Allgemeines Statistisches Archiv 89, 403–418 c Physica-Verlag 2005, ISSN 0002-6018
Using Analysis of Variance and Factor Analysis for the Reduction of High Dimensional Variation in Time Series of Energy Consumption By Carsten Schneider, Gerhard Arminger and Alexandra Schwarz Summary: In this paper the complexity of high dimensional data with cyclical variation is reduced using analysis of variance and factor analysis. It is shown that the prediction of a small number of main cyclical factors is more useful than forecasting all the timepoints separately as it is usually done by seasonal time series models. To give an example for this approach we analyze the electricity demand per quarter of an hour of industrial customers in Germany. The necessity of such predictions results from the liberalization of the German electricity market in 1998 due to legal requirements of the EC in 1996. Keywords: Cyclical structure, hierarchical analysis of variance, factor analysis, liberalized energy market.
1. Introduction and problem description In 1998 the German electricity market was deregulated by a new energymarket law (EnWG, 1998) which was a reaction to the European rule for the liberalization of electricity markets (Guideline 96/92/EG) in the EC1 . As a result, all customers, industrial as well as private consumers, can freely choose their energy provider. This opening of the market even to distributors without an own electricity network or without any power plants results in a serious difficulty: the whole physical infrastructure (network, transformation stations, etc.) remains in the possession of the original energy distributors with their formerly regional monopolies. All providers of power without an own power grid must have the possibility to use foreign networks and have to pay a certain amount for this use (M¨ uller, 1998). As electricity is not storable over time, it must be assured that the demand for and the supply of power are equal at any time (Laumanns, 2005). To guarantee the performance of the electricity networks, the regional providers need to know who will send how much power through their net at any point in time and will receive a fixed amount of money for every unit of power that is sent through their network (Fritz, K¨ onig, 2000). Therefore, electricity providers have to predict the demand of their customers, not only for all customers but for every regional network. Usually two groups of customers are considered seperately: There is a huge number of private Received: 03.11.04/ Revised: 30.06.05 1 The new Europeen Guideline (2003/54/EG) will be followed by a modified energymarket law in Germany but both of these modifications will probably not result in changes to the technical requirements to predictions of electricity demand.
404
CARSTEN SCHNEIDER et al.
customers in every region with a more or less stable standard profile of electricity consumption (Kraus, 2004, p. 175f). For these private customers individual data of demand are available only once a year. The second group of industrial customers is a smaller one, but these customers may have a much greater electricity demand and the daily or weekly figures of their individual consumption profiles are more heterogenous. Therefore most industrial customers own telemetric systems for their electric meters and new data of individual electricity consumption are theoretically available every quarter hour. The demand of power for all customers in each region has to be forecasted for every quarter of an hour. The prediction for the consumption of private customers uses well known profiles with regard to the weekday, the weather and some other indicators. The individual risk for a serious difference between the predictive profile and the actual consumption results in a higher amount for sending the power through the regional net (B¨ orninck, 2003, p. 24f). For the group of industrial customers prediction on a real data basis is possible and therefore this second group is the interesting one for using more complex statistical predictions. The time point is defined as quarter of an hour and the possibility of aggregating all customers belonging to one specific region is given. Therefore, the providers have to make at least four predictions for the four regional nets in Germany (RWE, EnBW, e-on, Vattenfall) (Specht, 2005, p. 30f). Additional difficulties in this aggregation are caused by possible changes in the portfolio of customers due to recruitment of new or loss of present industrial customers. To settle the details of using foreign physical net equipment the German electricity providers and some groups of industrial customers have made agreements in a ”federation of providers arrangement” in its latest version in (VDEW, 2001). Among other things as for instance the general amount for sending power through a foreign power grid (B¨ orninck, 2003, p. 2), a financial punishment for the case of a serious difference between consumption and supply for some time points is defined. In the case of underestimation, the regional net providers deliver the additional necessary electricity and are allowed to present a price uplift to the original electricity provider. In the case of overestimation, the compensation for the electricity not consumed by their own customers is lower than the usual buying price of the delivering provider. At this point the accuracy of the predicted power demand becomes a factor of business success: In the case of serious deviations between supply and consumption of the regional group of customers, the extent of the financial punishment soon becomes greater than the earnings from providing the power.
