An Approach of Electric Power Demand Forecasting using
Data-Mining Method: a case study of application of Data-Mining
technique to improvement of decision-making
Toshio Sugihara
Faculty of economics, Nagasaki University, Katafuchi 4-2-1, Nagasaki,JapanE-mail: sugihara@net.nagasaki-u.ac.jp

Abstract: In this paper, for the aim to build the management plan of stable electric power supply andaccommodation, the monthly electric demand prediction approach with dynamic and adaptive
mechanism connected to the business environment is proposed.
The proposed prediction adopts the Kalman-Filter as the basic prediction scheme and possess
two characteristics stated below. One is the state-space built with the principal component
time-series integrated by time-series PCA (Principal Component Method) from multi businessindices related the targeted time-series. The other is the self-organized auto-updating of thestate-space by structured neural network.
The proposed scheme shows considerably accurate prediction than any other models with single
variable time-series and the obvious effect appears to the high accuracy by adopting time-series
PCA as a Data-Mining technique. From these effects, the proposed prediction scheme might beconsidered to give an improvement to the stable electric power supply and accommodation. And
this prediction scheme can be applied to various management areas, so it might be considered to
be an effective method for decision-making support.

Keywords: Electric power supply; Knowledge extraction; Data-Mining technique;
Kalman-Filter processing; Self-organized state-space; Principal component time-series;
Batch-Sequential method

1. Introduction
1.1 Electric demand forecasting for improvement of decision-making
Needless to say, forecasting is an important element for decision-making support. The
common theme through decision-making is “selection and decision” and the forecastingis indispensable for the optimal realization of this theme. This is explained obviously by
the fact that the forecasting is positioned as a core method of DSS (Decision Support
System) which has been developed until now. In this paper, as the case study of the
improvement to reasonable and effective decision-making, an approach to the electric1power demand forecasting adapted to business variation is proposed and its effect isverified.
In modern days, high activity of industry as IT leading and high dependence to electric
power in daily life, require still more increase and higher stability of electric power. It isneedless to say that high accurate prediction of electric power demand has a decisiverole for these requirements.
In investigating of electric power demand, following 3-type demand patterns must be
considered corresponding to time span of demand. [10]･ long range demand under the influence of population, big plant, building, etc,
･ monthly demand required by industrial and economic activities, etc,
･ short range demand required by urgent need.
For the first pattern, in order to cope with building a new electric power station, thelong-range prediction based on yearly demand, is needed. For the last point, for theurgent demand or change of supply route caused by disaster, the short-range prediction
with hour range variation of demand is required. [12] On the other hand, the stablesupply of electric power, which is connected with business trend and daily life activities,
requires highly accurate supply plan at the power station and accommodates with eachother. For these requirements, the highly accurate monthly prediction, which isreflected by trend and seasonable variance, is indispensable.
We propose an adaptive and dynamic prediction approach, which is reflected bybusiness trend and seasonable variance, and show its effectiveness. By the reason forthe basic character of this approach, which has a dynamic adaptability to the businessenvironment, this approach is considered as a reasonable and effective decision-making
method to the stable supply of electric power.
1.2. Requirements to monthly demand forecasting of electric power
As above stated, the requirements to monthly prediction of electric power demand arelisted as follows.
･ prediction scheme adapted to the past demand time-series,･ prediction mechanism reflected to the movement of business environment.
The first requirement states that this prediction has to be a dynamic and adaptive
prediction scheme, which picks up the peculiar variation of the past time-series.
Obviously, this scheme is not based on the static scheme as ‘curve fitting’ or ‘regression
formula’ but based on the movement of time-series moment by moment and highprediction accuracy is required. About this scheme, a series of approaches based on the
structured neural network, which learned by the past variation of demand time-series
are investigated. [2,8] In this paper, the Kalman-Filter is adopted as the framework of2

