1. Definition

  • Time Series = data points collected or recorded in time order (usually equally spaced).
  • Formally:

$Y_t, \; t = 1, 2, \dots, T$

where $Y_t$​ is the observation at time $t$.

  • Key feature: order matters. Unlike regular tabular data, the sequence in time is crucial.

2. Examples

  • Finance: stock prices recorded daily
  • Economics: monthly unemployment rates, GDP growth
  • Weather: hourly temperature, daily rainfall
  • IoT / Sensors: electricity consumption every 15 minutes
  • Healthcare: heart rate over time

3. Components of a Time Series

Typical time series can be decomposed into:

  1. Trend
    • Long-term increase or decrease.
    • Example: gradual upward trend in housing prices.
  2. Seasonality
    • Regular, repeating patterns within fixed periods (day, week, year).
    • Example: retail sales spike every December.
  3. Cyclic patterns
    • Fluctuations that are not of fixed length (business cycles, economic expansions/recessions).
  4. Noise
    • Random variation not explained by trend/seasonality.

4. Types of Time Series

  • Univariate: only one variable over time (e.g., daily sales).
  • Multivariate: multiple variables measured together over time (e.g., sales + temperature + promotion spend).
  • Stationary: mean/variance constant over time, no trend/seasonality.
  • Non-stationary: mean/variance changes (common in real-world data).

5. Modeling Approaches

  • Classical Statistical Models
    • AR (AutoRegressive)
    • MA (Moving Average)
    • ARMA / ARIMA (AutoRegressive Integrated Moving Average)
    • SARIMA (Seasonal ARIMA)
    • Exponential Smoothing (Holt-Winters)
  • Modern ML/DL Models
    • Random Forest, XGBoost (applied to lag features)
    • RNN (Recurrent Neural Networks), LSTM, GRU
    • Temporal Convolutional Networks (TCN)
    • Transformers for Time Series (e.g., Informer, TimesNet)

6. Forecasting vs Analysis

  • Forecasting: predicting future values (e.g., tomorrow’s temperature).
  • Analysis: understanding structure, correlations, anomalies (e.g., trend detection, anomaly detection in sensor data).

7. Evaluation Metrics

Common metrics used for time series forecasting:

  • MAE (Mean Absolute Error)
  • MSE (Mean Squared Error)
  • RMSE (Root MSE)
  • MAPE (Mean Absolute Percentage Error)
  • WAPE (Weighted Absolute Percentage Error)
  • sMAPE (Symmetric MAPE)

8. Unique Challenges in Time Series

  • Autocorrelation: past values influence future values.
  • Non-stationarity: data properties (mean, variance) change over time.
  • Seasonality & Trend: models must capture repeating patterns.
  • Temporal dependency: train-test splits must respect time order (no future data leakage).
  • High dimensionality: multivariate series can be complex (e.g., multiple sensors).

Summary

  • A Time Series is data recorded sequentially over time.
  • Has components: trend, seasonality, cycles, noise.
  • Can be modeled using classical (ARIMA, Holt-Winters) or modern ML/DL methods (LSTM, Transformers).
  • Used for forecasting (future values) and analysis (patterns, anomalies).
  • Must handle autocorrelation, non-stationarity, temporal splits carefully.