Add local level example

[victorgarcia98]

To get familiar with the state space description, I wanted to implement the most simple SS model: the local level. Moreover, I show the StatsModels equivalent model next to it.

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[AlexAndorra]

Thanks @victorgarcia98 ! Just left a few comments in Review NB. A broader question is: what do you envision this example for? Which use-cases?
Also, be careful because you also edited pymc_extras/statespace/core/statespace.py, and it seems to be accidental as it's adding a fixed shape to parameters

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AlexAndorra commented on 2025-05-13T20:18:39Z
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Line #7.    mu = np.cumsum(rng.standard_normal(size=(100,1))*np.sqrt(true_var_nu**2), axis=0)
Line #8.    y = mu + rng.standard_normal(size=(100,1)) * np.sqrt(true_var_eta)

To follow your greek letter equations above, and since you're using the variances instead of the standard deviation, I think this should instead be:

Line #7.    mu = np.cumsum(rng.standard_normal(size=(100,1))*np.sqrt(true_var_eta), axis=0) # this is now eta, not nu, and not squared
Line #8.    y = mu + rng.standard_normal(size=(100,1)) * np.sqrt(true_var_nu) # this is now nu, not eta

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AlexAndorra commented on 2025-05-13T20:18:40Z
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Any reason you're rebuilding from scratch and not using the structural module?

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AlexAndorra commented on 2025-05-13T20:18:40Z
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Maybe overlay the reference value of the true params for var_eta and var_nu?