Time Series

Time Series#

Time-series methods apply to anything indexed by time: telemetry, market data, sensor streams, log counts, request rates. The operator’s question is usually one of three: what is the trend, is this point unusual, what will happen next.

Decomposition#

Every time series y(t) splits into:

  • Trend, the long-run direction.

  • Seasonality, periodic component (daily, weekly, yearly).

  • Cycle, longer non-seasonal periodicity (business cycle).

  • Noise / residual, what’s left.

The standard decompositions:

Method

Detail

Additive vs multiplicative

y = trend + season + noise vs y = trend * season * noise. Pick multiplicative when amplitude grows with level.

STL (Seasonal-Trend by Loess)

The default robust decomposition. Handles missing data and outliers.

X-13ARIMA-SEATS

The reference for economic / official statistics.

Smoothing#

Method

Detail

Moving average (SMA)

Mean over a sliding window. Lags by half the window.

Exponentially weighted moving average (EWMA)

Recent weights heavier; one parameter (alpha).

Holt-Winters / triple exponential smoothing

EWMA plus trend plus seasonality. The classic operator default for short forecasts.

LOWESS / LOESS

Local regression. Non-parametric, smooth, no period assumption.

Savitzky-Golay

Polynomial fit over a sliding window. Preserves peak shapes.

Kalman filter

State-space smoother; see Probabilistic.

Stationarity#

A series is stationary if its statistical properties (mean, variance, autocorrelation) don’t change with time. Many models (ARIMA) assume stationarity; the operator checks first.

  • ADF (Augmented Dickey-Fuller) test, null hypothesis is non-stationary. Low p-value => reject => stationary.

  • KPSS test, opposite null. Pair with ADF.

  • Differencing, y'(t) = y(t) - y(t-1). Removes a trend.

  • Log transform, removes multiplicative growth.

Forecasting#

Raw signal, the smoothed trend the operator fits on history, and the forecast horizon extending past the last observation.

        xychart-beta
    title "Trend + seasonality + forecast (illustrative)"
    x-axis [t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12]
    y-axis "value" 0 --> 120
    line [42, 58, 51, 68, 74, 65, 82, 88, 80, 95, 102, 94]
    line [45, 52, 58, 64, 70, 76, 82, 88, 94, 100, 106, 112]
    line [0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 108, 116]

Method

Use

Naive / seasonal naive

Last value or last seasonal value. The baseline every model must beat.

Holt-Winters

Short horizon, strong seasonality, no exogenous inputs. Cheap.

ARIMA / SARIMA

Autoregressive Integrated Moving Average. The classical model. SARIMA adds seasonality.

VAR

Vector AR. Multivariate time series with cross-dependencies.

State-space (Kalman, BSTS)

Latent-state models. Handles missing data and structural breaks.

Prophet

Facebook’s piecewise-linear trend + Fourier seasonality + holidays. Easy to use, opinionated.

Exponential smoothing state-space (ETS)

Modern Holt-Winters with information criteria for model selection.

Gradient-boosted trees (LightGBM, XGBoost)

Tabular forecasting with lag features. Often beats classical methods on rich feature sets.

Deep learning (N-BEATS, TFT, DeepAR, transformers)

Long-range forecasting at scale, when training data is abundant.

ARIMA in operator practice#

ARIMA(p, d, q): p is autoregressive order, d is the number of differences to make the series stationary, q is the moving-average order. Read ACF and PACF plots to pick p and q; or let auto.arima / pmdarima.auto_arima search.

import pmdarima as pm

model = pm.auto_arima(y, seasonal=True, m=24)        # m=24 for hourly daily seasonality
fcst, ci = model.predict(n_periods=72, return_conf_int=True)

Anomaly in time series#

Two flavours:

  • Point anomalies, one observation deviates strongly. Detect with z-score on the residuals after smoothing or decomposition; with isolation forest on lag-windowed features; with prediction-residual thresholds.

  • Change points, the data-generating process itself changes. Detect with CUSUM, Bayesian online change-point detection, PELT (ruptures library), or windowed two-sample tests.

For deep treatment see Anomaly Detection.

Implementations#

Tool

Detail

statsmodels

Python’s reference. ARIMA, SARIMAX, VAR, ETS, STL.

pmdarima

Auto-ARIMA in Python.

prophet

Facebook’s package. Easy default for series with strong seasonality.

tslearn / sktime / Darts

ML-shaped APIs over time-series problems.

Pandas

rolling, resample, shift, diff. The substrate.

Pitfalls#

  • Leakage in feature engineering. Future information must not enter past features.

  • Backtesting with proper time-series cross-validation; random k-fold splits are wrong.

  • Time-zone confusion. Always store UTC in storage; convert at the display boundary.

  • Holiday and calendar effects dominate many business series; build them in explicitly.

References#

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