FACTOR ANALYSIS AND CYCLICAL VARIATION
405
The task of predicting the demand of groups of industrial customers can be derived from the following features for each provider without an own network or for the providers conveying power to other networks: • Forecasts have to be made for every quarter of an hour of the following day. This results in an amount of 24 × 4 = 96 forecasts for each day. • Since the forecasts will be day-specific, the forecasts are usually made for the next 7 days. This leads to a total of 7 × 96 = 672 forecasts. • Forecasts have te be computed for each network, e.g. 4, which results in a total of 4 × 672 = 2678 forecasts which have to be actualized whenever new data become available which usually occurs daily. The time series of electricity demand has a complex structure that can be decomposed into the following components of variation: 1. Seasonal variation induced by the annual seasonal changes (temporal seasonality e.g. spring, summer, fall, winter). This kind of cyclical behaviour is of special interest for the private customers and does not have a strong relevance for industrial electricity consumption. Therefore the annual season is not regarded during the following procedure. 2. Variation induced by changes of demand for electricity current at different weekdays. The demand usually differs along the different workdays and the weekend respectively. In addition the demand usually changes if the workday is a holiday, for instance 25th of December (interday variation). 3. A cyclical component induced by changes of demand during the day. They can be caused by heavy workloads in the morning or in the evening in the case of private customers or by abrupt changes in the work process of industrial customers. This may be caused by the beginning or end of a shift, production interruption etc (intraday cyclical component). Figure 1 shows a typical pattern of interdaily and intradaily variation. The task of forecasting 2678 time series can be solved very efficiently if the complexity described above is systematically reduced by the following procedures of classification and dimension reduction which must be performed automatically. 1. The demand for the different weekdays is classified automatically into different ”day types” which show the same behaviour within each day type and different behaviour between day types. Since the effects of day types differ for workdays and holidays, a hierarchical analysis of variance is used for this classification. 2. Within each day type, the intraday cyclical variation described by 96 time points is reduced to a small number of factors which explain most of the correlation between the 96 time points. As only groups of industrial customers are regarded, the factors may be interpreted as a typical structure of shift (factor 1) and break times (factor 2) plus a basic or stand-by electricity demand (factor 3).
406
CARSTEN SCHNEIDER et al.
Figure 1. Example for interdaily and intradaily structure of electricity demand (customer 1).
As a result of the classification and dimension reduction procedures, forecasting needs to be performed only for the resulting factors of each day type. The predicted demand curve is then computed based on the forecasts of the factors using well known procedures of estimating the factor scores (Schneider, 2002). 2. Hierarchical Analysis of Variance for the Classification of Weekdays The seven weekdays and possible holidays within the week or at the weekend may possibly have different structures in their cyclical variation. Especially the weekends and the holidays may have a completely different daily cycle compared to the workdays. Moreover, Mondays and Fridays often look different than the other workdays. Figure 2 gives an example for the similarity of the daily electricity consumption for the five weekdays without holidays. In Figure 2 for every workday d, d = 1, ..., 5, at any time point i, i = 1, ..., 96, the average of observed electricity demand during the last four weeks j, j = −3, ..., 0, is calculated: (d)
y¯i
=
0 1 X (d) y (j) with d = 1, ..., 5 4 j=−3 i
(1)
where Monday is coded as 1, Tuesday as 2 and so on. With regard to stability and adaptivity of the following prediction process it is necessary to classify the weekdays on the basis of their typical daily
FACTOR ANALYSIS AND CYCLICAL VARIATION
407
Figure 2. Example for intradaily structure of electricity demand (customer 2) on different weekdays.
