Forecasting consumer price index in Ukraine with regression models and adaptive Kalman filter
The paper considers the problem of short term forecasting of consumer price index using regression models and adaptive Kalman filter. The main purpose of the study is constructing of high quality model for forecasting of consumer price index and application of Kalman filter for computing optimal est...
Saved in:
| Published in: | Системні дослідження та інформаційні технології |
|---|---|
| Date: | 2015 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
2015
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/123530 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Forecasting consumer price index in Ukraine with regression models and adaptive Kalman filter / І.V. Karayuz, P.I. Bidyuk // Системні дослідження та інформаційні технології. — 2015. — № 4. — С. 32-38. — Бібліогр.: 5 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860035205982584832 |
|---|---|
| author | Karayuz, І.V. Bidyuk, P.I. |
| author_facet | Karayuz, І.V. Bidyuk, P.I. |
| citation_txt | Forecasting consumer price index in Ukraine with regression models and adaptive Kalman filter / І.V. Karayuz, P.I. Bidyuk // Системні дослідження та інформаційні технології. — 2015. — № 4. — С. 32-38. — Бібліогр.: 5 назв. — англ. |
| collection | DSpace DC |
| container_title | Системні дослідження та інформаційні технології |
| description | The paper considers the problem of short term forecasting of consumer price index using regression models and adaptive Kalman filter. The main purpose of the study is constructing of high quality model for forecasting of consumer price index and application of Kalman filter for computing optimal estimates of states for the process under investigation. The basic results of the study are as follows: two modifications of the Kalman filter (ordinary and adaptive), directed towards estimation of covariances for stochastic state disturbances and measurement errors. Alternative short term forecasts are generated with regression models and Kalman filters. A comparative analysis of results achieved is given. The necessary statistical data was taken from Ukrainian economy in transition.
Роботу присвячено розв’язанню задачі короткострокового прогнозування індексу споживчих цін в Україні на основі регресійних моделей і адаптивного фільтра Калмана. Побудовано адекватну модель для прогнозування індексу споживчих цін і застосовано адаптивний фільтр Калмана для отримання оптимальних оцінок стану досліджуваного процесу і обчислення короткострокового прогнозу. Основні результати роботи: реалізація і застосування двох модифікацій фільтра Калмана (звичайний та адаптивний), орієнтовані на оцінювання коваріацій випадкових збурень стану та похибок вимірів. Альтернативні регресійні моделі та оцінки короткострокових прогнозів, отримані на основі фільтра. Надано порівняльний аналіз отриманих результатів. Для аналізу використано статистичну інформацію перехідної економіки України.
Работа посвященя решению задачи краткосрочного прогнозирования индекса потребительских цен в Украине на основе регрессионных моделей и адаптивного фильтра Калмана. Построена адекватная модель для прогнозирования индекса потребительских цен и использован адаптивный фильтр Калмана для получения оптимальных оценок состояний исследуемого процесса и краткосрочных прогнозов. Основные результаты работы: две модификации фильтра Калмана (обычный и адаптивный), ориентированные на оценивание ковариации случайных возмущений состояния и погрешностей измерений. Альтернативные регрессионные модели и оценки краткосрочных прогнозов получены при помощи фильтра. Дан сравнительный анализ достигнутых результатов. Для анализа использована статистическая информация переходной экономики Украины.
|
| first_indexed | 2025-12-07T16:53:33Z |
| format | Article |
| fulltext |
І.V. Karayuz, P.I. Bidyuk, 2015
32 ISSN 1681–6048 System Research & Information Technologies, 2015, № 4
УДК 519.766.4
FORECASTING CONSUMER PRICE INDEX
IN UKRAINE WITH REGRESSION MODELS
AND ADAPTIVE KALMAN FILTER
І.V. KARAYUZ, P.I. BIDYUK
The paper considers the problem of short term forecasting of consumer price index
using regression models and adaptive Kalman filter. The main purpose of the study
is constructing of high quality model for forecasting of consumer price index and
application of Kalman filter for computing optimal estimates of states for the proc-
ess under investigation. The basic results of the study are as follows: two modifica-
tions of the Kalman filter (ordinary and adaptive), directed towards estimation of
covariances for stochastic state disturbances and measurement errors. Alternative
short term forecasts are generated with regression models and Kalman filters.