the prediction. The Kalman-Filter is the real-time estimate/prediction scheme withmulti input/output. It processes the state-space, which constructs the system, and themeasurement-space, which observes the time-series and transforms to the state-space,
separately. The outline of the Kalman-Filter is described in Appendix 1.
The second requirement states that this prediction requires the scheme, whichreflected to the movements of business environment. For this reason, the movement ofbusiness environment, which influences electric power demand, should be adopted tothis prediction mechanism. Namely, this prediction has the basis that electric power
demand is under the movement of business environment, builds the framework which
extracts the integrated factors from the electric power and the related business indices.
The integrated factors time-series are sequentially extracted from the electric power
time-series and the related business indices time-series by PCA (Principal Component
Analysis) method, and are built as the explanatory variables of the state-space. The
outline of PCA is described in Appendix 2.
The concept of “Data-Mining” states “discover of new knowledge from accumulated
data” mainly. Namely, this is the concept, which does not verify the previously set
hypothesis, but discover any new facts or rules from accumulated data. [4,5,6] So, thevarious types of neural network are used frequently as the empirical modelingapproach based on the accumulated data. [13,17]
We use the time-series PCA as the integrating method. The component time-series
are integrated from the original time-series by PCA and independent with each other.
Under above definition of “Data-Mining”, the proposed prediction scheme can be basedon “Data-Mining” basically. Namely, the business environments are integrated andextracted from the some accumulated business indices. And the targeted electric
power demand time-series is predicted with these business environments. The
demand prediction approach investigated here, realizes the second of above statedrequirements from the aspect of “Data-Mining”.
Considering the first requirement, this proposed prediction scheme is higher adaptive
to quick varying time-series than slow varying. But according to the dynamic prediction
framework, the prediction of this scheme is limited comparatively to short prediction.
From the second requirement, this scheme has the reasonable reflections not only withthe targeted time-series, but also with the related environments. This states that the
selection of the indices time-series related with the targeted time-series is the key
matter of the prediction.
2. Adaptive electric power demand prediction

32.1. Electric power demand time-series and business indices time-series
As the targeted electric power demand in this study, domestic industry electric power
demand (monthly time-series) is adopted. This time-series is shown as Figure.1. The
sample size is 144, and the period is twelve years from April.1991 to March.2003. For
the reason that the evaluation of prediction is defined as the accuracy of the prediction
error, the prediction period is set in rapidly changed period(two years from April.1999 to
March.2000).
［Fig1. Monthly time-series of Domestic Industry Electric Power Demand(Fiscal 1991-2003), source Toyo-Keizai-Shinpo Corp, “Keizai-Tokei Nenkan”, 2003.］

Domestic Electric Power Demand (Fiscal1991-2002)

Demand(1000Kwh)

19000000
17000000
Monthly

15000000
13000000

Seasonable
Adjusted

11000000
9000000
7000000
1

25

49

73
97
Month

121

(1) Total picture
The total picture of the processing scheme is shown as Figure.2.
［Figure.2 Total picture］

4This scheme is composed of the pre-processing, the Kalman-Filter processing which isleading part, and the post-processing. In the pre-processing, the seasonable variation isseparated from the original time-series. The irregular variation is removed from theseparated trend leading time-series by smoothing (moving average method, etc), andthe smoothed trend leading time-series is successively input to the measurement
mechanism of the Kalman-Filter. The separated seasonal variable time-series istransformed to the seasonable monthly pattern. After it is input into the learned neuralnetwork, the prediction seasonal monthly pattern is generated.
In the measurement mechanism of the Kalman-Filter, the trend leading time-series
and the business indices time-series (seasonable adjusted) are processed by principal
component analysis and integrated into the some principal component time-series, thenbuild the state space of the Kalman-Filter. In the state space, the electric power
time-series and the principal time-series are input into the learned neural network. The
prediction of each variable is gained as the output from the self-organized state space.
The post-processing is the synthesis of the prediction value from the generated
prediction trend leading value and the prediction seasonable pattern. By thispost-processing, the prediction value of the electric power demand which contained theseasonable variation can be gained. [15](2) Prediction of the trend leading time-series
The four business indices relating with domestic industry electric power demand arelisted as follows. Their sample size and period are same and all seasonable adjusted.