pattern of energy consumption. This is used as preprocessing step (Harvey, Koopman, 1993). To reduce the number of weekdays from the maximum of 14 which arises from the fact that every weekday (Monday,...,Sunday) can either be a workday or a holiday, the combinations of any ”workday/weekday” or ”holiday/weekday” are classified into day types. Every day type consists of data with a homogeneous pattern within the 96 daily observations. In this context the interdaily variation is not regarded as a cyclical component but as a structure that can be simplified by classification of weekdays. Potential changes in the portfolio of industrial customers within a local network require a highly adaptive procedure (Schneider, 2002, p. 21ff). Therefore a hierarchical analysis of variance is chosen that has to be reinitialized if any changes in the portfolio of customers occur. Hence, the aim of this step is mainly the dimensional reduction with regard to a high adaptivity of the whole process. A preliminary step of reducing the number of possible day types has to take place because of a possible lack of data. Due to the rare occurence of certain holidays it is insufficient to start forecasts including seven weekdays as holidays separately. There occur e.g. too few ”monday and holiday”combinations to wait for enough observations for the calculation of this day type. Therefore an intuitive first classification results in merging all the weekdays with holidays and the holidays during the weekend to two day types (Harvey, Koopman, 1993). The basis of the following step are consequently the following nine days or (holiday)-day types respectively:
408
• • • • • • • • •
CARSTEN SCHNEIDER et al.
Monday + workday (W D M o) Tuesday + workday (W D T u) Wednesday + workday (W D W e) Thursday + workday (W D T h) Friday + workday (W D F r) Saturday + workday (W D Sa) Sunday + workday (W D Su) weekday + holiday (HD M o − HD F r) weekend-day + holiday (HD Sa + HD Su)
These preliminary nine day types may be further reduced by using a hierarchical analysis of variance. The first level of the hierarchy is the dichotomy workday versus holiday modelled as dummy variable in the second column of the design matrix in equation 2. The first column is a vector of ones for the regression constant. The second level of the hierarchy describes the weekdays modelled again as dummy variables, that is columns 3 to 8 for the workdays and column 9 for the holidays. The index of the regression coefficient corresponds to the column of the design matrix X0 . W D mo
β1 β2 β3 β4 β5 β6 β7 β8 β9
1 0 0 0 0 0 0 0 0
W D tu 1 W D we 1 W D th 1 WD fr 1 W D sa 1 W D su 1 HD mo 1 HD tu 1 HD we 1 HD th 1 HD f r 1 HD sa 1 1 HD su
= β′
0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 = X0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1
(2)
The values of the regression coefficients βd , d = 2, ..., 9 can be interpreted as the deviation of the average demand of the preliminary day types from the first day type defined as combination ”Monday and workday” for which always many observations are available. Please note that the regressor matrix in Equation (2) is written only for the case of one observation for each
FACTOR ANALYSIS AND CYCLICAL VARIATION
409
of the 14 possible combinations. In practical applications, this matrix is expanded to as many observations as available, that is usually over at least a couple of months. If any βd , d = 2, ..., 9, is not significantly different from zero, the corresponding day is merged into the first day type together with the reference day type (combination ”Monday and workday”). In this case, the merged day has a mean electricity demand of β1 - the same as a ”Monday and workday”. In a further step a similar matrix with less possible combination for the rest of the days has to be built. The days that do not merge ”Monday and workday” are compared next. The χ2 -tests for the single day comparisons have the following form: βˆ′ C ′ (C ′ Vˆ C)−1 C βˆ ∼ χ2r .
(3)
In this case C consists in the corresponding row(s) of X0 with rank r, e.g. testing for equal to ”Monday + workday” results ”Tuesday + workday” ˆ in C = 1 0 1 0 0 0 0 0 0 . V is a consistent estimation of the covariance ˆ The test statistic for a single day is χ2 − distributed with r dematrix of β. grees of freedom as βˆ is asymptotically normally distributed with expected value β and covariance matrix V (Amemiya, 1985, p. 30). The comparison can also be calculated for blocks of days if an F -Test is used instead of an χ2 -Test distribution (Baltagi, 2002, p. 82f). The number of day types can be reduced step by step using χ2 -tests for the differences between the daily means of electricity demand with regard to the multiple test problem concerning the level α of the various χ2 -tests (Schach, Sch¨ afer, 1978, p. 182ff). Note that the χ2 approximation is used instead of the F-test because the number of observations (rows of the design matrix X0 ) is usually greater than 100. The result is a number D of identified day types d, d = 1, ..., D. Every workday/weekday and holiday/weekday combination is allocated to exactly one day type. Figure 3 illustrates this classification for the data in Figure 2. This classification will be used for the further steps of the dimension reduction and prediction procedure. Figure 3 shows that the three weekdays Tuesday, Wednesday and Thursday have been merged to one day type, whereas Monday and Friday stay separate. Monday starts with a much lower electricity demand than all other workdays. This results from different times of starting the first shift after the weekend in the different firms belonging to this group of customers. A similar effect occurs on Friday where the demand decreases compared to the other days due to the beginning of the weekend. In this example the hierarchical analysis of variance results in reducing the five weekdays to three day types. Improvement of this procedure by rotation of the reference day has been tried but does not lead to better results in the sense of either less day
410
CARSTEN SCHNEIDER et al.