A comparative analysis of results achieved is given. The necessary statistical data
was taken from Ukrainian economy in transition.
INTRODUCTION
The problem of high quality forecasting for economic and financial processes re-
quires development and application of new modern techniques that are based on
systemic approach to development of respective computational software. Most
often this software is implemented in the form of modern decision support sys-
tems (DSS) that become popular as a solution instrument for many practical prob-
lems. One of the simplest definitions of DSS is as follows: DSS is computer based
information processing system that provides decision making user with any help
relevant to data collecting and storing, preliminary data processing, constructing
necessary mathematical models, generating alternatives, selection of the best solu-
tion, convenient presentation of intermediate and final results etc [1].
To improve substantially quality of forecasts for macroeconomic processes
in the frames of DSS it is useful to construct adaptive computing schemes for
a model and parameter estimation [2]. According to this scheme each step of data
processing is controlled by appropriate set of statistical parameters each of which
characterizes specific features of data, model as a whole, model parameters and
quality of forecasts estimates. A substantial help in forecasting linear and nonlin-
ear nonstationary processes can be provided by application of adaptive Kalman
filter that is useful for estimation of covariances for external stochastic distur-
bances and measurement noise (errors) [3]. The filter also provides a possibility
for computing optimal estimates for state vector of a system under study and qual-
ity short term forecasts.
The paper considers the possibilities for modeling selected financial and
economic processes with regression analysis approach and adaptive Kalman filter
using statistical data from Ukrainian economy in transition.
Forecasting consumer price index in Ukraine with regression models and adaptive Kalman filter
Системні дослідження та інформаційні технології, 2015, № 4 33
PROBLEM STATEMENT
The purpose of the study is as follows:
to construct mathematical models for selected financial and economic
processes providing acceptable quality of short term forecasts;
to transform the models into state space representation and apply adap-
tive Kalman filter to generate optimal state vector estimates and short term fore-
casts;
to perform comparison of the results achieved;
to develop software necessary for performing computational experiments
that incorporates possibilities for adaptive model structure and parameters estima-
tion.
PROPOSED APPROACH TO FORECASTING
We propose adaptive computing scheme that is distinguished with several possi-
bilities for adaptation using complex quality criterion. The statistical data col-
lected should be correctly prepared for model structure and parameter estimation.
The model structure and parameter estimation is a key issue for reaching neces-
sary quality of forecasts. It is proposed to define model structure as follows:
},,,,,,{ lzdnmprS , where r is model dimensionality (number of equa-
tions); p is model order (maximum order of differential or difference equation in
a model); m is a number of independent variables in the right hand side; n is
a type of nonlinearity; d is a lag or output reaction delay time; z is external dis-
turbance and its type; l are possible restrictions for variables. To perform auto-
matic search for the “best” model it is proposed to use the following criterion:
UDWR
NN eMAPEMSEe
N
SSE
eDV
)(ln)1(ln1ln),( |2||1| 2
,
where is a vector of model parameters; N is a power of time series used; 2R
is a determination coefficient; DW is Durbin-Watson statistic; MSE is mean
squared error; MAPE is mean absolute percentage error; U is Theil coefficient.
The power of the criterion was tested experimentally with a wide set of linear and
pseudolinear models with positive results.
