[ENH] Add partial dataset of NAB, Bayes Online for Anomaly Detection, and testing example notebook

[duydl]

Reference Issues/PRs

#6167, #3214

What does this implement/fix? Explain your changes.

Introduce dataset from Numenta Anomaly Benchmark into sktime. Implement online anomaly detection algos on the dataset.

Does your contribution introduce a new dependency? If yes, which one?

No

What should a reviewer concentrate their feedback on?

Did you add any tests for the change?

PR checklist

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• The PR title starts with either [ENH], [MNT], [DOC], or [BUG]. [BUG] - bugfix, [MNT] - CI, test framework, [ENH] - adding or improving code, [DOC] - writing or improving documentation or docstrings.

For new estimators

• I've added the estimator to the API reference - in docs/source/api_reference/taskname.rst, follow the pattern.
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[duydl]

Bayes CPD for Anomaly:

Initialization

Initialize self.max_length_probs[0, 0] to 1:
$$P(r_0 = 0) = 1$$
This probability matrix will store the probabilities for each potential run length ( r ).

Sequential Update for Each Data Point ( x_t )

The algorithm processes and updates probabilities for all potential run lengths for each new data point.

2.1. Predictive Probability for Each Run Length:
$$P(x_t \mid r_{t-1}, x_{1:t-1})$$
Calculated with observation_likelihood. The likelihood of observing ( x_t ) given the data model parameters for a specific run length ( r_{t-1} ).

2.2. Update Run Length Probabilities:
$$P(r_t = r_{t-1} + 1 \mid x_{1:t}) = (1 - H(r_{t-1})) \times P(x_t \mid r_{t-1}, x_{1:t-1}) \times P(r_{t-1} \mid x_{1:t-1})$$
( H(r) ) is the hazard function i.e., the probability of a change point at each run length. This is the probability of not having a change point and the run length updated accordingly.

2.3. Probability of a Change Point:
$$P(r_t = 0 \mid x_{1:t}) = \sum_{r=0}^{max_run_length} H(r) \times P(x_t \mid r, x_{1:t-1}) \times P(r \mid x_{1:t-1})$$
This step sums the probabilities across all previous run lengths, weighted by the hazard function, to compute the likelihood that ( x_t ) is a change point.

2.4. Normalization and Transfer:

$$P(r_t \mid x_{1:t}) = \frac{P(r_t \mid x_{1:t})}{\sum_{j=0}^{max_run_length + 1} P(r_j \mid x_{1:t})}$$
After updating, the probabilities are normalized and transferred from ([:, 1]) back to ([:, 0]).

Iterating Over All Data Points

The algorithm repeats these steps for each new data point, continuously updating the probability distribution over potential run lengths and adapting to new evidence as it comes in.

[fkiraly]

the distribution exists already in sktime.proba.t, no?

On first glance, I would have modelled this as a distribution fitter, inheriting from the "parameter estimator" template in param_est. However, it does not fit entirely the interface, so we can leave it as is for now.