of 52
8/10/2019 M343 Unit 15
1/52
FOR
REFEBENCE
ONLY
Mothemotics:
A Third
Level
Course
The
Open
University
Unit
l5
Time
series
ffiffiffiffiffiffiffiffiffi
W
W
n
ffiffi
fi
ffi
W
W
W
r
*Wffi
WW
ffieffiffi
ffiffiws@wt
ffiffi*ffimgws
ffifffiffiffireffi
wsE*&wtff
rsffi*@ffi
@re
WW
ffiEs ' GF6 ry
WW
WWffiffiW
ffimffiffiw
nnmffim
ffi
W
W
W
n
ffi
W
W
ffi
ffi
8/10/2019 M343 Unit 15
2/52
M343
APPLICATIONS
OF PROBABITIW
Mothemotics:
A Third
Level
Course
The
Open
Universifu
Unit l5
Time
series
Prepored by the
Course Teom
CONTENTS
Introduction
1
What
is
a
time
series?
1.1 Time
series data
1.2 Requirements
for
modelling
time series
2 Definitions
of stationarity
2.1 Strictly
stationary
processes
2.2
The
autocorrelation
function
2.3
Weakly
stationary
processes
(Video)
3
Autoregressive processes
of
order I
3.1
Definition
of
an
AR(1)
process
3.2
Stationary
AR(1)
processes
3.3
Aspects
of
stationarity
for
autoregressive
processes
4 More on
autoregressive
processes
4.1 Autoregressive
processes
of
order 2
(Audio)
4.2
General autoregressive
processes
5 Moving aYerage
processes
5.1 Moving
average
processes
of order
|
(Video)
5.2 Moving
average
processes
of
general
order
5.3
The
duality between AR
and
MA
processes
Objectives
Appendix:
Solutions to
questions
?
A
+
7
9
13
IJ
t4
15
t7
20
2l
2l
31
31
31
34
35
37
38
The
Open
Universitv Press
8/10/2019 M343 Unit 15
3/52
Statistics tables
The
recommended
book
of
statistics
tables
for
this course
is
H.
R. Neave,
Elementary
statistics
Tables
(George
Allen
&
Unwin, 1981).
In
this
unit,
these
tables
are referred
to
as Neaue.
Unit
titles
1
Random
Processes
2 Events
in
Time
3 Patterns
in
Space
4 Branching
Processes
5 Random
Walks
6
Markov
Chains
7
Birth
Processes
8
Birth
and
Death
Processes
9
Queues
10
Epidemics
11
More
Population
Models
12
Genetics
13 Renewal
Models
14
Diffusion
Processes
15
Time
Series
16
Problems, Problems,
Problems,
...
The
Open
University
Press,
Walton Hall,
Milton
Keynes.
First
published
1988.
Copyright
O
1988
The Open
University
A11 rights reserved.
No
part
of
this
publication
may
be
reproduced,
stored in a retrieval
system
or transmitted
in
any
form or
by any means,
without written
permission
from
the
publisher.
Designed
by the
Graphic
Design
Group
of the Open
University.
Produced
in
Great
Britain
by
Speedlith Photo
Litho
Limited,
Longford
Trading
Estate,
Manchester
M32
0JT.
ISBN
0 335
14299
0
This
text forms
part
of the
correspondence
element
of
an Open
University
Third
Level
Course.
Ior
general
availqbility
of supporting
material
referred
to in
this
text,
please
write
to:
o_p_en
U-niirersity
Pducational
Enterprises
Limited,
12
cofferidge
close,
Stony
stratford,
Milton
Keynes
MK11lBY,
Great Britain.
Further
information
on
Open University
courses may
be obtained from:
The Admissions
Office,
The
Open University,
P.O.
Box
48,
Milton
Keynes MK7
648.
1.1
8/10/2019 M343 Unit 15
4/52
ntroduction
-
.
::.:
::.irrst
commoniy
occurring
forms
of
statistical
data is the
time
series.
slp1r.
a
time
series
is
a sequence
of
observations
made
at
regular
time
ri: :-
\'tru have
met
such
sequences
before in
the
course,
and elsewhere.
For
:,
::-,:.
::e
ueekly
chicago
wheat
prices
discussed in[]nits
1
and 14
form
a time
\ld
the
monthly
UK
unemployment
figures
form
a
time
series,
which is
:-
::3:irred
graphically
in
the news media.
-
--:
:-':rstical
anaiysis
of data
from
time
series is
a
topic
of
importance.
The
main
--
such
an
analysis is
often
to
fit
a model
to
the data
that
will allow
forecasts
,
:'. :-ade
of future
values
in
the
series.
In
this
unit
some
of the most
important
-,:rirstic
models
used
for
this
purpose
are
introduced
and
developed;
but
it
is
.
:,rssib1e
in
one
unit to
describe
also
how
to
fit the models
to data,
or how
to
,:-m
lor
forecasting.
However,
after
studying this
unit,
you
should have
a
';nderstanding
of
how
the
models
work
that
will
stand
you
in
good
stead
if
.
-
Jlr
on
to
study
time
series model
fitting
and forecasting
elsewhere.
And
the
-::ls.
as well
as
being useful,
have some
fascinating theoretical
properties.
S:;tion
1 several
examples
of
time series data are
presented,
and the
kinds
of
that
they show are
discussed.
The idea
of a stationary time
series-
-
-_llilv
speaking, one
that 'looks
the
same
all
the way
along'-
is
introduced.
of
stationarity
are developed in
Section
2.
One
of
the key features
of
-:
series data is
that successive
observations
are usually correlated,
and
in
3 we consider
one
kind
of
stochastic
process,
the autoregressive
process
of
'::r
1.
which represents
this
dependence
between successive observations
in
a
simple
way.
In
Section 4,
the models
of
Section 3 are extended,
and
processes
of
order 2 and
higher orders
are defined. Another
kind
of
used to model
time
series data
-
the moving
average
process
-
is
in
Section
5, and
the relationship
between
autoregressive processes
and
average
processes
is discussed.
r
carry
out
some
of
the calculations
in
this unit,
you
need
to
be
able to
solve
first-
and
second-order
recurrence
relations
with
constant
coefficients.
You
be
familiar
with
the
techniques
required
from
(Jnits
5
and
6;
if
you want
to
yourself
of
them
you
may
find
it
helpful
to
read the
Handbook
entry
on
relations.
Throughout
this unit
the ideas
of stationarity
and
of
stochastic
processes
are used;
you
met these
first
in
Unit
2,
in
context
of
point
processes.
And
Section 2 contains
references
to
an
example
unit
6
on rainfall
in Melbourne,
though
you
do not
need
to remember
all
the
of this
example
in
order
to
study this
unit.
1 is
relatively
short,
while
the
other
four
sections
are all
about
average
in
Band
G
of
the video-cassette
is
associated
with
this unit.
you
will
gain
benefit
from
it
if
you
watch it
at
the
end
of Section
2,
before
the various
time
models
are introduced,
and again
in
Section
5, after they
have
been
studied.
is
also an
audio-tape
session;
this is
associated
with Subsection
4.1.
What
is
a time
series?
time
series is
a sequence
of observations
made
at
regular intervals
of time
-
erhaps
every minute,
or
every
week,
or
every
year.
The
kind
of
stochastic process
can
be used
to
model
time
series
data
thus appears
to
be
a
process
1,...) in
discrete
time,
where
the
time
unit
is
one minute,
or
one
week,
whatever
length
is
appropriate
for
the data.
what
about the
state
space?
It
be reasonable
for
the
state space
of the
process
to
match
the
data
-
if the
consist
of, say,
temperatures
taken
hourly in
a cold
store, then
a continuous
space
would
seem
appropriate,
while
if
the data
were counts
of
animals
livinga certain place,
a discrete
state
space
might
be
better.
