Geometry of markets : How to reproduce New Zealand market covariances in the JCM simulation model?

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LAPPEENRANTA UNIVERSITY OF TECHNOLOGY School of Engineering Science

Degree Programme in Computational Engineering and Technical Physics

Honoré NIYIMPA

GEOMETRY OF MARKETS: HOW TO REPRODUCE NEW ZEALAND MARKET COVARIANCES IN THE JCM SIMULATION MODEL?

Examiners: Associate Professor Tuomo Kauranne D.Sc. Matylda Jabªo«ska-Sabuka

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Lappeenranta University of Technology School of Engineering Science

Degree Programme in Computational Engineering and Technical Physics Honoré NIYIMPA

Geometry of markets: How to reproduce New Zeaaland market covariances in the JCM simulation model?

Master's thesis 2018

44 pages, 26 gures, 8 tables

Examiners: Associate Professor Tuomo Kauranne D.Sc. Matylda Jabªo«ska-Sabuka Keywords: Covariance, Geometry of markets

In this research, we tried to investigate how the New Zealand electricity market covariances can be reproduced using a stochastic system of dierential equations, namely the JCM model. The methods, employed in this study to analyze the data, are correlation analysis and distance covariance. Using Matlab, the data used to carry out simulations were taken from New Zealand Electricity Authority website, dating from 1 January 1999 to 31 March 2009. The research ndings have shown that the JCM model was able to reproduce some of the New Zealand electricity market covariances, as the model identied a number of strong relationships between nodes. It was also able to reproduce the fact that the South Island nodes are less correlated between them in contrast to the South Island nodes which have strong correlations between them. In short, the model can explain the New Zealand market covariances to certain extent.

The results also brought an other important property, symmetricity of correlation coecient, which the JCM model failed to verify yet it was easily veried in the original data.

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Acknowledgements

I would like to express my sincere gratitude to the Almighty God, who has helped me through my life, to my supervisors Associate Prof. Tuomo Kauranne and D.Sc.

Matylda Jabªo«ska-Sabuka for your excellent continuous support of my thesis. Your tremendous patience, motivation, guidance and immense advice helped me to decisively make this thesis. I could not have imagined having excellent supervisors like you.

Besides my suprrvisors, I would like to thank Prof. Verdiana Grace Masanja for her encouragement, insightful advices, good heart, and opportunities she has oered me.

My sincere thanks also goes to my fellow classmate who latter become my brother in law Janvier Ukwizagira for the stimulating discussions, for the sleepless nights we were working together, and for all the fun we have had in the last eleven years. Also I thank all my friends who have been with me.

I must express my very profound gratitude to my wife Damaris Niyimpa for provid- ing me with unfailing support and continuous encouragement through the process of writing this thesis. This accomplishment would not have been possible without her.

Thank you.

Last but not the least, I would like to thank my mother Anastasia Mukankomeje, for giving birth to me and supporting me spiritually throughout my life.

Lappeenranta, November 22, 2018.

Honoré NIYIMPA

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1 Introduction

Geometry of markets refers to a concept based mostly on using previous behaviour of the market to forecast future market dynamics. It illustrates a technical analysis to establish price patterns and forecast the assets' price movements. The latter patterns are investigated using appropriate models. In this thesis we use the Jabªo«ska-Capasso- Morale (JCM) model that has been found to be able to contribute to the explanation of why some extreme dynamics in nancial world, take place. The JCM model mainly states that market movements are not only rational but a mixture of rational and irrational decisions, Jabªo«ska(2011). For instance, using this model, Jabªo«ska and Kauranne (2012), studying animal spirits in population spatial dynamics were able to reproduce the natural fact that a subgroup of the size of 5% of the whole population can pull the rest of the population towards a specic direction and keep them closer to itself.

In this research work, the JCM model is being employed to reproduce the New Zealand electricity market covariances. Market covariance can be understood as a measure of the directional association between the returns on two tradable assets. When there is a positive covariance, it means that those assets' returns move together while a negative value of covariance means that they move inversely. The New Zealand electricity market is mainly composed in two big islands which produce and consume most of the energy produced across the whole country. Since New Zealand is a country of islands, this study motivation is to investigate its integrability as one market, to study the relationships between electricity price from trading nodes located on dierent islands and to investigate whether various nodes can interact each other.

