Estimating a mixed-profile MDCEV: case of daily activity type and duration

Abstract

Multiple Discrete Continuous Extreme Value (MDCEV) has become popular in the past years. Yet, the model suffers from an ‘empirical identification’ issue that is mainly due to inter-relations between two of its parameters, α and γ. This paper presents a hybrid optimization paradigm (named HELPME) to address this issue in a basic MDCEV formulation and take full advantage of the model by estimating a ‘mixed-profile.’ HELPME benefits from a coarse-to-fine search strategy, in which a customized Electromagnetism-like meta-heuristic precedes a gradient-based approach. The Atlanta Regional Travel Survey (2011) is used to empirically analyze performance of HELPME as well as significance of the accuracy gap between the mixed-profile, and α and γ profiles. As part of the results, it is observed that in-sample fit is significantly improved, percentage error of out-of-sample prediction is reduced up to 97% in a 90% confidence level, and bias of out-of-sample predictions are reduced up to 67%.

Keywords:

• Maximum-likelihood estimation
• Electromagnetism
• MDCEV
• HELPME

Acknowledgments

We appreciate Professor Chandra Bhat for sharing source code of the MDCEV model and its documentations. We would also like to thank three unanimous reviewers whose comments have helped a lot to strengthen arguments within the paper, as well as directions for future studies.

Notes

1. By definition, two unknown parameters of a model are theoretically unidentifiable if two distinct combinations of their values could be found, such that they both result in equal (not just similar) distributions of the outcome variable (Vij and Walker 2014 and Walker et al. 2007). For instance, variance of error term (σ) and coefficients of attributes (β) in an MNL formulation are known to be theoretically unidentifiable (Train 2009). That is, there is no way to estimate them both in that the probability distributions are shown to be functions of β/σ rather than β and σ.

2. In a hypothetical situation of α _k = 1 for all k, therefore, the model collapses to a simple MNL (Bhat 2008).

3. Available at https://www.dropbox.com/sh/lhq4cirywuv6f2p/AABpvZJ9C6bvOZNggyUrkf31a?dl = 0.

4. Zhang et al. (2013) evidenced that accuracy of EML is highly sensitive to the relative scale of a problem’s dimensions. The normalization step proposed in the present paper is intended to dampen the scale differences, especially between β parameters from one side, and α and γ parameters from the other.

5. Stubborn Movement is a simple direct-search routine developed as part of HELPME. see Section 3.4.

6. Synthetic data fail to truthfully represent real-life situations and, thereby, undermine practical findings such as magnitude of change in likelihood value, run time, and prediction power.

7. The algorithm is originally proposed to predict within a γ-profile framework. Pinjari and Bhat (2010b), also, suggest modifications using the bisection technique to make the algorithm compatible with cases where α _k and γ _k are both subject to change. We use the modified version for prediction of a mixed-profile. Instead of the bisection algorithm, though, the Newton’s algorithm is used to eliminate the need for defining proper upper and lower bounds for λ. Also, the λ from previous iterations is used as an initial value inputted to each iteration.

8. Previous studies have employed different approaches. Bhat et al. (2013), for instance, adopted the Fractional Split model proposed in Sivakumar and Bhat (2002).