In
fact, in
this
unit
(as
in
time
series
analysis)
only
a continuous
state
space is considered.
If
the
data
really
discrete,
the model
will
be
only
an approximation
-
but
then,
models
always
approximations
Unit l,Example 4.10;
Unit 14, Subsection
1.1.
In
earlier units
n
has
been used
for
the time variable in
discrete-time
stochastic
processes.
However,
it is
usual
to
use
t for the time variable
in time
series models, so r will
be
used
in
this
unit.
8/10/2019 M343 Unit 15
5/52
units
4,
5, 6,
13
Unit
15
Continuous
units
2, 7, 8, 9, 10, 13
Unit
14
To
summarize,
therefore,
in
this
unit
we shall
investigate discrete-time,
continuous
state space stochastic
processes
that
are
useful
for modelling time
series
data.
And
so the course
has
covered
all four
possibilities
shown
in
the table below.
1.1
Time
series data
To
progress
further
with
the
development
of
models,
we
must
look at some
time
series
data
to
investigate
the sort
of
properties that
the
processes must model.
This
subsection
consists
largely
of
a number
of
plots
of
time series
data which,
taken
together,
illustrate
most of
the
important
features
arising
in such data.
Figure
1.1 shows
the
annual
production
of lumber
in
the
USA
ftom
1947 to
1976'
Figure
1.2 shows
the annual
population of the
yellow-eyed
penguin in Dunedin,
New
Zealand,
from
1937
to
1952.
Lumber
(millions
of board
feet)
Pairs of
penguins
34000
32000
30000
Though the
values of
the series
in
Figures
1.1 and
1.2 are defined
onl;'
for whole
years
-
1947,
1948, ar.d
so on,
in
Figure
1.1
-
in the
graPhs
the
points corresponding
to
these
years
are
joined
with straight
lines.
This is a common
convention
for
time
series
plots,
and
it
will
be
used
throughout
this
unit.
i935
1910 1945
1950
1955
Year
Figure 1.2
Population of
the
yellow-eyed
penguin
(Megadyptes
antipodes)
in
Dunedin,
New
Zealand
(Source:
MST204 Unit 3)
100
t946
1956
1966
1976 Year
Figure 1.1
Annual US
production
of
lumber
(Source:
B. Abraham and J.
Ledolter, Statistical
Methods
for
Forecasting
(Wiley,
1983)
p.83)
In both
cases, the
graphs
appear
to
be
moving
irregularly up and down within a
fairly
restricted
range.
On
the
evidence of Figure
1..2,
for example, one would
probably
be
prepared
to
predict
that
the
penguin
population
in 1954
would
be
somewhere
near
90
pairs.
Both
these
plots
have the
property
that,
roughly
speaking,
they have
similar
characteristics
('look
the
same') all
the way along.
Other
timeseries
do
not look
like
these
two.
Figure
1.3 shows
the
production of
sulphuric
acid
in
the
UK,
measured
monthly,
from
July
1983
to November
1987.
Question
1.1 Describe
the
pattern
in
the
data
of
Figure
1.3.
Slow changes
in
the
overall
level
of
the
process,
like
those
in Figure
1.3, are
called
trends.
Another
common
pattern in
time
series
is
seasonal
uariation.
Figure
1.4
shows
the
monthly mean daily sunshine
in
Northern
lreland, from
January
1976
to
June
1986.
The series
moves
up
and down,
but not
entirely
irregularly:
there
is
a
pattern
related
to
the time
of
year,
with
sunshine
hours
not
surprisingiy
being
high in
summer
and
low
in
winter.
However, this
seasonal
variation is
not
completely
regular;
there
is random
variability
superimposed
on
the seasonal
pattern. Other
time series
show seasonal
variation
in addition
to
overall trends
in
the
level of the
process
-
see,
for
example,
Figure
1.5
which
shows
monthly
car
sales
in
Quebec
from
January
1960
to
December
1967.In
Figure
1.5
the trend
in
the
process
appears
to
be a
linear
increase.
70
8/10/2019 M343 Unit 15
6/52
iuutrr"t.-:
=trd
r
T '[lrilU[-jr1]
ilm uLk-,:
n mntr\
l:,
;
-
llll
,
1976
1978
198.1
1986
Year
Figure
1.4
Monthly
mean
daily
hours
of sunshine
in Northern
Ireland
(Source:
Annual
Abstract
of Statistics,
1987
{HMSO)
p.
a
)
Price
(f)
4
0
f-.-,
0
1987
Year
100% acid)
'
lg15 lq76
1t)77
1978
1979
Year
Figure
1.6
Monthly average
wholesale
price
for
10
England
and Wales
produced Baccara
roses
(Source:
Agriculture
Statistics,
England,
1976-77 and
1978
79,
Ministry
of
Agriculture.
Fisheries
and
Food)
1983
1984
1985
1986
Fgure
1.3
UK
production of
sulphuric
acid
(thousand
tonnes,
as
S".urse:
tr{onthly
D,igest
of Statis/ics,
January
1988
(HMSO) p'
57)
Hours of
f
:u
ns
n
il'lc
I
i0l
I
l
8l
I
j
6l
1
950
1962
t964
1966
Year
Figure
1.5
N{orrthly car
sales in
Quebec
(Source:
Abraham
and
Lcdoiter,
Statistic(il
Methods
for
Forecasting,
p.420).
Questier*
n.2 tr)escribe
the
pattcrn of
the time
series
in
Figure
1.6, which
shows
the
rnr-rrrflrly
averilge
wholesale
price
for
10 red
roses
in English and
Welsh
whuJ;r;:ii+
milrket
irorrr
January
1975
to
December
1979.
tr
8/10/2019 M343 Unit 15
7/52
$tofornnr,,-c
r.-rd
r
Ilnrudlu-d'n
mni{ltffi}i}:
XJEITIb:\
ll
I
_
llr
)
.l
i
lfil
i
i
l8f)-1
I
I
t
I
i60
FgEe
13
UK
production of
sulphuric
acid
(thousand
tonnes,
as
100%
acid)
,S..urce: Monthly
D,igest
of
Stati.sfics,
January
1988
(HMSO) p'
57)
1 983
Hours of
\unshine
i{l
1984
1986
Year
Figure
1.4
Monthly
mean
daily
hours of
sunshine
in
Northern
Ireland
(Source:
Annual
Abstract of Statistics,
1987
(HMSO)
p.
a
)
Price
()
4
198,i
1985
1986
1987
Year
"
1975 1976
197'1
1978
1919
Year
Figure
1.6 Monthly
average
wholesale
price for 10 England
and Wales
produced Baccara
roses
(Source:
Agriculture
Statistics,
England,
1976-77
and
1978-79,
Ministry
of
Agriculture,
Fisheries
and
Food)
(t
4
Car
sales
I
,
rhousands)
|
-301
l
j
l
2aI
1964
1966
Figure 1.5 Monthly
car sales
in
Quebec
(Source: Abraham
and
Ledcriter. Statistical
Methods
for
Forecasting,
p.420).
Questig*
1.2 Describe
the
pattern of the
time
series
in Figure
1.6,
which shows
the
motthlir
average
wholesale
price
for
10 red
roses
in
English and
Welsh
whcjl";rlai,j
niarkf,ts
from
January
1975
to
December
1979.
tr
8/10/2019 M343 Unit 15
8/52
The
patterns
shown
by
the time series
in Figures
1.7,1.8
and
1.9 are
particularly
interestine.
