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Prediction covariance matrix with Kalman filter

  1. Jun 28, 2011 #1
    Hello all.

    I have set up a model using the Kalman filter to estimate automobile prices. I'm having difficulty in figuring out how to formulate a prediction covariance matrix based on the model, i.e. given a set [itex] y_{new} = y_1, \ldots, y_N[/itex] of [itex]N[/itex] cars, finding the covariance matrix based on the predictions for the given set of cars.

    The model is set up in the following way:
    [tex] y_t = X_t \alpha_t + \epsilon_t [/tex]
    is the observation process, and
    [tex] \alpha_t = T\alpha_{t-1} + \nu_t [/tex]
    is the state process. Here [itex]T[/itex] is the identity matrix and the error terms, [itex] \epsilon, \nu[/itex] are uncorrelated mean zero processes, constant over time.

    From what I know, prediction variance for a single [itex]y_{new}[/itex] is calculated as
    [tex] Var(y_{\textrm{pred}}) = \sigma^2(X_t P_t X_t^T + 1) [/tex]
    where [itex]\sigma^2[/itex] is the model variance and [itex] P_t [/itex] is the covariance matrix for [itex] X_t [/itex], which is the characteristic coefficient matrix.

    But if I have multiple cars, [itex] y_{new} = y_1, \ldots, y_N[/itex], is it correct that the corresponding covariance matrix would be calculated as
    [tex] Var(y_{\textrm{new}}) = \sigma^2(X_t P_t X_t^T + I) [/tex]
    where [itex]I[/itex] is a [itex]N\times N[/itex] identity matrix, or am I missing something?

    Help much appreciated!
  2. jcsd
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