x1 : Gross Domestic Production (GDP)x 2 : Index of Industrial Production (Mining and Manufacturing) (MMI)5

x3 : Building Construction Starts
x 4 : Stock Average
The targeted domestic industry electric power demand is defined as the fifth variable.

x5 : Domestic Industry Electric Power Demand
The electric power demand is influenced extremely by the business environment
represented as the indices of industry and daily life. We selected the representative 4
indices as the business environment. Namely, GDP shows domestic total production inquarter of year and represents business trend. Other 3 indices are the indices whichreflect business trend at each time.
The second requirement stated in chapter 1.2 is represented as “the prediction
processing, which reflected to the movements of business environment”. The basic data,
which represented as the movement of business environment, are accumulated
time-series of the business indices. Therefore, for this aim to the prediction processing,
we investigate the processing framework using “Data-Mining” processing. For therealization of this point, following two methods are proposed.
The first is the method which reflects the movement of business environment toprediction mechanism. For solving this problem, the state-space of the Kalman-Filter
needs not to be built by the observed variables directly, but by the principal components
extracted from these observed variables. Namely, by the principal components
time-series, the fewer and non-correlative components are represented than original
observed variables. As a result, the variances of original variables are represented thevariances of the integrated fewer principal components. This states that theindependent variances are extracted from the total variances. The principal component
analysis of the time-series is processed in the measurement transformation mechanism
at each observation cycle, the state space is integrated and re-constructed from originaltime-series at each observation cycle.
The second is the update method of the state space at each update cycle. Generally, inthe field of economics/management analysis, the time update of the system almost
cannot be represented as logical model, but estimated by regression model, etc. As aresult, the estimated parameters are constant and the model is not dynamical.
Previously, we tried to introduce the self-organizing method using the neural network
for solving this problem. This method aimed to detect the structural changes of thesystem by the learned neural network composed of the space-state variables. And as aresult, it is regarded that the space state is updated. The prediction values are gainedby inputting the latest time-series values to the renewed state space. By this method, inthe case of the implicit relation of update mechanism in the economics/management

6

field, the update also can be processed and the time update of space state can be
represented. [9,14,16](3) Prediction of seasonable variation
As shown in Fig.2, the seasonable variation time-series is separated from the originaltime-series. The separating method is adopted CENSUS X.11. The extracted seasonable
time-series is divided into every month and is reconstructed as seasonable monthly
pattern. The prediction method to the monthly pattern is used structured type neuralnetwork. The training date sets are constructed as follows.
The 12 monthly patterns of the last/present year are signified as the input/output of
the neural network. The pair data are set into the neural network and the net is learned.
After learning, the monthly pattern of the present year is input into the learned
network and the output is gained as the prediction monthly pattern.
2.2. Prediction process of trend leading time-series(1) Extraction of the business environment by time-series PCATime-series Principal Component Analysis (PCA) generates the principal components
time-series from the original time-series, and the principal components time-series is
independent of each other.
As shown in Figure.3, this procedure is processed with the constant time period T1(batch) toward to time proceeding by moving one sample (Batch-Sequential Method).
[11]Here, {x ki } is the i -th business indices time-series at the time (k) and { y k } is thetargeted time-series at the same time respectively. And { p kj } is generated the j-thprincipal component time-series at the same time. The indexes and the processing
period are defined as follows.
index of the principal components : j = 1,2,...., M
index of the original variables : i = 1,2,...., Nprocessing period : T1
Considering that {x ki } is the business indices, the integrated component { p kj } canbe the common movement of business movements. Therefore, the time-series of theprincipal components are assumed to be represented the movement of the businessenvironment［Figure.3 Generation of principal component time-series
by Batch-Sequential Method］

