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: kiv@kpi.kharkov.ua

anton.repin@mail.ruPhone: +38(057)707-67-35Fax: +38(057)707-67-35