Visualization of the data set used – spanning over 1300 trips and 2 million records

Predicting Map-Matching Values using GPS Data from Navigation Software

Objective:• Map-Matching Values, or SnapWeights, are a measure of data point error. • Our aim is to model and predict this based on variables such as position,

velocity, and heading

Approaches:• Linear Regression of logit-transformed response with I.I.D. data assumed• Beta Regression of Independent responses that are Beta-Distributed with

different parameters• AR model, treating data as a Time Series.

Illustration of Data:

Overview

Time Series Models: AR/MA

• These models assume a linear dependence on previous data points and require stationarity and ergodicity

• AR(p): • MA(q): where all are White Noise error terms

• Different distributions for different trips. Generally appear stationary with no clear trend or seasonality.• Auto-correlations plotted. Generally slow decay indicates low probability of AR/MA process• Similarly, not likely high value in trying combination model ARMA

Linear Regression of Transformed SnapWeight

• Response variable transformed using a Logit Function and then linearly regressed on covariates such as speed, bearing, and acceleration

where

• To uphold I.I.D assumption, trips with similar distributions are to be combined• Similarity of distributions determined empirically using either Kolmogorov-Smirnov test or

Mutual Information function (less rigorous but so far less implementation problems)

• Not many covariates. Just a matter of training and then validating an assortment of covariate combinations using OLS and regularized methods such as ridge regression

• Initial runs on single trip data sets give very low prediction error values (R-squared)

Beta RegressionMotivation: • The SnapWeight is an indication of error rate and is constrained between 0 and 1.

We would like to respect this boundary.

Assumption:

Beta distribution, parameterized by mean (μ) and precision (ϕ):

Regression based on these assumptions:

Link Function (e.g. Logit, Cauchy,..)

Beta Regression

• Current Problems:– Huge dataset impedes the use of Summary()– Not sure about the effect of IDs and whether it’s necessary to separate them

beta_logit betareg(formula = SnapWeight ~ Speedx10, data = data.train.small)

Coefficients in two cases of regression formula:(Intercept) Speedx10 Headingx10 1.170e+00 1.034e-04 -4.888e-05

(Intercept) Speedx10 1.0773586 0.0001835

What’s Next

• Fit AR/MA models to many IDs and find most common p and q. Then validate using validation data set.

• Regress on larger aggregated data sets and then validate.• Do beta regression with the assumption of non-identical precision parameters, also

different link functions• Eliminate incomplete/irrelevant data from corpus, based on both intuition and regression

results• Compare different test values• Pick most likely models and apply to untouched Test data.

References

• [1] René A. Carmona. Statistical Analysis of Financial Data in S-PLUS. Springer, 2004.

• [2] Cribari-Neto, Francisco and Zeileis, Achim (2009) Beta Regression in R. Research Report Series / Department of Statistics and Mathematics, 98. Department of Statistics and Mathematics x, WU Vienna University of Economics and Business, Vienna.

• [3] Ferrari, Silvia and Cribari-Neto, Francisco (2004) Beta Regression for Modeling Rates and Proportions. Journal of Applied Sciences, Volume 31, Issue 7.