7

(2) Self-organizing of state-space
In case of building the Kalman-Filter, main role is the determination of the statetransition matrix. Generally, the dynamics of the system is presented with the group of
the differential equations in physical system and the transition matrix is formulated.
Otherwise, in the case of the targeted electric power demand time-series, its dynamical
movement almost cannot be presented deductively and gained by the estimation ofregression analysis from the past time-series. In this case, the transition matrix iscomposed of regression constants and is not presented with time variation. Therefore,
the Kalman-Filter cannot be applied to non-stationary system by the reason for its
theoretical framework.
To solve this problem, the self-organization of the state space is introduced as a copeto the dynamics of it. Namely, this method is based on that each element of the statetransition matrix is sequentially determined by the self-organization process with thestate variables and their correlations, at each cycle taking in the observed data.
Previously, we proposed the revised GMDH (Group Method of Data Handling) which
had linear correlation between the state variables, as the self-organizing method.［14］Here, the neural network is introduced, for the reason of taking in non-linear
correlation.
The targeted variable time-series and the principal component time-series taken in8the batch period (Batch-Sequential Method) T1 are noted as follows.
Principal component time-series ｛ pij ： i = 1,2,...., T1 : j = 1,2,..., M ｝
Targeted variable time-series ｛ y i ： i = 1,2,...., T1 ｝
Here, M is the number of principal components time-series, T1 is the sample
number of time-series in the batch period.
These time-series lumped together are noted as follow.
｛ xij ： i = 1,2,...., T1 : j = 1,2,..., m ｝(m = M + 1)

The neural network type self-organizing state space is structural type . The appliedneural network is trained using the data sets of neighboring time as training data at
constant interval. At each cycle of the Kalman-Filter processing, the state transition
matrix is renewed by the above stated process. By this training process, the learnedtransition matrix is generated and the prediction value is obtained by inputting thevalues of the latest state variables into the renewed state space. Namely, k is the
updating time, in order to generate the state transition matrix F k +1,k , the training
input/output time-series are set as follows.
・training input { x j ： j = k - T2 ,...., k - 1 } ( T2 is learning period)
・training output { x j ： j = k - T2 + 1,...., k } ( T2 is learning period)
The learning period is a part of the batch period, therefore T2 < T1 is obvious.
These procedure is showed as Figire.4.
［Fig4. Time variation of state space transition］

9

By taking in { x k }to*the learned neural network, { x k +1 } are gained as theprediction value.
2.3. Seasonable prediction and synthesis of prediction values
As shown in Fig.2, the seasonable variation time-series｛ S i ：i = 1,2,...., n ｝is separated
from the trend leading time-series by CENSUS X.11. The seasonal variation time-series
is divided into every month and the monthly seasonable pattern Ril is generated fromit.
｛ Ril ： i = 1,2,....,12 : l = 1,2,...., L ｝
Here, i ，l is the index of month, year respectively，and L is last year. n is the sizeof sample and L = n / 12 .
The generation method of the prediction seasonal pattern is structural neuralnetwork. The neural network is learned by the training data set and the training data
sets are composed as pair of input/output data. The l yearly seasonal pattern is inputof the training data and ( l +1) yearly pattern is output. After learning, L yearlypattern is taken in the neural network, then（ L +1）yearly prediction pattern is gained.
Next, the trend leading prediction time-series which gained in chapter 2.2 and theprediction seasonal pattern which is above stated, are synthesized into the prediction
time-series. The synthesizing method is the inverse process of the separation process
which stated in Figure.2.
3. Evaluation to the proposed scheme
3.1. Evaluation to the trend leading prediction
The targeted prediction time-series is Domestic Electric Power Demand Time-series
shown in Fig.1. The period of it is from Aplil.1991 to March.2003 and has 144 sample
size. We define the accuracy of the prediction error as the evaluation standard andselect the prediction period with the most rapid variation. Therefore, the prediction
period is from Aplil.1999 to March.2001. The previous 8 years (96 samples) time-series
is set as the initialization period for the calculation of the state space and the state noise.
The batch period （ T1 ） which generates the principal component time-series is 8 years(96 samples), it is moved by 1 sample toward time proceeding at each cycle updated in
the prediction period.
In the initial period, the state space of the Kalman-Filter is derived by the regression
calculation in its period. In the successive prediction period (2 years), the principal
component time-series is generated by the measurement mechanism of the
Kalman-Filter at each of the observation. The state space composed by the generated