Number
of lynx
pelts
(thousands)
182()
1840 i860 1880
1900 1970
1940
Ycar
t'igure
1.7
Variations in the
population
of lynx
in
the
Hudson Bay
area
of
Canada,
as reflected
by the number of
pelts
sold
to a company each
year
(Source:
MST204 Unit
3)
Number of
lemmings
(per
hectrrc)
1931
1933
1935
1931
[943
Year
Figure 1.8 Population density
of lemmings
in
the area
near
Churchill, Canada
(Source:
MST204 Unit 3)
1770 l7q0 l8l0 1830
l8-5{
Figure 1.9 Annual sunspot
numbers,
1770-1869
(Originally pubiished by Wcilfer)
1870
Year
Figures
1.7 and
1.8 show estimates
of the
populations
of
two
mammals,
lynx
and
iemrning,
in
parts
of
Canada,
while
Figure
1.9 shows
the
sunspot
numbers
counted
annually,
according
to
a standard
method,
over
a
period
of 100
years.
All these
series
exhibit
an
oscillatory
variation
over
time,
but
the
period
of
the
oscillation
does
not
appear
to
relate to the
seasons
of
the
year,
for
instance. Furthermore,
the
'peaks'
of
the oscillation are
not
always exactly
the
same number of
time
periods
apart.
For
instance,
in
Figure
1.9 the times between
peak
sunspot
years
range from
7
to
l1
years.
Thus
these series appear
to
show
a different type of
periodic
behaviour
from that
in
the series
in
Figures 1.4,
1.5
and
1.6. In those series,
the
periodic
variation
is
clearly
tied
to
the seasons,
and
the
peaks
and troughs
occur at
ioughly
the same
times every
year.
Therefore,
different
kinds
of
modei
are
needed
to
deal
with
the
two
different
kinds of
situation.
8/10/2019 M343 Unit 15
9/52
fi
.
"';irements
for
rnodelling tirne
series
-
-
::
--,c;ls
for
time
series
data,
as
in
most rnodelling,
it
pays
to
start
:.:i..rrv modeis
can
be developed
for
data
iike
those
in Figure 1.1,
'
r.-
,:end and
no
seasonal
variation,
tlien
it may be
possible
to
extend
,
-
-:
-:
:.,
take account of trend and seasonal
variation.
'
-
-
:
:' e
similar
characteristics
aiong the entire time
axis, like those
in
:
:-3
iometimes
described as
stationary"
They
can
be modeiled by
:
:.--,-esses
that
are also,
on
average,'the
same
ali
the way along the time
-
,.
-:-l.tic
process
defined
in
such
a way
that
its
properties
do not
vary
*
.
s
s:id.
ioosely,
to
be stationary;
definitions of what
is meant
by a
*:r-
:ltrCh&Stic
process
are developed
in Section
2.
-
:,-
-i
of one very simple
kind of
stationary
process is illustrated in
: ,.;
In
this simuiation, successive
values
of X,
are
independent
-..
:s iiom the
same distribution
in
this
case. a
normal
distribution.
0501001
Figure
1.10
A
series of
(simulated)
independent
observations
from
a normal distribution
with
mean 0 and
variance 1
::'rre
time series data can be
modelled
by
a
process
as
simple as this; but
most
are
:r.rie
complicated.
Look
againat Figure
1.1.
In
the
early
1950s,
the
data'stuck'for
. :eu'
years
around
the 38 000
level.
In the
late
1960s, the
data again
remained
,:irl,v-
constant for a
few
years,
this time
around 35000.
In
these data,
as
in
many
:iher
time
series, there
is a tendency
for
successive
values
of the
process
to
be
::iativeiy
close
to
one another.
Such a series
will
tend
to
look
'smoother'
than
a
.uccession of
independent values:
the differences
between
successive observations
*il1
be smaller
in
comparison
with
the
variability
of
the
series as a whole than
*'ould
be
the case for independent observations.
Therefore,
if
a
process
{X,}
is to
ce
a
suitable
model
for
these data, then
it must have the
property
that
successive
ralues
of
X,
are correlated
rather than independent.
This
type of correlation
is
:alled
autocorrelation because
it
involves the
correlation
of
part
of the
process
with
another
part
of
itself.
A
sequence
of independent
random
variables does
not
have
any autocorrelation.
Therefore
a
more complicated
model is needed
for
a series
like that
in
Figure
1.1.
Suitable
processes
that
involve autocorrelation
will
be
discussed
in
the remaining
sections
of
this
unit.
What
about
models
for
time
series that
are
not
stationary,
in that
they exhibit
trends or seasonal
variation
(or
both)?
It
is beyond
the scope
of
this
unit
to
describe
in
any detail
how to
set
up
such
models,
but two
basic
approaches
to
the
problem
will
be outlined.
In
the
first,
a
series
that
exhibits
trend and
seasonal
variation can be thought
of as
consisting
of a
slowly changing
level,
with
perhaps
a
seasonal
pattern
superimposed
on
it,
and
with
irregular variability
superimposed
on
these
two.
It may
be
reasonable to
model the values
in
such a series as
a
(deterministic)
trend
plus
a
(deterministic)
seasonal
pattern
plus
a
stationaty
irregular
(stochastic)
component.
Once the
trend and
seasonal components
have
been
removed, the
irregular
component
can
be modelled
using one
of
the
models
Autocorrelation was
introduced in
Subsection
4.3 of Unit
2.
8/10/2019 M343 Unit 15
10/52
for
stationary
series.
This
approach
is
illustrated
in
Figure
1.11,
which
shows
estimates
of
the
deterministic
trend
and
seasonal
variaiion
for
the
euebec
car
sales
{1i1
from
Figure
1.5,
together
with
the
remaining
stationary
irregular
component.
with
other
series
it
may
be more
appropriate
to
multiply
the
co#ponents
of
the
series
together.
Car
sales
(thousands)
0
Seasonal
vanatlon
10
0
-l0
Stationary
lrregular
componcnl
10
1060
1962
1964
1906
Year
Figure
1.I1
Trend,
seasonal
and
stationary
irregular
components
of the
euebec
car sales
series
from
Figure
1.5
Another
approach
to
modelling
non-stationary
time
series
data
involves
looking
at
the
differences
between
successive
observations
and
fitting
a stationary
model
to'
those.
Both
these
approaches
to
modelling
non-stationa
ry
data
involve
doing
something
to
the data
-
removing trend and
seasonal
components,
or taking
differences
-
nd
then
fitting
a
stationary
model
to
the
new
series.
So, even
if
tiuly
stationary
time
series
never
occurred,
it
would
still
be
important
to
develop
models
for
them,
and it is
to
such
models
that
we
now
turn
our
attention.
8
8/10/2019 M343 Unit 15
11/52
I
Definitions
of stationaritv
-
:':,,'
: ,
:he
basic
idea
of a
stationary
stochastic
process
was introduced;
in
"
".'"'
::, :,:rmai
definitions
of
stationarity
are developed.
A
stationary
point
rri
"r'
:
;*::
defined
inUnit
2.
In
Subsection2.1
this
definition
is
extended
to
s:-r:--
,,:,-nastic
processes
in
discrete
time;
the type
of
statiorrarity
defined
there
.",:,r
:r-"rr
stationarity.
Another
type
of
stationarity,
known
asweak
;i
';'
-..:s
of more
use
in
modelling
time series;
this type
of
stationarity
::i
::
.:e
.'oncept
of
the
autocorrelation
function
of
a
process,
which
was
iT
-
,:-:.i
lor point
processes
in
unit
2.
Autocorrelation
is
discussed
in
more
*:r
i:--
:-:rns
in
Subsection
2.2,
and
weak
stationarity
is studied
in
Subsection
2.3.
