Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.13, 2013 54 Application of Stochastic Lognormal Diffusion Model with Polynomial Exogenous Factors to Energy Consumption in Ghana Godfred Kwame Abledu(PhD) School of Applied Science and Technology, Koforidua Polytechnic, PO Box 981, Koforidua, Ghana * E-mail of the corresponding author: [email protected]Abstract. The main objective of this paper was the application of maximum likelihood ratio tests in lognormal diffusions with polynomial exogenous factors. The model described an innovation diffusion process considering at the same time disturbances coming from the environment of the system. Finally, the model was applied to energy consumption data in Ghana from 1999 to 2010. Maximum likelihood estimators (MLEs) were obtained for the drift and diffusion coefficients characterizing lognormal diffusion models involving exogenous factors affecting the drift term. The present paper provides the distribution of these MLEs, the Fisher information matrix, and the solution to some likelihood ratio tests of interest for hypotheses on the parameters weighting the relative effect of the exogenous factors. The results show that the total consumption of primary energy presents structural characteristics. The endogenous consumption pattern in Ghana, in absolute terms, also presents a clear upward trend. Key works: lognormal diffusions model, maximum likelihood estimators, endogenous actors, energy consumption . 1 Introduction The use of diffusion processes with exogenous factors and their trend is common in many fields. The reason of its application is the usual presence of deviations of the observed data with respect to the trend of some known homogenous diffusion process, in some time intervals. These factors are time dependent functions that allow, on one hand, a best fit to the data and, on the other hand, an external control to the process behaviour. The factors must be totally or partially known, that is, their functional form or some aspects about their time evolution must be available. The problem of estimating the parameters of the drift coefficient in these models has received considerable attention recently, especially in situations in which the process is observed continuously. The statistical inference is usually based on approximating maximum likelihood methodology. An extensive review of this theory an be found in Prakasa(1999), and related new work has been done by Kloeden et al. (1999), The usefulness of diffusion random fields in describing, for example, economic or environmental phenomena, has led to significant developments, particularly regarding inferential aspects. In that respect, from the contribution to theoretical foundations for diffusions given in Nualart (1983) and Ricciardi (1976), the lognormal diffusions involving exogenous factors affecting the drift term is considered. The maximum likelihood estimators (MLEs) for the drift and diffusion coefficients is obtained, which characterize these diffusions under certain conditions. Using these MLEs, techniques for estimation, prediction and conditional simulation of lognormal diffusions are developed. The study of variables that model dynamical systems has undergone a great development over the last decades, and a variety of statistical and probabilistic techniques has been worked out for this purpose. Among these, stochastic processes, and in particular diffusion processes, have been systematically employed. 2. Lognormal Diffusion Process Model (LNDP) The lognormal diffusion process with exogenous factors is defined as 0 : Xt t T T with infinitesimal moments 1 , A xt xh t xh t and 2 2 1 , A xt x 2 2 x 2 , where 0 0 and h is continuous function in 0 , t T containing the external information sources. So it is usual to take h as a linear combination of continuous functions. A class of two-parameter random fields which are diffusions on each coordinate and satisfy a particular Markov property related to partial ordering in 2 R are considered by Nualart(1983). Using this theory, Skiadas(2007) introduced a 2D lognormal diffusion random field as follows.
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Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
54
Application of Stochastic Lognormal Diffusion Model with
Polynomial Exogenous Factors to Energy Consumption in Ghana
Godfred Kwame Abledu(PhD)
School of Applied Science and Technology, Koforidua Polytechnic, PO Box 981, Koforidua, Ghana
The main objective of this paper was the application of maximum likelihood ratio tests in lognormal
diffusions with polynomial exogenous factors. The model described an innovation diffusion process considering
at the same time disturbances coming from the environment of the system. Finally, the model was applied to
energy consumption data in Ghana from 1999 to 2010.
Maximum likelihood estimators (MLEs) were obtained for the drift and diffusion coefficients characterizing
lognormal diffusion models involving exogenous factors affecting the drift term. The present paper provides the
distribution of these MLEs, the Fisher information matrix, and the solution to some likelihood ratio tests of
interest for hypotheses on the parameters weighting the relative effect of the exogenous factors.
The results show that the total consumption of primary energy presents structural characteristics. The
endogenous consumption pattern in Ghana, in absolute terms, also presents a clear upward trend.
Key works: lognormal diffusions model, maximum likelihood estimators, endogenous actors, energy
consumption
.