10

principal time-series is sequentially updated by self-organization of the neural network,
its convergence error is set as the state noise. The learning period （ T2 ） of the neuralnetwork is previous half year as the reason for avoidance of over learning.
(1) Principal component time-series (Data-Mining processing)In the initialization period, after principal component analysis to above stated fivetime-series, upper three eigenvalue and accumulated proportion are obtained asfollows.
eigenvalue accumulated proportion
first PC

（ p1 ）

0.03960.722

second PC （ p 2 ）0.0080

0.867

third PC （ p3 ）0.00470.953

We use three principal components which make up 95 percent of the accumulated
proportion. The factor loading of each｛ p1 , p 2 , p3 ｝from｛ x1 , x 2 , x3 , x 4 , x5 ｝
is as follows.
The meaning of ｛ p1 , p 2 , p3 ｝are considered as below stated.

x1

x2

p1

0.050

p2
p3

x3

x4

x5

-0.001-0.051 -0.063

0.175

0.006

0.024

0.045

0.035

0.065

-0.0010.011 -0.005 -0.017

0.064

・ p1 : makes up 72 percent of total variation, and is effected by｛ x5 ｝especially,
considered as the representative of electric power demand itself.
・ p 2 : makes up 14 percent of total variation, and effected by｛ x3 , x 4 ｝, consideredas the representative of business movement.
・ p3 : makes up 9 percent of total variation, and effected by｛ x1 ｝, uncertain factor.
(2) State-space and construction of the Kalman-Filter
The building of the state-space and the Kalman-Filter are as follows.
・state variables
first PC – third PC :｛ p1 ， p 2 ， p3 ｝
domestic electric power demand :｛ x5 ｝
Above listed are basic time-series and the next four combinations are applied to themodeling.
Model A : ｛ p1 ， x5 ｝

Model B : ｛ p1 ， p 2 ， x5 ｝

Model C : ｛ p1 ， p 2 ， p3 ， x5 ｝

Model D : ｛ x1 , x 2 , x3 , x 4 , x5 ｝

The model D does not adopt principal components but original 5 variables. In this11

model, the measurement matrix is unit matrix.
・state transition matrix
Each element is regression coefficient derived in the initial period (Fiscal
1991-1998). Considering independence with the principal component each other,
the relation formula of each component between k and（ k - 1 ）is represented inAppendix 3.
In the prediction period (Fiscal 1999-2000), the previous half year time-series aretaken in the neural network as the training series, the output is gained by the inputof the latest data. The number of elements in the input/output layers is the same
number of the state-variables of each model (A,B,C,D). And the intermediate layer
is one, its number of elements is the half of the number of the state variables. The
convergence error in learning is about 10-2.

・measurement matrix
The measurement matrix generates from the observed time-series to the statevariable time-series by the principal component analysis using batch-sequential
method, each element is the coefficient calculated by principal component analysis.
・state noise
In the initialization period, the RMS (Root Mean Square Error) value of theregression formula of each state variable is set as the state noise, corresponding
each model. The RMS value about ｛ x5 ｝ at each model is as follows.
Model A: 0.0080

Model B: 0.0145

Model C: 0.0075

Model D: 0.0074

In prediction period, the state noise is convergence RMS value derived by learningof the neural network. This is calculated and updated at each prediction cycle.
・measurement noise
The measurement noise is RMS value between each variable time-series and its
smoothed time-series (Moving Average Processing, averaging width is 3 samples,moving width is 1 sample). The RMS value of ｛ x5 ｝is as follows. This value is usedcommonly through all models and periods.

x5 : 0.0040(3) Evaluation to prediction accuracy
The error (RMS value) of the trend leading prediction applied by each model (A, B, C,D) is listed as Table1. In order to verify the efficiency of this scheme, two cases of thepredictions in the most rapid varying period (Fiscal 1999-2000) and in the most slow
varying period (Fiscal 1997-1998) are tested. These are listed in Table1. Adding thesecomparisons, the estimation error applied by AR (Autoregressive Model) is listed
12

