Prediction covariance matrix with Kalman filter

[mtal]

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!

 

[mmwave]

Thank you for sharing your model and question with us. Based on the information provided, it seems like you are on the right track with your calculation of the prediction covariance matrix. The formula you have mentioned, Var(y_{\textrm{new}}) = \sigma^2(X_t P_t X_t^T + I), is correct for calculating the covariance matrix for multiple cars in your model. The identity matrix, in this case, represents the covariance between the different cars in your set y_{new}.

However, it is important to note that the accuracy of your predictions and the covariance matrix will depend on the assumptions and parameters used in your model. It may be helpful to review and validate these assumptions and parameters to ensure the accuracy and reliability of your results.

Additionally, it may be beneficial to consult with a statistician or another expert in the field to review your model and calculations and provide further insights and guidance.

Best of luck with your research!