rho-square (0) : 0.4 AIC : 2904.51 BIC : 2978.41 Estimated parameters : 12 Time taken (hh:mm:ss) : 01:30:26.22 pre-estimation : 00:02:9.32 estimation : 00:57:25.51 post-estimation : 00:30:51.39 Iterations : 52 Min abs eigenvalue of Hessian : 14.54272 Some eigenvalues of Hessian are positive, indicating potential problems! Estimates: Estimate s.e. t.rat.(0) Rob.s.e. asc_1 -0.05781 0.076592 -0.7548 0.089621 asc_2 0.00000 NA NA NA mu_log_b_tc -2.57013 0.210190 -12.2276 0.219078 sigma_log_b_tc_inter 5.53322 0.779937 7.0944 0.706737 mu_log_v_tt -1.35153 0.025871 -52.2414 0.054621 sigma_log_v_tt_inter 0.61483 0.016733 36.7439 0.059487 sigma_log_v_tt_inter_2 0.02054 0.008918 2.3036 0.005540 sigma_log_v_tt_intra 0.53669 NaN NaN 0.042789 mu_log_v_hw -2.16078 NaN NaN 0.173999 sigma_log_v_hw_inter -0.92356 0.034377 -26.8654 0.086559 sigma_log_v_hw_v_tt_inter 0.43849 0.085597 5.1227 0.101018 v_ch 4.10368 NaN NaN 0.478431 gamma_vtt_business 2.34813 0.184735 12.7108 0.207332 Rob.t.rat.(0) asc_1 -0.6450 asc_2 NA mu_log_b_tc -11.7316 sigma_log_b_tc_inter 7.8293 mu_log_v_tt -24.7440 sigma_log_v_tt_inter 10.3355 sigma_log_v_tt_inter_2 3.7082 sigma_log_v_tt_intra 12.5426 mu_log_v_hw -12.4183 sigma_log_v_hw_inter -10.6697 sigma_log_v_hw_v_tt_inter 4.3407 v_ch 8.5774 gamma_vtt_business 11.3255 Overview of choices for MNL model component : alt1 alt2 Times available 3492.00 3492.00 Times chosen 1734.00 1758.00 Percentage chosen overall 49.66 50.34 Percentage chosen when available 49.66 50.34 Classical covariance matrix: asc_1 mu_log_b_tc asc_1 0.005866 0.001242 mu_log_b_tc 0.001242 0.044180 sigma_log_b_tc_inter -0.004781 -0.131350 mu_log_v_tt -1.1894e-04 0.001991 sigma_log_v_tt_inter -9.208e-05 0.002515 sigma_log_v_tt_inter_2 3.190e-06 -1.8153e-04 sigma_log_v_tt_intra -9.916e-05 0.002011 mu_log_v_hw -2.5566e-04 0.007590 sigma_log_v_hw_inter -3.664e-05 0.004908 sigma_log_v_hw_v_tt_inter -1.8814e-04 -0.001541 v_ch -5.5495e-04 0.003510 gamma_vtt_business -1.0250e-04 0.008236 sigma_log_b_tc_inter mu_log_v_tt asc_1 -0.004781 -1.1894e-04 mu_log_b_tc -0.131350 0.001991 sigma_log_b_tc_inter 0.608302 0.001626 mu_log_v_tt 0.001626 6.6930e-04 sigma_log_v_tt_inter -3.7261e-04 -6.0762e-04 sigma_log_v_tt_inter_2 4.219e-05 5.727e-05 sigma_log_v_tt_intra 7.1235e-04 -6.4198e-04 mu_log_v_hw -0.004727 -0.003201 sigma_log_v_hw_inter -0.005666 -0.001958 sigma_log_v_hw_v_tt_inter 0.008311 0.002221 v_ch -0.003638 -0.009596 gamma_vtt_business -0.008700 -0.005837 sigma_log_v_tt_inter sigma_log_v_tt_inter_2 asc_1 -9.208e-05 3.190e-06 mu_log_b_tc 0.002515 -1.8153e-04 sigma_log_b_tc_inter -3.7261e-04 4.219e-05 mu_log_v_tt -6.0762e-04 5.727e-05 sigma_log_v_tt_inter 