simultaneously. (The detail is shown in Appendix 4.) The reason why AR model isadopted as the comparison with the proposed scheme is that AR model is the dynamic
model with single time-series. In another description, the effectiveness by adopting
multi-variable method in the proposed scheme is also expected.
From the result of the trend leading, it is discovered that model B has the minimum
prediction error. In comparison with the estimation error derived by each model, theerror except model B are small uniformly, the auto regulation with the noise valance
involved in the Kalman-Filter is active. This fact shows that model B has the large
estimation error than any other and model B has better tracing power to the targetedtime-series. AR model built by single time-series has the estimation error with equal
order to model B, but the prediction error is extremely large. It is verified that thisprediction scheme with the Kalman-Filter and self-organized neural network has
considerably high efficiency to the prediction. Also, from the result showed in Figure.2,
the seasonable pattern by learning with neural network is efficient obviously.
［Table.1. Comparison prediction/estimation error derived by trend leading］＜Fiscal 1999−2000, most rapidly varying period＞
Model Name

Model A

Model B

Model C

Model D ＡＲ（２）Prediction error

0.03110.02570.1276

Estimation error

0.00010.00200.00010.00010.0241
AR(2): second order AR model with least AIC

0.0720------＜Fiscal 1997−1998, most slowly varying period＞
Model Name

Model A

Model B

Model C

Model D

ＡＲ（３）Prediction error

0.02620.02420.10760.0849------

Estimation error

0.00010.00260.00010.00010.0034
AR(3): third order AR model with least AIC

The prediction time-series of the trend leading derived by model B is showed in
Figure.5.
［Figure.5. Prediction value derived model B(Fiscal 1999-2000)］13

Ratio(1990 April Standard)

Trend and Prediction(B-Model) in 1999-2000
1.9
1.8

Prediction (b-Model)
Trend Timeseries

1.7
1.6
1.5
1 3

5

7

9 11 13 15 17 19 21 23
Month

The following are shown from the results of table.1 and figure.5.
・The prediction scheme using the Kalman-Filter and the learned state space by
self-organizing mechanism brings about 2.4 percent (the standard is the demand of
1991 fiscal) error accuracy, and it possess considerably accurate prediction than AR
model. And this scheme insures this accuracy to the rapid varying of the originaltime-series.
・The state space which built with the principal component time-series, exhibits higher
effect than with the original variables. The number of the principal components isneed to be increased until accumulated proportion reaches the determined standard,
but need not to be increased so many over the standard.
3.2. Evaluation to synthesized prediction
The error of the synthesized prediction is listed as Table.2, as well as the case of thetrend leading prediction. The synthesizing process is the multiplication the predicted
trend leading time-series with the predicted monthly pattern. Namely, this is theinverse process which separates the trend leading time-series from the originaltime-series.
［Table.2. Comparison prediction / estimation error derived by synthesized series］＜Fiscal 1999−2000＞
Model Name

Model A Model B

Model C Model D

Prediction Error（learned pattern）

0.07840.06290.16250.0838Estimation Error(average Pattern)

0.08330.07900.1687

0.0946The “learned pattern” is gained by taking in seasonable monthly pattern to thelearned neural network. The number of the elements in the input/output layers is

14

twelve. The intermediate layer is one, and the number of its elements is the quarter ofthe number in the input/output layer’s elements. The “average pattern” is gained byaveraging them over the same period which is the learning.
In Figure.6 as shown below, the prediction time-series of the synthesized time-series
with the trend leading derived by model B and the learned pattern is shown.
［Figure.6. The synthesized prediction time-series (Fiscal 1999-2000)
using learned pattern］

Ratio(Standard 1990 April)

Composed Timeseries in 1999.04-2000.03
2.4
2.2
2

Observed Timeseries

1.8

Composed Prediction(Learned)

