Multifractal detrended fluctuation analysis for clustering structures of electricity price periods

Physica A 392 (2013) 5723–5734

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Physica A journal homepage: www.elsevier.com/locate/physa

Multifractal detrended fluctuation analysis for clustering structures of electricity price periods Fang Wang a,b,∗ , Gui-ping Liao b , Jian-hui Li b , Xiao-chun Li c , Tie-jun Zhou a a

College of Science, Hunan Agricultural University, Changsha, 410128, PR China

b

Agricultural Information Institute, Hunan Agricultural University, Changsha, 410128, PR China

c

Orient Science and Technology College, Hunan Agricultural University, Changsha, 410128, PR China

highlights • • • •

Some multifractal parameters of electricity price signals are calculated by MF-DFA. Some trends in each signal are filtering by Fourier detrended fluctuation function. Some multi-dimensional spaces are constructed for clustering of price signals. Clustering result indicates that the discriminant accuracies are satisfactory.

article

info

Article history: Received 26 April 2013 Received in revised form 27 June 2013 Available online 26 July 2013 Keywords: Sub-periods electricity price Clustering Multifractal detrended fluctuation analysis Fisher’s linear discriminant algorithm

abstract A new model is proposed to investigate the structure of electricity price in different time periods. A popular method — the multifractal detrended fluctuation analysis (MF-DFA) method is employed to analyze the features achieved from three types of electricity price data after filtering some trends by Fourier detrended fluctuation function. Twelve multifractal parameters are calculated and selected as the characteristic indicators for comparison. Moreover, the minimum number of indicators is determined so that the discriminant accuracy reaches maximum based on Fisher’s linear discriminant algorithm (Fisher’s LDA) for each time period. These indicators form a multi-dimensional space, in which each point represents a price time series. This allows us to cluster the three price time periods, namely, the low price time periods, the average price time periods and the peak price time periods. Fisher’s LDA is employed to evaluate the discriminant accuracy on these three kinds of time periods. Our analysis is then applied to the data in California1999–2000 and PJM2001–2002 electricity markets to demonstrate the applicability of our methods. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The fundamental objective of electric power industry deregulation is to provide a competitive business environment. The operation of the electricity market is far more involved than for traditional markets since the electric commodity must be generated, distributed and consumed in real-time under strict physical laws and extremely high reliability requirements [1]. The complexity of the price’s structure make market administrator and participants attach great importance to the price distribution. As the electricity trading prices express the characteristic of price spikes, affine jump and cycle reply, the electricity system is considered a non-stationary system of nonlinear and weak relevant. A variety of models have been proposed, such as a mean-reversion and dynamic regression model [2], an affine jump model [3], the mix model above the

∗

Corresponding author at: College of Science, Hunan Agricultural University, Changsha, 410128, PR China. Tel.: +86 073184675042. E-mail address: [email protected] (F. Wang).

