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If the goal is anomaly detection on time series data, then over-fitting my forecast model on the data is just what I want. If the historical data is fit really well, then I should expect some error from this over-fit model to be large enough to signal an anomaly. Is this true? If not, why so?

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No. You can easily overfit without any presence of anomalies in the series. Think of FFT, if you don't smooth it, then the output is a perfectly overfitted series.

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