Time Series Fundamentals

A time series is a sequence of observations indexed by time. Forecasting differs from ordinary supervised prediction because order is part of the data-generating process: nearby observations can be dependent, seasonal positions can repeat, and future information is unavailable when the forecast is made.

The basic object is usually written as $y_1,\dots ,y_T$, with forecasts ${\hat{y}}_{T+h|T}$ for horizon $h$ made using information available up to time $T$. That conditioning bar is the discipline behind the whole section. Any feature, split, scaling statistic, or target transformation that uses observations after $T$ has leaked future information.

Most forecasting problems combine several structures. Trend, seasonality, cycles, and noise describe broad visible patterns. Autocorrelation and partial autocorrelation describe lag dependence. Stationarity asks whether those relationships are stable enough for classical models. Forecasting problem formulation fixes the target, horizon, granularity, update cadence, and decision that the forecast supports.

The simplest useful baselines are often naive: forecast the last observed value, the same seasonal position, or a moving average. These baselines are not throwaways. They reveal whether a complex model is learning temporal structure or merely matching an easy persistence pattern. More expressive methods - ARIMA, exponential smoothing, machine-learning regressors, and neural sequence models - should be compared to those baselines with backtesting.

Baselines to beat

Each baseline encodes one assumption about what persists. A model earns its complexity only by beating the baseline whose assumption matches the series.

Baseline	Forecast ${\hat{y}}_{T+h\mid T}$	Assumption it encodes
Naive	$y_T$	the series is a random walk; the last value is the best guess
Seasonal naive	$y_{T+h-m}$	the pattern repeats every $m$ steps (period $m$)
Drift	$y_T+h\frac{y_T-y_1}{T-1}$	a linear trend through the first and last observations continues
Moving average	$\frac{1}{k}{\sum }_{i=0}^{k-1}{y}_{T-i}$	recent level is stable and noise should be averaged out

A model that cannot beat seasonal naive on a strongly seasonal series is not learning the seasonality; it is usually overfitting noise.

A time-series workflow therefore starts by checking timestamp integrity, gaps, duplicates, aggregation level, calendar conventions, and known future covariates. Only then does model choice become meaningful. A daily sales forecast with censored stockouts, a temperature-driven load forecast, and an intermittent spare-parts forecast are all “time series,” but they require different assumptions and validation cuts.

References

• Hyndman & Athanasopoulos, FPP3: Getting started
• Hyndman & Athanasopoulos, FPP3: Time series graphics

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Section — Time-Series Forecasting

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