Figure 3. Result of calculation of day types (customer 2).
types or lower intraclass variation (variation within day types). Therefore a randomized rotation has not been implemented in the final automatic procedure. 3. Factor Analysis for the Reduction of Intraday-Variation For each day type d = 1, ..., D, there exists an intraday cyclical pattern of (d) electricity demand yij where i denotes the observation day (i = 1, ..., n) and j denotes the measurement time in quarters of an hour, j = 1, ..., m with m = 96. In the first step, the demands are standardized by using mean y¯j and standard deviation sj as follows: zij =
yij − y¯j . sj
(4)
This standardization is done for all days within a day type d. In the second step, the correlation matrix of the yij is computed using the standardized values: n 1 X zij zik . (5) rjk = n − 1 i=1 1 The result is summarized in the correlation matrix R = n−1 ZZ’. This is the 96 × 96 correlation matrix of the intradaily consumption demands for each day type d = 1, ..., D.
The basic idea of factor analysis is to reduce the m observed variables (demands at times j = 1, ..., 96) to as few as possible hypothetical (latent)
FACTOR ANALYSIS AND CYCLICAL VARIATION
411
variables that reproduce the stochastic dependencies and therefore the correlation matrix R of the standardized observations (Ost, 1984). The factor analytic model is described by the linear combination of Q, Q ≤ m factor scores piq and factor loadings ajq : zij =
Q X
ajq piq ,
i = 1, ..., n, j = 1, ..., m and q = 1, ..., Q
(6)
q=1
with piq : factor score for factor q concerning object i, ajq : factor loading of variable j for factor q . For identification it is assumed that the factor scores themselves are standardized, that is with mean values p¯q = 0 and standard deviations sq = 1, q = 1, ..., Q. Consequently the covariance (and correlation) matrix of the factors is the Q × Q identity matrix IQ . The properties of the factor model can more easily be seen using the matrix formulation Z = PA′ ,
(7)
where Z is the n × m matrix of standardized observations, P is the n × Q matrix of factor scores and A is the m×Q matrix of factor loadings. As a result of the standardization and non-correlation of the factors the correlation matrix R of observations may be written as R =
1 1 ZZ′ = AP′ PA′ = AIQ A′ = AA′ . n−1 n−1
(8)
Therefore R is only a function of the factor loading matrix A. This is the so called ”Fundamental Concept of the Factor Analysis” (Thurstone, 1941; Kim, Mueller, 1994) which is only applicable to explanatory factor analysis. Note that the factor loadings are only identified up to an orthogonal transformation T ∼ Q × Q with TT′ = IQ because the transformation of A into ′ A⋆ = AT yields the same result (A⋆ A⋆ = ATT′ A′ = AA′ ). From Equation 8 one can deduce immediately that the variance of each standardized observation variable may be written as the sum of squared factor loadings, that is 1 = σj2 =
Q X
a2jq .