There are several possibilities for adaptive model structure estimation: (1)
automatic analysis of partial autocorrelation for determining autoregression order;
(2) automatic search for the exogeneous variables lag estimate (detection of lead-
ing indicators); (3) automatic analysis of residual properties; (4) analysis of data
distribution type and its use for selecting correct model parameters estimation
method; (5) adaptive model parameter estimation with hiring extra data; (6) opti-
mal selection of weighting coefficients for exponential smoothing, nearest-
neighbor interpolation and some other techniques; (7) the use of adaptive ap-
proach to model type selection. The use of specific adaptation scheme depends on
volume and quality of data, specific problem statement, requirements to forecast
estimates, etc. In some cases it is possible to use logistic regression together with
linear regression to describe the data with nonlinearities. Application of the adap-
tive concept described provides the following advantages: (1) automatic search
І.V. Karayuz, P.I. Bidyuk,
ISSN 1681–6048 System Research & Information Technologies, 2015, № 4 34
for the «best» model reduces the search time for many times; (2) it is possible to
analyze much wider set of candidate models than manually; (3) the search is op-
timized thanks to the use of complex quality criterion; (4) in the frames of com-
puter system developed it is possible to integrate ideologically different methods
of modeling and forecasting and compute combined forecasts estimates that are
distinguished with better quality.
FILTERING ALGORITHM
Most of modern macroeconomic and financial processes on the post soviet terri-
tory are influenced by a substantial number of stochastic disturbances. Among
them are the following: low qualification of managerial staff; high dependence on
energy import; inconsistent and unstable laws; outdated technologies in industry
and agriculture, unstable and outdated education system, high inflation (hyperin-
flation in 90s) etc. It was also revealed that data taken from different sources very
often contain substantially different values for the same variables. It means that
the measurement errors are available that also should be taken into consideration
when a process model is developed. To take into consideration these disturbances
and to reduce influence of measurement errors linear optimal Kalman filter is ap-
plied. The filtering procedure for scalar case is given below [3].
Step 1. Mathematical model of linear dynamic system:
,)1()1()()1()()( kwkukCkxkFkx
where )()( kykx — process state vector (selected variable); 1)( kF — co-
efficient characterizing system dynamics; )1()1()( 1 kxkukC — influence
of possible control actions (regressor); )00(~)( kQ,Q,Nkw — external sto-
chastic disturbances; )()()()( kvkxkHkz — measurement equation;
1)( kH — measurement matrix; ))(0(~)( kR,Nkv — measurement errors;
initial conditions: ,ˆ}{ 00 xxE 00
T
00 }ˆ,ˆ{ PPxxE ; )(kP — posterior covari-
ance matrix for state vector estimation errors; )(kP — prior covariance matrix
for state vector estimation errors; )(kK — optimal Kalman filter coefficient; ini-
tial covariances:
.0}),({,0})0(),({,0})(),({ T
0
T
0
T xkvExkwEkvkwE
Step 2. State vector extrapolation:
)1()1()1()()1(ˆ)()(ˆ 11 kxkykukCkxkFkx .
Step 3. Extrapolation for covariance matrix of estimation errors:
.)1()1()1()()1()()( 2
1
T kQkPkQkFkPkFkP
Step 4. Scalar coefficient of the filter is recalculated adaptively using co-
variance of measurement errors as root mean squared for filtering errors:
}ˆ,ˆ{)( T
00 xxEkR . To compute the covariances an adaptive KF with moving
window (AKFMW) is proposed:
Forecasting consumer price index in Ukraine with regression models and adaptive Kalman filter
Системні дослідження та інформаційні технології, 2015, № 4 35
)()(
)(
)}()()()({)()()( 1TT
kRkP
kP
kRkHkPkHkHkPkK
.
Step 5. Computing of state vector estimate using the last measurement re-
ceived )(kz :
)(ˆ)()()()(ˆ)(ˆ kxkHkzkKkxkx .
Step 6. Computing the posterior matrix for state vector estimation errors:
)()]()([)( kPkHkKIkP .
Step 7. Go to step 2.
