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The 28 th Annual International Symposium on Forecasting June 22-25, 2008, Nice, France. FORECASTING METHODS OF NON-STATIONARY STOCHASTIC PROCESSES THAT USE EXTERNAL CRITERIA. Igor V. Kononenko, Anton N. Repin National Technical University “Kharkiv Polytechnic Institute”, Ukraine. INTRODUCTION. - PowerPoint PPT Presentation
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FORECASTING METHODS OF NON-STATIONARY STOCHASTIC
PROCESSES THAT USE EXTERNAL CRITERIA
Igor V. Kononenko, Anton N. RepinNational Technical University
“Kharkiv Polytechnic Institute”, Ukraine
The 28th Annual International Symposium on ForecastingJune 22-25, 2008, Nice, France
INTRODUCTION
While forecasting the development of socio-economic systems there often arise the problems of forecasting the non-stationary stochastic processes having a scarce number of observations (5-30), while the repeated realizations of processes are impossible.
To solve such problems there have been suggested a number of methods, in which unknown parameters of the model are estimated not at all points of time series, but at a certain subset of points, called a learning sequence. At the remaining points not included in the learning sequence and called the check sequence, the suitability of the model for describing the time series is determined.
PURPOSE OF WORK
The purpose of this work is to create and study an effective forecasting method of non-stationary stochastic processes in the case when observations in the base period are scarce
H-CRITERION METHOD(I. Kononenko, 1982)
Retrospective information
irГ , qr ,1 ni ,1
),...,,( ,12,11,1 nVector of values of the predicted variable
q – number of significant variables including the predicted variablen – number of points in the time base of forecast
H-CRITERION METHOD (2)
2/,2,1,
2/,22,21,2
2/,12,11,1
...
......
......
...
...
nqqq
n
n
LГ
nqnqnq
nnn
nnn
CГ
,22/,12/,
,222/,212/,2
,122/,112/,1
...
......
......
...
...
nqqq
n
n
Г
,2,1,
,22,21,2
,12,11,1
...
......
......
...
...
H-CRITERION METHOD (3)
The parameters of all formed models are estimated using the learning submatrix
LГ,ρηˆ jA jA – the vector of estimated parameters for j-th model
Tpj aaaA ,,, 21)(
L
ρ,,ρ,ρρ 21 N
– the vector of weighting coefficients considering the error variance or importance i1γ for building the model
H-CRITERION METHOD (4)
2/,1,
1,2
2/,12,11,1
....
......
......
.....
...
nqq
n
LГ
nqnq
n
nnn
CГ
,12/,
12/,2
,122/,112/,1
....
......
......
.....
...
For each j - th model at all points of past history we calculate
n
i
jjijii AFF
1
)()(11 ))ˆ,((
Calculating2
H-CRITERION METHOD (5) New learning and check submatrices are chosen.
The number of rows in ГL is decreased by one.The process of estimation of model parameters and calculation of 3 is repeated.
The learning submatrix is used as the check one and the check submatrix - as the learning one, 4 is calculated and similarly we continue using the bipartitioning.
The process is stopped after a set number of iteration g.
H-criterion:
gH 21
METHOD, THAT USES THE BOOTSTRAP EVALUATION(I. Kononenko, 1990)
Retrospective information
qj ,1 ni ,1
),...,,( ,12,11,1 n
Vector of values of the predicted variable
q – number of significant variables including the predicted variable n – volume of past history
,,ij
1L L – the number of a model in the set of test models
BN ,f iL
Testing the model
iiL
i,1 ,f BN
Ni – vector of independent variablesB – vector of estimated parametersi – independent errors having the same and symmetrical density of distribution
METHOD, THAT USES THE BOOTSTRAP EVALUATION (2)
1. The parameters of the model we estimate using matrix basing on the condition
n
ii
Li fF
1,1 ,minargˆ BNB
B
iF – loss function, BN ,f iL
i,1i n,1i
BN ˆ,,1 iL
ii fbias
Determining the deviation from points of
Numbers biasi form the BIAS vector
METHOD, THAT USES THE BOOTSTRAP EVALUATION (3)
2. We divide the matrix into two submatrices – learning submatrix L and check submatrix C.First n-1 columns of matrix are included in submatrix L and n-th column is included in C.
Estimating the parameters B of the test model and obtaining 0B̂
Calculating the deviation of the model from the statistics
20,10
ˆ,BNnL
nL fD
Let k=1, where k – number of iteration, which performs the bootstrap evaluation
METHOD, THAT USES THE BOOTSTRAP EVALUATION (4)
3. We perform bootstrap evaluation, which consists in the following.We randomly (with equal probability) select numbers from the BIAS vector and add them to the values of model.As a result we obtain “new” statistics which looks like the following
siLk
i biasf BN ˆ,,1
We divide the matrix k (a new one this time) into L,k and C,k.