(9)
q=1
If Q is set to m, the decomposition of R into Q factors in R = AA′ is called principal components analysis (Jolliffe, 1986). In this case the variance is completely explained by the Q factors (components). However, very often only a few components, for instance Q1 , suffice to explain a great proportion
412
CARSTEN SCHNEIDER et al.
of the existing variance. If these components are collected as the first Q1 factors, then the partial sum h2j,Q1 =
Q1 X
a2jq ≤ 1
(10)
q=1
is called ”Communality” (variance explained by the first Q1 factors). The remaining Q − Q1 factors are interpreted as specific factors comparable to the function of an error term in regression analysis. In the case of modelling the intradaily cyclical variation one aims to reduce the complexity of 96 data points down to only as few as possible common factors that determine the main features of the observed variation. To calculate the number of common factors one starts with h2j,Q1 = 1 and computes a principle components analysis with Q = m components as first solution. The algorithms for this principle components analysis are found in Jolliffe, 1986 or Ritsert et. al., 1976. The next aspect is to determine the right number of extracted factors for further use in a model for modelling or prediction of electricity demand. This question is equivalent to finding the optimal balance between the maximization of that part of the variance in the data that can be explained by the extracted factors (Q1 increasing) versus the simplification of the used model (Q1 decreasing). As seen in the last section, the communality in Equation (10) describes the part of the variance of the variable of interest explained by Q1 factors. On the other hand the part of the explained variance by the one special factor q concerning all variables is given by the squared eigenvalue: λ2q =
m X
a2jq ∀ q = 1, ..., Q .
(11)
j=1
Following the Kaiser-criterion (Arminger, 1979, p. 37) only factors q with eigenvalues λq > 1 are chosen. This means that the factor q explains a greater part of the variance in the data than a single variable does. All other factors with lower explanatory power and a corresponding eigenvalue λq < 1 are not extracted for the final model. After this identification of relevant factors the matrix A of factor loadings is reduced from Q to Q1 factors. Additionally the factors are now sorted by relevance which means that the loadings of the factor q with the greatest eigenvalue build the first column of the new matrix A⋆ ∼ m × Q1 . In this special example of explaining 96 daily time points by factor analysis the number of explanatory variables for a following prediction of electricity demand decreases from 96 to about 5 when using the criterion explained above. The factor extraction for modeling the daily demand pattern has to
FACTOR ANALYSIS AND CYCLICAL VARIATION
413
be calculated for every classified day type and will be based on a quarter of an hour average electricity demand over all days belonging to this day type. (d) In this case the factor values piq for every day of type d in the prediction period of length h, h = 1, ..., H, have to be calculated by solving (d) zij
=
Q1 X
(d)
(d)
a⋆jq piq + ηij .
(12)
q=1
If for example Monday and Tuesday belong to the same day type d, this will result in identical factor values for all Mondays and Tuesdays within the prediction period for all possible forecast horizons H. 4. Additional Aspects for the Prediction of Factor Values In the application of using factor analysis for reducing the number of cyclical dimensions in electricity demand some additional aspects have to be regarded. Especially the embedding of the factor analysis into a prediction framework leads to some special issues. Since the final aim of the dimensional reduction is making the calculation of the predictions of electricity demand easier and less time consuming, the factor analysis has to be reversed to give predictions of the demand for each quarter of an hour. For this purpose, the following steps are taken. First, the factor scores of the chosen model with Q1 factors are computed using the so called power algorithm proposed by Jolliffe (1986, p. 235ff). These factors have to be computed for every factor q = 1, ..., Q1 at (d) day i = 1, ..., n of day type d. The factor scores are denoted by piq . Second, the factor scores for future demand (on the factor analysis level) are calculated by using a regression model for the factor scores with two explanatory variables. The first input variable is the factor score of the last observation at day type d. The second input variable is the local mean factor score over the last seven days (one week) of day type d at day T , that is 7
(d)
p¯iq (T ) =
1 X (d) p (T − j) . 7 j=1 iq
(13)
The regression equation that is used for prediction is therefore given by (d)
(d)
(d)
(d)
piq (T ) = β1 piq (T − 1) + β2 p¯iq (T ) + ǫiq (T ) .
(14)
The results of the regression analysis are the estimated parameters βˆ1 and βˆ2 which in turn are used to define the forecasting equation (d) (d) (d) pˆiq (T + 1) = βˆ1 piq (T ) + βˆ2 p¯iq (T + 1) .
(15)
414
CARSTEN SCHNEIDER et al.