Basically, this investigation aims to analyze the ability of the JCM model to reproduce the New Zealand market dynamics. We are studying possible relationships between prices from some 183 nodes chosen across the country. The data used during simulations were collected from New Zealand Electricity Authority website, the set covers period from 1 January 1999 to 31 March 2009. The software, selected to carry out simulations and numerical analysis, is Matlab. Correlation analysis and distance covariance approaches were utilized to conclude on the ability of the model in question, the JCM model, to correctly reproduce market covariances.

This work is organised as follows: Section2discusses background and literature review.

In the next section, we present an overview of the New Zealand electricity wholesale market, the JCM model and dierent research works done on it (the JCM model) as well as New Zealand electricity market. Section3contains the methodology employed

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2 BACKGROUND AND LITERATURE REVIEW 7 in this study. In this section we detail two dierent approaches that later are used to conclude on the ndings. Here correlation analysis and distance covariance and / or distance correlation approaches are presented. Section 4 discusses the original data. In this part the data are analyzed using the two approaches, correlation analysis and distance covariance. Section 5 presents the results and discusses them. Here, the results are taken in two separate cases, the results obtained from correlation analysis and the results gotten from distance covariance analysis. Lastly, Section 6 concludes the study.

2 Background and literature review

2.1 Overview of the New Zealand electricity market

New Zealand (Maori: Aotearoa) is an island country that geographically comprises two main islands namely the North Island (Te Ika-a-Maui), and the South Island (Te Waipounamu) together with various smaller islands. In comparison, the South Island stands as the largest land mass which contains almost one quarter of the population (about 1.04 million). The New Zealand relief shows that the North Island is less mountainous with numerous volcanoes compared to the South. In terms of electricity consumption, the electricity consumption per capital is higher on South Island than in the north but the North Island consumes 62 % of the total energy of New Zealand, Qing (2013).

Considering its geographical situation in power generation, its electricity market devel- opment, New Zealand brings an insight to the study of electricity market spot prices.

Behavior of the power generated is based on renewable energy materials as it occupies around 70 % of the energy generated across the country. Renewable energy is devel- oping dramatically, Jabªo«ska (2011). In 2011, the South Island has produced 40.9%

of the whole New Zealand electricity and consumed 37.1%. The remaining generated power was from the North Island.The electricity generation in New Zealand is mainly based on hydroelectricity, Qing (2013).

Before the year 1987, New Zealand had a centrally run system of providers of generation, transmission, distribution and retailing. After the industry reform, the monopoized market became a competitive market in power generation and electricity retailing and various regulations were imposed on natural monopolies of transmission and distribu-

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tion, see Evens et al. (2005).

Currently the market is divided into 6 areas: regulation, generation, administration and market clearing, transmission, distribution, metering and retailing. The wholesale market for electricity operates under the Electricity Industry Participation Code (EIPC), and is overseen by Electricity Authority. Trades take place at approximately 248 nodes (grid injection points or grid exit points) across New Zealand every half hour. Generators make oers to supply electricity at 52 grid injection points (GIPs) at power stations, while retailers and major users make bids to buy electricity at 196 grid exit points (GXPs) on the national grid, see Electricity Authority report (2013).

Electricity is traded at a wholesale level in a spot market. The market operation is managed by several service providers under agreements with the Electricity Authority.

The physical operation of the market is managed by Transpower in its role as system operator. Generators submit oers through a Wholesale Information and Trading System (WITS). Each oer covers a future half-hour period and is an oer to generate a specied quantity at that time in return for a nominated price.

2.2 Jabªo«ska-Capasso-Morale (JCM) model

Mainly, this investigation is to analyze the JCM model performance and its ability to reproduce New Zealand electricity spot market covariance structure. The major insipi- ration for the JCM model came from Capasso-Bianchi model used to model population dynamics with the same model employed in price herding, see Capasso et al. (2003), Bianchi et al. (2003) and Capasso et al. (2005). The main components in this model were such that the movement of every agent, from the whole population, depends on the location of each agent with respect to the whole population and its interaction with its nearest neighbors, see Capasso et al. (2003), Bianchi et al. (2003) and Capasso et al. (2005).

dX_N^k(t) = [f(X_t^k) +h(k, X_t)]dt+σdW^k(t) (1) The functionf(X_t^k)stands for the forces acting on the whole population, in our context it can be viewed as a group of traders in the spot market where the price is taken as their measure of distance. h(k, X_t) stands for interaction between neighboring individuals in the population group of size N. σ stands for diusion coecient.

