Time Series Forecasting

The core assumption that observations are not independent but correlated with past values. Autocorrelation measures this relationship at different lags.

• AR (Autoregressive): Predicts future values using a linear combination of past values
• MA (Moving Average): Models the dependency between an observation and residual errors from a moving average applied to lagged observations
• ARIMA: Combines AR and MA components with differencing to handle non-stationary data
• Seasonal ARIMA (SARIMA): Extends ARIMA to capture repeating patterns at fixed intervals (e.g., hourly, weekly, yearly)

A stationary time series has constant statistical properties—mean, variance, and autocorrelation—over time. Most classical models require stationarity for reliable inference.

• Dickey-Fuller Test: A statistical hypothesis test where the null hypothesis assumes a unit root is present (non-stationary)
• Differencing: Subtracting the previous observation from the current one to stabilize the mean
• Log Transformation: Applied to stabilize variance in series with exponential growth patterns
• Decomposition: Separating a series into trend, seasonal, and residual components before modeling

Distinguishing between repeating patterns is critical for forecast accuracy. Seasonality refers to fixed-period fluctuations, while cyclicality involves non-fixed periods tied to economic conditions.

• Fourier Series: Used in Prophet and other models to capture multiple seasonal harmonics (e.g., weekly + yearly patterns)
• Dummy Variables: Binary indicators for specific periods like holidays or promotional events
• STL Decomposition: Seasonal-Trend decomposition using LOESS to handle complex, non-integer seasonality
• Calendar Effects: Adjustments for trading days, leap years, and moving holidays that distort regular patterns

Modern forecasting models incorporate external regressors beyond historical values. These exogenous variables capture causal drivers like pricing changes, marketing spend, or weather.

• ARIMAX: ARIMA with exogenous inputs, allowing linear regression on external features
• Dynamic Regression: Models where the relationship between exogenous variables and the target can vary over time
• Temporal Fusion Transformer: A deep learning architecture that uses attention mechanisms to select relevant exogenous features at each time step
• Transfer Functions: Modeling the delayed and decaying impact of an intervention (e.g., a price change takes 3 days to fully affect demand)

Point forecasts are insufficient for decision-making. Robust models produce prediction intervals that quantify the range of plausible outcomes.

• Conformal Prediction: A distribution-free framework that provides valid prediction intervals without assuming a specific error distribution
• Quantile Regression: Directly models specific percentiles (e.g., 10th and 90th) of the forecast distribution
• Bayesian Structural Time Series: Uses Markov Chain Monte Carlo to sample from the full posterior predictive distribution
• Bootstrapped Residuals: Generates prediction intervals by repeatedly sampling from historical forecast errors

The operational cadence of prediction generation varies by use case. Batch forecasting pre-computes predictions, while streaming models update with each new observation.

• Online Learning: Models like River or online ARIMA update coefficients incrementally without full retraining
• Backtesting: Simulating historical forecasting performance by walking forward through time to validate model stability
• Cold Start: Handling new products with no history using hierarchical models that borrow strength from similar items
• Feature Freshness: The latency between data generation and availability for inference, critical for intraday pricing adjustments

Time series models predict future product demand at the SKU-location-day granularity, forming the foundation for automated replenishment. By decomposing historical sales into trend, seasonality, and holiday effects, retailers can reduce stockouts by up to 30% while minimizing carrying costs.

• ARIMA and SARIMA models capture autoregressive patterns and seasonal cycles in stable product categories
• Prophet handles multiple seasonality patterns and holiday effects common in retail
• Temporal Fusion Transformers incorporate exogenous variables like weather, promotions, and competitor actions
• Forecasts feed directly into inventory-aware pricing engines to prevent margin erosion on overstocked items

Retailers use time series models to isolate the incremental lift generated by promotions, separating the promotional effect from baseline demand. This enables precise ROI measurement for each campaign and prevents unprofitable discounting.

• Causal impact models estimate counterfactual baseline sales using Bayesian structural time series
• Transfer function models quantify how promotional spend translates into demand over a lagged decay curve
• Cannibalization risk scoring integrates with forecasts to predict cross-product substitution effects
• Accurate uplift forecasts prevent margin dilution from over-discounting inelastic products

For products with finite shelf life, time series forecasting drives markdown optimization algorithms that determine the optimal discount trajectory. The goal is to maximize recovery value before inventory becomes unsellable waste.

• Survival analysis models estimate the probability of sale at each remaining shelf-life day
• Exponential decay functions model how demand responds to deepening discounts over time
• Forecasts incorporate sell-through rate and current inventory position
• Typical implementations reduce food waste by 20-25% while improving category margins

Time series forecasts serve as the foundational input for dynamic pricing algorithms. By predicting the demand curve at each price point, models enable revenue-optimal price adjustments that respond to real-time market signals.

• Price elasticity models use historical time series to estimate demand sensitivity at different price levels
• Competitive price indexing feeds competitor price movements as exogenous regressors
• Yield management systems combine demand forecasts with capacity constraints for perishable inventory
• Forecasts update at sub-daily frequencies during high-velocity events like flash sales or Black Friday

Beyond demand forecasting, time series models predict supplier lead times and transportation delays, enabling proactive supply chain orchestration. This dual forecasting approach aligns inbound supply with predicted outbound demand.

• Vector autoregression (VAR) models capture interdependencies between multiple supply chain time series
• Long Short-Term Memory (LSTM) networks learn complex temporal dependencies in logistics data
• Forecasts trigger safety stock recalculation and alternative sourcing decisions
• Integration with autonomous supply chain intelligence enables automated exception handling

Time series forecasting addresses the cold start problem for new product introductions by leveraging analogies from similar historical launches. This enables data-driven initial buy quantities and pricing strategies.

• Clustering-based analogies identify products with similar attribute profiles and launch trajectories
• Bass diffusion models estimate adoption curves based on innovation and imitation parameters
• Hierarchical forecasting pools information across product categories to improve sparse data estimates
• Forecasts inform dynamic assortment optimization decisions for initial shelf allocation

A time series is stationary if its statistical properties—mean, variance, and autocorrelation—are constant over time. Most classical models require stationarity.

• Augmented Dickey-Fuller (ADF) Test: A statistical test where a p-value < 0.05 suggests the series is stationary.
• Differencing: Subtracting the previous observation from the current one to remove trend. First-order differencing calculates y'(t) = y(t) - y(t-1).
• Seasonal Differencing: Subtracting the value from the same period in the previous season to remove periodic patterns.

Failing to achieve stationarity leads to spurious regressions and unreliable forecasts. Modern deep learning models are more robust to non-stationary data but still benefit from this preprocessing.

Rigorous evaluation requires multiple metrics to capture bias, accuracy, and scale-independent performance.

• MAPE (Mean Absolute Percentage Error): (1/n) * Σ |(Actual - Forecast) / Actual|. Intuitive but undefined when actuals are zero and penalizes under-forecasting more than over-forecasting.
• sMAPE (Symmetric MAPE): (2/n) * Σ |Actual - Forecast| / (|Actual| + |Forecast|). Bounded between 0% and 200%, addressing the asymmetry of MAPE.
• MASE (Mean Absolute Scaled Error): Scales the forecast error by the in-sample error of a naive seasonal model. A MASE < 1 indicates the model beats the naive baseline.
• Pinball Loss: Used for quantile regression, it asymmetrically penalizes over- and under-prediction to generate accurate prediction intervals.