0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.07.039

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two [4], and an artificial neural networks model [5]. However, the price evolution is a complex dynamic coupling process. In addition, the electricity prices in different periods show different characteristics. And there is a certain correlation among the prices in similar time intervals [6]. Hence, it is important to uncover the features of electricity prices for different periods. Many researchers pointed out that the complex kinetics feature can be uncovered by fractal theory [7–9]. As a new powerful tool for describing the real world, the multifractal theory [10] has attracted many scholars since its birth. Most important records in the real world are in the form of time signals and their multifractal properties have been extensively investigated [11–14]. In order to uncover the multifractal nature in power-law correlations of signals with non-stationarity, some multifractal methods have been proposed. As powerful trends analysis techniques, the detrended fluctuation analysis (DFA) reported in Ref. [15] and the multifractal detrended fluctuation analysis (MF-DFA) reported in Ref. [16] are used in different fields [17–27]. In this paper, we aim to study the electricity price structure. The electricity prices of low, average and peak time periods in California 1999–2000 electricity markets and PJM2001–2002 electricity markets are analyzed and efforts are also made to distinguish them in the three kinds of periods by some characteristic in each period. Different from Ref. [9], our focus is to discern the prices by feature parameters in each period which includes some natural time intervals rather than in each natural interval. Hereafter, one can forecast prices for different time periods based on the feature parameters proposed. The behavior of price forecasting for different periods has advantages of improving forecasting precision to compare with the whole time series forecast, and it can also cost savings and reduce operation difficulty to compare with forecasting for each natural interval. As the electricity prices are affected by system factors and random factors, some periodic trends (like the sinusoidal trend) always exist in price signals [28]. We will first eliminate the effect of the trends by Fourier detrended fluctuation analysis (F-DFA) [29], then we extract some characteristic of the prices in each period by some multifractal parameters calculated by the method of MF-DFA, which provide useful information of the prices in each period. An important procedure is to search the minimum number of indicators for each of the time periods. Thus, we place the prices of all time periods in some specified multi-dimensional spaces formed by some chosen indicators and expect to cluster the prices of each period together and separate one from another. Finally, we apply Fisher’s linear discriminant algorithm [30] to calculate the best discriminant accuracies of the clustering process by our methods. 2. Models and methods The traditional multifractal analysis is based on the standard partition function multifractal formalism, which describes stationary measurements [10]. In a regional electricity market, the electricity prices have abnormal fluctuations and apparent cyclical trends and thus are hardly stationary [2]. Hence, the standard MFA is not always a valid choice to analysis them. Fortunately, the method of MF-DFA [17], which is widely used in recent ten years, can deal with non-stationary series; it does not require the modulus maxima procedure, in contrast to the method of wavelet transform modulus maxima (WTMM) [31], the latter is also used to deal with the non-stationary series, but requires the modulus maxima procedure. MF-DFA is based on the identification of the scaling of the qth-order moments depending on the signal length and is more general than the standard detrended fluctuation analysis (DFA). Like the standard MFA, the method of MF-DFA can also describe multifractal nature of electricity prices signals. 2.1. Multifractal detrended fluctuation analysis (MF-DFA) The MF-DFA procedure consists of six steps [16], of which the first four steps are identical to the conventional DFA procedure (see Ref. [15]). We suppose that x(t ), t = 1, 2, . . . , N is a price time series. Noted that this series is of compact support, i.e. x(t ) = 0 merely for an insignificant fraction of the values. Step 1. We denotes the accumulated deviation series y(k) =

k  [x(t ) − x(t )],

k = 1, 2, . . . , N ,

(1)

t =1

where x(t ) is the average over whole time period of the price series. Step 2. Divide the accumulated deviation y(k) into Ns = [N /s] non-overlapping segments with an equal length s. The length N of the series is often not a multiple of the timescales s considered, and thus a short part at the end of the accumulated deviation could remain. To avoid omitting this part of the series, the same procedure is repeatedly conducted starting from the opposite end. As a result, 2 Ns segments can be obtained together. Step 3. We can obtain a local trend yv (i) by using least square method for each segment v . Since the detrending of the time series is done by the subtraction of the polynomial fits from the y(i). We can determine yv (i) as ys (i) = y(i) − yv (i),

i = 1, 2, . . . , N ,

(2)

linear, quadratic, cubic or higher m-order polynomials may be used in the fitting procedure (conventionally those methods could be called DFA1, DFA2, DFA3, . . . ). Obviously, in DFA mth-order DFA trends of order m and order m − 1 in the original series are eliminated.

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Step 4. Evaluate the local trend for each of the 2Ns segments by the least squares fit of the series, and determine the variance F 2 (s, v) =

s 1

s i=1

y2s [(v − 1)s + i],

v = 1, 2, . . . , Ns ;

(3)

and F 2 (s, v) =

s 1

s i=1

y2s [N − (v − Ns )s + i],

v = Ns + 1, 2, . . . , 2Ns .