However, the model above was not able to capture price spikes taken as an important dynamic character of market prices. After working on the performance of the

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2 BACKGROUND AND LITERATURE REVIEW 9 model, Jabªo«ska in her research work, realized that by considering the price as uid (liquid) one can improve the model's performance. Considering the equation (1), she incorporated a third term, called the momentum term. The added term stands for the market momentum which depicts the eect of some of the irrational decisions made by the agents. It was noticed that the momentum eect is frequently observed when a suciently big subgroup exhibits a behavior which diverges from the whole population mean. The later term was added to the model (1), and the resulting model explained the market dynamics even in utmost situations, Jabªo«ska (2011). The main model then became

dX_t^k = [γ_t(X_t^∗ −X_t^k) +θ_t(h(k, X_t)−X_t^k) +ξ_t(g(k, X_t)−X_t^k)]dt+σ_tdW_t^k (2) where k=1,2,3,...,N

• X_t^k is the price of trader k at time t

• X_t vector of all traders' prices at time t

• W_t^k is Wiener process value of trader k at timet

• X_t^∗ is the population mean at timet

• σ_t is standard deviation of Wiener increment at time t

• γ, θ and ξ are interaction forces taken as scalar weights.

• N_p%^k is the neighborhood of agentkformed by closestp%of the whole population.

Also h(k, Xt) = M(Xt)· [E(Xt) −M(Xt)] stands for global interaction and M(X) is the mode of the random variable X. E(X) is the classical expected value of the random variableX. The functiong(k, X_t) =max

kI {X_t^k−Xt} whereI ={k|_XkN_p%^k } represents the maximal distance from traderk to its furthest neighbor from its closest p%of the whole group. This JCM model form can be found in Jabªo«ska and Kauranne (2012).

The latter model is used to reproduce the New Zealand electricity spot market covariances. To achieve our objectives, the original data, from New Zealand electricity spot market, are used to estimate the parameters of the model equation (2). After the parameter estimates are found, the New Zealand electricity spot market data are reproduced through simulations. The features of the simulated data i.e. the data obtained using the equation (2), will be discussed against the features of the original data.

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2.3 Literature review

From the very start of its existence, the JCM model has been tested or employed in various ways with the main target to analyze its performance (stability, convergence and accuracy). In this work we review some literature that uses this JCM model as well as the New Zealand Electricity spot market and its structure.

Ukwizagira (2016), when working on animal spirits in nancial markets: agent based model (ABM), has investigated on how putting the idea from the JCM model into the ABM approach, may result in trustful results. In addition to the main idea from the JCM model, another concept, named greediness maximization, was introduced to check whether it can provide improved results. Using the silver metal market prices dating from 1.03.2000 to 1.03.2013, dierent types of ABM analogues of JCM models were tested. The model types of the JCM model were model type I, model type II, model type III and model type IV i.e. Type I: in JCM where the mean reversion term is the only acting force; Type II: JCM where the momentum eect tem was given zero weight and kept all other remaining forces acting; Type III: In JCM model the farthest neighbor eect term was removed from the model and kept the other terms; Type IV is the complete form of the JCM model. Lastly, the model type IV was used to assess the power of the ABM when forecasting future prices.

The research ndings have indicated that all model types were able to explain the main market trend dynamics. However, only a complete form of JCM model was able to mimic all features of the silver metal price dynamics even in disastrous events. After incorporating the maximal greediness idea, the results showed that the employed model kept tracking correctly the market price dynamics yet the original JCM model kept giving more accurate results than the one with maximized greediness. As conclusions, the ABM study conrmed the ndings obtained in Jabªo«ska (2011), which states that irrational decisions made by market participants are the causes of market failures. The ABM study also found that some of the rational decisions can be part of the reason for nancial crisis. Finally, the ABM was used to forecast the silver market price and the model was able to mimic the main market trends even though it could not forecast reasonably or correctly the market price.

Jabªo«ska and Kauranne (2012), worked on animal spirits in population spatial dynamics. They proposed a model which worked on a mathematical formulation of conserva- tive cohesive forces described in Couzin (2005). The main objective was to reproduce the fact that a subgroup of the size of 5% of the whole population can pull the rest of the population towards a specic direction and keep them closer to itself. The obtained

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2 BACKGROUND AND LITERATURE REVIEW 11 model was applied on nancial market dynamics and it was able to reproduce market prices even in catastrophic situations. As mentioned before, the main task here was to test whether the model proposed by Jabªo«ska (2011) and Jabªo«ska and Kauranne (2011), can reproduce the natural fact stated above. Using a two dimensional form of the above stochastic system of equations in Jabªo«ska (2011) and Jabªo«ska and Kauranne (2011), they were able to conrm that 5% of the whole population is the minimum size of the escaping subgroup to pull the rest of the population in a specic direction and keep them closer. They were able also to nd that any escaping subgroup can inuence the rest but it can not keep them very near to it.

