mtal
Jun28-11, 04:23 AM
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 y_{new} = y_1, \ldots, y_N of N cars, finding the covariance matrix based on the predictions for the given set of cars.
The model is set up in the following way:
y_t = X_t \alpha_t + \epsilon_t
is the observation process, and
\alpha_t = T\alpha_{t-1} + \nu_t
is the state process. Here T is the identity matrix and the error terms, \epsilon, \nu are uncorrelated mean zero processes, constant over time.
From what I know, prediction variance for a single y_{new} is calculated as
Var(y_{\textrm{pred}}) = \sigma^2(X_t P_t X_t^T + 1)
where \sigma^2 is the model variance and P_t is the covariance matrix for X_t , which is the characteristic coefficient matrix.
But if I have multiple cars, y_{new} = y_1, \ldots, y_N, is it correct that the corresponding covariance matrix would be calculated as
Var(y_{\textrm{new}}) = \sigma^2(X_t P_t X_t^T + I)
where I is a N\times N identity matrix, or am I missing something?
Help much appreciated!
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 y_{new} = y_1, \ldots, y_N of N cars, finding the covariance matrix based on the predictions for the given set of cars.
The model is set up in the following way:
y_t = X_t \alpha_t + \epsilon_t
is the observation process, and
\alpha_t = T\alpha_{t-1} + \nu_t
is the state process. Here T is the identity matrix and the error terms, \epsilon, \nu are uncorrelated mean zero processes, constant over time.
From what I know, prediction variance for a single y_{new} is calculated as
Var(y_{\textrm{pred}}) = \sigma^2(X_t P_t X_t^T + 1)
where \sigma^2 is the model variance and P_t is the covariance matrix for X_t , which is the characteristic coefficient matrix.
But if I have multiple cars, y_{new} = y_1, \ldots, y_N, is it correct that the corresponding covariance matrix would be calculated as
Var(y_{\textrm{new}}) = \sigma^2(X_t P_t X_t^T + I)
where I is a N\times N identity matrix, or am I missing something?
Help much appreciated!