I I
Strictll'
stationary
processes
:
-
:
--.
a
point
process
was defined
to
be stationary if
the
distribution
of
the
"r
"-:.::
-ri
erents
occurring
in
the
interval
(0,r]
is
the same as
the
distribution
of
':
-:r'rer
of events
in
the interval(r,t
*
r],
for all
r
> 0 and r > 0.
That
is,
a
r
::.rcess
is stationary
if its distributional
properties
are unaffected
by
shifting
'
: 1;
aris by
an amount
r.
An
extension
of this idea can be used
to
define what
,
:
.i
1r
as
s/rlcf
stationarity
for
general
stochastic
processes.
A
stochastic
process
. >.rid
to be
strictly
stationary if
the
probability
laws describing
the shifted
--
:; >
-Y,*,]
are the same
as those describing
the
original
process
{X,}.
To
be
-
-=
:recise. consider
the
original
process
at
a number
of
time
points
ty t2, ...,
tn.
'-:
trocess is
to
be
strictly
stationary,
then the
joint
distribution
of
the original
r'
-;i:
ai these
time
points
should
be the same
as the
joint
distribution
of
the
-
..:i
process
at
the
corresponding
shifted
time
points
11
+
r,
t2
+
r,
...,
tn+
r.
-
r .s stated
formally
for
a discrete-time
stochastic
process
in
the
followine
: .::-:;tOn.
Unlt
2,
Subsection
1.2
A similar
definition can be
given
for
proccsscs
in
continuous
time.
\
stochastic process
{Xr
t
:0,1,
...i
is
strictly
stationary
if
the
joint
::.trrbution
of X,,,
Xlrr,...,
X,"
is
the same
as ihe
joint
distribution
of
\,
-,.X,r*,,
..",
Xt,+,,
for all
positive
integers
n, aii
(non-negative
integer-
.:lued)
times
rr,
t2,
...,
t,,
and
all
positive
integers
r.
:::..re
investigating
how
this
definition
applies
to
the
kind
of
stochastic
process
-.:i
to
model
time
series,
we shall investigate
how
it
applies
to
a simple process
,:rch
you
met
earlier
in
the
course.
Erample
2.1
Melbourne
weather
'-:.-
L'nit
6 the sequence
of
dry
and
wet days
in Melbourne
was
modelled
by a two-
::.:te Markov
chain,
with
states
0 (wet)
and
1
(dry), and
transition matrix
01
o
fo.azo
o. r7l'l
P: I I
I
l0.600
i.400_l
1t
was
shown
that this Markov
chain
had a limiting
distribution
with
probabilities
-.o
:0.175
and
n1
:
0.225
(correct
to
three
decimal
places).
It
was
pointed
out
ihat
a finite,
irreducible,
aperiodic
Markov
chain
(such
as
this
one) has
a
unique
stationary
distribution,
and
that this
is the
same as the
limiting
distribution.
The
stationary
distribution
has
the following property:
if the
initial
state
of
the
Markov
chain
has
this
stationary
distribution,
then the
state
distribution
at
every
subsequent
time is
also the
stationary
distribution.
In
symbols, if
the
process
is
denoted
by
{X,; t
:
0,
7,...},
and
if
P(Xo:
0)
:
no
and
P(Xo
:
1)
:
nt,then,
for
all
integers
t> 0, P(4:
0)
:
z6 and P(X,: 1)
:
nr.Now, if
n
:
1 is
put
in
the
definition
of
a strictly
stationary
process,
then
this
definition
implies
that
the
distribution
of
X,
must
be the
same
for
all
integers
r
)
0, so if the
distribution
of
Xo is
the stationary
distribution,
then
this Markov
chain looks
as
if
it
misht
possibly
be strictly
staiionary.
Unit 6, Example 1.4
In
this
example
X,
is
discrete,
taking
only the
values
0 and 1.
8/10/2019 M343 Unit 15
12/52
I
Definitions
of
stationaritv
: :i,:-'
:
"
:he
basic idea
of
a stationary
stochastic
process
was introduced;
in
'
.
d
-
: :rrma1
definitions
of stationarity
are developed.
A
stationary
point
n
-:
','
defined
in
Unit
2.
In
Subsection
2.1
this
definition
is
extended
to
*i:-r:*r
':-;hastic
processes
in
discrete
time;
the type
of statiorrarity
defined
there
, :'-
r-:;i
sttttionarity.
Another
type of
stationarity, known
as weak
r
.;
'r-."...
i.
of
more
use
in
modelling
time
series;
this
type
of stationarity
lr
:: .:-
concept
of
thc
autocorrelation
function
of
a
process,
which
was
:
-
,:,:=i
1.rr
point
processes
inunit
2.
Autocorrelation
is
discussed
in
more
-::
i
:--
r::rls
rn
Subsection
2.2,
and
weak
stationarity
is
studied
in
Subsection
2.3.
: I
Strictll.'
stationary
processes
r
-
'
-'.
a
point
process
was
defined
to
be
stationary
if
the
distribution
of
the
r
"-
r,:.
.ri
events
occurring
in
the
interval
(0,r]
is the
same as the
distribution
of
':
--:-:er
ofevents
in
the interval(t,t
*
r], for allt> 0 and
r >0.
That
is,
a
r
.:
::.rcess is
stationary
if its
distributional
properties
are unaffected
by
shifting
'
:
r---i
aris by
an amount
r.
An
extension
of this idea can be used
to
define
what
,
, .
,i
n as
stricf
stotionarity
for
general
stochastic
processes.
A
stochastic
process
.
.iid
to
be
strictly st(itionary
if the
probability
laws
describing
the shifted
--
-;s. -Y,,.].
are
the same
as
those
describing the
originai
process
{Xr].
To
be
-
-;
rrecise.
consider the
original
process
at
a number of time
points
tt,
tz,
..., tn.
::
process
is
to
be
strictly
stationary,
then
the
joint
distribution
of
the
original
:- ,'s at
these
time
points
should be the
same as the
joint
distribution
of
the
',:.1
process
at
the corresponding
shifted time
points
tl
+
r,
t2
+
x,
...,
t,
+.t.
:.:
.s
stated
formally
for
a
discrete-time
stochastic
process
in
the
following
-
r
.:-litLrn.
Unir 2, Subsection 1.2
A
similar definition
can be
given
for
flrocesses
in
cont.inuous time.
\
stochastic process
t
X,; t
:
0,
1,
"..-|
is
strictly
stationary
if
the
joint
:r.tribution
of Xrr,
Xr.,
...,
X,,
is
the
same
zrs
the
joint
distribution
of
\,
-..X,r*,,
..., Xtn+,,
for
all
positive
integers
n,
aii
(non-negative
integer-
.;lued)
times
/1,
t2,
..
.,
tn, and
all
positive
integers
r.
3:rt1re
investigating
how
this
definition
applies
to
the
kind
of stochastic process
,:r'i
to
model
time
series,
we shall
investigate
how
it
applies
to a
simple
process
i.
:rr-h
you
met
earlier in
the
course.
Erample
2.1
Melbourne
weather
'-r-
L-nit
6 the
sequence
of
dry
and wet
days
in
Melbourne
was modelled
by a two-
,--:te
Markov
chain,
with
states
0 (wet)
and
1
(dry), and
transition matrix
01
o
lo.sze
o.
r74l
P:
I
t.
/
|
0.600
1.400
]
Ii
was
shown tfrat
tnls Markov
chain had
a limiting
distribution
with
probabilities
-o
:
0.775
and zr
:
0.225
(correct
to
three decimai
places).
It
was
pointed
out
that
a finite,
irreducible,
aperiodic
Markov
chain
(such
as this one)
has
a unique
stationary
distribution,
and
that
this is
the same
as the
limiting
distribution.