1.6
1.4
1.2
1

3

5

7

9 11 13 15 17 19 21 23
Month

The following are proved from the results of Table.2 and Figure.6.
・The learning of the seasonable monthly pattern contributes to total accuracy of theprediction time-series. The error accuracy of the synthesized prediction time-series
possesses about 5-6% (the standard is the demand of 1991 fiscal) and is also
considered to show considerable accuracy.
・ In comparison with the trend leading time-series, the prediction accuracy ofsynthesized time-series is lower considerably. This problem can be caused by the lessaccurate prediction of the seasonable variation pattern than the trend leading.
4. Conclusion and discussion
Until now, we tried to investigate the scheme of the demand prediction, which had thetime-varying state space by the learning of neural network and proved that thisapproach gave better accuracy than auto-regressive method, etc. Here, we tried to build
the state space by the principal components in place of the original variables and also
tried to change from the state-space built by the multi-variables with correlation eachother, to the state space built by fewer independent multi-variables. It is proved that thecommon variance extracted by the principal component analysis is effective

15

considerably. Considering the extracted principal components are represented as theindependent business movements, the state space is composed by fewer variables thanthe original variables. Summing up these points, this scheme is represented as follows.
･The proposed scheme based on the framework of the multi variable and dynamic
prediction supplies considerable accurate prediction and this prediction mechanism isconsidered to contribute to the decision-making support.
･ The adoption of the principal components time-series can extract common andindependent variables and the extracted principal components are gained from theaccumulated data. Therefore, this approach has a Data-Mining like character.
・The prediction accuracy is considerably lower in the synthesized time-series than in
the trend leading time-series. It can be needed that the seasonable monthly patternmust be predicted with high accuracy.
At last point, as well as the method which realizes to give high accuracy of theseasonable monthly pattern, the separating and synthesizing methods between thetrend leading time-series and the seasonable pattern shall be the core theme of theseapproach. [16]Appendix 1. Kalman-Filter
The Kalman-Filter is the rational, sequential and real-time estimation/prediction
processing scheme based on linear system theory.
The basic formulas of the Kalman-Filter are presented by the state vector and themeasurement vector. [1,3,14]

x k +1 = F k +1,k x k + v k( 1)

y k = H k x k + wk

( 2)

Here,

x k : state vector

F k +1,k : state transition matrixv k : state noise vector
y k : measurement vector

H k : measurement matrix

wk : measurement noise vector
k represents each time. The state noise and measurement noise are independent eachother. And they are assumed to be white noises which are non-correlative at each time.
Generally in processing the Kalman-Filter, the direct solution of state equation andmeasurement equations is not used but the sequential algorithm which updates error
covariance matrix of state noise at each observation cycle, is adopted.

16

Appendix 2. Principal Component Analysis
Principal Component Analysis integrates from multi variables with correlates eachother to independent multi components. The application PCA to time-series data is
described as follows.
At any time, the j-th component is presented as［7］p j = S lN=1 f jl xl( 3)Here, {xl } is l -th time-series,
index of the principal components : j = 1,2,...., M
index of the original variables : l = 1,2,...., Nand

S lN=1 f jl2 = 1
S lN=1 f il f jl = 0( 4)(i ¹ j )( 5)

M is determined by the proportion of the accumulated eigenvalue, which occupies
generally about 80-90 percent. The principal components are selected in the order ofhigh to low eigenvalue. f jl is a component corresponding to an eigenvalue, f j is
obtained as the linear combination of them. f jl is determined at each time (withconstant time period), as shown by formula (3), the time-series of the principal
component｛ p j ｝can be generated over the entire period.
Generally, M is far small than N ｛, p j ： j = 1,2,...., M ｝is considered to present thecommon variances of the original variables. Considering the independence of eachcomponent to the other components, the time-series of the principal component isassumed also to be independent.
Appendix 3. The relation formula of each components
The relation formulas of p1,k , p 2, k , p3,k between k and ( k - 1 ) are follows.

p1,k = - 0.0095 + 1.0155 p1,k -1( 6)

p 2, k = 0.1431 + 0.9208 p 2, k -1

( 7)

p3,k = 0.2926 + 0.6571 p3,k -1( 8)

For each model, the relation formulas of ｛ x5 ｝between k and ( k - 1 ) are as
follows.
Model A: x5,k = 0.0159 + 0.0407 p1,k -1 + 0.9607 x5,k -1( 9)

Model B: x5,k = - 0.0159 + 0.2221 p1,k -1 + 0.0616 p 2, k -1 + 0.7670 x5,k -1(10)

