Time Series Analysis — Forecasting the Future from Historical Data

What Is Time Series Data?

Time Series: Data collected sequentially over time.

Examples: Stock prices, GDP, Consumer Price Index (CPI), monthly retail sales, temperature readings

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The Four Components of a Time Series

$$\begin{aligned} \text{Multiplicative model}: Y_t &= T_t \times S_t \times C_t \times I_t \\ \text{Additive model}: Y_t &= T_t + S_t + C_t + I_t \end{aligned}$$

Component	Symbol	Description
Trend (T)	T	Long-term increase or decrease in the series
Seasonality (S)	S	Regular, repeating pattern within a year
Cycle (C)	C	Multi-year fluctuations (e.g., business cycles)
Irregular (I)	I	Random, unpredictable variation

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Moving Average Method

Forecasts the next period using the average of the past n periods.

$$MA(n) = (Y_t + Y_{t−1} + ... + Y_{t−n+1}) / n$$

Advantage: Smooths out short-term noise; reveals trend
Disadvantage: Assigns equal weight to recent and older data

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Exponential Smoothing

Assigns higher weights to more recent observations.

$$\hat{Y}_{t+1} = \alpha \times Y_t + (1−\alpha) \times \hat{Y}_t$$

• α: smoothing parameter (0 < α < 1)
• α → 1: heavily weights recent data (sensitive to fluctuations)
• α → 0: heavily weights past data (smoother forecasts)

Holt-Winters Method: Extends exponential smoothing to capture both trend and seasonality.

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Autocorrelation and Stationarity

Autocorrelation (ACF): The degree to which a time series is correlated with its own past values

Stationary Time Series: Mean and variance remain constant over time
→ Non-stationary series can be made stationary through differencing

Differencing:

$$\Delta Y_t = Y_t - Y_{t-1} \quad \text{(first-order differencing)}$$

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ARIMA Model

ARIMA(p, d, q): Autoregressive Integrated Moving Average

Symbol	Meaning
p	AR order (number of autoregressive terms)
d	Number of differences required for stationarity
q	MA order (number of lagged forecast errors)

• AR(p): Y_t = φ₁Y_{t−1} + … + φₚY_{t−p} + ε_t
• MA(q): Y_t = ε_t + θ₁ε_{t−1} + … + θ_qε_{t−q}
• ARMA(p,q): Combines AR and MA components
• ARIMA(p,d,q): Applies d rounds of differencing before fitting ARMA

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Forecast Accuracy Metrics

$$\begin{aligned} MAE &= \sum |Y_t - \hat{Y}_t| / n \quad \text{(Mean Absolute Error)} \\ MSE &= \sum (Y_t - \hat{Y}_t)^2 / n \quad \text{(Mean Squared Error)} \\ RMSE &= \sqrt{MSE} \quad \text{(Root Mean Squared Error)} \\ MAPE &= \sum |Y_t - \hat{Y}_t| / Y_t / n \times 100\% \quad \text{(Mean Absolute Percentage Error)} \end{aligned}$$

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Key Concept Cards

Four Components of a Time Series ★★★★★ : Trend (T), Seasonality (S), Cycle (C), Irregular (I). The foundation for decomposing and forecasting time series. Memory tip: T-S-C-I (Trend, Seasonal, Cyclical, Irregular)

Exponential Smoothing ★★★★☆ : Higher α → more weight on recent data. Variants: simple, double (Holt), triple (Holt-Winters). Memory tip: α → 1 = sensitive to recent data; α → 0 = slow to react

ARIMA(p,d,q) ★★★★☆ : Standard model for forecasting non-stationary time series. d = differencing order, p = AR terms, q = MA terms. Memory tip: d = stationarity, p = autoregression, q = moving average errors

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Practice Questions

Q. Sales data shows strong seasonality. Which forecasting method is most appropriate?

Holt-Winters exponential smoothing or SARIMA (Seasonal ARIMA). Both explicitly model seasonal patterns, allowing forecasts to reflect the repeating seasonal structure (e.g., holiday retail spikes, quarterly earnings cycles).

Q. What does ARIMA(1,1,0) mean?

One AR term (p = 1), one round of differencing (d = 1, treating a non-stationary series), and no MA terms (q = 0). The model applies first differencing to achieve stationarity, then fits an AR(1) model to the differenced series.