As proposed in Székely and Rizzo ( 2009), the distance covariance (dCov) coecient is seen as the weighted L2 distance between joint and product of marginal characteristic functions of random vectors. Let X and Y be two random variables from Rp and Rq respectively where p and q are positive integers. Let fX and fY be two characteristic functions ofX and Y respectively. Their joint function is fX,Y. The random variables X and Y are said to be independent if fX,Y =fXfY. Therefore, the dependence of X and Y can be studied by simply nding an appropriate norm to measure the distance between fX,Y and fXfY.
3 METHODOLOGY 17 The terms(Xi −Xj) and (Yi −Yj) can be thought of as the one-dimensional signed distances between the ith and jth observations. Székely et al. (2007) used centered Euclidean distancesD(Xi, Xj)in place of the terms above. And from they have dened the distance dependence in the following way:
Let us have a random sample (X, Y) = {(Xk, Yk) : k = 1, ..., N} from the joint
Therefore, distance covariancedCov(X, Y)is a nonnegative number dened by dCov(X, Y) = 1 Similarly, the distance variancedV ar(X) is a nonnegative number dened by
dV ar(X) = 1
and distance correlation coecient dCorr(X, Y) will be
dCorr(X, Y) = dCov(X, Y) pdV ar(X)p
dV ar(Y) (14)
Importantly, considering the measures above, the following properties hold:
• X and Y are independent if and only ifdCov(X, Y) = 0,
• distance variance of X can be calculated asdV ar(X) =dCov(X, X),
• 0≤dCorr(X, Y)≤1 for all X and Y,
• dCov(X, Y) is dened for random variables in arbitrary dimensions,
• If the elements inX and Y covary together, it means that X and Y will have a large distance correlation. Otherwise, they will have a small distance correlation.
In short, we run the JCM model for all nodes of New Zealand electricity market, with all parameters of the JCM model estimated at each of the nodes separately, the correlation structure of the simulated nodal prices is compared to the one obtained using original prices. The correlation between dierent types of nodes is tested and the existing signicant interaction between them.
4 Original data
The data employed in this study, came from New Zealand Electricity Authority website, we only focus on nal prices on both Islands dating from 1January 1999to 31March 2009. A spot price was the half-hour price wholesale market electricity and we have choosen obsevations concerning the trading period between 3 : 30and 4 : 00 pm which corresponds with the most volatile number 31. The entire data has two data subsets:
• North Island electricity prices from123 nodes.
• South Island electricity prices from 60nodes.
To start, Figure 1 and Figure 2(a), line plots of all nodes prices and mean price re-spectively, depicts that the electricity price in the New Zealand market between the years 1999 and 2009 has not been stable as at some dates the price has increased un-predictably. It clearly shows how the average market price kept changing with time.
However, the main trend of the market price remained steady.
Histogram of the original mean price from both Islands with number of bins equal to 60. Figure 2(b) shows that the spot price distribution shape is not following log-normal distribution even if it is heavily right skewed with rapidly decreasing tails.
It also indicates that the original average price ranges between 0 and 200 with some outlies at 600 and 700.
4 ORIGINAL DATA 19
Figure 1: Original data from both islands.
(a) (b)
Figure 2: Line plot in (a) histogram in(b) of original mean prices from both islands .
From Figure 3 (a) , the slowly decaying ACF shows that the New Zealand electricity price dynamics form a nonstationary process. The PACF, in Figure 3 (b), shows two signicant lags (the rst two lags) with values greater than0.5. It indicates exisitance of a serial correlation of the same lags which means that future prices will strongly depend on historical prices.
(a) (b)
Figure 3: ACF in (a) and PACF in (b) of original mean prices.
4.1 Correlation analysis and distance covariance analysis
Linear correlation analysis in Figure 4, shows that most of the nodes are strongly correlated with the bigest number of the most correlated pairs of nodes located in the South Island. It was found that the couples with indices (155,156),(141,142), (173,174) and (157,170) are the most correlated pairs from the South Island together with an other pair (34,35) from the North Island. These correlation coecents show that some nodes, in a stream of incoming information, behave exactly the same way that it will have the same impact on their prices. It may also suggest that asymmetric information strategy may not exist. Some power plants have correlation coecients which equal to 1.0. It probably reveals that such plants may be owned by the same shareholders.