(4)

Step 5. Obtain the qth-order fluctuation function by average over all segments

 F q ( s) =

2Ns 1 

2Ns v=1

1/q q/ 2

[F (s, v)] 2

,

(5)

in which the index variable q can generally take any real value except zero. For cases where q = 2, the standard DFA procedure can be retrieved. It is important to find that how the generalized q dependent fluctuation functions Fq (s) is correlated with the timescale s for different values of q. Therefore, Steps 2–5 need to be repeated for several timescales s. Obviously, Fq (s) will increase with s. Fq (s) depends on the DFA order m as well. By construction, Fq (s) is only defined for s ≥ m + 2. Step 6. Determine the scaling behavior of the fluctuation functions, which could be obtained by analyzing ln Fq (s) vs ln s for each value of q. In the cases where the series x(t ) are long range power law correlated, Fq (s) will increase, for large values of s, as a power law, Fq (s) ∝ sh(q) ,

(6)

the meaning of exponent h(2), can be referred to the Ref. [15], which is equivalent with the well-known Hurst index H [19]. Hence, the value of h(q) is called generalized Hurst index. In general, the value of h(q) is nonlinear dependent on the value of q. 2.2. Relation to standard multifractal analysis (MFA) For stationary, normalized records with compact support the multifractal scaling exponents h(q) defined in Eq. (6) are directly related to the scaling exponents τ (q) defined by the standard partition function based multifractal formalism. Suppose that the series xk of length N is a stationary, normalized sequence. Then the detrending procedure in Step 3 of the MF-DFA method is not required, because no trend needs to be eliminated. Hence, the MF-DFA can be replaced by the standard multifractal fluctuation analysis (MFA). The variance in Eqs. (3) and (4) for each segment v , v = 1, 2, . . . , Ns now can be redefined as F 2 (s, v) = [Y (v s) − Y (v − 1)s]2 .

(7)

Inserting this variance into Eq. (5) and using Eq. (6), we obtain



2Ns 1 

2Ns v=1

1/q |Y (v s) − Y (v − 1)s|

q

∝ sh(q) ,

(8)

for simplicity we can assume that the length N of the series is an integer multiple of the scale s, obtaining Ns = N /s and therefore F 2 (s, v) =

N /s 

|Y (v s) − Y (v − 1)s|q ∝ sqh(q)−1 .

(9)

v=1

This already corresponds to the standard multifractal formalism applied e.g. in Refs. [12,14]. In order to relate it to the standard box counting method [17], we adopt the definition of the profile in Eq. (1). Obviously, the term Y (v s) − Y (v − 1)s in Eq. (8) is identical to the sum of the xk within each segment v of size s. The sum is known as the box probability Ps (v) in the standard MFA for normalized series xk , Ps (v) ≡

vs 

xk = Y (v s) − Y (v − 1)s.

(10)

k=(v−1)s+1

In standard MFA, the mass exponent τ (q) is defined via the partition function ψq (ε) which is qth moment of probability Ps ,

ψq (ε) ≡

N /s  i =1

|Ps (v)|q s ∝ sτ (q) ,

(11)

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where the value of q is a real parameter as in the MF-DFA. One can find that the Eq. (11) is identical to Eq. (9), and obtain analytically the relation between the two sets of multifractal scaling exponents,

τ (q) = qh(q) − 1,

(12)

thus, we have shown that h(q) defined in Eq. (6) for the MF-DFA is directly related to the classical multifractal mass exponents τ (q) (for instance, we took q = −15–15 in our work). And the classical exponents are related to another some multifractal parameters, such as analogous specific heat Cq which can be used in many fields of fractal analysis. The expression of Cq is derived by the thermodynamic formulation of multifractal measures as Cq = −

∂ 2 τ (q) ≈ 2τ (q) − τ (q + 1) − τ (q − 1), ∂ q2

(13)

this multifractal parameter could respond to the characteristics of the price sequence well. The values of h(q) relate to the generalized multifractal dimensions as follows:

τ (q) qh(q) − 1 = , q ̸= 1 (14) q−1 q−1 ψ1,+ , q=1 (15) Dq = lim ε→0 ln s  where ψ1,+ = pi ̸=0 ln pi . We can obtain the generalized multifractal dimensions through a linear regression of ψ 1,+ versus ln s for q = 1. Another group of characterizing a multifractal series is Lipschitz–Hölder [10] exponent α(q) and the singularity spectrum f (α). α(q) is related to τ (q) by Dq =

α(q) = τ ′ (q),

(16)

about the multifractal spectrum f (α) versus α , we can use the Legendre transform [10]. f (α) = qα(q) − τ (q),

(17)

both the exponent α(q) and spectrum f (α) express the singularity of the price series. We can determine ∆α and ∆f as follows

∆α = αmax − αmin ,

∆f = f (αmax ) − f (αmin ).

