# Time Series Analysis

## Unit aims

This unit provides an introduction to time series analysis mainly from the statistical point of view but also covers some mathematical and signal processing ideas.

## Unit description

Time series are observations on variables collected through time. For example two well-known time series are daily temperature readings and hourly stock prices. Time series data are widely collected in many fields: for example in the pure sciences, medicine, marketing, economics and finance to name but a few. Time series data are different to the usual statistical data in that the observations are ordered in time and usually correlated. The emphasis is on understanding, modelling and forecasting of time- series data in both the time, frequency and time-frequency domains.

Time series specialists are valued by a wide range of organisations who collect time series data (see list above). This course will equip you with a formidable collection of skills and knowledge that are highly valued by employers. Alternatively, the course would give you a good grounding if you wished to develop time series methods for a higher degree (e.g. PhD).

## Relation to other units

As with units MATH30013 (Linear & Generalised Linear Models) and MATH30510 (Multivariate Analysis) this course is concerned with developing statistical methodology for a particular class of problems.

## Learning objectives

The students will be able to:

- carry out an initial data analysis of time-series data and be able to identify and remove simple trend and seasonalities;
- compute the correlogram and identify various features from it (eg short term correlation, alternating series, outliers);
- define various time-series probability models;
- construct time series probability models from data and verify model fits;
- define the spectral density function and understand it as a distribution of energy in the frequency domain;
- compute the periodogram and smoothed versions;
- analyse bivariate processes.

## Transferable skills

Use of R for advanced statistical time-series analyses.

Enhanced mathematical modelling skills

Problem solving

## Syllabus

(Approximate number of lectures in parentheses)

Simple descriptive techniques: times series plots; seasonal effects; trend; transformations; sample autocorrelation; the correlogram; filtering (2 lectures)

Probability models: stochastic processes; stationarity; second-order stationarity; autocorelation; white noise model; random walks; moving average processes; invertibility; autoregressive processes; Yule-Walker equations; ARMA models; ARIMA processes; the general linear process; the Wold decomposition theorem (6 lectures)

Model building: autocorrelation estimation; fitting an AR process; fitting an MA process; diagnostics (5 lectures)

Forecasting: naive procedures; exponential smoothing; Holt-Winters; Box- Jenkins forecasting; optimality models for exponential smoothing (4 lectures)

Spectral analysis: simple sinusoidal model; Wiener-Khintchine theory; the Cramer representation; periodogram analysis; relation between periodogram and autocovariance; statistical properties of the periodogram; consistent estimators of the spectral density - smoothing the periodogram (6 lectures)

Bivariate processes: cross-covariance and cross-correlation; cross-spectrum; cross-amplitude; phase spectrum; co-spectrum; quadratic-spectrum; coherence; gain (2 lectures)

ARCH modelling for econometrics. (3 lectures)

## Reading and References

The main text will be Chatfield (see below). The lecture course will closely follow this book, but the following will also be useful:

- C. Chatfield, The analysis of time series: an introduction, Chapman and Hall (1984).
- P. J. Diggle, Time Series: a biostatistical introduction, Oxford University Press (1990).
- G. Janacek,
*Practical Time Series,*Arnolds Texts in Statistics (2001).

**Unit code:** MATH33800

**Level of study:** H/6

**Credit points:** 20

**Teaching block (weeks):** 1 (1-12)

**Lecturer:** Professor Guy Nason

## Pre-requisites

MATH11300 Probability 1, MATH11400 Statistics 1, MATH10003 Analysis 1A, MATH11007 Calculus 1, and MATH11005 Linear Algebra and Geometry or equivalents from 2020/21.

## Co-requisites

None

## Methods of teaching

The teaching methods consist of

- Standard lectures.
- Regular problem sheets which will: develop theoretical understanding of the lectures and extra-lecture topics; relate the lectures to real practical problems arising in time-series analysis and signal processing. The students will develop a basic knowledge of time-series analysis within the R package.
- Detailed solution sheets will be released approximately two weeks after the problem sheets.

*Three problem sheets will count towards both assessment and credit points*. It will be made clear in the lectures and on the sheets which count for assessment and credit points. Other problem sheets will be set: they will be marked but it is not compulsory to hand these in (although it would obviously be to your benefit as you would receive feedback).

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 94% from a 2 hour 30 minute exam in January
- 6% from three homework assignments worth 2% each.

NOTE: Calculators of an approved type (non-programmable, no text facility) are allowed.

Statistical tables will be provided.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.