The statistical data for consumer price index (CPI) was taken from the state
statistic tables for the period from 01.01.2000 to 01.08.2014 [4] (Fig. 1). The
computations were performed using developed software [5].
Constructed autoregression AR(2) model without constant is as follows:
,)()2()1()( 21 kkyakyaky
where )(ky — volume of imported natural gas; )(kx — volume of consumed
gas. After estimation of the model parameters we have the equation:
)2(8,0)1(92,0)( kykyky .
Now test the distribution law for model residuals; respective histogram is
given in Fig. 2. According to statistics 2
2
1,05.0
2
1,
2
84,317,3
)(
1
r
k k
kk
np
npn
— the normality hypothesis is accepted.
In this expression n is a number of observations; r is a number of data in-
tervals; kn — is a number of measurements that are in a specific thk interval;
kp is a probability of appearing random value in thk interval of normal distri-
bution; is a level of significance.
Fig. 1. Consumer price index
Month
І.V. Karayuz, P.I. Bidyuk,
ISSN 1681–6048 System Research & Information Technologies, 2015, № 4 36
Model quality is characterized with the following statistics: 21,02 R ;
07,2DW ; 01,1482 e (Table 1). Three steps forecasts quality is given in Table 2.
T a b l e 1 . Quality of the forecasts
Statistics
Model type
MSE MAPE U K
AR(2) 148,07 0,66 0,01 –
KF ( 5,0i ) 36,39 0,36 0,01 0,73
KF ( 1r ) 22,04 0,26 0,01 0,61
AKF 24,49 0,27 0,01 0,60
AKFMW 100 24,01 0,27 0,01 0,61
AKFMW 50 23,15 0,26 0,01 0,62
AKFMW 20 22,90 0,26 0,01 0,64
AKFMW 10 24,15 0,26 0,01 0,65
AKFMW 5 27,76 0,26 0,01 0,68
AKFMW 3 30,86 0,26 0,01 0,70
T a b l e 2 . Three-step forecast
June 2014 July 2014 August 2014
Model type
Value % Value % Value %
Data 101,00 100,40 100,80
Model 103,74 2,71 103,72 3,31 103,70 2,88
KF ( 5,0r ) 102,60 1,59 102,58 2,18 102,57 1,75
KF ( 1i ) 103,22 2,19 103,20 2,79 103,18 2,36
AKF 103,26 2,23 103,24 2,83 103,22 2,40
AKFMW 100 103,29 2,26 103,27 2,86 103,25 2,43
AKFMW 50 103,32 2,30 103,31 2,90 103,29 2,47
AKFMW 20 103,39 2,37 103,38 2,97 103,36 2,54
Fig. 2. Residuals histogram for CPI model AR(2)
Forecasting consumer price index in Ukraine with regression models and adaptive Kalman filter
Системні дослідження та інформаційні технології, 2015, № 4 37
Table 2 (continued)
AKFMW 10 103,30 2,27 103,28 2,87 103,26 2,44
AKFMW 5 103,25 2,22 103,23 2,82 103,21 2,39
AKFMW 3 103,47 2,44 103,46 3,04 103,44 2,61
The Kalman filter characteristics of functioning are provided in Figs. 3–5.
Fig. 4. Prior covariance for state vector estimation error versus AKFMW window size
0 40 80 120 Рис
Windows size
2,03
1,93
1,83
1,73
1,63
1,53
1,43
1,33
Fig. 3 Posterior covariance for state vector estimation error versus AKFMW window size
Windows size
0 20 40 60 80 100 120 140 160
0,61
0,56
0,51
1,46
1,41
0,36
Fig. 5. RMSE of measurement noise versus AKFMW
R
M
S
E
o
f
m
ea
su
re
m
en
t n
oi
se
Windows size
І.V. Karayuz, P.I. Bidyuk,
ISSN 1681–6048 System Research & Information Technologies, 2015, № 4 38
Thus, the computational experiments performed support the idea that appli-
cation of optimal filtering provides a possibility for improvement of short term
forecasts using state space models. Different versions of optimal KF usually show
somewhat different results though in general final result of forecasting is quite
acceptable.