Hence, one week forecasts only need one step forecasts for each day type. Third, the process of factor analysis is reversed by computing the intraday demand vector from (d)
(d)
zˆi,j (T + 1) = A⋆ · pˆiq (T + 1) .
(16)
Again, this calculation must be executed separately for each day type d = 1, ..., D. Finally the task of computing a forecast for a 96 points profile per day has been reduced to computing Q1 factor scores for each day type which leads to much more easily calculated and more stable estimates. 5. Empirical Results of the Factor Analysis
day type 1 factor 1 factor 2 factor 3 factor 4 factor 5 factor 6 factor 7 factor 8 factor 9 factor 10 factor 11 day type 2 factor 1 factor 2 factor 3 day type 3 factor 1 factor 2 factor 3 factor 4 factor 5 factor 6 factor 7 factor 8
eigenvalue
% of expl. variance
cum. expl. variance
40.275 23.233 8.917 6.829 4.747 2.834 1.853 1.699 1.479 1.338 1.032
41.953 24.201 9.289 7.114 4.944 2.952 1.930 1.770 1.541 1.394 1.075
41.953 66.154 75.442 82.556 87.500 90.453 92.383 94.152 95.693 97.087 98.162
54.831 26.149 15.020
57.116 27.238 15.645
57.116 84.354 99.999
70.965 8.668 5.524 3.048 1.860 1.572 1.304 1.223
73.922 9.029 5.754 3.175 1.938 1.637 1.358 1.274
73.922 82.951 88.705 91.880 93.818 95.455 96.813 98.087
Table 1. Extracted factors per daytype for customergroup 1.
FACTOR ANALYSIS AND CYCLICAL VARIATION eigenvalue
% of expl. variance
cum. expl. variance
factor 1
39.587
41.237
41.237
factor 2
21.345
22.235
63.472
factor 3
16.140
16.812
80.284
factor 4
6.458
6.727
87.011
factor 5
2.844
2.962
89.974
factor 6
2.436
2.537
92.511
factor 7
1.941
2.022
94.533
factor 8
1.483
1.545
96.078
factor 9
1.299
1.353
97.431
factor 1
50.127
52.215
52.215
factor 2
34.219
35.645
87.860
factor 3
11.655
12.140
99.999
factor 1
48.804
50.838
50.838
factor 2
22.208
23.133
73.971
factor 3
15.448
16.091
90.063
factor 4
9.540
9.937
99.999
factor 1
61.748
64.320
64.320
factor 2
16.652
17.346
81.666
factor 3
12.354
12.869
94.536
factor 4
2.588
2.696
97.232
factor 5
1.485
1.547
98.779
factor 6
1.172
1.221
99.999
415
day type 1
day type 2
day type 3
day type 4
Table 2. Extracted factors per daytype for customergroup 2.
The dimensional reduction by hierarchical analysis of variance for calculating interdaily classes and factor analysis for the case of strong intradaily variation in electricity demand data is evaluated regarding two data sets. Both data sets consist of the demand of a group of customers over eight months with 96 observations per day. In a first step the day types are computed as described in Section 2. Afterwards the factors and factor loadings are calculated for the generated day types as described in Section 3 with respect to the steps described in Section 4. Finally, forecasts are computed for the electricity demand per quarter hour. In the following Tables 1 and 2 the factors with an eigenvalue of at least 1 are given (using the Kaiser-
416
CARSTEN SCHNEIDER et al.
criterion) for the resulting day types. For both groups of customers a sizable reduction of day types by using the automatized hierarchical analysis of variance is achieved. Instead of 9 days only 3 or 4 day types have to be regarded. Hence, the second step of dimensional reduction can already be done with much less computational effort. A reduction from 96 time points to not more than 11 factors depending on the day type can be shown for both customer groups. The part of the explained variance in the data is at least 97.4 % for customer group 2 at day type 1. For the other group of customers and other day types the part of the explained variance is even higher and goes up to 99.99 %. Therefore a relevant loss of information does not occur. This also means that the number of necessary predictions for future electricity demands decreases significantly. The following Table 3 summarizes the dimensional reduction using the factor analysis for the two data sets. all factors
at least 90% explained variance
number of
dimensional
number of
dimensional
factors
reduction
factors
reduction
day type 1
11
11.458
6
6.250
day type 2
3
3.125
3
3.125
day type 3
8
8.333
4
4.167
day type 1
9
9.375
6
6.250
day type 2
3
3.125
3
3.125
day type 3
4
4.167
3
3.125
day type 4
6
6.250
3
3.125
customer 1
customer 2
Table 3. Dimensional reduction for both customers based on 96 time points.
