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BUG: ETS multiplicative error models log-likelihood calculation #9137

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BUG: ETS multiplicative error models log-likelihood calculation #9137

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626bcb3
fixed likelihood overwrite for negative in-sample
ltsaprounis Jan 31, 2024
f7f8e18
update comment and change to -np.inf
ltsaprounis Jan 31, 2024
a1031c3
abs for yhat
ltsaprounis Feb 1, 2024
834b0e4
replace zeros with 10^-18
ltsaprounis Feb 1, 2024
cb4602a
change to 10^-32
ltsaprounis Feb 2, 2024
da2daf0
update comment
ltsaprounis Feb 2, 2024
1f905b0
update comment for multiplicative error models
ltsaprounis Mar 19, 2024
370c013
remove trailing whitespace
ltsaprounis Mar 19, 2024
97fc786
add missing parenthesis
ltsaprounis Mar 19, 2024
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15 changes: 8 additions & 7 deletions statsmodels/tsa/exponential_smoothing/ets.py
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Original file line number Diff line number Diff line change
Expand Up @@ -1160,13 +1160,14 @@ def _loglike_internal(
res = self._residuals(yhat, data=data)
logL = -self.nobs / 2 * (np.log(2 * np.pi * np.mean(res ** 2)) + 1)
if self.error == "mul":
# GH-7331: in some cases, yhat can become negative, so that a
# multiplicative model is no longer well-defined. To avoid these
# parameterizations, we clip negative values to very small positive
# values so that the log-transformation yields very large negative
# values.
yhat[yhat <= 0] = 1 / (1e-8 * (1 + np.abs(yhat[yhat <= 0])))
logL -= np.sum(np.log(yhat))
# In some cases, yhat can become negative or zero, so that a
# multiplicative model is no longer well-defined. Zero values
# are replaced with 10^-32 (a very small number). For more
# information on the derivation of the log-likelihood for the
# multiplicative error models see:
# https://openforecast.org/adam/ADAMETSEstimationLikelihood.html
yhat[yhat == 0] = 1e-32
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Really should leave the <=. == 0 is likely too strict, and if it really is ==0, then <=0 will catch all cases.

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Fun corner cases. FWIW, I don't think you need the clipping at all anymore. It's almost certainly not being triggered here, and if you do switch it to clip the negatives, then it becomes unstable again.

logL -= np.sum(np.log(np.abs(yhat)))
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I don't think reflecting negatives is a good idea. Better to replace with small numbers.

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Please note that the absolute value is not there to avoid negative numbers, but instead arises mathematically:

If $\epsilon_t \sim \mathcal{N}(0, \sigma^2)$ then $y_t = \mu_{y,t}(1+\epsilon_t) \sim \mathcal{N}(\mu_{y,t}, \mu_{y,t}^2 \sigma^2)$. We then end up with the following log-likelihood:

$\ell(\boldsymbol{\theta}, {\sigma}^2 | \mathbf{y}) = -\frac{1}{2} \sum_{t=1}^T \log(2 \pi \mu_{y,t}^2 \sigma^2) -\frac{1}{2} \sum_{t=1}^T \frac{\epsilon_t^2}{\sigma^2}$ . As you see, we have $\mu_{y,t}^2$ in the formula above, which after simplifications can be written as: $-\frac{T}{2} \log(2 \pi \sigma^2) -\frac{1}{2} \sum_{t=1}^T \frac{\epsilon_t^2}{\sigma^2} - \sum_{t=1}^T \log |\mu_{y,t}|$.

So, it is necessary to have it there for the correct calculation of the log-likelihood. The derivations are based on the Section 11.1 here: https://openforecast.org/adam/ADAMETSEstimationLikelihood.html

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Thanks for reviewing the PR @bashtage and thanks for the additional explanation on the likelihood derivation @config-i1.

Based on @config-i1's explanation and the other implementations, I think it makes sense to keep the np.abs(yhat) do you agree @bashtage?

return logL

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