Forecasting Fundamentals
Forecasting is the task of estimating future values of a variable based on its historical behavior and, in some cases, related external variables. It differs from general regression in that the observations are ordered in time and exhibit temporal dependencies. These dependencies — including trends, seasonality, and autocorrelation — must be explicitly modeled or accounted for to produce reliable forecasts.
Forecasting systems are used in a wide range of operational contexts, from inventory planning and demand forecasting in retail and manufacturing, to resource allocation in public health, to energy load forecasting in utilities. Each domain imposes specific requirements around forecast horizon, update frequency, and tolerance for prediction uncertainty.
Statistical Methods
Classical statistical forecasting methods have several decades of development and are well characterized in terms of their assumptions, failure modes, and diagnostic tools. They remain competitive for many univariate series and offer strong interpretability advantages.
ARIMA
The Autoregressive Integrated Moving Average (ARIMA) model captures linear temporal dependencies in a single series. The AR component models the relationship between an observation and a weighted sum of lagged observations. The MA component models the relationship between an observation and a weighted sum of lagged forecast errors. Differencing (the I component) removes non-stationarity by transforming the series to one where the statistical properties do not change over time.
Selecting ARIMA order parameters (p, d, q) requires diagnosing the autocorrelation function (ACF) and partial autocorrelation function (PACF) plots. Automated selection procedures such as auto-ARIMA iterate over candidate parameter combinations and select by information criterion, which reduces the manual diagnostic burden in production pipelines.
Exponential Smoothing
Exponential smoothing methods assign exponentially decreasing weights to older observations. Simple exponential smoothing applies when the series has no trend or seasonality. Holt's method extends this to trend, and Holt-Winters' method (also called ETS: Error, Trend, Seasonal) extends further to handle additive or multiplicative seasonality. ETS models are often competitive with ARIMA for series with clear seasonal structure and are straightforward to implement in both batch and streaming contexts.
Machine Learning Approaches
Machine learning methods for time series typically reformulate the forecasting problem as supervised regression: the target is a future value, and the features include lagged versions of the series, moving averages, time-derived encodings, and external covariates. This formulation allows tabular ML algorithms — including gradient-boosted trees and random forests — to be applied without specialized time-series machinery.
Feature Construction for ML Forecasters
The quality of an ML-based forecaster depends heavily on the feature set. Standard components include lag features at relevant intervals (daily, weekly, monthly), rolling window statistics (mean, standard deviation, min, max over a trailing window), calendar features (day of week, month, holiday indicator), and domain-specific covariates (weather, economic indicators, promotions). See the Data Requirements guide for a more detailed discussion of temporal feature extraction.
Deep Learning Forecasters
Recurrent neural networks, particularly Long Short-Term Memory (LSTM) architectures, were historically the primary deep learning approach to time series. More recently, Transformer-based architectures adapted for time series have demonstrated strong performance on long-horizon tasks. Temporal Fusion Transformers combine recurrent encoders with attention mechanisms and are designed to handle multiple related series with external covariates. However, the data volume required to train these architectures effectively limits their applicability in domains with short history or limited series count.
Time Series Decomposition
Decomposing a time series into trend, seasonal, and residual components aids both model diagnosis and feature construction. The classical additive decomposition assumes the observed value equals the sum of trend, seasonal, and residual. Multiplicative decomposition is appropriate when seasonal variation grows proportionally with trend level. Seasonal-Trend decomposition using LOESS (STL) provides a more robust decomposition that handles outliers and allows the seasonal component to vary over time, which the classical additive method does not.
Decomposition is particularly useful for identifying whether seasonality is fixed in period and amplitude — favoring ARIMA with seasonal terms or ETS — or whether it is complex and evolving — favoring ML-based or neural approaches that can model the seasonal pattern from raw features.
Multi-Step and Long-Horizon Forecasting
Many operational decisions require forecasts several periods ahead rather than just one. Multi-step forecasting introduces additional complexity because prediction errors accumulate over the horizon and the true future values of lagged features are unavailable.
Recursive Strategy
In the recursive strategy, a single-step-ahead model is applied repeatedly, using its own outputs as inputs for subsequent steps. This is simple to implement but propagates errors across the horizon.
Direct Strategy
In the direct strategy, a separate model is trained for each forecast horizon. This eliminates error propagation but requires training and maintaining multiple models and does not allow information sharing across horizons.
Evaluation Strategies for Forecasters
Time series evaluation must respect temporal ordering. As noted in the Implementation Workflows guide, k-fold cross-validation in its standard form is inappropriate because it allows future data to appear in training folds. Rolling window or expanding window back-testing evaluates the forecaster across multiple simulated prediction points, each using only data available up to that point.
Probabilistic forecasting evaluates not just point estimates but the calibration of prediction intervals. A well-calibrated 90% prediction interval should contain the true future value in approximately 90% of evaluation windows. Coverage metrics and interval width provide complementary views of forecast uncertainty management.
Operational Use in Canadian Contexts
Forecasting systems are deployed across Canadian public and private sectors in contexts including provincial health resource planning, utility demand forecasting, transportation throughput modeling, and municipal budget projection. Each context involves domain-specific temporal patterns — for example, Canadian healthcare demand exhibits strong flu-season seasonality, while energy demand in northern regions is strongly influenced by temperature and daylight hours — that must be incorporated through appropriate feature engineering or model selection. For how forecast outputs connect to downstream action logic, see Decision Support Platforms.