differensering

Differensering, or differencing, is a time series transformation that replaces a sequence with the differences between consecutive observations. Its main purpose is to remove or reduce non-stationarity arising from trends or seasonality, making the data better suited for statistical modeling that assumes a stationary process.

In first-order differencing, the differenced series is defined as y_t' = y_t − y_{t−1}. This operation often removes a linear trend and yields a series whose statistical properties are more stable over time. The differenced data represent changes from one period to the next rather than the original levels, which changes the interpretation of any subsequent model.

Seasonal differencing targets recurring seasonal patterns. For a seasonal period s, the differenced series is y_t' = y_t − y_{t−s}. This can remove regular seasonal effects, such as monthly or quarterly cycles, when present. Higher-order differencing, such as second-order (differences of differences), is possible but used with caution, as it can over-correct the series.

In practice, differencing is a standard step in autoregressive integrated moving average (ARIMA) modeling. After fitting a model to the differenced (stationary) series, forecasts are transformed back to the original scale, typically by cumulative summation, to yield predictions in levels. While effective, differencing can reduce interpretability and, if over-applied, distort relationships or inflate noise. Analysts often balance differencing with other approaches like detrending or seasonal adjustment and rely on unit-root tests and diagnostic checks to decide the appropriate order.