732A34 Time series analysis

Fall semester 2012

• 6 ECTS-credits

• Course tutor and examiner: Anders Nordgaard

• Course web: www.ida.liu.se/~732A34

• Course literature:

• Cryer J.D., Chan K-S.: Time Series Analysis – With Applications in R. 2nd ed. ISBN 978-0-387-75958-6.

• Complementary handouts

Organization of this course:

• Weekly “meetings”: Mixture between lectures, (computer) exercises and seminars

• A great portion of self-studying

• Assignments (every second week)

• Individual written exam

Access to a computer is necessary.

Optimal: Bring your own laptop to the meetings

Examination

The course is examined by

1.Assignments (3 in total)

2.Final written exam

Assignments will be marked Passed or Failed. If Failed, corrections must be done for the mark Pass.

Written exam marks are given according to ECTS grades.

The final grade will be the same grade as for the written exam.

Communication

Contact with course tutor is best through e-mail: Anders.Nordgaard@liu.se.

Office in Building B, Entrance 27, 2nd floor, corridor E (the small one close to Building E), room 3E:485.

Normal working hours: When teaching

E-mail response almost all weekdays and occassionally in weekends

All necessary information will be communicated through the course web. Always use the English version. The first page contains the most recent information (messages)

Solutions to assignments should be e-mailed.

Note! Course tutor is away from Linköping on

3-4 September

12-14 September

25-29 September

Assignments

A number of exercises will be given as assignments to be individually carried out.

There will not be any supervision for these assignments since they are part of the examination, but they can be carried out in the computer rooms or at home. No other statistical software than R will be needed.

The solutions to the assignments should be submitted in forms of written reports. The core text of these reports may contain graphs and tables, but the latter should be constructed from scratch (i.e. no copying and pasting from R or other software). Besides such components the text should be completely your own and easy to read. Direct outputs from the software (except graphs) can only be included in form of attachments.

In the marking of these reports, emphasis will be put on the English language. It will not be sufficient to simply give short answers to the detailed questions of the exercises.

Time series

Sales figures jan 98 - dec 01

051015202530354045

jun-97

jan-98

jul-98

feb-99

aug-99

mar-00

okt-00

apr-01

nov-01

maj-02

• What kind of patterns can visually be detected?

• Is the development stable or non-stable?

Concentrations of Total Phosphorus (ug/l), Råån, Helsingborg, County of Skåne, Sweden

Monthly measurements1980-2001

0

100

200

300

400

500

600

700

800

900

1000

1980-01-15

1981-01-15

1982-01-15

1983-01-15

1984-01-15

1985-01-15

1986-01-15

1987-01-15

1988-01-15

1989-01-15

1990-01-15

1991-01-15

1992-01-15

1993-01-15

1994-01-15

1995-01-15

1996-01-15

1997-01-15

1998-01-15

1999-01-15

2000-01-15

2001-01-15

• What kind of patterns can visually be detected?

• Is the development stable or non-stable?

• Non-independent observations (correlations structure)

• Systematic variation within a year (seasonal effects)

• Long-term increasing or decreasing level (trend)

• Irregular variation of small magnitude (noise)

Sales figures jan 98 - dec 01

051015202530354045

jun-97

jan-98

jul-98

feb-99

aug-99

mar-00

okt-00

apr-01

nov-01

maj-02

Characteristics:

Concentrations of Total Phosphorus (ug/l), Råån, Helsingborg, County of Skåne, Sweden

Monthly measurements1980-2001

0

100

200

300

400

500

600

700

800

900

1000

1980-01-15

1981-01-15

1982-01-15

1983-01-15

1984-01-15

1985-01-15

1986-01-15

1987-01-15

1988-01-15

1989-01-15

1990-01-15

1991-01-15

1992-01-15

1993-01-15

1994-01-15

1995-01-15

1996-01-15

1997-01-15

1998-01-15

1999-01-15

2000-01-15

2001-01-15

• Economic indicators: Sales figures, employment statistics, stock market indices, …

• Meteorological data: precipitation, temperature,…• Environmental monitoring: concentrations of nutrients and pollutants

in air masses, rivers, marine basins,…• Sports statistics?• Electromagnetic och thermal fields

Where can time series be found?

Time series analysis

Estimate/Investigate different parts of a time series in order to

–understand the historical pattern

–judge upon the current status

–make forecasts of the future development

–judge upon the quality of data

Method This course?