Uwamariya (2012) compared two model equations, Jabªo«ska-Capasso-Morale (JCM) model proposed by Jabªo«ska (2011) and Kalman Dynamics (KD) models, Shcherbacheva (2011) on capturing the gold market price dynamics. The KD model in this investigation, used a part of JCM model and enhanced by Kalman Filtering Techniques. To compare the simulated results, from the two proposed models, tables, graphical rep- resentation and histograms were used. As results, the two models were found capable of capturing the gold market dynamics. The author found no signicant dierence between the JCM and KD model results as they move closer to the original price of gold.

It was concluded that human psychology and market momentum may be signicantly present in gold market as it is the leading ideology in the JCM model also used in our current KD approach.

Niyigaba (2013) similarly to Uwamariya (2012) assessed the performance of the two models the JCM and the Kalman Dynamics when applied to EUR/USD currency exchange rate. Using the data collected from the 6th September 2004 to 16th March 2012, he made simulations which brought him to the conclusion that none of the two models can be chosen as the best compared to the other. He also rearmed that they, JCM and KD, successfully explained the EUR/USD exchange rate dynamics.

Lebedeva (2015) worked on Forecasting Financial Weather - Can We Forsee Market Sentiment? using quantitative analysis of nodal prices in New Zealand electricity spot market. The aim was to rearm that the price oscillations from one node can aect the remaining node prices in New Zealand electricity spot market. Mainly, this investigation relied on many features such as natural factors, location of power plant generation, generation types (geothermal, bio-energy, hydroelectricity, fossil fuel energy, wind energy, etc.) of the electricity, which can alter correlations between various nodal electricity prices. Correlation analysis and numerous illustrations were used to better understand the properties of the data. The original data from the New Zealand electricity market employed to make the analysis included: Oer prices, sampled from

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the 29th of March 2002 to the 31st of March 2009, and nal price from the 1st of January 1999 to the 31st of March 2009.

As a result, correlation analysis concluded that the most correlated nodes, from the North and the South Islands, are the nodes which are closer to each other in contrast to those which are the least correlated located on opposite sides of the islands. A hierarchical tree depicted that geographical location of the nodes strongly inuences correlations between nodal prices, particularly with the nal prices. However, oer price fails to inuence correlation between nodes. Using correlation distances of South and North Island nodes, it was found that the correlation coecient depends on their geographical positions, but not on their generation types. Moreover, after visualizing the correlation coecients, the dependence on geographical position persists with nal prices but failed with oer prices. This can be caused, for instance, by power leakage during transmission.

Lewis et al (2006) in assessing the integration of electricity markets using Principal Component Analysis: network and market structure eects, the New Zealand electricity market (NZEM) has served as a study case. They were mainly checking whether the NZEM is a national market or is a set of local markets i.e. to investigate the degree of integration of NZEM (to see if NZEM price dynamics at all nodes can be explained by one factor). The employed approach assessed the number of major factors which drives the market price and give reliable information on where problems may probably arise. The data used in their research were of the years 1997-2004 inclusive. The data indicated that the market was not always integrated and they realized that the New Zealand spot market could be vulnerable to segmentation in between morning and evening peaks. However, they noticed that the NZEM was not vulnerable to segmentation during the peak hours. Additionally, they found that there was some regular separation in the market in the period of 1999 to 2000 because on the 1stApril 1999 a generator, in the center North Island, got replaced and this has shown signicant eects. After this generator issue was xed in 2000, the spot market separation ended and the market became integrated (or one-market status).

3 Methodology

In this work, the main task is to assess the strength of the JCM model in reproducing the New Zealand electricity market price movements. To be able to reasonably discuss the power of the JCM model, the author proposed to use two dierent statis-

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3 METHODOLOGY 13 tical methods mostly employed to analyze time series covariances. They are used to investigate the relationship or association between random variables. Those methods are namely correlation analysis and distance correlation or distance covariance analysis (dCorr and dCov respectively) methods. The chosen approaches are applied to original data as well as simulated data (obtained using the JCM model) to carry out the analysis. The reason the author has proposed the second method, is that it was found capable of investigating a non linear association between two or more random variables, see Székely and Rizzo (2009), whereas the correlation approach is mostly limited to identifying a linear relationship between variables.

