Tidserie

Tidserie, or time series, is a sequence of data points indexed in time, often measured at regular intervals such as seconds, minutes, days, or years. Each observation y_t represents the value of a variable at time t. Time series can also be irregular, but most common methods assume a consistent sampling frequency. The data are typically ordered so that temporal dependence may exist, meaning observations closer in time are often more related than distant ones.

A time series usually comprises several components: a long-term trend reflecting persistent increase or decrease, seasonal or periodic patterns repeating at fixed times of the year or other cycles, occasional cyclic fluctuations, and a residual or irregular component containing random variation and measurement error. These components can combine additively or multiplicatively, and various decomposition methods aim to separate them.

Modeling and forecasting rely on distinguishing stationary from non-stationary series. Stationarity implies constant mean and variance over time, which is important for many forecasting approaches. Differencing, detrending, or transformation can help achieve stationarity. Common forecasting methods include classical decomposition with Holt-Winters, exponential smoothing state-space models, ARIMA and SARIMA models, and more flexible approaches like Prophet or Kalman filter-based state-space models. Multivariate time series use models such as vector autoregression (VAR) to capture interdependencies between multiple series.

Practical considerations include handling missing data, irregular sampling, outliers, and structural breaks. Evaluation typically uses time-ordered training and test splits, cross-validation adapted to time series, and metrics such as MSE, RMSE, MAE, or MAPE. Time series are used across finance, economics, meteorology, energy, engineering and many scientific disciplines. Software implementations exist in R, Python, and other statistical packages.