Addendum: Predictive form of the FPM model

The article Oustaloup et al. (2021) has shown that the Fractional Power Model (FPM), $A+B{t}^m$, enables well representing the cumulated data of COVID infections, thanks to a nonlinear identification technique. Beyond this identification interval, the article has also shown that the model enables predicting the future values on an unusual prediction horizon as for its range. The objective of this addendum is to explain, via an autoregressive form, why this model intrinsically benefits from such a predictivity property, the idea being to show the interest of the FPM model by highlighting its predictive specificity, inherent to non-integer integration that conditions the model. More precisely, this addendum establishes a predictive form with long memory of the FPM model. This form corresponds to an autoregressive (AR) filter of infinite order. Taking into account the whole past through an indefinite linear combination of past values, a first predictive form, said to be with long memory, results from an approach using one of the formulations of non-integer differentiation. Actually, as this first predictive form is the one of the power-law, $t^m$, its adaptation to the FPM model, $A+B{t}^m$, which generalizes the linear regression, $A+Bt$, is then straightforward: it leads to the predictive form of the FPM model that specifies the model in prediction. This predictive form with long memory shows that the predictivity of the FPM model is such that any predicted value takes into account the whole past, according to a weighted sum of all the past values. These values are taken into account through weighting coefficients, that, for $m>-1$ and a fortiori for $m>0$, correspond to an attenuation of the past, that the non-integer power, $m$, determines by itself. To confirm the specificity of the FPM model in considering the past, this model is compared with a model of another nature, also having three parameters, namely an exponential model (Liu et al. (2020); Sallahi et al. (2021)): whereas, for the FPM model, the past is taken into account globally through all past instants, for the exponential model, the past is taken into account only locally through one single past instant, the predictive form of the model having a short memory and corresponding to an AR filter of order 1. Comparative results, obtained in prediction for these two models, show the predictive interest of the FPM model.