The
stationary
distribution
has
the
following property:
if
the
initial
state
of the
Markov
chain has
this
stationary
distribution,
then
the state
distribution
at
every
subsequent
time
is also
the stationary
distribution.
In symbols,
if the
process
is
denoted
by
lX,;
t
:
0,
1,
...],
and
if
P(Xo
:
0)
:
z6 and
P(X.
:
l)
:
rr,rhen,
for
all
integers
t
>
0, P(Xt:0):
no
and
P(.Xt: l):
nr. Now, if
n:
1 is
put
in
the
definition
of a
strictly stationary
process,
then this
definition
implies
that
the
distribution
of
x,
must
be the same
for all
integers
/
)
0, so if
the
distribution
of
Xo
is
the stationary
distribution,
then
this Markov
chain looks
as
if
it mieht
possiblv
be
strictlv
staiionarv.
Unit 6,
Example
1.4
In
this
example
X,
is
discrete,
taking
only the values
0 and
1.
8/10/2019 M343 Unit 15
13/52
Question
2.1
Could this Markov
chain be
a
strictly
stationary
process
if the
distribution of
the
initial
state were
not
given
by no
and
nr?
n
Showing
that
the
definition
is
satisfied when n
:
1 is
not
sufficient
for
the
process
to be strictly
stationary; the
definition
must
be satisfied
for all
n.
A
complete
proof
of this will not
be
given
here; instead,
the
principles
underlying the
proof
will
be
sketched.
Let
t, t
t
I
and
t
+
i
+7
be
three
integer
times
in
increasing
order
(so
that
I and
j
are
positive
integers). Then,
if this
process
really is
strictiy
stationary,
it
must
be
the case that,
for
example,
P(Xt+t+:
:
l, Xt+i:
0,
X,
:
1)
:
P(.Xt+i+
j+,:
I,Xt+i+,:0,Xra":
I),
(2.1)
for
any
positive
integer
r.
Are these
two
probabilities
really equal?
Consider
first
the
probability
on
the left-
hand side. It is
the
probability
of the
following
event:
that
the
process
is
in
state 1
on
day
/,
then
in
state 0 I days later, and
finally
in
state
1
after another
j
days.
Since the
process
is a
Markov
chain, this
probability
is equal to
P(X,:
I)
x
P(X,*t:01X,:
1)
x
P(Xt+i+i:
llX,*;:0)
:
P(X,:
t)pf).p8)'.
Similarly, the
right-hand
side of
Equation
(2.1)
can
be
written
as
P(X,
-,
:
ll
P'ito
P'd\
But,
as
you
have
already
seen,
X,
has the same
distribution
for
all
r, so
P(X,*,: 1)
:
P(X,: 11
for
any
positive integer
t.
Therefore,
the
two
probabilities
in
Equation
(2.1)
arc equal, and we have
P(Xt+i+:: l,Xt+i:
0,Xr: 1)
:
P(Xt+i+
j+t:
l,Xt+i+t:0,Xra,:
l),
for
any
positive
integer
r.
Now
the sequence
of
states
101
in
Equation
(2.1)
could be replaced
by any other
sequence
100,
for
example.
So a
condition
similar to
Equation
(2.1)
holds for
any sequence
of three states. In
order
to
prove
that
the
process
is strictly
stationary,
it
is necessary
to
extend
the argument
to
cover sequences of any
number
of states,
not
just
three; and this
can be
done
by
a
similar
argument
to
that
used above.
So this
Markov
chain is strictly
stationary.
tr
If a
stochastic
process
{X,}
is
strictly stationary, then the distribution
of
X,
is the
same
for all
integers
/,
so the
mean
of
X,
is
the same for all r;
that
is,
E(X):p
fort:0,1,....
Question
2.2
Calculate
p
for
the Melbourne
weather
process,
given
that
the
initial
distribution
is the
stationary distribution.
tr
In statistical
terms, as
you
saw
in
Section 1,
what
makes a time
series
different
from
a
random
sample
is
that
successive
observations
are
correlated.
How
can
this
correlation be
measured? IfX and Yarc
any
two
random
variables, the
degree of
relationship
between
them can
be
quantified
by their covariance:
C(x,Y): El(X
-
p)(Y- p)l:
E(XY)
-
Hxtty,
where
p"
and
trt,
are the expected
values of
X and Y
Therefore
the relationship
between successive
observations
Xrand
X,*,
in
a time series
(or
any other
stochastic
process)
can be measured
by
the covariance
of these random
variables:
C(XbXt+):
El(X,
-
H*,)(Xr*r
-
ltx,*,)1.
Now, in
general,
the covariance
of, say, Xrand
Xrneed
not
be the
same as
that
of, say,
Xro
and Xrr.
But
suppose
that
the
stochastic
process
{X,}
is
strictly
stationary. Then
the
joint
distribution
of
X,
and X, is
the same
as
that
of
Xro
and
Xrr,
so
that
the
covariance
of
X,
and
X,
is the
same as
that
of
Xro
and
X,
1in
The sequence of states 1 0 1
is
being
used merely as an example to
demonstrate the ideas involved in
;
proof
of stationarity: there is
nothing special about it. Another
sequence, such as 00
1,
would
have
done
just
as well.
10
8/10/2019 M343 Unit 15
14/52
ffi
m" Thus
the
covariance
of
X,
and Xr*,
is the
same
for
all integer
varues
of
iff}:fllh
ir
n usual
to
write
this covariance
as
C|X"X'*t)
:7r.
ft*crip
I represents
the fact
that
the random
variables
on
the
left
are
one
ddti'.e apart;
this
difference
is
called
the tag. The
covariance y,
is
called
an
ffiry[ri'r4ce
because
it involves
the
covariance
between
randcm
variables
from
,k
parts
of
the
same
process,
and
hence
y,
is
called
the autocovariance
at
h
f
of
the process
{X,}.
h$
Zl
Melbourne
weather,
continued
mm
ft
Melbourne
weather process,
made
station
ary by
choosing
the appropriate
'rkrfrution
for
Xs,
the
value
of
y,
is
given
by
"y1:
C(XyX2):
E(X:X)
-
E(X)E(XI).
fu
frxr)
and
E(Xr)
are both
equal
to
,u,
the mean
of
the
process,
which
you
ffi
to
&
0.225 in
Question
2.2,
so
i'r:E(XtX)-0.2252.
,lr
I,
and X,
can each
take
the
values 0 and 1,
the
product
XrX,
can also
take
fr
ralues
0
and
1,
and
it
takes
the value
1 only
when X,
:
Xz
:1.
Therefore
E(X$): I
x
P(X1:
1,Xz:
1)
+
0
x P(X,
and
X2
not
both
1)
:
P(Xr:
l,Xz: l)
:
P(Xz:
llXr:
1)P(x1
:
1).
Frm
the
transition matrix
for
the
Markov
chain, we have
P(X2
:
llxr:
1)
:
p,.,
:0.4,
ul since
the chain is
stationary,
P(X1
:1):nr:0.225.
llence
and
E(Xd):
0.4
x
0.22s
:
0.09
7r
:
0.09
-
0.2252
:
0.039.
tr
The covariance
that
has
just
been calculated
is called
the
autocouariance
at
tag 1
of
the process
{x,}.
other
autocovariances
can be
defined
in
a
similar
way.
Suppose
that
{X,;
t:0,1,...}
is
a
strictly
stationary process.
Then
the
joint
distribution
of
-\
and
X,*.
is
the same
as
the
joint
distribution
of
Xo
and
X" for
any
positive
integers
t and.c,
so
that
C(Xt,Xt+"):
C(Xs,X")
for
all
positive
integers
t
and r.