Model C: x5,k = 0.0051 - 0.4072 p1,k -1 - 0.1929 p 2, k -1 + 0.1517 p3,k -1

17

+ 1.4715 x5,k -1(11)

Model D: x5,k = 0.0245 + 0.0319 x1, k -1 - 0.0782 x 2 ,k -1 + 0.0175 x3,k -1- 0.0003 x 4 ,k -1 + 1.0024 x5,k -1(12)

Appendix 4. Comparison with AR modelThis model is autoregressive model with a single time-series (domestic electric power
demand time-series only). To apply AR model, the applied time-series must be madeinto stationary time-series, this process fits the curve to original time-series, subtracts
it from original time-series in 10 years (fiscal 1991-2000) and the residual time-series isgenerated. The residual time-series is stationary and can be applied by autoregressive
approach. The optimal AIC (Akaike’s Information Criteria) order is 2 in 1991-2000 and
3 in 1991-1998.
Reference
1. Aoki,M. (1987) State Space Modeling of Time Series, Springer-Verlag
2. Araki,H. Kimura,I. Arizono,I. and Ohta,H. (1996) ‘Demand Forecasting based on
Differences of Demands via Neural Networks’, J.JIMA, Vol.47, No.2, pp.59-68.
3. Arimoto,S. (1977) Kalman Filter, Sangyo Toshyo
4. DasguptaC,G. Disdensa,G,S. and Ghose,S. (1994) ‘Computing the predictive
performance of a neural network model with some traditional market responsemodels’, International Journal of Forecasting, Vol.1, pp.235-244.
5. Dutta,S. Shekhar,S. and Wong,W,Y. (1994) ‘Decision Support in non-conservative
domains: generalization with neural networks’, Decision Support System, Vol.11,
No.5, pp.527-544.
6. Joseph,P.B. (1996) Data Mining with Neural Networks, McGraw-ill
7. Kariya,T. (1986) ‘Multi Variate Time-series Variation Factor Analysis Model’,

Keizai-kenkyu, Vol.37, No.1
8. Kimura,I. Arizono,I. and Ohta,H. (1993) ‘An application Neural Network to Demand
Forecasting’, J.JIMA, Vol.44, No.5, pp.401-407.
9. Kitagawa,G. Gersch,W. (1985) ‘A smoothness prior time-varying AR coefficientmodeling of non-stationary covariance time series’, IEEE Transactions on

Automatic Control, Vol.AC-30, No.1, pp.48-56.
10. Kunisawa,N. (1994) ‘A demand estimation (prediction in electric industry)’,Communications of the Operations Research of Japan, Operations Research, Vol.39,

18

No.9, pp.463-469.
11. Nisimura,T. (1978) ‘An application Kalman Filter Theory to Vehicle System’, System

and Control, Vol.22, No.1, pp.10-19.
12. Ono,K. Onoda,T. (1992) ‘Two approach for the prediction of the tomorrow maximum
electric power’, Communications of the Operations Research of Japan, Operations

Research, Operations Research, Vol.37, No.9, pp.417-427.
13. Quaddus,M,A. Khan,M,S. (2002) ‘Evolution of artificial neural networks in businessapplications: an empirical investigation using a growth model’, Int. J. Management

and Decision Making, Vol.3, No.1, pp.19-34.
14. Sugihara,T. (1996) Economic time-series analysis adopting adaptive models,Kougaku Toshyo
15. Sugihara,T., Tsubone,H. (1999) ‘An Adaptive Demand Forecasting Approach for
Supply Chain Management’, Proc.4-th. Inter Symposium of Logistics, pp.255-260.
16. Sugihara,T. (2002) ‘Adaptive Demand Forecasting Approach based on principal
components time-series’, Int. J. Management and Decision Making, Vol.2, No.3,
pp.151-164.
17. Vojinovic,Z. Kecman,V. and Seidel,R. (2002) ‘Modeling empirical data and decision
making with neural networks’, Int. J. Management and Decision Making, Vol.3,
No.2, pp.180-202.

19