This method is also known as Holt's method, after Charles C. Holt and his paper from 1957. It's called double exponential smoothing because it's based on two smoothing parameters -- Alpha (for level) and Beta (for trend). The algorithm solves the primary issue of simple exponential smoothing, as now the forecasts can account for the trend in historical data. The ExponentialSmoothing class from the Statsmodels library accepts a trend parameter, with the possible values being add and mul. If you can't decide which type of trend is present in your data, test both and see which results in the lowest error.

We consider partially observable Markov decision processes (POMDPs) with a set of target states and every transition is associated with an integer cost. The optimization objective we study asks to minimize the expected total cost till the target set is reached, while ensuring that the target set is reached almost-surely (with probability 1). We show that for integer costs approximating the optimal cost is undecidable. For positive costs, our results are as follows: (i) we establish matching lower and upper bounds for the optimal cost and the bound is double exponential; (ii) we show that the problem of approximating the optimal cost is decidable and present approximation algorithms developing on the existing algorithms for POMDPs with finite-horizon objectives. While the worst-case running time of our algorithm is double exponential, we present efficient stopping criteria for the algorithm and show experimentally that it performs well in many examples of interest.