Add num_steps_per_season parameter in TimeSeasonality #509
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Awesome start! You have exactly the right idea. I think we can be a bit more clever with the math is all.
For the actual implementation, I don't think we can do anything clever -- we'll need to expand the transition matrix to s * d and repeat the rows (and also repeat the starting values). The replicates should all be from the same symbolic value, so the user still just provides s inputs.
| .. math:: | ||
| \underbrace{\gamma_0, \gamma_0, \ldots, \gamma_0}_{r\ \text{times}}, | ||
| \underbrace{\gamma_1, \gamma_1, \ldots, \gamma_1}_{r\ \text{times}}, | ||
| \ldots, | ||
| \underbrace{\gamma_{s-1}, \gamma_{s-1}, \ldots, \gamma_{s-1}}_{r\ \text{times}}, | ||
| \ldots, |
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This is correct, but this can be written more nicely using a floor operator on the index:
$
\gamma_t = -\sum_{i=1}^{s-1} \gamma_{\left\lfloor \frac{t}{d} \right\rfloor - i} + \omega_{\left\lfloor \frac{t}{d} \right\rfloor}, \quad \omega_j \sim \mathcal{N}(0, \sigma_\gamma)
$
I suggest to use d for "duration" instead of r for the number of steps, but I'm fine with being overruled if you have a strong opinion.
| To give interpretation to the :math:`\gamma` terms, it is helpful to work through the algebra for a simple | ||
| example. Let :math:`s=4`, and omit the shock term. Define initial conditions :math:`\gamma_0, \gamma_{-1}, | ||
| example. Let :math:`s=4`, :math:`r=1`, and omit the shock term. Define initial conditions :math:`\gamma_0, \gamma_{-1}, | ||
| \gamma_{-2}`. The value of the seasonal component for the first 5 timesteps will be: | ||
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| .. math:: |
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Also update this block with a second example that has s=3 and d=2. Show that the floor math works out and you get the expected pattern.
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