6. Conclusions The results imply a great simplification for the computation of predictions of the electricity demand. In comparison to the forecasting of 672 separated time points per customer and week in this case only 22 predictions for customer group 1 (11 + 3 + 8) and also 22 for the second group of customers are necessary due to the results of the classification of weekdays and the factor analysis. The degree of dimensional reduction in this example therefore is up to 96% while the loss in explanatory power due to the factor analysis is less than 3%. Simultaneously the possibility for computing reliable predictions increases as the classification of weekdays allows an earlier begin
FACTOR ANALYSIS AND CYCLICAL VARIATION
417
of automatized forecasts. This is of special interest regarding the required prediction for new customers or after changes in the portfolio of customers belonging to one regional network as well as shorter forecasting horizons for the other groups of customers.
References Amemiya, T. (1985). Advanced Econometrics. Blackwell, Oxford. Arminger, G. (1979). Faktorenanalyse. Teubner, Stuttgart. Baltagi, B. (2002). Econometrics (3rd. Edition). Springer, Berlin. ¨ rninck, S. (2003). Auswirkungen der Strommarktliberalisierung auf das deutBo sche Energienetz. Logos, Berlin. EnWG (1998). Energiewirtschaftsgesetz vom 24.04.1998. Bundesgesetzblatt. ¨ nig, S. (2000). Der liberalisierte Strommarkt – eine Einf¨ Fritz, W., Ko uhrung. In Wettbewerb im liberalisierten Strommarkt (Kahmann, M., K¨ onig, S., eds.), Springer, Berlin. Harvey, A. C., Koopman, S. J. (1993). Forecasting hourly electricity demand using time-varying splines. Journal of the American Statistical Association 88 425, 1228–1236. Jolliffe, I. T. (1986). Principle Component Analysis. Springer, New York. Kim, J.-O., Mueller, C. W. (1994). Introduction to factor analysis. In Factor analysis and related techniques (Lewis-Beck, MS., ed.), Sage, London. Kraus, M. (2004). Lexikon der Energiewirtschaft. Verlag Deutscher Wirtschaftsdienst, K¨ oln. Laumanns, U. (2005). Technische Grundlagen der Energiepolitik. In Grundlagen der Energiepolitik (Reiche, D., ed.), Verlag Peter Lang, Frankfurt/Main. ¨ller, L. (1998). Handbuch der Elekrizit¨ Mu atswirtschaft. Springer, Berlin. Ost (1984). Faktorenanalyse. In Multivariate statistische Verfahren (Fahrmeir, L., Hamerle, A., eds.), 575-662, de Gruyter, Berlin. Ritsert, J., Stracke, E., Heider, F. (1976). Grundz¨ uge der Varianz- und Faktorenanalyse. Campus, Frankfurt/Main. ¨fer, T. (1978). Regressions- und Varianzanalyse. Springer, Schach, S., Scha Berlin. Schneider, C. (2002). Kostenoptimale Prognose von Lasten in der Energiewirtschaft. Eul, Lohmar. Specht, H. (2005). Stromliefervertr¨ age im liberalisierten Energiemarkt. VDE Verlag, Berlin. Thurstone, L. L. (1941). Factorial Studies of Intelligence. University of Chicago Press, Chicago. ¨tswerke e.V. (2001). Verb¨ VDEW-Vereinigung deutscher Elektrizita andevereinbarung II+.
418
CARSTEN SCHNEIDER et al.
Carsten Schneider Gerhard Arminger Alexandra Schwarz Fachbereich B - Wirtschaftsstatistik Bergische Universit¨ at Wuppertal 42097 Wuppertal Germany
