Fractional differentiation

An event classifier that should know how far a residual sits from equilibrium cannot be trained on a wandering level, and it also cannot be trained only on a one-step change if that level is the signal. Integer differencing is one end of a continuum, not a guarantee that the feature is stationary or that the feature process is Markov. This note derives the weights of $(1-L)^d$, the fixed-width truncation used in practice, and the limits of picking $d^*$ from an ADF grid, then freezes that choice inside the training window.

• derive the weight recursion for $(1-L)^d$ and compute the first weights;
• state what a first difference removes, and what it does not imply about stationarity or the Markov property;
• apply a fixed-width fractional difference to a short series;
• choose a candidate $d^*$ with an ADF grid and name the limitations of that choice;
• freeze the weights inside the training window of the event classifier.

1. Weights of $(1-L)^d$
2. What a first difference does not do
3. Fixed-width window
4. Choosing $d^*$ from an ADF grid
5. Feature policy for the event classifier