(18)

The index ∆α is considered as a measure indicator which is used to indicate absolute magnitude of price volatility. The bigger of ∆α is, the smaller even distribution of probability measure is, and the more violent price fluctuations will usually be expected. ∆f spectra is the Hausdorff dimension of the measure object, and the Hausdorff dimension is an indicator used to measure degree of confusion. Specifically, if ∆f > 0, it shows that the spot price running above the mean price is more than the price running below the mean price, the price shows strong performance, and the multifractal spectrum presents the ‘‘left hook’’ shape (See Fig. 6(a)); on the contrary, if ∆f < 0, the price shows weak performance, and the multifractal spectrum presents the ‘‘right hook’’ shape (See Fig. 6(b)). In this regard, both ∆α and ∆f are also important multifractal parameters, which are employed in our work. 2.3. Fisher’s linear discriminant algorithm Fisher’s linear discriminant [30] is a method commonly used in statistics, pattern recognition and machine learning to find a linear combination of features which characterize or separate two or more classes of objects or events. The resulting combination may be used as a linear classifier, or more commonly, for dimensionality reduction before later classification. Fisher’s LDA is employed to find a classifier in the parameter space for a training set. The given training set W = {x1 , x2 , . . . , xn } is partitioned into n1 ≤ n training vectors in a subset W1 and n2 ≤ n training vectors in a subset W2 , where n1 + n2 = n and each vector xi is a point in the M-D space and thus W = W1 ∪ W2 . An eigenvector µ = (u1 , u2 , . . . , uM )T for the M-D space should be found such that yi = µT x, (i = 1, 2, . . . , n) can be classified into two types in the space of real numbers. In this regards, we denote

 1   xi , kj =   n  j x  i∈Wj  (xi − kj )(xi − kj )T , Cj =   x  i∈Wj  Cµ = C1 + C2 .

j = 1, 2 j = 1, 2

(19)

Then the eigenvector µ is evaluated as Cµ−1 (k1 − k2 ). Obviously, Cµ is a symmetric matrix. Therefore, Fisher’s discriminant

rule becomes assigning x to W1 if (k1 − k2 )T Cµ−1 [x − 21 (k1 − k2 )] > 0 and to W2 otherwise.

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Fig. 1. Mean and standard deviation of price in 96 sub-periods. Table 1 Three kinds of price periods in the California1999/2000 electricity markets and PJM2001/2002 electricity markets. Periods

California 1999(20)

California 2000(20)

PJM2001(16)

PJM2002(16)

Low price (23 intervals) Average price (25 intervals) Peak price (24 intervals)

1–5:00, 24:00 6–13:00, 21:00 14–18:00

1–4:00, 23–24:00 5–6:00, 9–10:00, 22:00 12–20:00

1–4:00, 24:00 7–10:00, 21:00 13–18:00

1–6:00 9–13:00, 22:00 17–20:00

Tips: The remaining 24 intervals do not belong to any time periods.

The whole price data is used as the training set herein. The discriminant accuracies for resubstitution analysis can be defined as ncWj R Wj = , j = 1, 2, (20) nj where ncWj is the number of correctly discriminating Wj elements in the training set. 3. Data, results and discussions In order to investigate distinction problem of price periods and reflect the universality of the method, we consider the sub-periods of California1999–2000 and PJM2001–2002 price data (see Fig. 1) for our study, which contain 96 time intervals. According to the mean price and the variance of price in each interval, the low price, the average price and the peak price can be determined (see Table 1). As an illustrative example, there are some differences of the prices among the three types of price periods in PJM2001 electricity market, which are shown in Fig. 2. As expected, a spurious correlation may be detected if the price time series is non-stationary, so direct calculations of multifractal parameters, such as mass exponent, H o¨ lder exponent and the singularity spectrum, etc., do not give reliable results. We can verify that the chosen price time series are non-stationary by the sample autocorrelation function defined as follows: N −s

 ρs =

(x(t ) − x¯ )(x(t + s) − x¯ )

t =1 N 

,

(21)