3.1 Variance, covariance and correlation

Given real valued random variables X and Y, variance of X can be dened as the measure of spread around its center of location or expected value or mean. However, covariance of X and Y stands for the measure of the relation between X and Y. A positive covariance indicates a positive relation between X and Y i.e. the two variables move in the same direction. Contrary, if the covariance is negative, it shows an inverse relation between X and Y which means the variables move in opposite direc- tions. Another important measure of the relation between two variables, is correlation.

Correlation between X and Y also indicates the degree at whichX and Y are related.

A correlation coecient has a value between -1 and 1. A positive correlation means that the two variables are positively correlated whereas a negative one indicates that the two variables are inversely correlated. If a correlation coecient equals 0, then X and Y are uncorrelated.

Mathematical representations of variance, covariance and correlation coecient are given as follows:

• The variance of a random variable X with expected value (center of location) X¯ is dened as

V ar(X) =

N

P

i=1

(X_i−X)¯ ²

N (3)

• The covariance between two random variables X and Y with the respective expected values X¯ and Y¯, is formulated as

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Cov(X, Y) =

N

P

i=1

(X_i−X)(Y¯ _i−Y¯)

N −1 (4)

• The correlation coecient between two random variables X and Y is dened as

ρ(X, Y) = Cov(X, Y)

pVar(X)Var(Y). (5) The sample correlation coecientr between two samplesX andY is dened as

r =

N

P

i=1

(X_i−X)(Y¯ _i−Y¯) sN

P

i=1

(X_i−X)¯ ²

N

P

i=1

(Y_i −Y¯)²

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Where, N is the number of scores in sample, X¯ and Y¯ are the sample means

or simply r = S_XY/S_XS_Y. , where S_XY is the sample covariance, S_X and S_Y are the sample standard deviations.

3.1.1 Covariance matrix

Given random column vectorsX and Y where their entries are random variables with nite variances. Therefore, the variance-covariance matrix, simply called the covariance matrix can be formulated as below.

Let us haveX ∈Rⁿ, then covariance matrix V ar(X) = Σcan be dened as the matrix whose (i, j)entries are the covariances. i, j = 1,2, ...., n

Σ_i,j=cov(X_i, X_j)=E[(X_i−E(X_i))(X_j−E(Xj))]=E(X_iX_j)−E(X_i)E(X_j)

where the operator E denotes the expected (mean) value of its argument.The general form will be

V ar(X) =Σ=E(XX^T)−E(X)E(X^T) (7) Finally, one can write the matrix form as:

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3 METHODOLOGY 15

V ar(X) =







σ_X²₁ σX1X2 · · · σX1Xn

σ_X₁_X₂ σ²_X₂ · · · σ_X₂_X_n ... ... ... ...

σ_X₁_X_n σ_X₂_X_n · · · σ_X²_n







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where,

• The covariance matrix is square, symmetric and should always be positive denite, i.e. all of the eigenvalues must be positive

• The inverse coveriance matrix Σ⁻¹, if it exists, is called the concentration or precision matrix.

3.1.2 Correlation matrix

The correlation matrix ofX ∈Rⁿ is then×nmatrix whose(i, j)entry iscorr(Xi, Xj). If the measures of correlation used are product-moment coecients, the correlation matrix is the same as the covariance matrix of the standardized random variables X_i/σ(X_i) for i= 1, . . . n.

This applies both to the matrix of population correlations (in which case σ is the population standard deviation), and to the matrix of sample correlations (in which case σdenotes the sample standard deviation). Consequently, each is necessarily a positive- semidenite matrix. Moreover, the correlation matrix is strictly positive denite if no variable can have all its values exactly generated as a linear function of the values of the others. The correlation matrix is symmetric because the correlation between X_i and X_j is the same as the correlation between X_j and X_i.

By denition, the correlation matix is witten as follows:

Corr(X) = (diag(Σ))⁻¹²Σ(diag(Σ))⁻¹² (9)

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Corr(X) =







1 _σ^σ^X¹^X²

X1σX2

· · · _σ^σ^X^1Xn

X1σ_Xn σX2X1

σX2σX1

1 · · · _σ^σ^X²^Xn

X2σ_Xn

... ... ... ...

σ_XnX₁ σ_XnσX1

σ_XnX₂ σ_XnσX2

· · · 1







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This matrix has the following properties

• diag(Σ) is the matrix of the diagonal elements of Σ

• Each element on the principal diagonal of a correlation matrix is equal to the unit.

• Each element out of the diagonal falls between−1and+1inclunding these values.

