Faromoment

Faromoment is a theoretical construct in time-series analysis described as a forward-looking generalization of ordinary moments. It aims to quantify the influence of future values of a process by weighting forthcoming observations according to a specified kernel. The term blends “forward” and “moment” to reflect its emphasis on the forward horizon.

Definition and formulation: For a discrete-time sequence f_t, the faromoment of order p with a forward weighting kernel w_d (d ≥ 0) is FM_p(t) = sum_{d=0}^{∞} d^p w_d f_{t+d}, provided the sum converges. A common choice is exponential weighting w_d = (1−α) α^d with α in (0,1), yielding FM_p(t) = sum_{d=0}^{∞} d^p (1−α) α^d f_{t+d}. In continuous time, for a function f(t), the analogue is FM_p(t) = ∫_0^∞ τ^p w(τ) f(t+τ) dτ with w(τ) = λ e^{−λ τ}.

Special cases and properties: FM_0(t) reduces to a forward-average of f, while FM_1(t) captures a forward-looking first moment. The choice of horizon and decay, governed by α or λ, determines how strongly near-future versus far-future values influence the measure. Linear properties hold: FM_p is linear in f, and changing the kernel changes the weighting without altering the basic moment order p.

Applications and scope: The concept is proposed as a tool for forward-looking analysis in forecasting, risk assessment, and signal processing, offering a perspective that prioritizes upcoming values over historical data. It is not a standard measure in published literature and remains primarily a theoretical or exploratory construct.

See also: moments, moving average, exponential smoothing, forward-backward analysis.