Classical decomposition (Yes)

Time series regression Yes

Exponential smoothing No

ARIMA modelling (Box-Jenkins) Yes

Non-parametric and semi-parametric analysis No

Transfer function and intervention models Yes

State space modelling No

Heteroscedastic models: ARCH, GARCH Yes

Advanced econometric methods: Cointegration No

Spectral domain analysis No

Data mining techniques No

Methodologies

Decomposition

yt yt

A time series can be thought of as built-up by a number of components

Number of employees, private sector 1993:1-2008:4, (1994:1=100)

90

100

110

120

130

140

150

Quarter

Ind

ex

What kind of components can we think of? Long-term? Short-term? Deterministic? Purely random?

Decomposition – Analyse the observed time series in its different components:Trend part (TR)Seasonal part (SN)Cyclical part (CL)Irregular part (IR)

Cyclical part: State-of-market in economic time seriesIn environmental series, usually together with TR

Multiplicative model:

yt=TRt·SNt ·CLt ·IRt

Suitable for economic indicators Level is present in TRt or in TCt=(TR∙CL)t

SNt , IRt (and CLt) works as indices Seasonal variation increases with level of yt

161412108642

16

14

12

10

8

6

4

2

161412108642

10

9

8

7

6

5

4

3

2

1

Additive model:

yt=TRt+SNt+CLt +IRt

More suitable for environmental data Requires constant seasonal variation SNt , IRt (and CLt) vary around 0

Number of employees, private sector 1993:1-2008:4, (1994:1=100)

90

100

110

120

130

140

150

Quarter

Ind

ex Additive or multiplicative

model?

Example 1: Sales figures, additive decomposition

sales figures jan-98-dec-01

0

10

20

30

40

50

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

observed (blue), deseasonalised (magenta)

0

10

20

30

40

50

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

observed (blue), estimated trend (green)

0

10

20

30

40

50

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46-10

0

10

20

30

40

50

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

observed TR SN fitted IR

A more theoretical description

A time-series is a special case of a stochastic process.

A stochastic process is a family of random variables coupled with a deterministic index variable t:

t can be continuous or discrete. Y can be continuous- or discrete-valued.

RtYt ,

Examples

•Yt = Number of events (e.g. number of telephone calls arrived) up to time t. Point process (usually modelled as a Poisson process)•Yt =Number of customers in a queue at time-point t (Birth-and-death process)•Yn = The number of offspring in generation n of a population starting with an initial population Y0. Markov chain (Yn depends only on Yn – 1 )•Assume you score +1 if you toss a coin and get “heads” and –1 if you get “tails”. Let Yn = The sum of scores after n tosses. Random walk•Yt = The temperature outdoors at time point t (infinitesimal resolution)

When t is discrete a stochastic process is called a sequence and constitute a model for an observed time series.(Sometimes the sequence itself is referred to as the time series)

Mean (value) function:

(Auto)covariance function:

(Auto)correlation function:

,2,1,0, tYE tt

,2,1,0,,,, stYYEYYCov stststst

sstt

st

st

ststst

YVarYVar

YYCovYYCorr

,,

,,

,,

Example Random walk

22

2

1

2

111,

11

1

11

2

21

00

i.i.d.00

:1For

000

,3,2,

and 0 with s variablerandom

(i.i.d.) ddistributey identicall andt independen of sequence,,

ee

jiji

jiji

t

ii

ststttst

tttt

ttt

ett

tteE

eEeEeEteeEeE

eeeeeeE

st

eEeEeeEYE

teYY

eY

eVareE

ee

98.025

24;94.0

9

8;2.0

25

1;71.0

2

1

:1For

,min,min

,min

25,249,825,12,1,

22

2

,,

,,

2,

2,

s

t

st

st

st

st

st

tYVar

st

st

ee

e

sstt

stst

ettt

est

Note that t,s and t,s are symmetric functions, i.e.

tsst

tsst

,,

,,

Stationarity

A stochastic process is said to be

strictly stationary if the joint probability distribution of

is the same as the joint probability distribution of

for any set of time points (t1, … , tn ) no matter of the value of k

nttt YYY ,,,

21

ktktkt nYYY ,,,

21

A stochastic process is said to be

weakly stationary (or second-order stationary) if

) ofnt (independe and allfor

allfor (constant)

,0, tkt

t

kktt

t

stationary non-stationary

Roughly: Constant mean and constant variance

White noise

A stochastic process that is a sequence of independent and identically distributed (i.i.d.) random variables e1, e2, … is called a white noise process.

By definition a white noise process is strictly stationary

Independent random variables

ststt ,0, and

00

02

,0 st

steVar etst

Of interest in the construction of models for general processes