In
other
words,
if
r,
and
t2areintegers
with
tz) tt
>
0,
then
the
covariance
between
x,,
and
Xr,
does
not
depend
directly
on
the
values
of
r,
and /r,
but
only
on
the
difference
tz
-
tt
between
them.
This
difference
is
called
the lag.
Thus
if
we
write
C(Xe,
X,)
:
y"
for
all
positive
integers
z, then
C{X^,
Xe)
:
ltz-tt.
(2.2)
The
covarianca
"-
c(xo,x") is
called the autocovariance
function
at
lag r
of the
process
{Xr}.
(The
quantity
y"
is called
the autocovariance
fuinction
because it can
be
thought
of as a
function
of
the lag
z.)
So
far,7"
has
been
defined
only
for
.c:1,2,...,
but
it is
useful
to extend
the
deflnition
to non-positive
integers.
If
r
:
0 is
put
in
the definition
above,
then
we
obtain
c(xo,xd: ys,
but
the
covariance
of
a
random
variable
with
itself is
just
its
vadance,
so that
lo:
v(xd.
Since
{x,}
is
a
strictly
stationary process,
we
have
V(X,)
:
V(Xo)
:
yo
for r
:
0, 1,
...
.
What
about
negative
values
of c?
Consider
the covariance
C(Xn,Xr). Attempting
to
use Equation (2.2)
to find
this
covariance
in
terms
of
the
autocovariance
function
leads
to
C(X4,X):Tz-+:t-2.
But
C(Xo,Xr):
C(X2,Xa):
lz,
so
?-z
:
y2.
And,
in
fact,
|
-":
l,
for
all
integers
r.
The
prefix'auto'has
Greek
origin
and means
'self'.
p
_
o[0.826
0.t741
I
10.600
0.4001
8/10/2019 M343 Unit 15
15/52
8/10/2019 M343 Unit 15
16/52
1r
..,.
{
*--i
:1e
covariance
of
X,
and X,*, is
the same
for
all
integer
values
of
-:-ri
:'.
n'rite
this covariance
as
-
i 1 \-.,
'l'
w r";*::
-
represents
the fact
that
the random
variables
on
the
left
are one
lr:
*:
::art:
this
difference
is called
thelag. The
covariance y1
is called
an
,iitrru,r
-,r"
j:-i'oecause
it
involves
the
covariance
between
random
variables
from
ri
iri-r:'-
r:rli trf
the
same
process,
and
hence
7,
is called
the autocovariance
at
lilm
'::;crcess{Xr}.
il,rrurnmrnul
1l
\Ielbourne
weather,
continued
'
-
1;
\f
li'rourne
weather
process,
made
stationary
by
choosing
the appropriate
".
,
-
-
-:
:
ior Xo,
the
value
of
7,
is
given
by
:
C(XrXz):
E(XrXz)
-
E(X)E(Xr).
'*
u
:
f
_
'
and
E(X)
are both
equal
to
p,
the
mean
of
the
process,
which
you
r-n: .,
:e
0.225
in
Question
2.2, so
.:E(X6z\-0.22s2.
,
.
.:.1
,Y:
can
each take the values 0 and 1, the
product
XrXrcan
also take
,r
:
-is
0 and
1,
and it
takes the value
1
only when
Xr:
Xz:1. Therefore
Et-Y,Xr):
1
x P(X,
: l,Xz:
1)
+
0 x
P(X, and X,
not
both
1)
:
P(Xr
:
l,Xz: 1)
:
P(Xz: llXt
:
1)P(X1
:
1).
;:
-
:
::e transition
matrix
for
the
Markov
chain.
we
have
prX,
:11X, :
t):
prr:
e.4,,
*ri r :tlJe
the
chain is
stationary,
P(X1: l):
nt:0.225.
-
:
':-
E(XJ):0.4
x
0.225
:0.09
...:
I
r
:
0.09
-
0.2252:
0.039.
n
--.r
"Lrvariance
that
has
just
been calcuiated
is
called the autocouariance
at
lag 1
of
-:3rocess
{X,}.
Other
autocovariances
can be defined
in
a
similar
way.
Suppose
'-
:i .Y,:
r
:
0,
1, . . .
]
is
a strictly
stationary
process.
Then the
joint
distribution
of
i .nd,Y,*,
is
the
same as the
joint
distribution
of
Xo
and
X, for
any positive
:..::ers
t and
r,
so
that
C(Xt,Xt+"):
C(Xs,X")
for all
positive
integers
t and r.
:.
..ther
words,
if
r,
and
t,
are integers
with
tz) tt
>
0, then
the
covariance
:':.'''\een
X,,
and
X,,
does
not
depend
directly
on
the
values
of /,
and
rr,
but
only
:. the
difference
t2
-
r,
between
them. This
difference is called
the lag.
Thus
if
we
,,:te
C(Xo, X,):
y, for
all
positive
integers
r,
then
C(Xt,,
X,)
:
ltz_,t.
(2.2)
The
prefix'auto'has
Greek origin
and means 'self'.
p
_
o[0.826
o.t74l
I
10.600
0.400
1
The
covarianc
)"
-
C(Xo,X,)
is
called the autocovariance
function
at
lag
r
of
the
rrt'rcSS
{X,}.
(The
quantity
7.
is
called
the autocovariance
function
because
it
can
r:
thought
of as
a
function
of the lag
r.)
So
far,
y,
has
been
defined
only for
r:
,2,...,
but
it
is useful
to extend
the
lefinition
to
non-positive
integers.
If r
:
0
is
put
in
the
definition
above,
then
we
,rbtain
c(xo,xo):
yo,
but
the covariance
of
a
random variable
with
itself is
just
its
iariance,
so
that
lo:V(Xo).
Since
{X,}
is a
strictly
stationary
process,
we have
V(.X)
:
V(Xd
:
yo
for /
:
0, 1,
... .
What
about
negative
values
of
r?
Consider the
covariance
C(Xa,Xr).
Attempting
to
use Equation
(2.2)
to
find
this
covariance
in
terms
of
the autocovariance
function
leads
to
C(X4,X)
:
niz-+:
I,z.
But
C(Xo,X)
--
C(X2,X+):
lz,
so
I-z
:7r.
And,
in
fact,
.i
_":
l,
for
all
integers
r.
8/10/2019 M343 Unit 15
17/52
Question
2.3 Find
the
values
of
ys
and
y-1for
the
(stationary)
Meibourne
weather
process.
n
Example
2.1
Melbourne
weathero
continued
To find
the
values
of
7"
for ali
integers
r
for
the Melbourne
weather process,
we
begin
by using
the
same
approach
that
was
used to flnd
yr^
For
positive
integers
r,
y,:
C(X6,X")
:
t'(X.
X")
*
E(X}E(X"),
and
E(xo)
and
E(x,)
are
both
equal
to
the
process
mean,e.225.
Just
as before,
E(X,X"):
I
x
P{XoX,:1)
*0
x
p(Xo&:0)
:
p(Xo
X,:
I)
:
P(X": 1'Xo:
1)
:
P(X,:
llxo:
1)P(Xo:
1).
Now
P(X,:
llxc
:
1)
is
the
probability
that
the
process
will return
to
state
1
at
time
r
given
that
it
started there
at
time 0; this
return
probabiiity
is
written
pf)r.
Denoting
the
n-step
transition
matrix
for
the
process
by
P('),
and
using
the fact
that
P('+1)
-
P(n)P,
it
can be
shown that,
for
n:0,1,.".,
pli')
:0.226pP'
+
a.174.
This is
a
first-order
recurrence
relation
of the form
ilnrt
:
cun
* d
for n
-0,
1,...,
which has the
solution
d(l
-
c")
u":
1_ c
4-
t"Llo,
prcrvided
c
11.