1 Introduction
The use of diffusion processes with exogenous factors and their trend is common in many fields. The
reason of its application is the usual presence of deviations of the observed data with respect to the trend of some
known homogenous diffusion process, in some time intervals. These factors are time dependent functions that
allow, on one hand, a best fit to the data and, on the other hand, an external control to the process behaviour. The
factors must be totally or partially known, that is, their functional form or some aspects about their time
evolution must be available. The problem of estimating the parameters of the drift coefficient in these models
has received considerable attention recently, especially in situations in which the process is observed
continuously. The statistical inference is usually based on approximating maximum likelihood methodology. An
extensive review of this theory an be found in Prakasa(1999), and related new work has been done by Kloeden
et al. (1999),
The usefulness of diffusion random fields in describing, for example, economic or environmental
phenomena, has led to significant developments, particularly regarding inferential aspects. In that respect, from
the contribution to theoretical foundations for diffusions given in Nualart (1983) and Ricciardi (1976), the
lognormal diffusions involving exogenous factors affecting the drift term is considered. The maximum
likelihood estimators (MLEs) for the drift and diffusion coefficients is obtained, which characterize these
diffusions under certain conditions. Using these MLEs, techniques for estimation, prediction and conditional
simulation of lognormal diffusions are developed.
The study of variables that model dynamical systems has undergone a great development over the last
decades, and a variety of statistical and probabilistic techniques has been worked out for this purpose. Among
these, stochastic processes, and in particular diffusion processes, have been systematically employed.
2. Lognormal Diffusion Process Model (LNDP)
The lognormal diffusion process with exogenous factors is defined as 0:X t t TT with infinitesimal
moments 1 ,A x t xh txh t and 2 2
1 ,A x t x2 2x2 22 2, where 00 and h is continuous function in 0 ,t T
containing the external information sources. So it is usual to take h as a linear combination of continuous
functions. A class of two-parameter random fields which are diffusions on each coordinate and satisfy a
particular Markov property related to partial ordering in 2R are considered by Nualart(1983). Using this
theory, Skiadas(2007) introduced a 2D lognormal diffusion random field as follows.
Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
55
Let 2: , 0, 0, RX z z s t I S T 2, 0, 0, RI S T, 0, 0,, , , be a positive-valued Markov random field,
defined on a probability space , ,A P, ,A P,, , where X(0,0) is assumed to be constant or a lognormal random
variable with 0In 0,0E XIn 0,0Inn 0 and 2
0In 0,0Var X 2In 0,0InIn 2
0 . The distribution of the random field is
determined by the following transition probabilities:
1 2, , | , , ,P B s h t k x x x z1 2, | , , ,1 21 2 z, | , , ,, | , , ,1 21 21 2 1, | , ,P X s h t k B X s t k x X z1, | , ,1h t k B X s t k x X z, | , ,, | , ,1
where , , , 0z s t I h k, , , 0s t I h k, , ,, , , , 2
1 2, , Rx x x 2R and B is a Borel subset. It is assumed that the transition
densities exist and are given by
1 2( , ( , ) | , , , )g y s h t k x x x z1 21 2, ) | ,, ) ,1 21 21 2
1 2
2
; ,
2; ,; ,
In1 1exp
22
yx
z h kx x
z h kz h k
m
y 2
; ,z h k; ,; ,
2222
1 mI yxIn yx
1exp
2 2exp
1 mIn yx
; ,z h k; ,; ,z h k; ,; ,z h k; ,; ,z h k111 z h kz h kz h kmz h kmIn1 2x x1 21 21 2x x1 21 2
Inx x
In1 2x x1 21 2 ; ,1 2
2222222 z h kz h k; ,; ,z h k; ,; ,z h kz h k
for 2Ry 2R , with
; , ,s k t k
z h ks t
m a d dds t
as k ts k t ks k t
s t
s k t
da , d, , 2
; , ,s k t k
z h ks t
B d dd2
; ,z h k; ,; ,s t
s k t
s tB
s k ts k t ks k t k
Bs k t
s t
s k t
d d,B d,B
and aa , BB being continuous functions on I. Under these conditions we can assert that :X z z II is a
lognormal diffusion random field. The one parameter drift and diffusion coefficients associated are given by:
1 1 1
1:
2a z x a z B z x
111 1 11 1 11 1 1:1 1 11 1 1
1B
1
21 1 1
2B1 1 11 1 1B
2221 1 1a1 1 11 1 1a B1 1 1B z1 1 11 1 1B z xxxx
1B
111111 1 1
21 1 11 1 1B1 1 11 1 1B
21 1 1
21 1 11 1 1a z1 1 11 1 1a z ,
2 2
1 1:B z x B z x1 11 11 1
2 2B2 2
1 11 1:2 22 2
1 11 1
2 22 2
1 1B z x2 22 2
1 11 1
2 2 2
1:
2a z x a z B z x
112 2 22 2 22 2 2:2 2 22 2 2
1B
1a
22 2 2
2B2 2 22 2 2B
2222 2 2a2 2 22 2 2a B2 2 2B z2 2 22 2 2B z xxxx
1B
111112 2 2
22 2 22 2 2B2 2 2B
22 2 2
22 2 22 2 2a z2 2 22 2 2a z ,
2 2
2 2:B z x B z x2 22 22 2
2 2B2 2
2 22 2:2 22 2
2 22 2
2 22 2
2 2B z x2 22 2
2 22 2
where
10
, ,t
a s t a s r dr00
1 ,a s t1 ,, r dr0
,t
a s,, , 10
,t
B s t Brdt00
1 ,B s t1 , t0
t
Brdt
20
, ,t
a s t a t d00
2 ,a s t2 ,, ,t d,0
t
a ,, , 10
, ,t
B s t B t dt00
1 ,B s t1 ,, , t dt,0
t
B
for all , , Rz s t I x, , Rs, , .
