**Duration:**

**Number of lectures per week:** 3

**Assessment:** a small number of class exercises will be compulsary and will
carry 10% of the marks

**End-of-year Examination:** One 3-hour examination

**Description: **

Remarks: This course will run in 1994/5 and in 1996/7. The course has been
substantially revised in 1994/5. Students will be expected to be able to
carry out a number of computer based exercises.

**Introduction:**

Stochastic Processes are widely used to model processes which change
randomly in time. They form the basis of most forecasting procedures.
Additionally they provide the basis for the understanding phenomena from
congestion to evolution. Generalisations allow the modelling of processes
which change randomly in space. These have gained wide application in image
processing, in exploration geology and in environmental studies. Further
generalisation of the concept of 'space' have recently allowed much wider
application and research in stochastic models is very active indeed. This
course will provide an introduction to the general concepts such as
stationarity and Markov processes and will subsequently concentrate on the
models underlying linear processes as used in time series modelling, in
spatial regression and in geostatistics.

**Michaelmas:**

Introduction to Stochastic Processes - sequences of events governed by
probabilistic laws. Random walks. Markov Chains. Markov Processes.
Branching Processes. Brownian Motion. Applications in industry, science
and sociology.

**Hilary and Trinity:**

Linear models with general covariance structure. Conditional and marginal
distributions, correlation and partial correlation. Linear transformations.
Mahalanobis distance. Review of the theory of optimal estimation and
prediction. Application to linear regression through weighted and general
least squares. Application to multivariate data. Discussion of residuals.
Time series models. Application to forecasting. The variogram, the
autocorrelation function and the partial autocorrelation function in the
identification of models. Residuals. Introduction to ARIMA processes and to
state-space models. This part of the course will be illustrated by data on
global warming and on currency price variations.
Regression with spatially correlated errors. Economic data will be used for
illustration. Geostatistical models. Kriging. Data from soil science will
be the basis for the examples discussed.

**TEXTS:**

Cox, D.R. and Miller, H.D., "The Theory of Stochastic Processes", Methuen.
Bhat, U.N., Ëlements of Applied Stochastic Processes", Wiley.

Karlin, S. and Taylor, H.M. Ä First Course in Stochastic Processes", The
Academic Press.

Taylor, H.M. and Karlin, S., Ïntroduction to Stochastic Modelling", The
Academic Press.

Chatfield, C., Änalysis of Time Series - An Introduction", Fourth ed.,
Chapman and Hall.

Cryer, J.D., "Time Series Analysis", Duxbury Press, Boston.

Raftery, A.E., "Time Series Analysis", European Journal of Operation
Research, Vol. 20, pp 127-137, 1985.

Ripley, B.D "Spatial Statistics", Wiley, in particular Chap 4.

Hohn, M.E. "Geostatistics and the Petroleum Industry" Van Nostrand Reinhold Matheron, G "The Theory of Regionalised Variables and Its Applications",

Lecture Notes, published by Ecole Nationale Superieure des Mines de Paris
Journel, A and Huijbergt, Ch. "Mining Geostatistics", Academic Press.

Haslett, J and Hayes, K. Residuals for the Linear Model with Arbitrary Covariance Function; research paper Department of Statistics, TCD, 1994.

Jun 10, 1998