Predict Commuting Transportation Choice

• Introduction
• Dataset
• Bayesian Model Set Up
• Result and prediction
  • Posterior Distribution
  • Posterior prediction
• Acknowledgement

Introduction

Individuals choice of transportation methods, such as car, transit, rail, share-ridding etc, is related to various characteristic (cost of different transportation methods, distance, etc). The interest lies in the ability to predict individual’s choice of transportation methods based on their characteristic. This prediction task is equivalent as a binary or multicast classification problem. In this project, we create a Bayesian multi linear regression model to predict the individual choice of transportation methods. We create a hierarchical structure to allow randomness in the parameters with the idea to accommodate mechanisms in the underlying process.

Dataset

The dataset used in this project is a random sample of 543 individual workers in the Boston Metropolitan area. The prediction variable we use are

• TC_CAR: Travel cost associated with car
• TC_TR: Travel cost associated with public transit
• TT_TR: Total travel time in car
• OVTTRDI: Over vehicle transportation time in public transit per distance
• NUMVEH: Number of vehicle within household
• INCOME: Annual household income

The response variable we wish to predict with the model is

• Commuting choice: 1 for driving; 0 for public transit

We wish to ingestive the grouping effect of household income on family level prediction variables such as number of car own. The income group is set up with the following descriptive table. Low income $\leq$ \$50000; Medium income \$50001-\$1000000; High income $>$ \$100000.

Mode	Low	Medium	High
DA	103 (85.12 %)	149 (87.65 %)	217 (86.11 %)
TR	18 (14.88 %)	21 (12.35 %)	35 (13.89 %)

In the project, the number of vehicle are grouped with different income level. The income level is set from the annual income with Low income.

Bayesian Model Set Up

The proposed Bayesian Model start with the data distribution. I use a simple approach where the decision is given by Bernoulli distribution with probability of success $p_{ij}$

\[y_{ij}\sim\mathrm{Bern}(p_{ij}), \quad i=1,\cdots,N,\quad j=1, 2, 3.\]

Here, the $i$ stands for the index of each data (individual) and $j$ represent different income group. The success probability $p_{ij}$ is linked with regressive variables through a logit link function as

\[\begin{gather*} \mathrm{logit}(p_{ij})=\mu_{ij}\\ \mu_{ij}\sim\mathrm{N}(\mathbf{A}^T_{i}\boldsymbol{\alpha}+\mathbf{D}^T_{ij}\boldsymbol{\beta}_{j},\sigma^2). \end{gather*}\]

Noticed that a normal distribution is used to introduce the random ness in the general linear model (the larger the linear regression give only means the sucess probability will have higher chance to be larger). $\mathbf{A}^T_{i}$ is the matrix of covariates associated with the alternatives, whereas $\mathbf{D}^T_{ij}$ concerns the matrix of covariates associated with the individual’s characteristics, grouped by their respective income $j$. Notice that while the parameter $\alpha$ is fixed, whereas $\beta_{j}$ is allowed to vary based capturing the random effect of the income levels. This set up should allow us to investigate if the decision-maker’s parameters for the utility of commuting by car is affected differently according to their income. The prior distribution is assume to be as follow

\[\begin{gather*} \alpha_{l}\sim\text{N}(\mu_{\alpha,l},\sigma^2_{\alpha,l}), \quad l=1,...,L\\ \beta_{k,j}\sim\text{N}(\boldsymbol{\mu}_{\beta}, \boldsymbol{\sigma^2}_{\beta}),\quad k=1,...,K\\ \mu_{\alpha,l}\sim\text{N}(\mu_{0}, \sigma^2_{0})\\ \sigma^2_{\alpha}\sim\text{IG}(q_{\alpha},r_{\alpha})\\ \mu_{\beta,j}\sim\text{N}(\mu_{\beta_{0}}, \sigma^2_{\beta_{0}})\\ \mu_{\beta_{0}}\sim\text{N}(\mu_{0}, \sigma^2_{0})\\ \sigma^2_{\beta,j}\sim\text{IG}(q_{\beta},r_{\beta})\\ \sigma^2\sim\text{IG}(q,r),\\ \end{gather*}\]

To find posterior distribution, we set up Markov Chain Monte Carlo (MCMC) sampling using JAGS package. The MCMC is set with 50000 iterations, and burn-in period of 10000 iterations. The trace plots below demonstrate the convergence of the chain.

Result and prediction

Posterior Distribution

Using the MCMC sampling, we were able to obtained the posterior distribution for all the regressive parameters and the corresponding marginal distribution. The mean, standard deviation and 95% credible interval for each parameter is shown in the table below.

Description	Parameter	Estimate	Std.Error	2.5% - 97.5% CI
ASC	$\alpha_0$	3.347	0.878	[1.733, 5.184]
TC_CAR	$\alpha_1$	-1.828	0.300	[-2.508, -1.324]
TC_TR	$\alpha_2$	1.472	0.588	[0.375, 2.604]
TT_TR	$\alpha_3$	2.126	0.599	[1.117, 3.432]
OVTIRDI	$\alpha_4$	2.650	0.836	[1.159, 4.479]
NUMVEH_LOW	$\beta_{1,1}$	0.691	0.349	[0.032, 1.392]
NUMVEH_MED	$\beta_{2,1}$	0.791	0.312	[0.218, 1.432]
NUMVEH_HIGH	$\beta_{3,1}$	1.078	0.456	[0.359, 2.153]

The posterior distribution can also be visually inspect through violin plot.

Posterior prediction

Based on the MCMC, it is able to give posterior prediction result.

First we visualized the posterior prediction by plotting the prediction curves of the Bernoulli coefficient $p_i$ for three different income groups along with their corresponding 95\% CI. Notice that the posterior predictive mean probability of an individual choosing CAR mode aligns with the observed data distribution.

Due to the high-dimensional nature of the predictions, it is difficult to visualize change in the posterior predictive variable across the full range of predictor values. Therefore, the posterior predictive mean and CI were calculated along a specific line in the predictor space.

The Receiver Operating Characteristic (ROC) plot was used to evaluate the performance of the proposed Bayesian model. The ROC curve displays the True Positive Rate versus the False Positive Rate across various classification thresholds, where each threshold determines the class based on the predicted probability.

In a ROC plot, a perfect model is represented by a curve hugging the top-left corner, as it would achieve a True Positive Rate of 1 and a False Positive Rate of 0. A random model (i.e., guessing the outcome) would follow the diagonal line. Therefore, the closer the curve is to the top-left corner, the better the model’s performance. The ROC plot for the proposed model shows a curve close to the top-left corner, indicating good predictive ability.

More specifically, the model prediction reach accruacy of 93% accuracy when the threshold is 0.5 (i.e. $>$ 0.5 predict label as 1; $\leq$ 0.5 predict as 0).

Acknowledgement

We thank Danilo Inoue on collaborating this project.