2.7998e-04 1.1273e-04 sigma_log_v_tt_inter_2 1.1273e-04 7.954e-05 sigma_log_v_tt_intra -4.5620e-04 5.899e-05 mu_log_v_hw -0.002947 3.3665e-04 sigma_log_v_hw_inter -0.002026 4.140e-05 sigma_log_v_hw_v_tt_inter 0.002554 1.1960e-04 v_ch -0.009500 7.8907e-04 gamma_vtt_business -0.003841 1.7311e-04 sigma_log_v_tt_intra mu_log_v_hw asc_1 -9.916e-05 -2.5566e-04 mu_log_b_tc 0.002011 0.007590 sigma_log_b_tc_inter 7.1235e-04 -0.004727 mu_log_v_tt -6.4198e-04 -0.003201 sigma_log_v_tt_inter -4.5620e-04 -0.002947 sigma_log_v_tt_inter_2 5.899e-05 3.3665e-04 sigma_log_v_tt_intra -1.1569e-04 -0.002699 mu_log_v_hw -0.002699 -0.009153 sigma_log_v_hw_inter -0.001459 -0.004173 sigma_log_v_hw_v_tt_inter 0.001273 0.004244 v_ch -0.007885 -0.031990 gamma_vtt_business -0.002576 -0.011966 sigma_log_v_hw_inter sigma_log_v_hw_v_tt_inter asc_1 -3.664e-05 -1.8814e-04 mu_log_b_tc 0.004908 -0.001541 sigma_log_b_tc_inter -0.005666 0.008311 mu_log_v_tt -0.001958 0.002221 sigma_log_v_tt_inter -0.002026 0.002554 sigma_log_v_tt_inter_2 4.140e-05 1.1960e-04 sigma_log_v_tt_intra -0.001459 0.001273 mu_log_v_hw -0.004173 0.004244 sigma_log_v_hw_inter 0.001182 -6.6336e-04 sigma_log_v_hw_v_tt_inter -6.6336e-04 0.007327 v_ch -0.018055 0.001614 gamma_vtt_business -0.003709 -2.0003e-04 v_ch gamma_vtt_business asc_1 -5.5495e-04 -1.0250e-04 mu_log_b_tc 0.003510 0.008236 sigma_log_b_tc_inter -0.003638 -0.008700 mu_log_v_tt -0.009596 -0.005837 sigma_log_v_tt_inter -0.009500 -0.003841 sigma_log_v_tt_inter_2 7.8907e-04 1.7311e-04 sigma_log_v_tt_intra -0.007885 -0.002576 mu_log_v_hw -0.031990 -0.011966 sigma_log_v_hw_inter -0.018055 -0.003709 sigma_log_v_hw_v_tt_inter 0.001614 -2.0003e-04 v_ch -0.019720 -0.031798 gamma_vtt_business -0.031798 0.034127 Robust covariance matrix: asc_1 mu_log_b_tc asc_1 0.008032 0.004864 mu_log_b_tc 0.004864 0.047995 sigma_log_b_tc_inter -0.017017 -0.106590 mu_log_v_tt 4.5419e-04 -0.003110 sigma_log_v_tt_inter 5.7743e-04 -0.003355 sigma_log_v_tt_inter_2 -3.367e-05 4.4933e-04 sigma_log_v_tt_intra 4.3670e-04 -0.003167 mu_log_v_hw 0.001749 -0.013986 sigma_log_v_hw_inter 7.6222e-04 -0.004695 sigma_log_v_hw_v_tt_inter -1.9034e-04 0.004686 v_ch 0.002493 -0.054564 gamma_vtt_business 0.002287 -0.012279 sigma_log_b_tc_inter mu_log_v_tt asc_1 -0.017017 4.5419e-04 mu_log_b_tc -0.106590 -0.003110 sigma_log_b_tc_inter 0.499477 0.002507 mu_log_v_tt 0.002507 0.002983 sigma_log_v_tt_inter 0.001076 0.002945 sigma_log_v_tt_inter_2 8.004e-05 -1.3850e-04 sigma_log_v_tt_intra 9.8392e-04 0.002090 mu_log_v_hw 6.2987e-04 0.007434 sigma_log_v_hw_inter -0.005810 0.002241 sigma_log_v_hw_v_tt_inter 0.007139 0.001086 v_ch 0.026082 0.014297 gamma_vtt_business -0.008154 