*Currently on leave froB the University of New South Hales
ESTHiATION IN ONE OmENSIONAL BILATERft.L PROCESSES
by
C. A. i'kGILCHRIST*
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics ;H'Ileo Series
June, 1974
;')0.
930
Estimation in One Dimensional Bilateral Processes
C. A. McGi1christ
University of North Carolina*
SUf1r,1ARY
A method of fitting a bilateral, autoregressive, moving average process
is described with emphasis placed on the ca.se in which only short sequences of
observations a.re available.
Such a process is fitted to individual plant
yields in a monoculture plot.
Keywords:
1.
Bilateral, autoregression, moving average, plant competition.
INTRODUCTION
,A
sequence of observations
Xl' X , ...
2
forms a one-
dimensional bilateral process if the observation at position
observations at either side of
t.
depends on
A model of the form,
+ ~ X
+
+ ~ X
+ ~ X + ... +~ X
l' t-l
..•
'l'p t-p
't' -1 t+l
'¥ -T--t+r
xt
t
=
may be described as a bilateral, autoregressive, moving average process of
order p, q, r, s •
and
4>i' 6j ,
0
2
In the model the
are unknown parameters.
are independent
N(O,
0
2
)
variates
The model is analagous to those of
Box and Jenkins (1970) and using a similar notation to these authors we call
the model the
operator
*
BX
t
(p, r,
= Xt _l
0,
q, s)
model.
' the model may be
Letting
v~itten
on leave from the University of New South Wales
B be the backward shift
-2(1 + cP1B + .•• +
<p
p
BP + <p
eq Bq
+
-1
B-1 + ..• + 4>
e-1 B~l
+ ... +
-r
=
B"·r)X
t
e-5 B-S)Et
'
or more simply
Usually the above model is not sufficient to describe
because all means are ~ero.
a deterministic trend
experL~ental
data
It is enough for our purposes to add to the model
Pd_l(t) , which is a polynomial in
t
of order
d - 1 .
Thus
and this model is referred to as the
(p, r, d, q, s)
cations, including the one discussed in this paper,
model.
d
In many appli-
equals
1
indicating
a process with unknown mean.
The problem which motivates this study is that of fitting a bilateral
process to the dry weight of individual plants growing in monoculture plots.
In a particular plant competition studytmonoculture plots of each species
contained 20 plants; two rows each containing 10 plants.
For purposes of
estimating appropriate variance components (which turn up in other analyses
that need not concern us here), it is enough to consider the average
plant weight) for each column.
yields
X ' t
t
= 1,
2,
£n
(dry
The basic data we handle are such average
, 10 , and each plot contributes such a series.
Bilateral models for one and two dimensional processes have previously
been considered by T'Thittle (1954) and "!atern(l96o).
\'Thittle reconnnended
fitting the one dimensional, bilateral, autoregressive process by fitting a
unilateral process wl1t1Lthe same spectral demdtv.function.
The parameters of
-3--
the unilateral process derived in this way from a stationary bilateral process
are such that the unilateral process is nonstationary and attempts at fitting
Such a model using procedures in
McGilcr~ist
(1974), which are specifically
suited to small samples, produced nonsense results.
In the following sections we advance a direct method of fitting the
bilateral process and, like the method of McGilchrist (1974), the procedure is
intended particularly for small samples.
Since the general model is as easily
handled as any simple model we discuss the general case.
2.
REl110VAL OF TREND
In a short time series the order of a polynomial trend
can be obtained by simply graphing the series.
The usual Box-Jenkins procedure
for elimation of trend is used, viz. the difference operator
On
Pd_l(t)
- B)~t
= (1
.. B)dPd..l(t) + (1 - B)d</l-l(B)S(B)e:
= </l-l(B)(l
Thus
of
</l(B)Y
*
e (B)
encing there are
T + d
3.
