# Time series analysis

## Introduction

A set of ordered data points taken at successive intervals in time is known as time series. Time in terms of years, months, days, or hours can be considered as reference points for ordered data. The dependence introduced by the sampling data over time restricts the applicability of many conventional statistical methods that require random samples. The analysis of such data is commonly referred to as time series analysis. The primary objective of time series analysis is to develop statistical models that provide plausible description for sample data observed over time.

A time series depicts the relationship between two variables, one of them being time, e.g., the numbers of international airline passengers $$(x_t)$$ in different months $$(t)$$. This relationship can be portray with the help of a time series plot taking time at the $$X-axis$$ as shown in the following figure:

## Components of Time Series

The factors affecting the data points in a time series, known as the components of time series. These are:

• Long-Term Trend (Secular Trend)
• Short-Term Fluctuations (Periodic Changes)
• Seasonal Variations
• Cyclic Variations
• Random/Irregular Movements

### Long-Term Trend

This is the core component of a time series. It shows general tendency of the data to increase or decrease over a long period of time. It does not have to be linear. Sometimes it is referred as changing direction, when it might go from increasing trend to a decreasing trend. There is an upward long-term trend in the numbers of monthly airline passengers during 1949-1960 shown in Figure 1.

### Short-Term Fluctuations

It prevents the smooth flow of the series in one direction and tend to repeat themselves over a period of time. These components do not influence the time series continuously but act in a regular sporadic way.

#### Seasonal Variations

A seasonal pattern occurs due to the seasonal factors such as time of the year or the day of the week. These factors operate in a regular and periodic manner over a span of less than a year and have the same pattern year after year. Thus seasonal variations in a time series will be there if the data are recorded quarterly (every three months), monthly, weekly, daily, hourly and so on. Although in each of the above cases, the frequency of the seasonal variations are different. Seasonality is always of a fixed and known frequency. The numbers of monthly airline passengers during 1949-1960 shown in Figure 1 shows seasonality.

#### Cylic Variations

The rises and falls in time series data with a period of more than one year is known as cyclic variations. The frequency of cyclic variations are not fixed. These upswings and downswings are normally due to the business cycle, which may also be referred to as the four-phase cycle composed of prosperity (period of boom), recession, depression and recovery. It usually lasts from seven to eleven years.

### Random/Irreguar Movements

These components are not accounted for by long-term trend and short-term fluctuations. These fluctuations are purely random, unpredictable and are due to numerous non-recurring and irregular circumstances which are beyond the control of human but at the same time are a part of or systems such as floods, famines, earthquakes, wars, etc.