CONCLUSIONS
An adaptive modeling concept is proposed based on application of simultaneous
model structure and parameters estimation strategy and adaptive Kalman filter.
Using statistical data for the consumer price index in Ukraine an autoregression
AR(2) model was constructed for short term forecasting. To compute optimal
state vector estimates two versions of Kalman filter have been applied: ordinary
optimal KF and adaptive KF. The latter version was constructed with hiring mov-
ing data windows for estimation of measurement noise covariances. The moving
window size has been varied in a wide range from 3 to 100. The best forecasting
results were received with ordinary optimal KF with noise variance ;5,0R and
lower results were received with model AR(2) without filtering procedure.
The best one step ahead forecast was received with KF that uses 0,1R ,
what can be explained by higher information quality at this period of time or on
purpose decreasing of CPI. Somewhat lower results of forecasting with adaptive
filter could be explained by the process stationarity at the period of time examined.
During such periods the adaptation is not required.
Testing of the forecasting system with wide set of macroeconomic and stock
price data showed that it is possible easy to reach a value of mean absolute per-
centage error of about 3–4 % for short term forecasting. The use of dynamic and
static estimates allows for generating necessary forecasts estimates depending on
specific problem statement.
In the future studies it will be reasonable to incorporate into the software de-
veloped new forecasting techniques such as neural networks, probabilistic Bayes-
ian networks, and immune algorithms that cover a wide class of nonlinear proc-
esses. Further steps towards automation of a model development and selection
procedure are also necessary.
REFERENCES
1. Holsapple C.W., Winston A.B. Decision Support Systems (a knowledge based ap-
proach). — New York: West Publishing Company, 1994. — 860 p.
2. Бідюк П.І. Адаптивне прогнозування фінансово-економічних процесів на
основі принципів системного аналізу // Наукові Вісті НТУУ «КПІ», 2009. —
№ 5. — С. 54–61.
3. Gibbs B.P. Advanced Kalman filtering, least-squares and modeling. — New York:
John Wiley & Sons, Inc., 2011. — 627 p.
4. Державна служба статистики України. — Електрон. дані. — http://www.ukrstat.
gov.ua/.
5. Grewal M.S., Andrews A.P. Kalman filtering: theory and practice using Matlab. —
Hoboken: John Wiley & Sons, Inc., 2008. — 575 p.
Надійшла 15.03.2015
From the Editorial Board: the article corresponds completely to submitted manuscript.
|
| id | nasplib_isofts_kiev_ua-123456789-123530 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1681–6048 |
| language | English |
| last_indexed | 2025-12-07T16:53:33Z |
| publishDate | 2015 |
| publisher | Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України |
| record_format | dspace |
| spelling | Karayuz, І.V. Bidyuk, P.I. 2017-09-06T14:51:48Z 2017-09-06T14:51:48Z 2015 Forecasting consumer price index in Ukraine with regression models and adaptive Kalman filter / І.V. Karayuz, P.I. Bidyuk // Системні дослідження та інформаційні технології. — 2015. — № 4. — С. 32-38. — Бібліогр.: 5 назв. — англ. 1681–6048 https://nasplib.isofts.kiev.ua/handle/123456789/123530 519.766.4 The paper considers the problem of short term forecasting of consumer price index using regression models and adaptive Kalman filter. The main purpose of the study is constructing of high quality model for forecasting of consumer price index and application of Kalman filter for computing optimal estimates of states for the process under investigation. The basic results of the study are as follows: two modifications of the Kalman filter (ordinary and adaptive), directed towards estimation of covariances for stochastic state disturbances and measurement errors. Alternative short term forecasts are generated with regression models and Kalman filters. A comparative analysis of results achieved is given. The necessary statistical data was taken