0.005110 sigma_log_v_tt_inter sigma_log_v_tt_inter_2 asc_1 5.7743e-04 -3.367e-05 mu_log_b_tc -0.003355 4.4933e-04 sigma_log_b_tc_inter 0.001076 8.004e-05 mu_log_v_tt 0.002945 -1.3850e-04 sigma_log_v_tt_inter 0.003539 -1.4427e-04 sigma_log_v_tt_inter_2 -1.4427e-04 3.069e-05 sigma_log_v_tt_intra 0.002387 -1.6058e-04 mu_log_v_hw 0.008889 -7.3157e-04 sigma_log_v_hw_inter 0.002779 -4.0943e-04 sigma_log_v_hw_v_tt_inter 0.001339 3.5992e-04 v_ch 0.016326 -0.002022 gamma_vtt_business 0.006440 -8.7747e-04 sigma_log_v_tt_intra mu_log_v_hw asc_1 4.3670e-04 0.001749 mu_log_b_tc -0.003167 -0.013986 sigma_log_b_tc_inter 9.8392e-04 6.2987e-04 mu_log_v_tt 0.002090 0.007434 sigma_log_v_tt_inter 0.002387 0.008889 sigma_log_v_tt_inter_2 -1.6058e-04 -7.3157e-04 sigma_log_v_tt_intra 0.001831 0.006998 mu_log_v_hw 0.006998 0.030276 sigma_log_v_hw_inter 0.002592 0.012621 sigma_log_v_hw_v_tt_inter -1.3403e-04 -0.003211 v_ch 0.014696 0.063681 gamma_vtt_business 0.005894 0.025981 sigma_log_v_hw_inter sigma_log_v_hw_v_tt_inter asc_1 7.6222e-04 -1.9034e-04 mu_log_b_tc -0.004695 0.004686 sigma_log_b_tc_inter -0.005810 0.007139 mu_log_v_tt 0.002241 0.001086 sigma_log_v_tt_inter 0.002779 0.001339 sigma_log_v_tt_inter_2 -4.0943e-04 3.5992e-04 sigma_log_v_tt_intra 0.002592 -1.3403e-04 mu_log_v_hw 0.012621 -0.003211 sigma_log_v_hw_inter 0.007493 -0.004424 sigma_log_v_hw_v_tt_inter -0.004424 0.010205 v_ch 0.025088 -0.024103 gamma_vtt_business 0.013109 -0.008398 v_ch gamma_vtt_business asc_1 0.002493 0.002287 mu_log_b_tc -0.054564 -0.012279 sigma_log_b_tc_inter 0.026082 -0.008154 mu_log_v_tt 0.014297 0.005110 sigma_log_v_tt_inter 0.016326 0.006440 sigma_log_v_tt_inter_2 -0.002022 -8.7747e-04 sigma_log_v_tt_intra 0.014696 0.005894 mu_log_v_hw 0.063681 0.025981 sigma_log_v_hw_inter 0.025088 0.013109 sigma_log_v_hw_v_tt_inter -0.024103 -0.008398 v_ch 0.228896 0.060810 gamma_vtt_business 0.060810 0.042986 Classical correlation matrix: asc_1 mu_log_b_tc asc_1 1.000000 0.07718 mu_log_b_tc 0.077177 1.00000 sigma_log_b_tc_inter -0.080043 -0.80123 mu_log_v_tt -0.060023 0.36616 sigma_log_v_tt_inter -0.071848 0.71507 sigma_log_v_tt_inter_2 0.004670 -0.09684 sigma_log_v_tt_intra NaN NaN mu_log_v_hw NaN NaN sigma_log_v_hw_inter -0.013914 0.67922 sigma_log_v_hw_v_tt_inter -0.028698 -0.08563 v_ch NaN NaN gamma_vtt_business -0.007244 0.21211 sigma_log_b_tc_inter mu_log_v_tt asc_1 -0.080043 -0.06002 mu_log_b_tc -0.801230 0.36616 sigma_log_b_tc_inter 1.000000 0.08059 mu_log_v_tt 0.080586 1.00000 sigma_log_v_tt_inter -0.028552 -1.40363 sigma_log_v_tt_inter_2 0.006066 0.24820 sigma_log_v_tt_intra NaN NaN mu_log_v_hw NaN NaN sigma_log_v_hw_inter -0.211333 -2.20201 