Comparing
the recurrence
relations
(2.3)and (2.4),
we have
u":
pY1,
c
:
Q.226 and
d
:
0.114.
To
use
Formula
(2.5)
to
write
down
the
solutron
of
the
recurrence
reiation
(2.3),
we need
uo; this is
p ol,
which is
equal
to
1 by
See Unit
6,
page
l0
definition. Therefore
0.t74(1_0
:?61
_L
o
))^n
P.i,i:u,
_0.226
4.174
:
ffi(1
-
0.226\
+
0.226'
:
0.225
(l
-
0.226\
+
0.226
:0.775
x
0.226'
+
0.225.
Question
2.4
Show thar,
for
positive
lags
z. the value
of the
autocovariance
function
l,
for the
Meibourne
weather process
is
given
b;r
T,:
4.775
x
0.225
x
0.226'.
Check that
this
formula
gives
1,
:
0.039,
as already
caiculated.
fl
From
Solution
2.3,
I'o
:
0.225
-
A.2252
:
A.225(l
-
0.225):
A.225 x
A.775,
so the
formula
for
y, given
in
Question
2.4 also
works
for
t
:
0.
And
since
1t":l
,,
this
formula
can be extended
to
cover
all
(integer) lags,
including
negative
lags,
by
writing
'it":
0.775
x
0.225
x
0.2261"t,
where
lrl
denotes the
absolute
value of r.
n
{2.3)
See Unit 6, Solution 2.5;
a stmllar
argument is required
to obtain
this
recurrence relation.
(2.4)
(2
5)
See
Unit 6,
page
17.
This
method
was used
tn Unit
6 to
derive
the stationary probabilities
zo
and z, for
a two-state
Markov
chain. As
n+
a), the return
probability
p(tl
4
0,225,
the
value
of n1.
(.2.6)
8/10/2019 M343 Unit 15
18/52
Question
2.3
Find
the
values
of
yo
and
y_,
for
the
(stationary)
Melbourne
weather
process.
tr
Example
2.1
Melbourne
weather,
continued
To
find
the
values
of
7"
for
ali
integers
r
for
the Melbourne
weather process,
we
begin
by
using
the
same
approach
that
was
used to find
yr.
For
positive
integers
r,
1t,:
C{Xs,X")
:
E(x.
X,)
-
E(X}E(X,),
and
E(xo)
and
E(x,)
are
both
equal
to
the
process
mean,0.225.
Just
as before,
E(XyX,)
:
I
x
P(Xo
X,
:
1)+
0
x
P(X,X,
:
0)
:
F(Xo X.
:
1)
:
P(X,:
1,Xo:
1)
:
P(X,:
liXo:
1)P(Xo:
1).
Now
P(x.
:
llxc
:
1)
is
the
probability
that
the
process
will return
to
state
1
at
time
r
given
that
it started
there
at time
0; this
return probabiiity
is
written
pf)r.
Denoting
the
n-step
transition
matrix
for
the
process
by
P('),
and
using
the fact
that
P('+1):
P(nrP,
it
can be
shown that,
for
n:0,1,...,
Pfrnt):
8-226pPt
+
a.174.
This is
a
first-order
recurrence relation
of the form
un+1
:
cu^
I tl
for
n:0,
1,...,
which has the
solution
d(I
-
c')
un:-,
1
C,uo.
I
-('
provided
c
11.
Comparing
the recurrence
relations
(2.3)and
(2.4),
we have
un:
pYt,
c
:
Q.226 and
d
:
0.114.
To
use
Formula
(2.5)
to
write
down
the
soluiron
of
the
recurrence
reiation
(2.3),
we need
ao; this is
pt0J,
which is
equal
to
1 by
definition.
Therefore
0.t74(1
-
0.216')
p'{'i
:u,:-
I
_
02N'
+0.226"
4.174
:
afr(I
-
0.226")
+
0.226'
:
0.225
(l
*
0.226)
+
0.226
:0.775
x
0.226,
+
0.225.
Question
2"4
Show that,
for
positive
lags
r,
the value
of the autocovariance
function
y"
for the Metbourne
weather process
is
given
bv
"1":
4.775
x
0.225
x
0.226'.
Check that
this
formula
gives
i,,
:
0.039,
as alreacly
caiculated"
n
From
Solution
2.3,
ta
:0.225
-
A.2252
:
A.225(.1
-
0.225\
:
A.225 x
0.775,
so
the
formuia
for
ir,
given
in
Question
2.4 also
works
for
r
:
0.
And
since
)t":1,,,
this
formula
can be extended
to
cover
all
(integer)
lags,
including
negative
lags,
by writing
"t,
:
A.l7
5
x
0.225
x
0.2251"t,
where
lri
denotes the
absolute value
of t.
n
(.23)
See Unit 6, Solution 2.5;
a similar
argument is required
to obtain
this
recurrence relation.
(2
4)
See
Unil 6,
page
17
See Unit
6,
page
10.
This
method
was used in
Unit
6
to
derive
the stationary probabilities
no
and z, for
a two-state
Markov
chain. As
n4
u:),
the return
probability
pY)t
r
0.225,
the value
of nr.
(2
6)
8/10/2019 M343 Unit 15
19/52
T'
The
autocorrelation
function
:.:rgating
the
relationship
between
random
varrables'
it
is sometimes
-itoco,rsiaertheircorrelationthantheircovariance.Thisisthecasefor
'iiabies
associated
with
stochastic
processes,
particularly
for
those-used
'':me
series.
The
correlation
p(X,Y)
of
two
random
variables
X and
Yis
C8.Y\
.\
Yt:
u/lvWvrvt)
: i
:
0,
1,
. . .)
is
a stochastic
process
in
discrete
time'
then
the
correlation
,In
and
X"
is
given bY
c(xo.
x,)
.,t
.\
^.
X,)
u/1v1xol
vtx,t)
-
.
rrLrcess
is strictly
stationary,
then
C(Xo,
X")
:
y",
and
V(Xl
:
V(X")
:
)o'
so
:
-
-:ielation
between
Xn and
{
is equal
to
y",fy6'And
then'
since
the
process
is
--:.-.
stationary,
the
corielation
between
X,aid
X,*"
is also
given-by
7'/y6'
for
l::-ger,>0.Thrscorrelationisknownastheautocorrelationfunctionofthe
-: s
at
lag
t,
and
is
denoted
bY
P,,
so
(21)
['t
-
ltl
lO'
.,
:,-,rrelationJunctionis
often
abbreviated
to
acf'
but
not
in
this
course')
1,1,rt:don
2.5
Find
the
autocorrelation
function
p"
of
the
Melbourne
weather
--:;utocorrelationfunction,andestimatesofitfromdata'areofkeyimportance
-
.:e
modelling
and
analysis
of
time
series'
:"3
Weakly
stationarY
Processes
S:
tar
in
this
section we have
concentrated
on
one
particular definition of
.."lionarity
-
strict
stationarity'
Not
surprisingly'
there
is-another
type
of
.::rionarity
called
weak
stationarity.
A stochastic
process
{xt}
it
weakly
stationary
.-
rhe
mean
of
X,
does
noi
a"p.nO
t"
t
u"d
the
covariance
between
X"
and
X"
:.p.nd,
only
onthe
lag
tz-'tt'
This
definition
is
stated
formally
below'
A
stochastic
process
{Xr;
t
:0,
1,
. .
.}
is weakly
stationary
if
it satisfies
the
followine
two
conditions.
