Multiple Behavior Models

Human behavior, in all but the simplest tasks, is not as simple as a single dynamic model. The next most complex model of human behavior is to have several alternative models of the person's dynamics, one for each class of response. Then at each instant we can make observations of the person's state, decide which model applies, and then use that model for estimation. This is known as the multiple model or generalized likelihood approach, and produces a generalized maximum likelihood estimate of the current and future values of the state variables [20]. Moreover, the cost of the Kalman filter calculations is sufficiently small to make the approach quite practical.

Intuitively, this solution breaks the person's overall behavior down into several ``prototypical'' behaviors. For instance, we might have dynamic models corresponding to a relaxed state, a very ``tight'' state, and so forth. We then classify the behavior by determining which model best fits the observations.

Mathematically, this is accomplished by evaluating a dynamic model in the form of a Kalman filter:

$$\hat{\bf X}_k = {\bf X}_k + {\bf K}_k({\bf Y}_k - {\bf h}({\bf X}^{*}_k, t))$$ (12)

The measurement innovations process for the model (and associated Kalman filter) is then

$$\Gamma_k={\bf Y}_k - {\bf h}({\bf X}^{*}_k, t)$$ (13)

The measurement innovations process is zero-mean with covariance ${\cal R}$.

The measurement innovations process is, intuitively, the part of the observation data that is unexplained by the dynamic model. The behavior model that explains the largest portion of the observations is, of course, the model most likely to be correct. Thus, at each time step, we calculate the probability Pr⁽ⁱ⁾ of the m-dimensional observations ${\bf Y}_k$ given the i^th model using Equation 2 and choose the model with the largest probability. This model is then used to estimate the current value of the state variables, to predict their future values, and to choose among alternative responses.

Note that when optimizing predictions of measurements $\Delta t$ in the future, Equation 13 must be modified slightly to test the predictive accuracy of state estimates from $\Delta t$ in the past.