differencing

Differencing is a transformation applied to a sequence of data points that creates a new series of differences between consecutive observations. The simplest form is the first difference: Δy_t = y_t − y_{t−1}. Repeated application yields higher-order differences, such as Δ^2 y_t = Δ(Δy_t). Differencing is commonly used to remove trends or seasonality and to stabilize the mean of a time series.

In time series analysis, differencing is used to achieve stationarity, a condition in which statistical properties do not depend on time. The order of differencing, d, indicates how many times the operation is applied. The first difference often removes linear trends; seasonal differencing with lag s, Δ_s y_t = y_t − y_{t−s}, targets repeating seasonal patterns. The goal is to apply enough differencing to obtain stationarity without over-differencing, which can obscure structure or induce spurious dynamics.

In numerical analysis, differencing denotes finite difference schemes for estimating derivatives. Forward difference approximates f'(x) by [f(x+h) − f(x)]/h; backward difference uses [f(x) − f(x−h)]/h; central difference uses [f(x+h) − f(x−h)]/(2h). These approximations underpin numerical differentiation and are also used to discretize differential equations in finite difference methods, offering various orders of accuracy depending on h and the scheme.

In statistical practice, differencing is used with unit root tests (such as the augmented Dickey-Fuller test) and in model specification to ensure residuals are stationary. Caution is advised: too much differencing can remove meaningful long-run information and distort relationships; diagnostics should guide the choice of order.