The random field :Y z z II defined as InXY z zzInX is then a Gaussian diffusion random
field with aa and BB being, respectively, the drift and diffusion coefficients, and 1a1a , 2a2a , 1B1B and 2B2B being the
corresponding one parameter drift and diffusion coefficients. Furthermore, if
, , , , ,z z I z s t z s tz , , , ,I z s t z s t, , , , , then
00 0
: ,s t
Ym z E Y z a d drdr: E: Y z 0000 00 00 0
s t
a0 00 00 00 0
d, d d,a
2 2
00 0
: ,s t
Y z Var Y z B d drdr2 2:Y z2 22 2:2 22 22 2:2 22 2::2 22 22 22 22 2
00
2 22 22 22 2
00 00 00 0
s t
B d, d,B
2, : ,Y Yc z z Cov Y z Y z z zY Y:::Y Y: ,Cov Y z Y z::Y YY Y: ,: 2
Y Y z z2
Y Y
It is also assumed that the conditions usually considered for estimation of the drift and diffusion coefficients in
the one-parameter case hold, that is, 0InX 0,0 1P InX 0,0 0 10 and 2
Y z Bst2
Y z BBstB , ,z s t I,s t I, .
Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013
56
3. Inference in the Lognormal Diffusion Process Model (LNDP)
Let :X z z II be a lognormal diffusion random field. Data 1X= ,...,t
nX z X z are
assumed to be observed at known spatial locations 1 1 1,z s t1 1 11 1 11 1 1s t1 1 11 1 11 1 11 1 1 , 2 2 2,z s t2 2 22 2 22 2 2s t2 2 22 2 22 2 22 2 2 , …, ,n n nz s t In n n,s t I,n n nn n n, . Let
1 2x , ,...,t
nx x x1 2, ,..1 21 2, ,....1 21 21 2 be a sample. Let us consider the log-transformed n-dimensional random vector,
1 2Y , ,...,t
nY z Y z Y z1 2,1 21 2,,1 21 21 2 1In ,..., In In Xt
nX z X z1In ,..., In In X1
t
n,..., In ,..., n 1 , and the log-transformed sample
1 2, ,..., In xt
ny y y y1 2 ... In xt
ny y y1 21 2, ,...,...1 21 21 2 . We denote 1 2m , ,...,t
Y Y Y Y nm z m z m zY Y Y1 21 21 21 2 and
2
, 1,...,Y Y i j
i j nz z
i j n, 1,...,, 1,Y Y i
i j n, 1,, 1,
2
Y Y i jz zY Y i jY Y i j, 1,i j n, 1,, 1,
In order to estimate the MLEs for the drift and diffusion coefficient using exogenous factors, it is
supposed that the drift coefficient aa of Y is a linear combination of several known functions,
1 ,..., :ph z h z z II , with real coefficients 1,..., p1,..., p :
1
,p
a z h z z IIp
1
zp
h z z,h z z,a zp
Defining for ,z s t II ,
0 1f z 1 , 0
,s t
of z h r d drf z drdr
0 oh
0 oh r dr d,
s t
h0 o
h , 1,..., pp1,..., p
the mean of Y is given by
0 00
1 0
, ,p ps t
Yo
m s t h r d dr f z0 00 00 0
p pp pp p
1 01 0
0 0
p p
0 00 0
p p
dr f zdr f zp pp p
01 0
0 o1 01 0
0 o0
p ps tp pp p
h r dh r d,00 o
Thus, denoting 0 1F= f ,f ,..., f p , with 1 2f = , ,...,t
nf z f z f z1 2f = nf z f z f z1 21 2, ,...,.1 21 21 2 for 0,1,..., p, p0,1,... , and
0 ,...,t
p
t
p0 ,..., the following is obtained:
0 0 1 1= f f ... f FY p pm f0 0 10 0 1Y p p0 0 10 0 10 0 1f ... f F0 0 1 10 0 1 1Y p p0 0 1 1 ... f... f0 0 1 10 0 1 1
Let us write
1 1 1 2 1 2 1 1
1 2 1 2 2 2 2 2
1 1 2 2
...