The measures discussed above, covariance and correlation, are mostly chosen to analyze linear relationship between nodes. However, this research work is investigating the level at which the JCM model is capable at reproducing the New Zealand electricity market covariances. As it was previously mentioned, the added approach studies any other possible non linear association between nodes. That method is disctance covariance (dCov) method or distance correlation (dCorr) method.

3.2 dCov and dCorr approaches

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 R^p and R^q respectively where p and q are positive integers. Let fX and fY be two characteristic functions ofX and Y respectively. Their joint function is f_X,Y. The random variables X and Y are said to be independent if f_X,Y =f_Xf_Y. Therefore, the dependence of X and Y can be studied by simply nding an appropriate norm to measure the distance between f_X,Y and f_Xf_Y.

Let

Cov(X, Y) = 1 N²

N

X

i=1 N

X

j=1

1

2(X_i−X_j)(Y_i−Y_j) (11)

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3 METHODOLOGY 17 The terms(X_i −X_j) and (Y_i −Y_j) can be thought of as the one-dimensional signed distances between the i^th and j^th observations. Székely et al. (2007) used centered Euclidean distancesD(X_i, X_j)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) = {(X_k, Y_k) : k = 1, ..., N} from the joint distribution of random vectors X in R^p and Y in R^q , dene dX_ij = |X_i −X_j|_p , dX¯_i. =

1 N

N

P

i=1

dXij, dX¯.j = _N¹

N

P

j=1

dXij, dX¯.. = _N¹2

N

P

i=1 N

P

j=1

dXij and D(Xi, Xj) = dXij −dX¯i. − dX¯_.j+dX¯_.. Similarly, dene dY_ij =|Y_i−Y_j|_p, and D(Y_i, Y_j) =dY_ij −dY¯_i.−dY¯_.j+dY¯_..

Therefore, distance covariancedCov(X, Y)is a nonnegative number dened by dCov(X, Y) = 1

N²

N

X

i=1 N

X

j=1

D(X_i, X_j)D(Y_i, Y_j) (12) Similarly, the distance variancedV ar(X) is a nonnegative number dened by

dV ar(X) = 1 N²

N

X

i=1 N

X

j=1

D²(X_i, X_j) (13)

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.

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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 respectively, 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 unpredictably. 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.

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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.

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(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-

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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).

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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.

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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 . . . ... . . .

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Figure 7: Distance covariance visualisation in 3D.

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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.

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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.

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Figure 10: Simulated mean price histogram from both islands.

.

Looking at Figure 11, ACF and PACF respectively, the price dynamics form a non stationary 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 between −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-

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(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.

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(a) Simulated price matrix in 3D. (b) Correlations matrix in 3D.

Figure 12: Simulated price from both Islands and their correlation matrices in 3 dimension.

(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.

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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

(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

(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

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(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.

Figure 17: Comparison of two least correlated nodes with indexes 103 and 154.

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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 simulated 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

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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

(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

(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.

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(a) Histograms. (b) Line pot comparison .

Figure 20: Comparison of two most correlated nodes with indexes 16 and 35 from JCM.

Figure 21: Comparison of two least correlated nodes with indexes 1 and 64 from JCM.

Figure 22: Comparison of two least correlated nodes with indexes 59 and 165 from JCM.

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5.2 Distance covariance analysis

As mentioned before, correlation analysis mostly studies linear association between variables. However, it can cetainly fail to identify any non linear association between variables. Due to this limitation, the author brought in another approach which is capable of investigating any possible association between nodes. As it was conrmed by correlation analysis, distance covariance method has shown strong association between many pairs of nodes as all calculated distance covariance values are clearly signicant (greater than zero).

Table 8 displays the distance covariance coecients of the least correlated power plants taken from the Table 4. The highlighted coecients are the ve least correlations in question, and these distance covariance coecients are also more signicant which may prompt a side thinking. It may suggest that between those nodes with poor correlation coecients, there is possibly anther sort of association than linear one. i.e. another type of association rather than linear association between those pairwise nodes. Generally, it is clear that in Figure 23, the distance covariance coecients of the simulated prices are varying between 0 and 0.9. Certainly, the JCM model tried to capture the main features from the original prices to some extent.

Table 8: Distance covariance matrix from simulated prices

Nodes . . . 6 . . . 32 . . . 64 . . . 128 . . . 165 . . . 1 . . . 0.26 . . . 0.31 . . . 0.29 . . . 0.36 . . . 0.33 . . . ... . . . 15 . . . 0.23 . . . 0.19 . . . 0.19 . . . 0.20 . . . 0.12 . . . ... . . . 59 . . . 0.26 . . . 0.25 . . . 0.32 . . . 0.26 . . . 0.24 . . . ... . . .