We estimate the unknown parameters basing on L,k as earlier and calculate the model deviation from C,k
2,1ˆ, k
nLk
nLk fD BN
n,1i n,...,2,1s
METHOD, THAT USES THE BOOTSTRAP EVALUATION (5)
4. If k<K-1 then we suppose that k:=k+1 and return to step 3 (where K – number of bootstrap iterations), otherwise proceed to step 5
5. Evaluating
1N
0k
Lk
L DD
6. If L<z then we suppose that L:=L+1 and move to step 1(where z – number of models in the list), otherwise we stop.
The model with minimal DL is considered to be the best one.
METHOD, THAT USES THE BOOTSTRAP EVALUATION (6)
ANALYSIS OF METHODSMathematical models
3x2xy 2
3x6xy 2
3x8x2y 2
3x16xy 2
11x6xy 2
11x2xy 2
27x16x2y 2
27x8x2y 2
Additive noise
),0(N~ 2
9)10yy(3.010
1i
10
1iii
2)()(F
The loss function
ANALYSIS OF METHODS (2)Relative PMAD, evaluated at the estimation period
10
1
10
1
ˆi
ii
ii zyz
iii yz
PMAD, evaluated at the estimation period
10
1
10
1
ˆi
ii
ii yyyE
PMAD, evaluated at the forecasting period
d
ii
d
iiid yyyE
11
10
11
ˆ1
Relative MSE, evaluated at the forecasting period
d
iii
md yz
dD
10
11
2ˆ1
MSE, evaluated at the forecasting period
d
iii
td yyd
D10
11
2ˆ1
ANALYSIS OF METHODS (3)
21212121
21212121
21212121
21212121
21211221
12211221
12121221
12121212
12121212
12121212
1R
12121212
12121212
12121212
12121212
12211212
12211221
12212121
21212121
21212121
21212121
2R
2111111111
1211111111
1121111111
1112111111
1111211111
1111121111
1111112111
1111111211
1111111121
1111111112
3R
1221112112
1112121111
2121221121
1211111121
2121221212
2121222111
1222121112
1111112111
2112221122
2121211212
4R
N
1kkN
1
N
1kkEN
1E
N
kdkd E
NE
1
11
1
N
1k
mdk
md D
N
1D
N
1k
tdk
td D
N
1D
Average (across noise realizations):
ANALYSIS OF METHODS (4)
We compared the characteristics with the two-sample t-test assuming the samples were drawn from the normally distributed populations:
Q2Q,NVvP
N
ssv
22
21
21 1000N
5,2Q 96,1, QNV
ANALYSIS OF METHODS (5)
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
1 2 3 4 5 6 7 8 9 10
g
%
Matrix R1 Matrix R2 Matrix R3 - CV Matrix R4 - Random Bootstrap
PMAD, evaluated at the forecasting period
RESULTS OF ANALYSIS
When the number of partitions increases in case of using matrices R1 and R2 we observe the downward trend of PMAD with some
fluctuations in this trend that depend on the ways of data partition
The partition according to the cross-validation procedure, in which the check points fall into the observation interval, produces significantly less accurate forecasts. The comparison of the efficiency of different partitions with randomly generated matrix R4 has shown that the reasonable choice of partition sequences permits to get a more accurate longer-term forecast
RESULTS OF ANALYSIS (2)
The method that uses bootstrap evaluation produces the more accurate forecast than the cross-validation procedure
The comparison of two suggested methods enables to state that the method that uses bootstrap evaluation makes it possible to obtain more accurate longer-term forecasts as compared with H-criterion method only in case of a small number of partitions. Otherwise the usage of selected matrices R1 or R2 permits to get more accurate forecasts. Nevertheless, the method that uses bootstrap evaluation turned out to be more accurate than the H-criterion method when using matrix R4
PRODUCTION VOLUME OF BREAD AND BAKERY IN KHARKIV REGION
0
50
100
150
200
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008
Tho
usan
d to
ns
Prehistory Actual Data Forecast
Mean relative error for 2003-2006 is 5,91 %
COMBINED USE OF TWO METHODS
In the real-life problems the method that use the bootstrap evaluation might turn out to be more accurate in some cases.
It is recommended to use the given methods together.
In such case every result obtained by means of these methods must be assigned some weight on the basis of the a priori estimates of the methods accuracy.
The final forecast will be received in the form of the weighted average value of individual forecasts.
Igor V. Kononenko – Professor, Doctor of Technical Sciences, Head of Strategic Management Department;
Anton N. Repin – Post-graduate of Strategic Management Department
STRATEGIC MANAGEMENT DEPARTMENTNATIONAL TECHNICAL UNIVERSITY “KHARKIV POLITECHNIC INSTITUTE”
21, Frunze St., Kharkiv, 61002,E-mail: [email protected]
[email protected]: +38(057)707-67-35Fax: +38(057)707-67-35