[kalman] One comment on the code

[longye-tian]

Hi @jstac

I had a code-related question in the first Kalman filter lecture.

The exercise asks readers to compare

$$ | x_t - A x_{t-1} |^2 $$

with

$$ | x_t - \hat x_t |^2, $$

where $\hat x_t$ is the Kalman filter prediction of $x_t$.

In the solution, the loop currently does

for t in range(1, T):
    kn.update(y[:,t])
    diff1 = x[:, t] - kn.x_hat.flatten()
    diff2 = x[:, t] - A @ x[:, t-1]

The reason I am worried about the timing is that Kalman.update(y) appears to call both prior_to_filtered(y) and filtered_to_forecast(), according to the QuantEcon documentation:

def update(self, y):
        """
        Updates x_hat and Sigma given k x 1 ndarray y.  The full
        update, from one period to the next

        Parameters
        ----------
        y : np.ndarray
            A k x 1 ndarray y representing the current measurement

        """
        self.prior_to_filtered(y)
        self.filtered_to_forecast()

Since filtered_to_forecast() advances the filtering distribution to the next predictive distribution, kn.x_hat after kn.update(y[:, t]) seems to represent the next prior rather than the prediction being compared with $x_t$.

If that is right, then diff1 is comparing $x_t$ with a one-period-ahead object.

Would it be more consistent with the exercise to update the filter using the previous observation before comparing with $x_t$, e.g.

for t in range(1, T):
    kn.update(y[:, t-1])
    diff1 = x[:, t] - kn.x_hat.flatten()
    diff2 = x[:, t] - A @ x[:, t-1]

Then both errors are comparing predictions of $x_t$.

What do you think? I can put up a PR if this interpretation is correct.

Best,
Longye

[jstac]

Hi @longye-tian , good catch! Yes, please put up a PR with a fix.

Much appreciated!