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Lecture 1
Good afternoon!
Lecturer: Dr. Natalia Janson
Department of Mathematical Sciences
Loughborough University
Loughborough
Office: W205
Tel: (01509) 22 2874
E-mail: n.b.janson@lboro.ac.uk
Module: MAC272 “Time Series Analysis”
2 lectures and 1 tutorial per week during weeks 1-12 of Semester 2, 2004/05
Coursework: 20%
Exams: 80%
Time Series Analysis by N. Janson
Time Series Analysis
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Lecture 1: Introduction
1.
2.
3.
4.
Processes, state variables
Signals and their examples
Time series: definition
Aims of Time Series Analysis
Time Series Analysis by N. Janson
What do we study?
Whatever is going on around us are processes occurring in certain systems. Some
obvious examples are:
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•the change of weather (system: Earth atmospehere)
•the change of illumination during the day (system: Earth atmospehere)
•the daily change in exchange rates (system: financial market)
•the change in monthly amount of beer drunk by a certain person (system: person)
In lay terms: process is the change in time of the state of the system.
Note: the state of the same system can be characterized by one or several variables.
Examples:
•weather at the current moment can be characterized by air temperature, humidity,
wind velocity, atmosphere pressure, etc.
•state of the person can be characterized by his/her body temperature, average heart
rate, average respiration frequency, blood pressure, appetite, etc.
One may record and observe the change in time of several, or of just one variable
characterizing the system state. The recorded dependence of some variable in time
is also called a realization.
Time Series Analysis by N. Janson
Marketing example:
wine sales of a certain company
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System: company
State variable: monthly wine sales
months
Data are taken from http://home.vicnet.net.au/~norca/Red_Wine.htm
Time Series Analysis by N. Janson
A medical example:
Human Electrocardiogramme (ECG)
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voltage
System: cardiovascular system of a human
Process: heart beats
State variable: voltage between two points on the human body.
~ 1 sec
time
Measures electrical activity of a human heart.
Time Series Analysis by N. Janson
A biological example:
position of a point on the surface of Isolated
Frog’s Heart
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coordinate
System: frog’s heart
State variable: position of a point on its surface
position of this point is recorded
time
Time Series Analysis by N. Janson
A mechanical example
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System: mechanical system
State variable: position of the load
Time Series Analysis by N. Janson
System, Process and Signal
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State variable 1
Signals
System
State variable 2
Time Series Analysis by N. Janson
Time Series
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Time series: a collection of observations of state variables made sequentially in time.
Univariate (bivariate, multivariate) time series: collection of observations of one
(two, several) state variables, each made at sequential time moments.
Note: the order of observations is important!
Notations
•continuous signal a(t)
•time series a(ti)=a(iDt)=ai, i=1,2,…,L
•sampling step Dt
•length of time series L
•sampling frequency fs=1/Dt
Synonims:
•Time series, (experimental) data, sampled signal, discretized signal
•Sampling rate (step), discretization rate (step)
•Time Series Analysis, Data Analysis, Signal Processing, Data Processing
Remark:
Mathematically, “time series” is not a SERIES, but a SEQUENCE!
Time Series Analysis by N. Janson
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Pressure, au
Example of time series:
blood pressure of a rat
Time Series Analysis by N. Janson
Aims of Time Series Analysis
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1. Description
Describe (characterize) a generating process using its time series.
2. Explanation
If time series is bi- or multi-variate, then it may be possible to use variations in one
variable to explain the variations in another variable.
3. Prediction (forecasting)
Use the knowledge of the past of the time series to predict its future.
4. Control
To change deliberately the properties of the process by influencing it and
observing the changes introduced by our intervention. One can then learn to make
the needed effort to achive control.
Time Series Analysis by N. Janson
Example of description
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Assume the time series shows the tendency to repeat itself with
some accuracy. ECG shows a sign of periodicity.
Then one can assume that the process is inherently rhythmic, and can estimate
the average or most probable rhythm in it.
The average rhythm of heartbeats can be estimated from estimating the
rhythm of ECG.
For information:
Average heart rate of a healthy
Human is ~ 1 sec.
Time Series Analysis by N. Janson
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Example of explanation
Three signals are
measured
from the same ill human
simultaneously:
Electrocardiogramme
(ECG),
pressure, respiration.
Floating of average
level of ECG and
especially of pressure
are caused by breathing.
Time Series Analysis by N. Janson
Example of prediction
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Weather forecast
A lot of experimental data are measured during a certain time interval.
The data are being analysed, the tendencies are being revealed.
From what is available by the current moment the future weather is predicted.
Time Series Analysis by N. Janson
Example of control 1
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Balancing a tray.
Time Series Analysis by N. Janson
Example of control 2
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A sailing boat is being navigated in windy weather. It needs to go in the
particular direction, and this direction is governed by the angle between the wind
and the sail. The wind is occasionally changing its direction. The sailor
needs to adjust the angle between the sail and the wind in such a way that the
direction of motion is kept as constant as possible.
System: atmosphere interacting with the sail
Process: change of the direction of sail
Signal: angle between the sail and the wind.
Time Series Analysis by N. Janson
Example of control 3
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Imagine rainy, windy weather, and the wind changes its direction all the time.
A girl is holding an umbrella. In order to protect the umbrella from breaking, its
roof should be held perpendicular to wind.
System: atmosphere interacting with the umbrella
Process: changing of the direction of the wind
The girl’s brain “measures” (without perhaps the girl realizing it) the angle between
the stick of umbrella and the wind.
Signal: the angle a between the umbrella stick and
the wind
If this angle a deviates from zero, the girl turns the
umbrella in order to reduce angle a to zero.
Time Series Analysis by N. Janson
How time series can arise
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1.
Given a continuous signal, one can sample its values at equal time intervals.
Example: sampled human electrocardiogramme
2.
The value of the state variable aggregates (accumulates) during some time interval.
Example: daily rainfall
3.
Some processes are inherently discrete.
Example: trains arriving to the station at discrete time moments
Kinds of processes
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•
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Random (stochastic) process
Deterministic process
Mixed
Time Series Analysis by N. Janson
Outline of the course
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Assumption: The process is random (stochastic)
We assume that the process obeys probabilistic laws, or that the number of
influencing factors is too large to be taken account for. We do not assume the
existence of deterministic model governing the behaviour of the system considered.
This is the most general assumption that can be applies to all processes observed.
To be able to judge about the properties of random processes from observing
their time series, one should know the theory of random processes in the first
instance. We will therefore start from the theory of random processes.
After we grasp the ideas of the theory of random processes, we will learn how
to extract the necessary information from the time series. We will mostly consider
univariate time series.
Time Series Analysis by N. Janson
Homework
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Problem Sheet 1
1. Give examples of situations in which time series can be used for
explanation, description, forecasting and control.
Time Series Analysis by N. Janson