(x(t ) − x¯ )2

t =1

where s is the time lag. The sample autocorrelation  function is a joint hypothesis of 0-value for s > 0. And it can be tested by s Ljung–Box Q statistics [32], where QLB = N (N + 2) t =1 (ρs2 /(N − s)). If xt is a stationary time series, the values ρs will tend to zero rapidly with increasing s. To the contrary, decreasing velocity of ρs is much more sluggish for a non-stationary time series. Fig. 3 shows that the values ρs of all the price time series in the chosen six time intervals in the PJM 2001 electricity market do not tend to zero rapidly with increasing time lag s. Table 2 shows that all of the Ljung–Box Q statistics of the price time series in chosen six time intervals in the four electricity markets are bigger than the critical value. In this regard, we reject the null hypothesis that the values ρs of Eq. (21) come from some stationary time series and consider the time series as non-stationary. Now, let us determine whether the data sets have a sinusoidal trend or not. Generalized Hurst exponents h(q) in Eq. (6) can be obtained by analyzing double log plots of Fq (s) versus s for each q. As an example, the 17th time interval in PJM2001

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Fig. 2. Prices of three types of price period in the PJM2001 electricity market.

Fig. 3. Sample autocorrelation function of price series in six chosen time intervals. Table 2 Ljung–Box Q statistics with 100 time lags of six chosen time intervals in the four electricity markets. Sample

Time lag

2nd period

4th period

9th period

10th period

17th period

18th period

Ca.1999 Ca.2000 PJM.2001 PJM.2002

100 100 100 100

7666.8383 4122.4243 1716.4094 1742.4531

7423.1731 5482.0419 1671.5859 2163.3656

4054.3796 2753.5441 2765.4063 1594.3987

3987.9370 2769.8081 1367.9030 1095.9236

1902.9686 2029.6148 691.0854 1209.7819

2088.1592 2212.7517 543.6089 1304.6998

Tips: the critical value of L–B. Q statistics of lags 100 with significance at the 5% level and 1% level are 124.3421 and 135.8067, respectively.

electricity market is analyzed to show that there are two crossover timescales s× in the double log plots, which is shown in Fig. 4 (here, we took q = −2). It is important to note that there is different of number of the crossovers existed in different time intervals. In order to cancel the sinusoidal trend in MF-DFA1, we apply the F-DFA method to the price time series. In practice, we truncate some of the first coefficients of the Fourier expansion of the series. Then, by inverse Fourier transformation, the noise without a sinusoidal trend is removed, which is also shown in Fig. 4. After filtering off the sinusoidal trend by Fourier filter, for the remaining new signals, the multifractal parameters of price series discussed in the above chosen time intervals can be calculated by MF-DFA1. For instance, some multifractal parameters of the prices series in three kinds of periods of low, average and peak in PJM2001 electricity market are shown in Figs. 5–9. There are differences in the generalized Hurst exponents h(q) for different values of q among the low, the average and the peak price time periods (see Fig. 5). However, comparing with h(−3), h(−2) and h(−1), the broken lines of h(1), h(2)

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Fig. 4. The MF-DFA1 functions Fq (s) for the price series of 17th time interval in PJM2001 electricity market versus the timescale s in a double log plot. Square symbol denotes the original time series and spherical symbol denotes the filtering series.

Fig. 5. The generalized Hurst exponent h(q) in low, average and peak time periods for different q (q = −3–3 except 0).

Fig. 6. The values of Dq in low, average and peak time periods for different q (q = −3–3 except 0).

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Fig. 7. The values of Cq in low, average and peak time periods for different q (q = −3–2).

