Time Series Analysis

Walmart stores 10 years of sales data for every product in every store - 400 million rows. Three days before a hurricane, Pop-Tart sales spike 7x. Predicting this in advance means stocking shelves correctly and not losing USD 50 million in revenue. This is what time series models do at production scale.

• **Amazon Forecast** uses DeepAR (a global LSTM model) for demand forecasting across 30 million products - accuracy improved 36% over ARIMA baselines
• **Uber** forecasts driver demand in 30-minute slots for each city zone using Prophet with custom seasonality components
• **ECG monitoring**: cardiac arrhythmia detection through time series anomaly analysis is the foundation of the Apple Watch ECG feature

ARIMA

In 1970, Box and Jenkins published a book that changed forecasting forever. Their ARIMA model (AutoRegressive Integrated Moving Average) describes a time series with three parameters: **AR(p)** - dependence on p previous values, **I(d)** - differencing order to achieve stationarity, **MA(q)** - moving average of q past forecast errors. Amazon uses ARIMA for short-term demand forecasting across 30 million SKUs every day.

Key concepts: stationarity (mean and variance don't change over time - tested with the Augmented Dickey-Fuller test), ACF/PACF (autocorrelation / partial autocorrelation - for choosing p and q), AIC/BIC (model selection criteria). Parameter selection: auto_arima from pmdarima automatically searches (p,d,q) combinations by AIC. ARIMA limitation: linear method, poorly captures complex nonlinear patterns and multiple seasonality.

Facebook Prophet

In 2017, Facebook open-sourced Prophet - a time series model that an analyst without a statistics background can apply in 10 lines of code. The approach: instead of ARIMA parameters, Prophet decomposes the series into **trend** (piecewise linear or logistic growth), **seasonality** (Fourier series for yearly, weekly, daily patterns) and **holidays** (custom user-defined events). It is robust to missing data and automatically detects trend change points.

Prophet model: y(t) = trend(t) + seasonality(t) + holidays(t) + error. Trend: linear (with changepoints) or logistic (for capacity-limited growth). Seasonality: Fourier series up to order N (yearly: N=10, weekly: N=3). Changepoints: automatically detected trend-rate change points (default: 25 in 80% of the series). Bayesian estimation via Stan (MCMC or MAP) provides forecast uncertainty for free.

Seasonality and Decomposition

Ice cream sales peak in summer. Website traffic drops on Sundays. A heartbeat repeats every 0.8 seconds. All of these are **seasonality**: predictable repeating patterns with a fixed period. Classical decomposition splits a series into three components: trend (T), seasonality (S) and remainder (R). Additive model: y = T + S + R (seasonal amplitude is constant). Multiplicative model: y = T * S * R (amplitude grows with the trend - typical for business data).

Decomposition methods: (1) STL (Seasonal and Trend decomposition using Loess) - robust to outliers, handles multiple seasonalities, recommended by default; (2) X-13ARIMA-SEATS - standard used by government statistical agencies; (3) MSTL (Multiple STL) - for hourly data with simultaneous daily and weekly seasonality. statsmodels includes seasonal_decompose and STL. Common seasonal periods: 12 (months), 7 (days of week), 24 (hours), 52 (weeks).

Forecasting and Model Evaluation

Time series forecasting is not just model fitting. The critical error: evaluating quality on the same data the model was trained on. Due to autocorrelation, standard cross-validation gives overly optimistic estimates. The correct approach: **walk-forward validation** (rolling training/test window). The model is always trained on the past and tested on the future, exactly as in production deployment.

Forecasting quality metrics: MAE (mean absolute error, interpretable in original units), RMSE (penalizes large errors more heavily), MAPE (percentage error, undefined when values are zero), SMAPE (symmetric MAPE, range 0-200%). Baseline models for comparison: Naive (forecast = last observation), Seasonal Naive (forecast = same period from previous season), Simple Exponential Smoothing. If a complex model can't beat the baseline, the problem is in data quality or feature engineering.

Key Ideas

• **ARIMA(p,d,q)** is the statistical workhorse: AR captures dependence on past values, I differences to stationarity, MA accounts for past forecast errors
• **Prophet** decomposes series into trend + seasonality + holidays via Fourier series and Bayesian changepoint detection - interpretable and robust to missing data
• **Walk-forward validation** is mandatory for honest evaluation: the model is always trained on the past and tested on the future, mirroring real production conditions

Related Topics

Time series analysis intersects with several Data Science domains:

• Ensemble Methods — Gradient boosting (LightGBM, XGBoost) is applied to time series via feature engineering: lag features, rolling averages, seasonal indicators
• Causal Inference — Causal Impact (Google) uses Bayesian structural time series to measure causal effects of interventions (ad campaigns, product changes) with counterfactual reasoning

Вопросы для размышления

• If two analysts fit ARIMA to the same series and choose different (p,d,q) parameters, who is right? How do you objectively compare the models?
• Prophet automatically detects changepoints. What happens if training data includes a single anomalous month due to a pandemic?
• Walk-forward validation always trains on all available history. When can data from three years ago harm forecast quality rather than help it?

Связанные уроки

• stat-13-time-series