The Cube Analyst Suite

1. Conceptual Differences Between Cube Analyst and Cube Analyst Drive Austen C. Duffy, Ph.D. Computational Mathematician, Citilabs

2. The Cube Analyst Suite • The Cube Analyst Suite consists of two OD matrix estimation programs, each possessing some unique features and capabilities. • Cube Analyst- The original matrix estimation program capable of a wide variety of static type estimation problems. • Cube Analyst Drive – The new matrix estimation program designed for dynamic estimation, but possessing some nice static estimation features as well.

3. Matrix Estimation w/ the Analyst Suite

4. Cube Analyst vs. Cube Analyst Drive: The Basics Cube Analyst Cube Analyst Drive Methodology: Estimation Based on Data Assimilation Techniques Problem Types: Dynamic Matrices and Large Static Matrices Supporting Programs: Highway, Avenue, any dynamic assignment program • Methodology: Statistics Based Estimation using Maximum Likelihood Method • Problem Types: Small Static Matrices, Public Transit • Supporting Programs: Highway, PT, TRIPS- MVESTE, MVESTL, etc.

5. Maximum Likelihood Method (Analyst) • The maximum likelihood method applies a statistical model utilizing maximum likelihood estimators (MLE’s) to a given data set (e.g. observed counts) to provide estimates for model parameters (e.g. OD matrix) that maximize a likelihood function. • “How likely is it that our OD matrix matches the observed count data”

6. Data Assimilation (Analyst Drive) • In Data Assimilation, we seek to minimize the squared error between a forecasted analysis (e.g. simulated volume) and a set of data observations (e.g. traffic counts) weighted by the reliability of the observations (e.g. confidence values), while simultaneously seeking to preserve a defined structure within our background solution (e.g. Input OD matrix).

7. Cube Analyst vs. Cube Analyst Drive: Mathematical Models Cube Analyst Cube Analyst Drive Analyst Drive’s primary objective aims to minimize the distance between the simulated volume AX and the count data b… i.e we wish to find a solution where AX-b=0 Secondary objective: Preserve the background matrix structure • Analyst Applies the maximum likelihood method. • Solution estimates obtained via likelihood functions represented by PDF’s • Essentially minimizes a sum of logarithmic function terms representing screenline counts, trip origins, trip destinations and prior matrix along with a term for the derived cost matrix • Derivation and details available in the Cube Analyst Reference Guide

8. Cube Analyst Drive: Static EstimationSingle Class Estimation Problem (Multiple Class in Parallel)

9. Cube Analyst vs. Cube Analyst Drive: Multiple Static User Classes Cube Analyst Cube Analyst Drive Multiclass static and dynamic estimation capabilities Static capabilities allow users to estimate multiple classes from fewer count sets New file format allows for multiple user class data to be stored in a single ICP file, requiring only a single Highway run. • Analyst does not allow the user to estimate Multiple User Classes • Must perform a separate Analyst run for each user class • May require multiple Highway runs to produce individual files (e.g. ICP) for each class

10. Estimating Multiple User Classes with Fewer Count Sets in Analyst Drive Screenline Preprocessing Aggregate Estimation Volume contributions from multiple class sets are summed and matched against a single count set Numerically Inferior (fewer data points) More theoretically sound as it does not require any assumptions on end behavior • Given multiple matrices to estimate with fewer count sets, Analyst Drive will preprocess the input screenline files to produce an individual count set for each, and run the single class estimation problem in parallel • Numerically Superior (more data points) • Requires The assumption that the proportionality of class counts does not change with the evolving matrices

11. Cube Analyst Drive: Static EstimationAggregate Multiple Class Estimation Problem

12. Cube Analyst Drive: Dynamic EstimationSingle Class Estimation Problem (Multiple Class in Parallel)

13. Differences In The Use Of Confidence Values Analyst Analyst Drive Uses Confidences values to determine the weight a particular value contributes to the problem. → Less Reliable data contributes less to the cost function, and hence is not as important in the estimation. • Uses confidence values to statistically provide value ranges for an estimated entry. • → Less Reliable Data Provides for a larger variance and hence a wider range of possible values.

14. Cube Analyst vs. Cube Analyst Drive: Numerical Methods Cube Analyst Cube Analyst Drive Objective Function is minimized via a conjugate gradient method which does not require a Hessian matrix Parallel Computation at various levels • Objective Function is minimized using the quasi-Newton BFGS method which requires an approximation to the Hessian matrix (expensive to compute, large storage penalty) • Sequential Computation

15. Cube Analyst vs. Cube Analyst Drive: Capabilities

16. Cube Analyst vs. Cube Analyst Drive: Capabilities Cont.

17. Analyst vs. Analyst Drive: Files • Analyst Drive does not accept all files that the original Analyst does • New Format ICP files developed for Analyst Drive allow for multiple user classes and provide up to 80% space savings • Matrix files contain multiple tabs for multiple user classes (static) and multiple time periods (dynamic)

18. Cube Analyst vs. Cube Analyst Drive: Block Diagrams Cube Analyst Cube Analyst Drive

19. Cube File Compatibility

20. Cube File Compatibility Cont.: Analyst Specific Files

21. Cube File Compatibility Cont.: Analyst Drive Specific Files

22. So How Do The Solutions Compare? • Both Analyst and Analyst Drive should produce similar solutions, but will not be identical. • Because Analyst is statistics based, it is able to provide a solution within a range of values (similar to a confidence interval) w/ sensitivities. • Analyst Drive is better able to preserve the structure of the background matrix.

23. Analyst vs. Analyst Drive Use Summary Analyst Analyst Drive Dynamic Matrices Large Static Matrices Multiple Class Estimation FAST! Data Assimilation Method: Better at producing reliable matrices which preserve the defining structure of the background matrix. • Good for small static problems with a wide range of input options • Public Transit Networks. • Maximum Likelihood Method: Better at using poor input to generate reasonable matrices ‘from scratch’.

24. References • Cube Analyst Drive Reference Guide - Version 1.01.12 • Cube Analyst Reference Guide – Version 6.1.0

25. Where to Find Reference Guides

26. Questions?