Fig. 8. Multifractal spectrum and real hour-time price in 3rd time interval ((a) and (c)) and in 17th time interval ((b) and (d)) in PJM2001 electricity market.

and h(3) occupy a wider dynamic range, which demonstrates a better ability to distinguish time periods. As expected, the effect of distinguishing the three kinds of periods by Dq is identical to the h(q) shown in Fig. 6, since the two indicators have a linear relationship in Eq. (14) for each q except 1. Similar to Fig. 6, there is a difference of effect in distinguishing the three kinds of periods by using the values of Cq for different q, shown in Fig. 7. In practice, the values of C0 , C1 and C2 occupy a more wider dynamic range than C−3 , C−2 and C−1 , which explains that the component of C0 , C1 and C2 can distinguish those periods better than the other values of Cq for q = −3, −2 and −1. Comparing the multifractal singularity spectrum and the real-time trend of the price series in the 3rd and the 17th time interval in the PJM2001 electricity market in Fig. 8, we find that the price above the mean price is more than below for the 3rd time interval. It shows a stronger performance in this time interval, which well agrees with the multifractal delta spectrum ∆f = 0.1115 > 0. For the 17th time interval, the price above the mean price is less than below clearly, and a weaker performance of the price is expressed, which coincides with the multifractal delta spectrum ∆f = −0.4689 < 0.

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Fig. 9. The values of αmax , αmin , ∆α and ∆f in low, average and peak time periods in PJM2001 electricity market. Table 3 Discriminant accuracies and minimum number of indicators for the three kinds of prices based on Fisher’s LDA. Periods

No. of samples Accuracies No. of indicators

Low price

Average price

Peak price

RW1

RW2

RW1

RW2

RW1

RW2

23 100% 3

49 95.92% 7

25 88% 4

47 91.49% 7

24 100% 4

48 95.83% 5

Apart from the difference existing in shape of multifractal spectrum, there is a different distribution span of the multifractal spectrum and exponent α(q) between the two time intervals. Also from Fig. 8 (a) and (b), we can find that the span of exponent α(q) of the 3rd time interval (∆α = 0.3972) is less than the 17th time interval (∆α = 0.6239). It shows that the price in 3rd time interval is more even than the price in 17th time interval. One can obtain a homologous conclusion from Fig. 8 (c) and (d). In fact, the standard deviations of the price in the two time intervals are 5.30 and 38.73 respectively. Yet, we cannot distinguish the low and the average periods by the indicator ∆α , since the broken lines of ∆α for those two kinds of periods are intertwined, as shown in the upper left sub-graph of Fig. 9. Fortunately, the other three multifractal parameters of ∆f , αmax and αmin shown in Fig. 9 are good indicators that can distinguish the three kinds of price periods well. To summarize, in order to discern the prices in three kinds of periods and cluster the prices in the same period together, twelve multifractal indicators are selected shown in Figs. 5–9, namely, h(1), h(2), h(3) calculated by Eq. (6); C0 , C1 , C2 calculated by Eq. (13); D1 , D2 , D3 obtained from Eqs. (14) and (15); αmax , αmin obtained from Eq. (16) and ∆f obtained from Eq. (18). These twelve indicators can be used as candidates to construct parameter multi-dimensional spaces. In each parameter space, one point represents a time interval. Similar points in the same kinds of price period are expected to gather as a group and be separated from the others. We hope to find the least number of indicators which gives the maximum discriminant accuracy. Table 3 gives a quantitative assessment of clustering on the three kinds of price periods. It shows the best discriminant accuracies and the least number of indicators based on Fisher’s LDA. It is found from these accuracies that the multifractal parameters extracted from the price intervals do express the characteristic of different price periods. This clearly demonstrates that our method is applicable. We take the low price as an example to explain how to calculate the discriminant accuracies of RW1 and RW2 . On the one hand, we only need three indicators which can completely distinguish the low price points from the others. In particular, in the three-dimensional space of {αmax , αmin , ∆f }, we denote all the low price time intervals as W1 , the average and peak price time intervals as W2 . The number of points which belong to W1 are recorded and the discriminant accuracy RW1 = 100% can be reached by Eq. (20). On the other hand, in order to obtain the best distinguishing effect, seven parameters are needed to construct a seven-dimensional space, there are 47 points of average prices and peak prices belonging to W2 . It is worth

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Fig. 10. The discriminant accuracies versus the number of indicators. Table 4 Discriminant accuracies and minimum number of indicators for low and average prices based on Fisher’s LDA. Periods

