Maximum-Likelihood

Maximum-likelihood estimation is used extensively in the statistical analysis of traffic data. The idea underlying maximum-likelihood estimation is that different statistical distributions generate different samples, and any one sample is more likely to come from some distributions than from others.

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Y1, Y2, . . . , Y6, and there are two possible distributions that could generate these numbers.

It is clear from Fig. 8B.1 that distribution A is much more likely to generate these six numbers than distribution ′. The objective of maximum-likelihood estimation is to estimate a coefficient vector, say B, that defines a distribution that is most likely to generate some observed data. To show how this is done, consider the Poisson regression of trip generation discussed in Section 8.4.2. The maximum-likelihood function can be written as a simple product of the probabilities of a Poisson distribution with coefficients B generating observed household trip generation. This is, for a given trip type,

( ) ( )i i

L P T= ∏B (8B.1)

where

B = vector of estimable coefficients,

L(B) = likelihood function,

Ti = number of vehicle-based trips of a specific type (shopping, social/recreational, etc.) in some specified time period by household i, and

P(Ti) = probability of household i making T trips.