One can say that 80% of the most correlated nodes were sampled from the South Is-land with correlation coecient equal to1, see also Table 1, and20%from the Noth Is-land. On the other hand, the least correlated couples with indices(103,146),(103,165), (103,135),(103,153),(103,154)are formed by nodes mostly from the South Island with respect to one node from the north with index 103. We can see that all correlation coecients, for the least correlated pairs, are around 0.65 which is also a signicant positive correlation. In short, any incoming information will have a clear impact on all 183 nodes prices and they will highly likely change in the same direction. The later, association clearly observed in these prices from nodes may justify the New Zealand electricity wholesale market integration.
In Figure:4(b), from all possible pairewise associations between nodes, correlation
coef-4 ORIGINAL DATA 21
Table 1: Top ve most and least correlated nodes from original pricesd Most correlated Least correlated
Nodes correlation coe. Nodes correlation coe.
(155 , 156) 1 (103 , 146 ) 0.65
(141 , 142) 1 (103 , 165 ) 0.65
(173 , 174 ) 1 (103, 135) 0.65
(34 , 35) 1 (103 , 153) 0.65
(157, 170) 1 (103 , 154) 0.65
cients between prices asciallate between0.6and 1. One can conclude that the prices, sampled from the chosen nodes, depict strong linear relationship between them. When we look at the corrrelation matrices of North and South Islands separetely, see Figure 5 and Figure 6, it can be noticed that the South Island electricity prices, from the sampled nodes, tend to be more associated (correlated) than those form North Island.
As mentioned before, during simulations the original matrix of prices were formulated in the manner that the rst123 nodes (1st block) were from the North Island and the last60nodes (2nd block) were from South Island. Looking at the results, after we have interchanged the blocks' order and begin with those 60 nodes (1st block) from South Island then123 nodes (2nd block) from the north Island, the smallest correlation coef-cients remained the same, see Table 2. It can bee easily seen that the pair (155,156) before blocks permutation becomes(32,33)after permutaion,(34,35)becomes(94,95), (157,170) becomes (3,47), (103,146) becomes (23,163), (103,154) becomes (31,163). It means that the rst ve least correlated nodes did not change after interchanging blocks positions. This property agrees with correlation coecient symmetricity. It states that for any real valued random variablesX and Y,Corr(X, Y) =Corr(Y, X), see Wilhelm Brenig (1989).
Table 2: The symmetricity in the data correlation coecients.
Before blocks permutation After blocks permutation Nodes correlation coe. Nodes correlation coe.
(155 , 156) 1 (32 , 33 ) 1
(34 , 35) 1 (94 , 95 ) 1
(157 , 170 ) 1 (3, 47) 1
(103 , 146) 0,65 (23 , 163) 0.65
(103, 154) 0.65 (31 , 163) 0.65
(a) Original data in 3D. (b) Correlation matrix in 3D.
Figure 4: Original data and its correlation matrices in 3 dimension.
(a) North original data in 3D. (b) North correlations matrix in 3D.
Figure 5: North original data and North correlation matrices in 3 dimension.
4 ORIGINAL DATA 23
(a) South original data in 3D. (b) South correlations in 3D.
Figure 6: South original data and South correlation matrices in 3 dimension.
Using distance covariance approach, for instance in the Table 3, we have only focused on the ve least correlated power plants in the original price and veried whether the distance covariance approach could detect any association between them too. Figure 7 contains an illustration of the distance covariance coecients in three dimenions. We can see that they oscillate between 0.8 and 1, which may suggest, it is highly likely that there is a non linear association between some power plants. It is not in the scope of this work to gure out what kind of non linear association it would be.
Table 3: Distance covariance matrix of the original data
Nodes . . . 135 . . . 146 . . . 153 154 . . . 165 . . . ... . . . 103 . . . 0.8794 . . . 0.8793 . . . 0.8791 0.8792 . . . 0.8790 . . . ... . . .
Figure 7: Distance covariance visualisation in 3D.
5 COMPARISON WITH SIMULATED DATA 25
5 Comparison with simulated data
As it was previously explained, the simulated electricity prices of New Zealand market, obtained by running the JCM model, are for123nodes from North Island and 60 nodes of the South Island. The simulation results were plotted in Figure 8. It is clear that the simulated prices follow the main trend of the original price in the long run. However, in a short time window, the model did not produce very convincing results. Similarly to the original price, the simulated market price has been unstable. The results show that the model explained certainly the main market trend, see Figure 8. Additionally, spikes or extreme variations observed in the original prices, appeared in the simulated electricity price too. Figure 9 demonstrates the dynamics of overall average simulated price of both islands.