No. of sample

Low

Average

Accuracy

No. of indicators

Low Average

23 25

23 1

0 24

100% 96%

6 4

mentioning that the best seven-dimensional space is not unique. In fact, there are eleven combinations composed of seven parameters giving the discriminant accuracy RW2 = 95.92%. As expected, the discriminant accuracy increases as the number of indicators increases. For instance, Fig. 10 shows the discriminant accuracies RW2 of the average prices are increasing with respect to the number of indicators. When we use only one parameter to discern the average prices, there is only one way to choose the indicator. That is the parameter of h(2) and the accuracy is only 68.09%. When we use two parameters to discern them, the only combination of {αmax , h (2)} can be chosen and the accuracy is 76.60%. However, there are five combinations if we put the points into a threedimensional or four-dimensional space, respectively. Such as {αmax , αmin , C0 } and {αmax , ∆f , h(3)} for a three-dimensional space; {αmax , αmin , ∆f , C0 } and {αmax , ∆f , C2 , h(3)} for a four-dimensional space. The discriminant accuracies are 82.98% and 85.11%, respectively. When we put those points into a five-dimensional or a six-dimensional space, there are two combinations, namely, {∆f , D1 , C1 , C2 , h(2)} and {∆f , D1 , C1 , C2 , h(3)} for a five-dimensional space; {αmax , ∆f , D1 , C1 , C2 , h(2)} and {αmax , ∆f , D1 , C1 , C2 , h(3)} for a six-dimensional space. The discriminant accuracies are 87.23% and 89.36%, respectively. For the seven-dimensional space, there are three ways to construct, which are {αmax , ∆f , D1 , C1 , C2 , h(1), h(2)}, {αmax , ∆f , D1 , C1 , C2 , h(1), h(3)} and {αmax , ∆f , D1 , C1 , C2 , h(1), h(2)}. At this time, the discriminant accuracy RW2 reaches the maximum 91.49% since it does not increase as we increase the number of indicators. In this regard, ‘‘seven’’ is the minimum number of space dimension we can find. Similar to the previous clustering process, we apply Fisher’s LDA and the twelve multifractal parameters calculated by the proposed method to gather each type of price period together and separate one price period from another, which are shown in Tables 4–6. The purpose of this experiment is also to search the minimum number of indicators by which the discriminant accuracies reach the maximum. From the three tables, we can conclude that the accuracies show a better distinguishing effect based on Fisher’s LDA when the calculated multifractal parameters are used. As expected, the low prices and the peak prices can be completely separated from each other due to the obvious difference in the two kinds of price periods shown in Fig. 2. Furthermore, we need only two indicators to make the accuracy reach the maximum, shown in Table 5. 4. Conclusion Price prediction plays an important role in electricity market, which directly affects the IPPs’ and the grid company’s decision-making. As reconstructing the price structure is very difficult in electricity price prediction, distinguishing the price patterns in different time periods becomes particularly meaningful. In this paper, we have proposed a procedure to successfully distinguish the electricity prices among different time periods. Applications of our procedure to the California 1999–2000 electricity markets and PJM2001–2002 electricity markets show that there are huge different multifractal features between the different time periods, which can provide a theoretical basis of identifying the periods of price peak and valley for the market managers. Thus they can accurately intervene in the market to avoid causing widespread power outages due to the high tariffs as the California electricity crisis and then promote the steady and sound development of the electricity market.

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Table 5 Discriminant accuracies and minimum number of indicators for low and peak prices based on Fisher’s LDA. Periods

No. of sample

Low

Peak

Accuracy

No. of indicators

Low Peak

23 24

23 0

0 24

100% 100%

2 2

Table 6 Discriminant accuracies and minimum number of indicators for average and peak prices based on Fisher’s LDA. Periods

No. of sample

Average

Peak

Accuracy

No. of indicators

Average Peak

25 24

25 0

0 24

100% 100%

5 4

Discriminant and identification the electricity prices among different periods are a preliminary work, the next step of work should be predicting the future price. The process of clustering the time periods with same multifractal features together can help electricity traders improve forecasting precision to compare with the whole time series forecast, and it can also cost savings and reduce operation difficulty to compare with forecasting for each natural interval. Thereby, the market participants can reduce the risks of unknown future prices and improve profitability. Acknowledgments The authors wish to thank the reviewers and the main editor Dr. H. E. Stanley for their comments and suggestions, which led to a great improvement to the presentation of this work. 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