Univariate Time Series Analysis - Seasonal Patterns and Methods

Topic 4: Univariate Methods II - Seasonal Patterns

• Seasonal Patterns

  • Defined as recurring patterns that occur within a known fixed time frame or periodicity.

  • Commonly related to the calendar year, such as quarters or months (e.g., influenced by weather).

  • May also apply to other time frames like days of the week or hours of the day.

  • Each season or subperiod has a known and fixed duration.

• Agenda Overview

  • 4.1 Time-Series Components

  • 4.2 Purely Seasonal Series

  • 4.3 Decompositions

  • 4.4 Holt-Winters Methods

4.1 Time-Series Components

• General Concept

  • Decompose original data into three components:

  1. $X<em>t = f(T</em>t, S<em>t, E</em>t)$

     • $X_t$: Original time-series data.

     • $T_t$: Trend cycle component capturing long-term changes.

     • $S_t$: Seasonal component capturing recurrent patterns.

     • $E_t$: Random error component representing random fluctuations.

  • Note: The cyclical component is combined with the trend.

• Types of Decompositions

  • Additive Model:
    $X<em>t = T</em>t + S<em>t + E</em>t$

  • Mixed Additive-Multiplicative Model:
    $X<em>t = (T</em>t + E<em>t) imes S</em>t$

  • Purely Multiplicative Model:
    $X<em>t = T</em>t imes S<em>t imes E</em>t$

  • The multiplicative model can be converted to an additive model via log transformation.

4.2 Purely Seasonal Series

• Characterization

  • No trend cycle component present.

  • For an additive model, it follows:
    $X<em>t = S</em>t + E_t$

• Forecasting Method

  • Generate forecasts for each season separately.

  • Example: One would forecast April 2025 temperature using data solely from April in previous years.

  • Methods include simple average, moving average, and exponential smoothing.

• Seasonal Exponential Smoothing (SES)

  • Formally defined as:
    $F^{(m)}<em>t = F</em>t + \beta(X<em>t - F</em>t^{(m)})$

  • Here, $\beta$ denotes the seasonal smoothing parameter.

4.3 Decompositions

• Additive Decomposition: Main Steps

  1. Estimate trend by averaging out seasonality.

  2. Calculate a detrended series by subtracting trend from original data.

  3. Estimate seasonal components via seasonal averages of the detrended series.

  4. Obtain the deseasonalized series and forecasts by adding back the seasonal component to the deseasonalized forecast.

• Formal Approach for Additive Decomposition

  1. Use moving averages to estimate trends:
     $T_t = MA(t - 1, m)$

  2. Detrended series:
     $X^D<em>t = X</em>t - T_t$

  3. Seasonal means by averaging the detrended series across each season.

4.4 Holt-Winters Methods

• Basic Idea and Notation

  • Integrate Local Level Exponential Smoothing (LES) with seasonal component.

  • Notation:

  • $L_t$: Local level at time t

  • $T_t$: Slope of linear trend at time t

  • $S_t$: Seasonal component at time t

  • $m$: Number of seasons

4.4.1 Additive Holt-Winters Method

• Forecasting Equations

  1. One-step-ahead forecast:
     $F<em>{t+h}^{(1)} = L</em>t + T<em>t + S</em>{t+h-m}$

  2. Forecast error:
     $e<em>t = X</em>t - F_t^{(1)}$

• Updating Schemes

  • Level update:
    $L<em>t = L</em>{t-1} + T<em>{t-1} + \beta imes e</em>t$

  • Trend update:
    $T<em>t = T</em>{t-1} + \beta imes e_t$

  • Seasonal update:
    $S<em>t = S</em>{t-m} + au imes e_t$

4.4.2 Multiplicative Holt-Winters Method

• Techniques for forecasting and updating are similar, using multiplicative adjustments.

• Ensure to adjust for seasonal fluctuations when they're relative to the level of the series.

This structured note encapsulates various aspects of univariate methods for seasonal time-series analysis, highlighting key components, methods, and forecasts techniques while detailing decomposition methodologies.