Figure 8: Simulated data from both Islands.
Figure 9: Simulated mean price from both Islands.
The histogram of the simulated price from both islands with a number of bins equal to 60, in blue, see Figure 10. It comrms that the most of the mean prices over the time, through all nodes were under 200. The shape of its distribution shape shows that the simulated market price does not form a normal series with mean equals to 88.6 and median76.7. It is right skewed with rapid cut o tails since the mean is greater than the median. In green, it is obviously clear that the nal price, from JCM model, is not normally distributed and it is heavily right skewed with rapid decreasing tails since its mean 69.86 is greater than its median 60.7. Apparently, the most simulated nal prices are concentrated between 0and 150.
5 COMPARISON WITH SIMULATED DATA 27
Figure 10: Simulated mean price histogram from both islands.
.
Looking at Figure 11, ACF and PACF respectively, the price dynamics form a non sta-tionary process as it can be explained by the slowly decaying autocorrelation function and the partial autocorrelation function with more than two signicant lags i.e. the future prices depend on historical prices. In short, the simulated prices contain a serial correlation clearly observed in the original New Zealand electricity prices.
5.1 Correlation analysis results
In this subsection, we discuss correlation analysis results. To start, correlation matrix of the simulated price indicates that correlation coecients between nodes vary be-tween −0.2 to 1, see Figure 12 which contains all combined islands results. This also can be seen when the island results were presented in two separate gures, Figure 13 and Figure 14. As previously seen, original prices seemed to all have positive
correla-(a) (b)
Figure 11: ACF in (a) and PACF in(b) of simulated nal prices.
tions varying between them. However, in the simulated results, it is not the case as negative correlations exist. Looking at Table 4 of the ve most correlated pairs from the simulated electricity prices, we can see that one part comes from North Island with indices (16,35), (6,8),(94,107). Also the remaining pairs include a mixed pair i.e. it contains Northen and Southern Islands nodes,(70,170), and the pair solely formed by southern nodes,(165,175).
However, the main part of ve least correlated pairs from the simulated electricity prices are from North Island, with indices (1,6),(1,64) and(15,32) and another part is of mixed pairs with respective indices (59,128),(59,165), see Table 5. Another identied feature is that, considering both islands separetely ( i.e. calculating pairwise correlation coecients between Northen Island nodes only or correlation coecients between south island nodes), the South Island nodes seem to be more strongly correlated between them than the level at which the North Island nodes are correlated between them, see Figures 13 and 14.
Lastly, the original electricity prices in Figures 5 and 6, stand more rmly correlated than the simulated ones. However, as previously seen, the simulated prices conserve the feature that the North Island nodes are less correlated than the ones in the South Island.
5 COMPARISON WITH SIMULATED DATA 29
(a) Simulated price matrix in 3D. (b) Correlations matrix in 3D.
Figure 12: Simulated price from both Islands and their correlation matrices in 3 di-mension.
(a) Simulated prices North Island in
3D. (b) Correlations matrix in 3D .
Figure 13: North simulated prices and their correlation matrices in 3 dimension.
(a) Simulated prices South Island in
3D. (b) Correlations matrix in 3D.
Figure 14: South simulated price and its correlation matrices in 3 dimension.
From Table 1, we have randomly picked the pairs of the most correlated nodes from original prices, one is from South Island with indices(157,170)(Figure 15) and another one is from North Island with indices (34,35) (see Figure 16) and two least correlated pairs from with indices (103,165),(103,154) (Figures 17 and 18), show the behaviours of the most and the least correlated nodes. Comparing original and simulated prices, from all those gures, they behave in the same manner and their histograms are right skewed with rapid decreasing tails. It depicts that even a single nodal price, from original or simulated series, is not stationary.
Table 4: Top ve most and least correlated nodes from JCM model Most correlated Least correlated
Nodes correlation coe. Nodes correlation coe.
(16 , 35) 0.90 ( 59 , 128 ) -0.14
(6 , 8) 0.90 (1 , 6 ) -0.14
(70 , 170) 0.91 (1 , 64 ) -0.11
(94 , 107) 0.92 ( 15 , 32 ) -0.10
(165 , 175) 0.93 (59 , 165 ) -0.10
Table 5: Top ve most and least correlated nodes from JCM model after interchanging the block position in data matrix
Most correlated Least correlated
Nodes correlation coe. Nodes correlation coe.