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Figure 23: Distance covariance coecients matrix in 3D.

5.3 Comparison

The New Zealand electricity prices in two separate sets i.e. the original and simulated prices, were plotted in Figure 24. Firstly, it is clear that the JCM model has tried to reproduce the electricity price by capturing the main market trend observed in the original electricity price, despite some cases where it failed to work it out. The mean prices from both original and simulated prices together with the nal price from simulated price, they were plotted in Figure 26. One can see that the average prices and nal price have not been stable over time. It is clear that all dynamics agree with the main market trend as they did not show any signicant dierence.

We have tried to study the propagation of errors between the original price mean and the simulated mean from JCM model by taking the dierence between them in Figure 25. it was noticed that the dierence does not follow a gaussian distribution since some moments like mean, −13 and median, −8.7 are not equal, see Figure 25(a) and in Figure 25(b). Interestingly, from Figure 25(a), the majority of the prices are closer

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5 COMPARISON WITH SIMULATED DATA 37 to the mean as the error distribution have a leptokurtic shape .i.e. most of mean errors are clustered around the mean zero. This leptokurtic shape, with the mean 0, gives an information that the average price (of simulated series) is closer to the true market price. Therefore, from these plots, one can draw conclusions that there was not error accumulation and that the model has produced quite good results.

Figure 24: Simulated price against original price from both islands.

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(a) Dierence mean visualization. (b) Dierence mean Histogram.

Figure 25: Dierence visualization between original price and simulated price from JCM.

Figure 26: Final price from JCM, simulated mean price agaist original mean prices from both islands.

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6 CONCLUSIONS 39

6 Conclusions

This research work aimed at testing the ability of the JCM model which was used to reproduce the New Zealand electricity market covariances. To better understand the correctness in the simulated results, two dierent approaches were selected to analyze the original and simulated prices. Correlation analysis and distance covariance methods helped to compare the covariances in original prices against covariances in the simulated prices. Our ndings indicate that the JCM model can reproduce the New Zealand electricity market covariances to a certain extent. However, it still has a room for improvements as some features from original prices did not appear in the simulated prices.

Mainly, the JCM model managed to explain the main market trend with some extreme spikes obeserved in the original prices. The model was able to show that the North Island's pairs are less correlated between them compared to the pairs of nodes from the South Island found to have strong correlations between them too. The model also was able to reproduce a signicant number of associations between nodes. Nevertheless, to some nodes the JCM model was not able to reproduce the association between them. Another important property, the JCM model was not able to maintain, was the symmetricity of the correlation coecient. The latter property was clearly maintained in the original prices but failed to be preserved in the simulated results.

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References

[1] Anna Shcherbacheva: Applying uid mechanics and Kalman ltering to forecasting electricity spot prices. 2011: Masters Thesis, Lappeenranta University of Technol- ogy, Finland.

[2] Annamaria Bianchi, Vincenzo Capasso and Daniela Morale: Estimation and Pre- diction of A Nonlinear Model For Price Herding. In :Complex Models and Inten- sive Computational Methods for Estimation and Prediction, 2005. (C. Provasi, Ed) Padova.

[3] Bianchi, A., Capasso, V., Morale, D., Sioli, F.: A Mathematical Mosl for Price Herding. Applicatin to Prediction of Multivariate Time Series. MIRIAM report (condential), (2003). Milan.

[4] Lebedeva Nadezhda:Forecasting Financial Weather - Can We Forsee Market Sen- timent? Quantitative Analysis of Nodal Prices in the New Zealand Electricity Spot Market. 2015: Masters Thesis, Lappeenranta University of Technology, Finland.

[5] Lewis Evans, Graeme Guthrie and Steen Videbeck :Assessing the Integration of Electricity Markets using Principal Component Analysis: Network and Market Structure Eects*, 2006: Victoria University of Wellington, New Zealand.

[6] Lewis T. Evans and Richard B. Meade: Alternating Currents or Counte- Revolution?, Contemporary Electricity Reform in New Zealand, 2005: Victoria University of Wellington.

[7] Matylda Jabªonska and Tuomo Kauranne: Animal Spirits in Population Spatial Dynamics, 2012: Lappeenranta University of Technology, Finland

[8] Matylda Jabªonska: From Fluids Dynamics to Human Psychology.What Drives Financial markets Towards Extreme Events, 2011: Doctoral thesis, Lappeenranta University of Technology, Finland.