2
t
t
The coefficients
observations in each time series so that after differ-'
T observations
average model with zero mean.
a
- B)dS(B)E
= e*(B)€t ,where (1 - B)de(B) = e*(B) •
t
are related to the coefficients of e(B) •
We assu,,"t1e
and
acting
removes it so that
= (1
Yt
(1 - B)d
Yt
which follow an autoregressive, moving
The coefficients to be estimated are still
¢i'
•
ESTIMATION The method of maximising the liklihood conditional on fixed
border elements is used.
The border elements are the first
p
values of
Y
t
-4and the last
r
values.
Let
= e*(B)Et
E
t
=P
' t
then the border elements and the
+ J.~ p + 2, •.. ~ T - r
determine the
Be.
Y
t
according to the
following equation,
1
~-l
1
4>1
...
0
4>_r
...
Ep +l - 4> 1Y1 -
Yp +l
Yp +2
~
-
Y
PP
E2p - <P1Y2p-l
4>p
E2p+l
=
(3.1)
4>_r
E _
T 2r
1
0
¢p
...
E.p_-2r+
. 1 -
¢-l
4>1 1
YT-r
-1 YT·-2r+2
...
T-r -4> -1 YT··r+l
E,.
Let
We introduce the following notation.
type
~
be an
A
_n
given on the left-hand side of (3.1).
n x n
•. ¢
y
-r T
matrix of the
Let
••. ~ F + ] ~ n = 1, 2~ ... , T - r - p
p n
a 2~l_n
= Var
e_n
Now since
Cov(E , E +T )
t
t
=a
2 q+d-T
I
j=
-s
**
8 j 8 j +ITI
,
0
= alTI
0
ITI s q + d +
S
,
ITI
S
,
ITI
S
q + d +
, ITI
>
q + d + s
S ,
>
q + d +
-5= a./a
then for
2
and
~
k=q+d+s
o
Let
If we delete the
obtain a
E
t
T - P - r
:n
v' = [Y + ' Y + '
p l
p 2
...
) Yp+n ]
Z = </>(B)Y
t
t
z'
_n = [Zp+l' Zp+2'
...
, Zp+n]
terms from the vector on the right-hand side of (3.1) we
dimensional vector in which the first
p
and last
r
terms are linear combinations of the border elements and the remaining terms
are zero.
Let this vector be denoted by
Jl.
Thus (3%'1) becomes
and the liklihood is the probability density function of
elements held fixed.
E(y
_T-p-r )
Y
_T·-p-r
with border
Since
= -:"'"'T-p-r
A:l
Jl,
_
Var(Y_T-·p-r )
= a2
-1
L
('
~ ..p-.r _T·-p-r -s.-p-r
f- l
the liklihood function is
L
-1
= ( 2wa 2)-~(T-p-.r) I-:"'"'T-p-r
A_
~
L
_T~p-r
(A~'
~"'T-p-r
).-ll·-~exp ( 2
-Q=
-~-p-r
/ 20 2)
,
-6where
2
Q,f-p-r
= (~T-p-r
=
Now
A: l
- ~~-p-r ~
(~-p-r ~T-p-r
)
I·
r-
1
L
)
~
(v
A:l
~~-p-r _T-p-r ~~-p-r :T-p-r - ~~-p-r~
~)' f;=p.-r(~T-p-r
Y.
_ -p-r =L
~~-p-r _T-p-r
ZT
-~
2
so that
~)
:T-p-r -
Q;j;-p-r
= Z_T-p-r
I
\-1.
Z
_T-p-r_T-p-r
L
and
On maximising with respect to
d2
0
2
we find
2
= 0"'T-p-r
/(T-p-r)
In what follows we choose estimates by minimising
RT-p-r
= (T-p-r)in(~l-p-r
and we do so by finding an
parameter values.
The
algorit~~
minim~~
4. COMPUTATION OF lIKFlIHOOD
.Q,nl~_p_rI2
_L-p-r I
+ inllm
to evaluate
for any specific
R
- T··p-r
is found by standard iterative procedures.
The method of computing
involve approximations and is particularly suited to
R
T-·p-r
~all
does not
samples.