(96 , 176) 0.94 ( 39 , 128) -0.18
(10 , 38 ) 0.94 ( 38 , 127 ) -0.17
(95 , 114) 0.94 ( 127, 176) -0.14
(62 , 112) 0.96 (26 , 127) -0.14
(95 , 157) 0.96 (100, 163) -0.13
5 COMPARISON WITH SIMULATED DATA 31
(a) Histograms . (b) Line plot comparison.
Figure 15: Comparison of two most correlated nodes with indexes 157 and170.
Value
0 200 400 600
Number of Counts
0 200 400
Pairwise Simulated Histogram:34 35
Row34 Column35
Value
0 200 400 600 800
Number of Counts
0 200 400
Pairwise Real data Histogram:34 35
Row34 Column35
(a) Histograms. (b) Line plot comparison .
Figure 16: Comparison of two most correlated nodes with indexes 34 and 35.
(a) Histograms. (b) Line plot comparison .
Figure 17: Comparison of two least correlated nodes with indexes 103 and 154.
(a) Histograms. (b) Line plot comparison .
Figure 18: Comparison of two least correlated nodes with indexes 103 and 165.
Similarly, in Table 4, we have randomly chosen two most correlated nodes from the sim-ulated prices, one is from the South Island with indices(165,175)whereas another was taken from North Island with indices(16,35) see Figure 19 and Figure 20 respectively.
The least correlated nodes with indices (1,64) and (59,165) represented in Figure 21 and Figure 22, were also randomly chosen. Comparing the least and most correlated pairs of nodes in the simulated and original data, one can see that all gures behave in the same manner since their histograms are right skewed with rapid in decreasing tails. Another clear common feature for both sets (original and simulated data sets), they are not stationary as their line plots indicate. Also one can see that the electricity price in the New Zealand market has not been stable as at some dates the price has been changing unpredictably.
Let us interchange the positions of the nodes (blocks ) in the original matrix which is (N,S) system (i.e the rst 123 columns of the data matrix are nodes from the North Island and the last 60 columns are nodes from South Island). By interchanging the blocks we get (S,N) system (i.e the rst 60columns of the matrix which are 60nodes from South Island and the remaining 123 columns are 123 nodes from North Island) then run the JCM using the obtained matrix. This is to test how the JCM model preserves the symmetricity of correlation coecient which states that for any real valued random variables X and Y, Corr(X, Y) = Corr(Y, X). The test found that the simulated results seem to be violating this symmetricity of correlation coecient as it can bee seen in Table 6 where we take some pairs from the (N,S) system and nd their correlations in the (S,N) system and in Table 7 we take some pairs of nodes in the (S,N) system and try to see how much they have changed in the (N,S) system. One can easily see that those pairs of nodes are the same, only their indices have changed due to permutations (use the same reasoning as demonstated on original data in the
5 COMPARISON WITH SIMULATED DATA 33
previous Section 4) but their correlations are totally deferent.
Table 6: The symmetricity in the simulated correlation before and after permutation Before permutation; (N,S) system After permutation in (S,N) system
Nodes correlation coe. Nodes correlation coe.
(16 , 35) 0.90 ( 76 , 95 ) 0.74
(6 , 8) 0.90 (66 , 68 ) 0.66
(165 , 175) 0.91 (42 , 52 ) 0.80
(1 , 6) -0.14 ( 61 , 66 ) 0.26
(15 , 32) -0.10 (75 , 92 ) 0.44
Table 7: The symmetricity in the simulated correlation after and before permutation After permutation;(S,N) system Before permutation;(N,S) system
Nodes correlation coe. Nodes correlation coe.
(10 , 38) 0.94 ( 133 , 161) 0.54
(62 , 112) 0.96 (2 , 11) 0.58
(95 , 157) 0.96 (34 , 36 ) 0.36
(39 , 127) -0.18 ( 4 , 162 ) 0.19
(100 , 163) -0.13 (40 , 103 ) 0.41
(a) Histograms. (b) Line plot comparison.
Figure 19: Comparison of two most correlated nodes with indexes 165 and 175 from JCM.
(a) Histograms. (b) Line pot comparison .
Figure 20: Comparison of two most correlated nodes with indexes 16 and 35 from JCM.
(a) Histograms. (b) Line plot comparison .
Figure 21: Comparison of two least correlated nodes with indexes 1 and 64 from JCM.
(a) Histograms. (b) Line plot comparison .
Figure 22: Comparison of two least correlated nodes with indexes 59 and 165 from JCM.
5 COMPARISON WITH SIMULATED DATA 35