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  1. When can we construct prediction interval and we can't: What factor govern this?

  2. One of the way to constrained forecast to an interval is using a log transformation. For ex: if we want forecast to be between (a, b) we would use y=log((x−a)/(b−x)) transformation.

What is the impact of this transformation on prediction intervals of final forecast?

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  1. The only situation I can think of where we can't construct a prediction interval would be when we have observed only a single historical data point, and don't have any contextual information. In this situation, we simply don't have any idea about the variability of the series, and can't say whether the next observation might by close to or far away from that single value.

    However, usually we have observed multiple actuals, and then we can indeed construct PIs. They may not be very useful or well-calibrated, but we can at least construct them. Similarly in situations where the focal time series is very short, but we have contextual information. For instance, if a product has been newly introduced, then we may indeed have only a single data point, but we typically know about the demand for similar products in the past and can at least say something about PIs for the new product.

  2. Since your transformation is monotonic, calculate PIs of the transformed unconstrained forecast and back-transform the PI endpoints. This will preserve the coverage probability. Now your only challenge is to get well-calibrated PIs on the transformed scale.

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