Some of
the computing procedure is the same as that reported in T'JTcf1ilchrist (1974) but
results are given here for completeness.
Let
Computation of
dimensional vector and
elements of
p_n •
rev(p)
_n
First find
I.
~n
=
[
0, .•.• 0, sk' .•. , s2 ]
be an
n
stand for the vector obtained by reversing the
Yt
= (1
- B)dx , Zt
t
= ~(B)Yt
and
-7-
e*(B)
= (1 -
B)de(B) •
0
2 _
-
~
,.1
=q
k
Then for
r- l
Z
l Z
~k ~k ~k
+ d + s
~
'~k
find directly
l
= LrP
~k ~k
•
g~n 7 n2
"'n'
The following recurrence formulae give successively
T - P -
r
= (81
_ pl' g )-1 , b
~n
~n
~nlfT_p_rl
Computation of
+ 1 , . •• ,
1n lAm
Computation of
~n
ILkl
~n
is calculated bv the usual
Further
2
~.l.-p-r
= rev(p')g
n
Firstly
fk'
subroutines when inverting
(iii)
= k'
.
c2
n
(ii)
n
I:
Let
a_n ,d_n
be tuo
n
dimensional
vectors,
. ..
let
m
= max(r,
,
p) .
<I>
=
-r ,0, ••.• 0] ,d_n
[4>1' <l>2~ ... ,·lfl p , 0" .•• , 0] ,
·-1
First compute directly A
~m
and
and
and then successively
Ie
n
= [1
- rev(d'l)q
~n
~n
r- l
-8-
~he
proof of these results for the unsymmetric band matrix
~JlHTTLE
is given in
(1974).
McGilchrist
5.
A
_n
I
S METHOD
To indicate what goes wrong with \'!hittle i s method 1'ie
consider only the simplest case,
a, S are unknown autoregressive parameters.
where
and
then
X
t
may be generated for
Given border elements
together with the autoregressive parameters,
t = 2;, 3, ... , T .. 1.
However;, the above
equation (5.1) may be rearranged as
or equivalently
T .
Now equations (5.1) and (5.2) are equivalent and (5.2) describes a unilateral
process of order 2 with parameters related to the parameters of the bilateral
process.
lsi
<
lal
For stationarity of the bilateral process we require
< 1 ,
1 and these conditions imply that (5.2) is nonstationary.
However, the two processes are equivalent only if the border elements
Xl' X
2
of (5.2) are such as to generate the border elements
Xl' X
T
of
.. -9-
(5.1) and hence the distribution arising from (5.2) which is equivalent to
the distribution arising from (5.1) is a complicated conditional distribution.
This approach is not used here.
6. EXAMPLES
for the
To test the method a sample of 100 observations were generated
(I, 1,1, 0, 0)
process with
liklihood surface with a minimum at
~-l
=
.7, ~l
= .2.
Contours of the
(.6, .3) are shown in Figure 1. The two
contours shown correspond to 90% and 99% confidence regions.
We now come to the problem described in the introduction.
The data
given in Table 1 are dry weight of vegetative part in grams of three varieties
of oats each grown in monocu1ture.
This data is provided by Dr. B. R.
Trenbath, Australian National University.
Each variety of oats has two
monoculture plots and within each plot there are two rows of plants
plants in each row.
with 10
The same rectangular planting configuration and the same
spacing is used in each plot.
taken in order, data sets
For convenience we call data from the six plots
1,2, ... ,6 .
As described in the introduction we analyse
~n
(dry weight) and it is
sufficient to deal with the average of these values over the columns of each
plot resulting in six univariate time series, each of length 10.
It is
Unlikely that we can fit an extensive model to such short series so we fit a
(1, 1, 1, 0, 0)
process to each of the six sets of data.
The likelihood
Surfaces are evaluated and the minimum and 90% confidence contours are given
for each data set in Figures 2{a) and (b).
