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Time Series Analysis (MATH 33800)

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Administrative Information

  1. Unit number and title: MATH 33800 Time Series Analysis
  2. Level: H/6
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 1994
  6. Unit Organiser: Li Chen
  7. Lecturer: Dr L. Chen and Dr A. Kovac
  8. Teaching block: 1
  9. Prerequisites: MATH11300 Probability 1, MATH 11400 Statistics 1 and the first year core units (MATH11006 Analysis 1, MATH 11007 Calculus 1, MATH 11005 Linear Algebra & Geometry)

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.

General Description of the Unit

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 MATH 35110 (Linear Models) and MATH 30510 (Multivariate Analysis) this course is concerned with developing statistical methodology for a particular class of problems.

Teaching Methods

The teaching methods consist of

  • 30 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).

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.

Assessment Methods

The final assessment mark for Time Series Analysis will be calculated as follows:

  • 5% from THREE satisfactory completed homework assignments.
  • 95% from a 2½-hour written examination in April consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators of an approved type (non-programmable, no text facility) are allowed. From 2012-13 ONLY calculators carrying a 'Faculty of Science approved' sticker will be allowed in the examination room.
    Statistical tables will be provided.

Award of Credit Points

Credit points are gained by:

  • either passing the unit (getting a final assessment mark of 40 or over),
  • or getting a final assessment mark of 30 or over, and also handing in satisfactory attempts at three homework assignments.

Transferable Skills

Use of R for advanced statistical time-series analyses.
Enhanced mathematical modelling skills
Problem solving

Texts

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

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

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)