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Introduction

• A time series is an ordered sequence of observations.

• The ordering of the observations is usually through time, but may also be taken through other dimensions such as space.

• Time series analysis deal with relationship between observations that are separated by k units of time or space (lagged observations).

• We are interested to know how the present depends upon the past.

• Time series occur in a variety of fields.

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Examples

• In agriculture, we observe annual crop production and prices.

• In economics, we observe daily stock prices, weekly interest rates, monthly price indices, quarterly sales and yearly earnings.

• In engineering, we observe sound, electric signals and voltage.

• In meteorology, we observe hourly wind speed, daily temperature and annual rainfall.

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Type of Time Series Data

• A time series that can be recorded continuously in time, is said to be continuous. For example, electrical signals and voltage.

• A time series that is taken only at specific time intervals is said to be discrete. For example, interest rates, volume of sales etc.

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Objectives

• Understanding and description of the generating mechanism.

• Modeling and inference.

• Forecasting and prediction.

• Optimal control of a system.

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Important note

• The basic nature of a time series is that its observations are dependent or correlated, and the order of the observations is therefore important.

• Statistical procedures and techniques that rely on independence assumptions are no longer applicable.

• The statistical methodology available for analyzing time series is referred to as time series analysis.

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Time versus Frequency Domain

• Time series approach, which uses autocorrelation and partial autocorrelation functions to study the evolution of a time series through parametric models, is known as frequency domain analysis.

• An alternative approach, which uses spectral functions to study the nonparametric decomposition of a time series into its different frequency components, is known as frequency domain analysis.

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Graphical Methods

• The preliminary goal is to describe the overall (macro) structure of data and to identify memory type of time series.

• There are two types of memories:(1) Short memory – immediate past gives some information about immediate future but less information about long-term future.(2) Long memory – past gives (potentially) more information about future (long term). Includes series with trends or cycles (seasonality).

• A basic useful graphical tool is a Time Series Plot. We plot the data versus time.