Short note: Time series, regression

Time series data

A time series is a sequence of data values measured at successive, usually equally spaced, times: daily stock price, monthly rainfall, hourly electricity load, ECG signal. The order of the values matters.

Components of a time series

1. Trend (T): long-term increase or decrease.
2. Seasonal variation (S): regular pattern repeating in fixed periods (festival sales).
3. Cyclic variation (C): long-term, non-fixed wave-like movement (business cycles).
4. Irregular/random variation (I): unpredictable noise. Models: additive Y = T + S + C + I or multiplicative Y = T × S × C × I.

Mining tasks: trend analysis (moving average, smoothing), similarity search, forecasting (ARIMA), classification and clustering of series, anomaly detection and sequence/subsequence pattern discovery (e.g. with dynamic time warping).

Regression analysis

Regression models the relation between a dependent (response) variable y and one or more independent (predictor) variables x, to predict continuous values.

• Simple linear regression: y = w₀ + w₁x, with w₁ = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)², w₀ = ȳ − w₁x̄ (least squares: minimises the sum of squared errors).
• Multiple linear regression: y = w₀ + w₁x₁ + w₂x₂ + …
• Non-linear (polynomial) and logistic regression (for class prediction).

Example: x = 1, 2, 3, 4 and y = 2, 4, 5, 8: x̄ = 2.5, ȳ = 4.75; Σ(xᵢ − x̄)(yᵢ − ȳ) = 9.5 and Σ(xᵢ − x̄)² = 5, so w₁ = 9.5/5 = 1.9, w₀ = 4.75 − 1.9 × 2.5 = 0, giving y = 1.9x. For x = 5 the prediction is 9.5.

Regression is used on time series to fit trends and forecast future values.