[9] Morale Daniela, Vincenzo Capasso and Oelschlager Karl: An interacting particle system modelling aggregation behavior: from individuals populations, 2005: Jour- nal of Mathematical Biology 50(1), 49-66

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2016 : Masters Thesis, Lappeenranta University of Technology, Finland

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REFERENCES 41 [11] Jean De Dieu Niyigaba : Simulating The Currency Markets With Computational Market Dynamics, 2013 : Masters Thesis, Lappeenranta University of Technology, Finland.

[12] Denise Uwamariya : Simulating the Dynamics of the Gold Market using Com- putational Market Dynamics 2012: Masters Thesis, Lappeenranta University of Technology, Finland

[13] Wilhelm Brenig :Symmetry Properties of Correlation Functions, 1989: A Chapiter from the book published by Springer.

[14] Prabhanjan Tattar, Tony Ojeda, Sean Patrick Murphy, Benjamin Bengfort and Abhijit Dasgupta: Practical Data Science Cookbook, Second Edition , June 2017:

A book published by Packt

[15] Arjun K. Gupta: Advances in Multivariate Statistical Analysis, 31 July 1987, Netherlands: A book Published by Springer

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35: An article published by the Institute of Mathematical Statistics.

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List of Tables

1 Top ve most and least correlated nodes from original pricesd . . . 21

2 The symmetricity in the data correlation coecients. . . 22

3 Distance covariance matrix of the original data . . . 23

4 Top ve most and least correlated nodes from JCM model . . . 30

5 Top ve most and least correlated nodes from JCM model after interchanging the block position in data matrix . . . 30

6 The symmetricity in the simulated correlation before and after permutation . . . 33

7 The symmetricity in the simulated correlation after and before permutation . . . 33

8 Distance covariance matrix from simulated prices . . . 35

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LIST OF FIGURES 43

List of Figures

1 Original data from both islands. . . 19

2 Line plot in(a)histogram in(b)of original mean prices from both islands . 19 3 ACF in (a)and PACF in (b) of original mean prices. . . 20

4 Original data and its correlation matrices in 3 dimension. . . 22

5 North original data and North correlation matrices in 3 dimension. . . 22

6 South original data and South correlation matrices in 3 dimension. . . . 23

7 Distance covariance visualisation in 3D. . . 24

8 Simulated data from both Islands. . . 25

9 Simulated mean price from both Islands. . . 26

10 Simulated mean price histogram from both islands. . . 27

11 ACF in (a)and PACF in(b)of simulated nal prices. . . 28

12 Simulated price from both Islands and their correlation matrices in 3 dimension. . . 29

13 North simulated prices and their correlation matrices in 3 dimension. . 29

14 South simulated price and its correlation matrices in 3 dimension. . . . 29

15 Comparison of two most correlated nodes with indexes 157 and170. . . 31

16 Comparison of two most correlated nodes with indexes 34 and 35. . . . 31

17 Comparison of two least correlated nodes with indexes 103 and 154. . . 31

18 Comparison of two least correlated nodes with indexes 103 and 165. . . 32

19 Comparison of two most correlated nodes with indexes 165 and 175 from JCM. . . 33

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20 Comparison of two most correlated nodes with indexes 16 and 35 from JCM. . . 34 21 Comparison of two least correlated nodes with indexes 1 and 64 from

JCM. . . 34 22 Comparison of two least correlated nodes with indexes 59 and 165 from

JCM. . . 34 23 Distance covariance coecients matrix in 3D. . . 36 24 Simulated price against original price from both islands. . . 37 25 Dierence visualization between original price and simulated price from

JCM. . . 38 26 Final price from JCM, simulated mean price agaist original mean prices

from both islands. . . 38

Geometry of markets : How to reproduce New Zealand market covariances in the JCM simulation model?

Acknowledgements

Contents

1 Introduction

2 Background and literature review

2.1 Overview of the New Zealand electricity market

2.2 Jabªo«ska-Capasso-Morale (JCM) model

2.3 Literature review

3 Methodology

3.1 Variance, covariance and correlation

3.2 dCov and dCorr approaches

4 Original data

4.1 Correlation analysis and distance covariance analysis

5 Comparison with simulated data

5.1 Correlation analysis results

5.2 Distance covariance analysis

5.3 Comparison

6 Conclusions

References

List of Tables

List of Figures