...M : =
... ... ... ...
...
n n
n n
Y
n n n n n n
s t s s t t s s t t
s s t t s t s s t tB B
s s t t s s t t s t
1 1 1s1 1 11 1 11 1 1 2 1 2 1 1n n1 11 1s t t s s t t1 1 1 2 1 2 1 11 1 1 2 1 2 1 1n nn n1 11 11 1s ts t1 1 11 1 11 1 1s1 1 11 1 1
1 2 1 2 2 2 2 2n n2 22 2t s t s s t t1 2 1 2 2 2 2 21 2 1 2 2 2 2 2n nn n2 22 22 2M : Y BM : =
1 2 11 2 11 2 11 2 1
......
n n ns tn n nn n n1 1 21 1 2s1 1 2s n n n n n n1 1 2 21 1 2 ...t s s t t1 1 2 21 1 2 2 ...n n nn n n1 1 2 21 1 21 1 2 2 ...1 1 2n n n1 1 21 1 21 1 21 1 21 1 2s t1 1 21 1 21 1 21 1 21 1 2s t
F1 t -1F1 t -11 t -1B 1 t -1 t -1 t -F M F1 t -11 t -1B 1 t -F M1 t -1 t -F M FB (7)
5. Hypotheses Testing
In order to test the hypothesis, the vector 0 1, ,...,
t
p..,t
p0 1,.0 10 1is split as follows (Skiadas, 2007;
Anderson, 2003):
1
2
111
22
where 11 is 1 1p 1 and 22 is 2 1p 1, with 1 2 1p p p1 2 1p p1 21 2 . The hypothesis of interest is
0 1 1:H0 1 10 1 1
1 1 1:H1 1 11 1 1
where 11 is 1 1p 1 fixed vector. The total region and the region associated with the null hypothesis are,
respectively,
2, : 0 R pB B R p, 20 R:::
2 1
1 1, : : 0 R pB B 2R p
1 1, : : 0 1
1 1 : 01 1 ::1 11 1: 1 1:: 1 11 11 11 1
Under these hypotheses,
122 2* 1
1
max x; , 2 | M| exp2
nnn
i
i
nL B B x; ,; ,,ax xx
n
2222i 1
* 1 exp* 1
i
* 1* 1* 1 exp* 1
i
n
2222
1* 1* 11* 1* 1* 11
M|M| 2* 1* 1* 1122 2 * 12 2 * 1
n
22 2 * 1* 1* 1* 1M* 1* 1* 1* 12* 1* 1* 12n
|* 1* 1* 12
|||n
* 1* 1n
* 1
122 2* 1
1
max x; , 2 | M| exp2
nnn
i
i
nL B B xax xxx ,,,
n
2222
n
i 1
* 1 exp* 1n
i
* 1* 1* 1* 1n
* 1* 1 exp* 1
i
n
2222M|* 1* 1* 1M|* 1* 12 2 * 12 2 * 1
n
22 2 * 1||* 1* 1* 1M* 1* 1* 1* 12* 1* 1* 12n
|* 1* 1* 12
|1* 1* 11* 1* 11
| 2* 1* 112
nn* 1* 1
n* 1
and the likelihood ratio statistic for testing 0H is
2 2* *
* *
max
max
n n
L B B
L B BL
Bmax
max
L
L
L
L
BBB
BBBB
2 2* *2 22 2n n
* *2 2n n2 2* *2 22 2
B2 22 2* *2 22 2* *2 22 2* *2 22 2
B B2 22 2
B BB2 2
B B
B BB B* *B B* ** ** *B B
B B
For obtaining the distridution of this statistic, let us denote 1A=F M Ft 1M F1
and 1C=F M In Xt 1M In X1
and
consider the following partitions: 1 1
11 21 11 1 1 1 2
1 2 1 121 22 2 2 1 2 2
A A F F M F F M FA= M F | F
A A F F M F F M F
t t t
t t t
111 1F1 1t t tF M 1 11 11 1t t t1 11 1M 1 11 11 1t t tFt t tFFt t tA A F1 1F MF M 1 11 1M 1 11 11 1FFF F MF M 1 11 1M 1 11 11 1FFF M1 11 1F MF MM F F M1 11 1M F FM F F1 11 11 1FFF11 21A A11 211 2