sigma_log_v_hw_v_tt_inter 0.124484 1.00312 v_ch NaN NaN gamma_vtt_business -0.060381 -1.22125 sigma_log_v_tt_inter sigma_log_v_tt_inter_2 asc_1 -0.07185 0.004670 mu_log_b_tc 0.71507 -0.096840 sigma_log_b_tc_inter -0.02855 0.006066 mu_log_v_tt -1.40363 0.248201 sigma_log_v_tt_inter 1.00000 0.755438 sigma_log_v_tt_inter_2 0.75544 1.000000 sigma_log_v_tt_intra NaN NaN mu_log_v_hw NaN NaN sigma_log_v_hw_inter -3.52282 0.135039 sigma_log_v_hw_v_tt_inter 1.78326 0.156673 v_ch NaN NaN gamma_vtt_business -1.24244 0.105071 sigma_log_v_tt_intra mu_log_v_hw asc_1 NaN NaN mu_log_b_tc NaN NaN sigma_log_b_tc_inter NaN NaN mu_log_v_tt NaN NaN sigma_log_v_tt_inter NaN NaN sigma_log_v_tt_inter_2 NaN NaN sigma_log_v_tt_intra NaN NaN mu_log_v_hw NaN NaN sigma_log_v_hw_inter NaN NaN sigma_log_v_hw_v_tt_inter NaN NaN v_ch NaN NaN gamma_vtt_business NaN NaN sigma_log_v_hw_inter sigma_log_v_hw_v_tt_inter asc_1 -0.01391 -0.02870 mu_log_b_tc 0.67922 -0.08563 sigma_log_b_tc_inter -0.21133 0.12448 mu_log_v_tt -2.20201 1.00312 sigma_log_v_tt_inter -3.52282 1.78326 sigma_log_v_tt_inter_2 0.13504 0.15667 sigma_log_v_tt_intra NaN NaN mu_log_v_hw NaN NaN sigma_log_v_hw_inter 1.00000 -0.22543 sigma_log_v_hw_v_tt_inter -0.22543 1.00000 v_ch NaN NaN gamma_vtt_business -0.58396 -0.01265 v_ch gamma_vtt_business asc_1 NaN -0.007244 mu_log_b_tc NaN 0.212114 sigma_log_b_tc_inter NaN -0.060381 mu_log_v_tt NaN -1.221254 sigma_log_v_tt_inter NaN -1.242437 sigma_log_v_tt_inter_2 NaN 0.105071 sigma_log_v_tt_intra NaN NaN mu_log_v_hw NaN NaN sigma_log_v_hw_inter NaN -0.583960 sigma_log_v_hw_v_tt_inter NaN -0.012650 v_ch NaN NaN gamma_vtt_business NaN 1.000000 Robust correlation matrix: asc_1 mu_log_b_tc asc_1 1.00000 0.2477 mu_log_b_tc 0.24773 1.0000 sigma_log_b_tc_inter -0.26867 -0.6884 mu_log_v_tt 0.09278 -0.2599 sigma_log_v_tt_inter 0.10831 -0.2575 sigma_log_v_tt_inter_2 -0.06781 0.3702 sigma_log_v_tt_intra 0.11388 -0.3378 mu_log_v_hw 0.11219 -0.3669 sigma_log_v_hw_inter 0.09825 -0.2476 sigma_log_v_hw_v_tt_inter -0.02102 0.2117 v_ch 0.05814 -0.5206 gamma_vtt_business 0.12308 -0.2703 sigma_log_b_tc_inter mu_log_v_tt asc_1 -0.268670 0.09278 mu_log_b_tc -0.688432 -0.25992 sigma_log_b_tc_inter 1.000000 0.06493 mu_log_v_tt 0.064934 1.00000 sigma_log_v_tt_inter 0.025587 0.90650 sigma_log_v_tt_inter_2 0.020441 -0.45767 sigma_log_v_tt_intra 0.032536 0.89422 mu_log_v_hw 0.005122 0.78220 sigma_log_v_hw_inter -0.094971 0.47410 sigma_log_v_hw_v_tt_inter 0.100000 0.19678 v_ch 0.077138 0.54711 gamma_vtt_business -0.055650 0.45127 sigma_log_v_tt_inter sigma_log_v_tt_inter_2 asc_1 0.10831 -0.06781 mu_log_b_tc -0.25747 0.37020 sigma_log_b_tc_inter 0.02559 0.02044 mu_log_v_tt 0.90650 -0.45767 