E(X,)
takes
the
same
value,
p,
for
all
r'
For
all
non-negative
integers
t1
and
t2,the
value
of
C(Xt1,Xt2)
depends
only
on
the
lag
tz-
tr
The
autocorrelation
function
for
a
stationary
point
process
was
defined
rn
Unit
2,
Subsection
2.3'
The
definition
given
here
for
general
orocesses
does
not
correspono
ixactlv
to
the
deflnitionin
Unit
2,
becauie
that
deflnition
had
to
take
account
of
the
time
interval
over
which
the
Process
was
observed
on
each
occasron.
There
are
several
important
points
about
this
definition'
First,
Condition
2 applies
t o
the
case
t
r
:
t
z
:
t,
in
which
case
the
covariance
is simply
the
variance
of
X"
and
the
definition
says
that
this
variance
is
the
same
for
all
t'
That
is,
the
variance
of
the
process
is
constant
as
well
as
its
mean.
This
is clearly
a
basic
requirement
for
a
definition
of
stationurity'
S""o"aly,
if the
covariance
C(Xo'X')
is
denoted
by
1'.,
then
it
must
be
true
for weakly
stati'onary
processes
(as
for
strictly
stationary
processes)
that
y-.:
y,
for
all
t'
Thirdly,-it
is
important
to
realize
what
the
definition
does
not
,uy.
tt says
nothing
about
any
features
of
the
joint
distribution
of
the
X,
other
than
their
firit-order
moments
(i.e.
their
means)
and
second-order
moments
(i.e.
variances
and
covariances).
Thus,
for
instance,
it
is
quite
possible
to
have a
weakly stationary
process
in
which
the
shapes
of
the
distributions
of,
say,
X, and
Xo
aie
quite ditrerent,
though
the
means
and
variances
of the distributions
13
8/10/2019 M343 Unit 15
20/52
must
be
the same.
Thus
a
process
can be
weakly stationary
without
being
strictly
stationary.
on
the
other hand,
for
practical
purposes,
strictly
stationary processes
may
be
assumed
to
be
weakly
stationary
as well.
It is
the
weak
definition
of stationarity
that
is
used most
in
modelling
time
series.
There
are
basically
two
reasons
for
this.
First,
many
of the
properties
of stationary
processes
that
are
useful in
modelling
time series
data depend
only
on first-
and
second-order
moments,
so
it
is
unnecessary
to
use the
strict
definition
of
stationarity
for
these
properties
to
hold.
Secondly,
when
faced
with
time
series
data, it
is
not
usually possible
in
practice
to
estimate much
more
than
the
first
and
second
moments
of the
underlying
random
variables. Thus
it is
just
not
practicable
to
check
whether
the
process
is
strictly
stationary.
Question
2.6
Is
the Melbourne
weather
process
weakly stationary?
n
A Markov
chain
is
not
the sort
of
process
that
is commonly
used
to
model
stationary
time
series.
In
fact,
the most commonly
used
models
do
not,
in
general,
possess
the
Markov property;
they are
built
up
from sequences
of independent
identically
distributed
random
variables.
Now
one does want
successive
observations
in
a stationary
time
series
model
to
be
identically
distributed
(or
at
any rate
to
have
constant mean
and constant
variance),
but
a model in
which all
the observations
are
independent
will
not usually do, because,
as
you
saw
in
Section 1,
time series
very often
exhibit
a
kind
of
'smoothness'in
which
successive
observations
are relatively
close to one another,
compared
to
the
overall level
of
variability
in
the
process.
We shall consider
two
different ways
of
getting
the
required dependence
into
the model.
These
two
ways lead
to
mouing
aDerage
processes
and
to
autoregressiue
processes.
Band
G
of the video-cassette
demonstrates
how
simuiations
of
a
simple moving
average process
and
two
simple
autoregressive processes
can be
built up from
sequences
of independent
normal
random
variables
with
zero
mean
and constant
variance,
which
in
the
context
of
time
series modelling
are sometimes
called white
noise.
(The
term 'white
noise'
can
mean
different
things
in
other
contexts.
In
particular,
a continuous-time
white
noise process
can
be defined
in
terms
of
Brownian
motion
(unit
14),
and
this
continuous-time
process can be
approximated
by
sequences
of independent normal
random
variables.)
This is
a suitable
point
at which
to
watch Band
G of the video-cassette.
you
may
find
it
helpfui
to
watch it again
after reading
Subsection
5.1,
by
which
time
you
will
have
studied
the
definitions
and
properties
of the
processes
simulated
on
the
video-cassette.
3
Autoregressive
processes
of
order
1
In
this
section
and the next
we
study
a class
of
processes
known
as autoregressiue
processes,
which are
widely used
in
modelling
time series
data. In an
autoregressive
process,
each successive
observation
is a linear
function
of
previous
observations,
plus
a
further
random
term.
In this
section we discuss
autoregressive processes
of
order
1,
in
which the
observation
at time
r is
given
by
a
multiple
of the
observation
at time
r
-
1,
added to
a random
term. Autoregressive processes
of
higher
order,
in
which
the observation
at
time
r
is
defined
in
terms
of more
than
one
previous
observation,
are left
until
Section
4.
Under
some rather
special
clrcumstances,
a
process
can
in fa;:
be
strictly
stationary
without
bein,i
weakly stationary;
but this is
of
1i::,:
or no
practical
importance.
Any
strictly stationary process
that
has
finite
variances
and autocovariane:.
is
weakly stationarv.
HT
R"T?T'|
8/10/2019 M343 Unit 15
21/52
;Jdrliromr
j.l
-{
foraging
animal
-
::-
movement
of
an animal
that
spent
its
waking time foraging
for
food
unit
14, Example
2.2
i
'r
-'"::
bank
was
modelled. The
path
of
the animal
was approximated by
"
-
-.-
:rr)tion,
and a simulation of the animal's
path
was obtained as a
'r"
'
:
:
::iacement
of the animal
from
a reference
point
at time f
-
1. Then
'
--
,-
time
interval
from
time
t
-
1
to
time
r
the animal
moves
by
a random
..
:
-
-::r.ni.
Z, say, so
that its
displacement
at
time r is
'f,:Xr-r*Zr.
(3
1)
":
-
"i.e
assumptions
of ordinary
Brownian
motion, Z, has
a
normal distribution
:"n
0 and fixed
variance o2"(say).
Moreover, displacements in
different
unit
a
:
:-:i:\.als
are independent. Thus
the
animal's movement is being modelled
by
.:
: :r u'alk
with normally distributed
steps.
::-r.-ess demonstrates
how
a stochastic
process,
{X,},
can be
built
up
from a
,
"
-::-r- ',2,.,
2r,...
l
of
independent
identically distributed
random
variables.
-
rurcsfion
3.1
S:orv
that
the sequence
{Zt,2r,...}
for
the
foraging animal, treated
as a
,.-.;hastic
process,
is
weakly
stationary.
r,\':ite
down
its
autocorrelation function.
r
-.
the
process
strictly
stationary?
-
::-.
context
of
time series
models, a sequence
of
independent
identically
: ,::r:uted
random variables
hke
{Zr}
is called a
purely random
process
or
white
:xr,:,ne.
Such a
process
is
strictly
stationary,
whatever
the
distribution
of Zr,
and
is
i,::{i}'
stationary
as long as
the
variance
is finite.
But
clearly
it
is of
little
use
for
:-,
Je11ing
time
series
that exhibit
dependence.
',i
:.rt
about the
process
{X,}
defined
by
Equation
(3.1)?
There is
clearly
-.:endence
between successive
observations
where the animal
is
this
minute
-::ends
on where
it
was
a minute ago
-
but
a
major difficulty is that
this
process
.i-. is
not
stationary.
In Section 1
of Unit
14,
you
saw
that
the
variance of X, is
::rportional to
r,
rather than being constant.
So
this
process
may
be
a
use