from Ukrainian economy in transition. Роботу присвячено розв’язанню задачі короткострокового прогнозування індексу споживчих цін в Україні на основі регресійних моделей і адаптивного фільтра Калмана. Побудовано адекватну модель для прогнозування індексу споживчих цін і застосовано адаптивний фільтр Калмана для отримання оптимальних оцінок стану досліджуваного процесу і обчислення короткострокового прогнозу. Основні результати роботи: реалізація і застосування двох модифікацій фільтра Калмана (звичайний та адаптивний), орієнтовані на оцінювання коваріацій випадкових збурень стану та похибок вимірів. Альтернативні регресійні моделі та оцінки короткострокових прогнозів, отримані на основі фільтра. Надано порівняльний аналіз отриманих результатів. Для аналізу використано статистичну інформацію перехідної економіки України. Работа посвященя решению задачи краткосрочного прогнозирования индекса потребительских цен в Украине на основе регрессионных моделей и адаптивного фильтра Калмана. Построена адекватная модель для прогнозирования индекса потребительских цен и использован адаптивный фильтр Калмана для получения оптимальных оценок состояний исследуемого процесса и краткосрочных прогнозов. Основные результаты работы: две модификации фильтра Калмана (обычный и адаптивный), ориентированные на оценивание ковариации случайных возмущений состояния и погрешностей измерений. Альтернативные регрессионные модели и оценки краткосрочных прогнозов получены при помощи фильтра. Дан сравнительный анализ достигнутых результатов. Для анализа использована статистическая информация переходной экономики Украины. en Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України Системні дослідження та інформаційні технології Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи Forecasting consumer price index in Ukraine with regression models and adaptive Kalman filter Прогнозування індексу споживчих цін в Україні з використанням регресійних моделей і фільтра Калмана Прогнозирование индекса потребительских цен в Украине с использованием регрессионных моделей и фильтра Калмана Article published earlier |
| spellingShingle | Forecasting consumer price index in Ukraine with regression models and adaptive Kalman filter Karayuz, І.V. Bidyuk, P.I. Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи |
| title | Forecasting consumer price index in Ukraine with regression models and adaptive Kalman filter |
| title_alt | Прогнозування індексу споживчих цін в Україні з використанням регресійних моделей і фільтра Калмана Прогнозирование индекса потребительских цен в Украине с использованием регрессионных моделей и фильтра Калмана |
| title_full | Forecasting consumer price index in Ukraine with regression models and adaptive Kalman filter |
| title_fullStr | Forecasting consumer price index in Ukraine with regression models and adaptive Kalman filter |
| title_full_unstemmed | Forecasting consumer price index in Ukraine with regression models and adaptive Kalman filter |
| title_short | Forecasting consumer price index in Ukraine with regression models and adaptive Kalman filter |
| title_sort | forecasting consumer price index in ukraine with regression models and adaptive kalman filter |
| topic | Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи |
| topic_facet | Прогресивні інформаційні технології, високопродуктивні комп’ютерні системи |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/123530 |
| work_keys_str_mv | AT karayuzív forecastingconsumerpriceindexinukrainewithregressionmodelsandadaptivekalmanfilter AT bidyukpi forecastingconsumerpriceindexinukrainewithregressionmodelsandadaptivekalmanfilter AT karayuzív prognozuvannâíndeksuspoživčihcínvukraínízvikoristannâmregresíinihmodeleiífílʹtrakalmana AT bidyukpi prognozuvannâíndeksuspoživčihcínvukraínízvikoristannâmregresíinihmodeleiífílʹtrakalmana AT karayuzív prognozirovanieindeksapotrebitelʹskihcenvukrainesispolʹzovaniemregressionnyhmodeleiifilʹtrakalmana AT bidyukpi prognozirovanieindeksapotrebitelʹskihcenvukrainesispolʹzovaniemregressionnyhmodeleiifilʹtrakalmana |