These confidence contours will
subsequently be used to estimate within plot variability for the type of dial1el
analysis reported in
~1cGi1christ
and Trenbath (1971).
-10-
REFERENCES
Box, G. E. P. and G. M.
Jenkins~
Time series analysis; forecasting and
control, Holden-Day, San Francisco. (1970).
~1atern,
B., "Spatial variation" ~ Reports of the Forest Research Institute of
Sweden, vol. 49, (1960).
McGi1christ, C. A., "Estimation in short time series" ~ submitted to the
Journal of the Royal Statistical
Society~
(1974).
HcGilchrist, C. A., IIInverse and determinant of a band matrix" ~ submitted to
the International Journal of Computer Mathematics ~ (1974).
McGilchrist, C. A. and B. R.
Trenbath~
llA revised analysis of plant competition
experiments", Biometrics" vol. 27, pp. 659-71, (1971).
Hhittle, P., "On stationary processes in the plane" 5 Biometrika ~ vol. 41,
pp. 434--49 ~ (1954).
-li-
TABLE 1.
Dry weight (9) of vegetative part of oat plants
Variety Plot
Dry
~!eight
_ _ _ _ _ _ _ _
1
1
2
2
1
2
3
1
2
'---.
9.16
8.40
9.93
7.08
2.48
3.42
21.99
15.77
4.55
10.05
3.93
7.87
4. 56
4. 94
10. 79
8. 22
2. 16
II. 58
13. 99
7. 10
6. 85
8. 21
5. 23
7. 43
7.23
4.56
11.92
7.37
4.99
9.99
16.35
14.84
12.84
11.00
18.96
13.21
4.94
10.32
9.36
7.65
5.93
14.09
9.49
11.58
7.75
8.70
10.54
10.97
3.03
7.25
6.23
7.08
4.04
6.56
11.87
13.97
8.70
14.69
9.64
7.43
' • • _. _ _·0. _ _ ·'
2.26
5.71
8.51
7.65
4.99
15.62
13.70
8.29
7.75
8.70
9.64
6.57
._.
..._._._
6.48
6.09
6.23
7.08
6.86
17.86
8.29
13.09
5.01
4.10
11.41
14.98
__ --- --_ .•_- ---
.,_._--~._._.
.~
. , ._ _ . _
•
••••
_ _ •••
7.25 10.70 7.25
6.48 9.55 8.78
6.52 7.08 4.81
7.65 5.95 8.22
9.37 11.58 15.34
5.30 8.12 3.10
19.03 8.29 19.61
11.58 12.19 8.29
11.90 19.32 4.55
7.30 4.10 3.16
20.29 14.98 18.09
18.52 11.88 8.77
_. - _.. _ _---_._---'
-12-
Figure 1
90% and 99% confidence contours of likelihood surface for 100 values
generated by (1, 1, 1, 0, 0) process with ~-l = .7 , ~1 = .2 .
Minimum is marked with a cross.
1-----------:----,
o
1
-13-
Figure 2 (a)
Minima and 90% confidence contours of likelihood surfaces for data sets
1, 2, 3. Minimum is marked with the number of the data set and arrows
point to corresponding confidence contours.
1-------:-,- - - l i r - - - ; - \ - - - - - - - - ,
•
\
\
\
•
\
\
\
\
.,
\
<P1
•
\
\
,
•
,
•
,
"
•
•
" ........ .......
"
""
,
•
\•
-1
\•
1
-14-
Figure 2(b)
~inima
and 90% confidence contours of likelihood surfaces for data sets
4, 5, 6. Minimum is marked with the number of the data set and arrows
point to corresponding confidence contours.
1
I
I
...... -... .,
I
......... .......
,
,
I
.
Cf>t
,
. ........ . ........ .
, . ......
.,
\
\
\
\
\
\
\
0
,
\
\
'. ..
"
\
" ,
"
\
,,
\
\
•
,
\
•
,
"
,,
•
\
",
•
,
",
", ,
\.
,
•
\
-1
,
,,
o
,,
1