The Four Components Of Time Series

1. Core Concept Overview & Intuition

A time series is a sequence of observations recorded in chronological order. Because such data are ordered in time, successive values are generally dependent, and the analyst's task is to separate the systematic movements from the accidental ones. The systematic movements are captured by four components, and the central intellectual move of this topic is to recognise that any observed series can be regarded as a composition of these four forces acting simultaneously.

The four components are:

• Secular trend (or simply trend): the long-run general tendency of the data to increase, decrease, or remain constant.
• Seasonal variations: periodic, regular movements whose period is less than one year.
• Cyclic variations: oscillatory movements whose period of oscillation is more than one year.
• Random or irregular variations: fluctuations not accounted for by trend, seasonal, or cyclic effects.

The intuition is that of decomposition. A series such as monthly store sales rises over the years (trend), peaks in festive months (seasonal), drifts with the multi-year business cycle (cyclic), and is disturbed by unforeseen shocks (random). The analyst's goal is to identify which component is at work and to isolate, study, analyse, and measure each component independently.

Two modelling conventions govern how the components combine. Under the additive model the components act with equal absolute effect regardless of the level of the trend; under the multiplicative model the components act proportionally to the general level of the series. The choice between them is made by inspecting the time plot: if the amplitude of seasonal swings grows as the trend level grows, the multiplicative description is more appropriate; if the amplitude stays roughly constant, the additive description is better.

2. Mathematical Formalization & Core Definitions

> Definition (Time Series). A time series is a set of observations $X_t$ ordered in time $t$, where the ordering carries information that is essential to the analysis.

> Definition (Secular Trend). The trend $T_t$ is the general tendency of the data to increase or decrease over a long period of time.

Upward tendencies are seen in data on population, agricultural production, and currency in circulation. Downward tendencies are seen in data on deaths from epidemics as a result of advancement in medical sciences, better medical facilities, literacy, and higher standards of living. Some series fluctuate about a constant reading that does not change with time, for example temperature readings of a particular place operating about a constant room temperature.

> Definition (Seasonal Variations). Seasonal variations $S_t$ are periodic and regular movements in a time series with period of less than one year.

Seasonal variations are attributed to two causes:

• Natural forces: production of commodities that depend on seasons, and prices of agricultural products which go down during harvesting and go up during planting.
• Man-made forces: sales and profits of some departmental stores increase during festivities.

Examples of series exhibiting seasonal rhythms include issue of books from a library, daily traffic density, hourly number of telephone messages in a switchboard, and marriages.

> Definition (Cyclic Variations). Cyclic variations $C_t$ are oscillatory movements in a time series with period of oscillation more than one year.

Examples include the review of production and the review of salaries.

> Definition (Random or Irregular Variations). Random variations $e_t$ are fluctuations that are not accounted for by trend, seasonal, or cyclic variations; they are purely random, unforeseen, unpredictable, and beyond human control.

Examples include wars, earthquakes, floods, epidemics, and famine.

> Definition (Additive Model). According to the additive model, a time series is expressed as

$$
X_t=T_t+S_t+C_t+e_t
$$

> where $X_t$ is the time series value at time $t$, $T_t$ is the trend value at time $t$, $S_t$ is the seasonal value at time $t$, $C_t$ is the cyclic value at time $t$, and $e_t$ is the random value at time $t$.

The additive model implies that seasonal forces (in different years), cyclic forces (in different cycles), and random forces (in long-term periods) operate with equal absolute effects irrespective of the long-term level of the trend value.

> Definition (Multiplicative Model). According to the multiplicative model,

$$
X_t=T_t\cdot S_t\cdot C_t\cdot e_t
$$

> This holds if we assume that the various components in a time series operate proportionally to the general level of the series.

> Remark (Properties of the Multiplicative Model).
> 1. $S_t$, $C_t$, and $e_t$, instead of assuming positive and negative values, are indices fluctuating above or below unity, and the geometric means of $S_t$ in a year, $C_t$ in a cycle, and $e_t$ in a long period are unity.
> 2. The multiplicative model can be converted to an additive model of logarithmic values of the original time series, namely

$$
\ln X_t=\ln T_t+\ln S_t+\ln C_t+\ln e_t
$$

> The time plot should be examined to see which model is likely to give the better description.

> Definition (Objectives of Time Series Analysis). The objectives of analysing a time series are Description, Explanation, Prediction, and Control.

• Description: the first step is usually to plot the data and obtain simple descriptive measures of the main properties of the series. Features such as trend and seasonal effects can sometimes be easily seen from such a plot, and the graph enables one to look for unusual observations or outliers which do not appear consistent with the rest of the data.
• Explanation: when observations are taken on two or more variables, it may be possible to use the variation in one time series to explain the variation in another, leading to a deeper understanding of the process that generates the given time series.
• Prediction: given an observed time series, one may want to forecast future values of the series, a very important task in sales forecasting and in the analysis of economic and industrial time series.
• Control: when a time series measures the quality of a manufacturing process, the aim may be to control the process. A stochastic model is fitted to the series, future values are predicted, and the input process variables are adjusted so as to keep the process on target.

> Definition (Goals of the Analysis of Time Series). The main problem in time series analysis is to
> 1. identify the component at work (trend, seasonal, cyclic, or random effects), and
> 2. isolate, study, analyse, and measure the components independently.