sigma_log_v_tt_inter 1.00000 -0.43775 sigma_log_v_tt_inter_2 -0.43775 1.00000 sigma_log_v_tt_intra 0.93770 -0.67736 mu_log_v_hw 0.85880 -0.75889 sigma_log_v_hw_inter 0.53967 -0.85376 sigma_log_v_hw_v_tt_inter 0.22277 0.64309 v_ch 0.57364 -0.76301 gamma_vtt_business 0.52215 -0.76390 sigma_log_v_tt_intra mu_log_v_hw asc_1 0.11388 0.112189 mu_log_b_tc -0.33780 -0.366890 sigma_log_b_tc_inter 0.03254 0.005122 mu_log_v_tt 0.89422 0.782203 sigma_log_v_tt_inter 0.93770 0.858802 sigma_log_v_tt_inter_2 -0.67736 -0.758888 sigma_log_v_tt_intra 1.00000 0.939922 mu_log_v_hw 0.93992 1.000000 sigma_log_v_hw_inter 0.69978 0.838011 sigma_log_v_hw_v_tt_inter -0.03101 -0.182665 v_ch 0.71787 0.764969 gamma_vtt_business 0.66439 0.720199 sigma_log_v_hw_inter sigma_log_v_hw_v_tt_inter asc_1 0.09825 -0.02102 mu_log_b_tc -0.24757 0.21172 sigma_log_b_tc_inter -0.09497 0.10000 mu_log_v_tt 0.47410 0.19678 sigma_log_v_tt_inter 0.53967 0.22277 sigma_log_v_tt_inter_2 -0.85376 0.64309 sigma_log_v_tt_intra 0.69978 -0.03101 mu_log_v_hw 0.83801 -0.18266 sigma_log_v_hw_inter 1.00000 -0.50598 sigma_log_v_hw_v_tt_inter -0.50598 1.00000 v_ch 0.60580 -0.49871 gamma_vtt_business 0.73045 -0.40099 v_ch gamma_vtt_business asc_1 0.05814 0.12308 mu_log_b_tc -0.52058 -0.27033 sigma_log_b_tc_inter 0.07714 -0.05565 mu_log_v_tt 0.54711 0.45127 sigma_log_v_tt_inter 0.57364 0.52215 sigma_log_v_tt_inter_2 -0.76301 -0.76390 sigma_log_v_tt_intra 0.71787 0.66439 mu_log_v_hw 0.76497 0.72020 sigma_log_v_hw_inter 0.60580 0.73045 sigma_log_v_hw_v_tt_inter -0.49871 -0.40099 v_ch 1.00000 0.61304 gamma_vtt_business 0.61304 1.00000 20 worst outliers in terms of lowest average per choice prediction: ID Avg prob per choice 23205 0.3596644 15174 0.3606908 76862 0.3776255 16178 0.3804148 14802 0.3850604 22580 0.3864620 22820 0.3934131 15056 0.3970924 22278 0.4003175 16489 0.4023864 82613 0.4024181 18219 0.4052207 80546 0.4054516 20323 0.4114678 20063 0.4145488 22961 0.4178511 14353 0.4195294 17645 0.4218831 21922 0.4228660 12534 0.4263598 Changes in parameter estimates from starting values: Initial Estimate Difference asc_1 0.000 -0.05781 -0.05781 asc_2 0.000 0.00000 0.00000 mu_log_b_tc -3.000 -2.57013 0.42987 sigma_log_b_tc_inter 0.000 5.53322 5.53322 mu_log_v_tt -3.000 -1.35153 1.64847 sigma_log_v_tt_inter 0.000 0.61483 0.61483 sigma_log_v_tt_inter_2 0.000 0.02054 0.02054 sigma_log_v_tt_intra 0.000 0.53669 0.53669 mu_log_v_hw -3.000 -2.16078 0.83922 sigma_log_v_hw_inter 0.000 -0.92356 -0.92356 sigma_log_v_hw_v_tt_inter 0.000 0.43849 0.43849 v_ch 0.000 4.10368 4.10368 gamma_vtt_business 0.000 2.34813 2.34813 Settings and functions used in model definition: apollo_control -------------- Value modelName "Apollo_example_16" modelDescr "Mixed logit model on Swiss route choice data, WTP space with correlated and flexible distributions, inter and intra-individual heterogeneity" indivID "ID" mixing "TRUE" nCores "3" debug "FALSE" workInLogs "FALSE" seed "13" HB "FALSE" noValidation "FALSE" noDiagnostics "FALSE" panelData "TRUE" analyticGrad "TRUE" Hessian routines attempted -------------- numerical second derivative of LL (using numDeriv) Scaling used in computing Hessian -------------- Value asc_1 0.05780845 mu_log_b_tc 2.57012709 sigma_log_b_tc_inter 5.53322469 mu_log_v_tt 1.35153219 sigma_log_v_tt_inter 0.61482658 sigma_log_v_tt_inter_2 0.02054457 sigma_log_v_tt_intra 0.53668707 mu_log_v_hw 2.16077559 sigma_log_v_hw_inter 0.92356460 sigma_log_v_hw_v_tt_inter 0.43848835 v_ch 4.10368449 gamma_vtt_business 2.34813289 apollo_randCoeff ---------------- function(apollo_beta, apollo_inputs){ randcoeff = list() randcoeff[["b_tc"]] = -exp( mu_log_b_tc + sigma_log_b_tc_inter * draws_tc_inter ) randcoeff[["v_tt"]] = ( exp( mu_log_v_tt + sigma_log_v_tt_inter * draws_tt_inter + sigma_log_v_tt_inter_2 * draws_tt_inter ^ 2 + sigma_log_v_tt_intra * draws_tt_intra ) * ( gamma_vtt_business * business + ( 1 - business ) ) ) randcoeff[["v_hw"]] = exp( mu_log_v_hw + sigma_log_v_hw_inter * draws_hw_inter + sigma_log_v_hw_v_tt_inter * draws_tt_inter ) return(randcoeff) apollo_probabilities -------------------- function(apollo_beta, apollo_inputs, functionality="estimate"){ ### Function initialisation: do not change the following three commands ### Attach inputs and detach after function exit apollo_attach(apollo_beta, apollo_inputs) on.exit(apollo_detach(apollo_beta, apollo_inputs)) ### Create list of probabilities P P = list() ### List of utilities: these must use the same names as in mnl_settings, order is irrelevant V = list() V[['alt1']] = asc_1 + b_tc*(v_tt*tt1 + tc1 + v_hw*hw1 + v_ch*ch1) V[['alt2']] = asc_2 + b_tc*(v_tt*tt2 + tc2 + v_hw*hw2 + v_ch*ch2) ### Define settings for MNL model component mnl_settings = list( alternatives = c(alt1=1, alt2=2), avail = list(alt1=1, alt2=1), choiceVar = choice, V = V ) ### Compute probabilities using MNL model P[['model']] = apollo_mnl(mnl_settings, functionality) ### Average across intra-individual draws P = apollo_avgIntraDraws(P, apollo_inputs, functionality) ### Take product across observation for same individual P = apollo_panelProd(P, apollo_inputs, functionality) ### Average across inter-individual draws P = apollo_avgInterDraws(P, apollo_inputs, functionality) ### Prepare and return outputs of function P = apollo_prepareProb(P, apollo_inputs, functionality) return(P)