UNIVERSITY OF GHANA EXTREME VALUE MODELLING OF THE WATER LEVELS OF THE AKOSOMBO DAM BY ERIC OCRAN (10244344) THIS THESIS IS SUBMITTED TO THE UNIVERSITY OF GHANA, LEGON IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE AWARD OF MPHIL STATISTICS DEGREE. JUNE, 2015 University of Ghana http://ugspace.ug.edu.gh i DECLARATION I hereby declare this submission is my own work, except where otherwise stated, under the supervision of Dr. E. N. N Nortey and Dr. K Doku-Amponsah. Candidate: Eric Ocran (10244344) ……………………………. Signature …………………………… Date Principal supervisor: Dr. E. N. N Nortey Co-supervisor: Dr. K Doku-Amponsah ……………………………. ………………………… Signature Signature …………………………… ………………………… Date Date University of Ghana http://ugspace.ug.edu.gh ii ABSTRACT The Akosombo dam is Ghana‟s main source of energy. Considering the current increase in the demand for electricity in the country, where an increase in demand implies more pressure on the dam, it is therefore very important to investigate the tail behaviour of the water levels of the dam. This is necessary because the level of water in the dam determines the amount of electricity generated. The study investigated this problem using Univariate Extreme Value Theory. Both the Generalized Extreme Value distribution and the Generalized Pareto distribution were fitted to the Akosombo data. The Generalized Extreme Value distribution performed better than the Generalized Pareto distribution. Maximum likelihood estimation method was employed in the estimation of the model parameters. The study indicated that the water levels cannot fall below 226.00ft., which is the critical water level of the Akosombo dam. However, there is evidence of the dam attaining water levels below the minimum head operation level. We therefore recommended that, other forms of power generation should be explored to reduce the dependence on the Akosombo dam due to the fact that the dam can attain water levels below the minimum head operation. We also recommended that to prevent flooding during periods of excess water and abundant inflow, the high of the reservoir should be extended to retain the excess water which is spilled out. University of Ghana http://ugspace.ug.edu.gh iii DEDICATION To God almighty who protects and grants me breath and grace. And To my parents, Mr and Mrs Johnson Ocran and brothers who love and supported me. And To my Lectures and Colleagues who enriched my knowledge. University of Ghana http://ugspace.ug.edu.gh iii ACKNOWLEDGEMENT “WITH GOD ALL THINGS ARE POSSIBLE”. I thank my Heavenly Father, Jehovah Jireh, for protecting me through this journey. Without His mercy, I would not have reached where I am now. This work has benefited greatly from the guidance, ideas and critical review of my supervisors, Dr. E. N. N Nortey and Dr. K Doku-Amponsah. To them I say thank you and God richly bless you. I equally wish to express my sincere gratitude to Mr Richard Minkah for providing me with books and articles and helping me with my R codes. I am also grateful to Mr Abeku Asare -Kumi for his help with my R codes and also his guidance. This work could not have been completed without the financial support of the Carnegie Corporation of New York under the Next Generation of Academics in Africa through the Office of Research, Innovation and Development (ORID), University of Ghana. I am sincerely grateful for their support. I am also grateful to the Volta River Authorities for providing me with the needed data for the study. God bless you all. Finally, thanks to my family for their support, my friends and colleagues for their encouragement and guidance. To you all may the almighty God bless you abundantly. University of Ghana http://ugspace.ug.edu.gh iv TABLE OF CONTENTS TITLE PAGE DECLARATION ............................................................................................................................. i ABSTRACT .................................................................................................................................... ii DEDICATION ............................................................................................................................... iii ACKNOWLEDGEMENT ............................................................................................................. iii TABLE OF CONTENTS ............................................................................................................... iv LIST OF FIGURES ..................................................................................................................... viii LIST OF TABLES ......................................................................................................................... ix CHAPTER ONE ............................................................................................................................. 1 GENERAL INTRODUCTION ....................................................................................................... 1 1.1 Introduction ........................................................................................................................... 1 1.2 Background of the Akosombo Dam ...................................................................................... 2 1.3 Problem Statement ................................................................................................................ 3 1.4 Objectives of the Study ......................................................................................................... 5 1.5 Rationale of the Study ........................................................................................................... 5 1.6 Scope of the study ................................................................................................................. 5 1.7 Limitations of the study......................................................................................................... 6 LITERATURE REVIEW ............................................................................................................... 7 2.1 Historical Review (the Birth of Extreme Value Theory) ...................................................... 7 2.2 Some Application of Extreme Value Theory ........................................................................ 8 2.3 Extreme Value Analysis Models ......................................................................................... 10 2.4 Parameter Estimation .......................................................................................................... 13 University of Ghana http://ugspace.ug.edu.gh v CHAPTHER THREE ................................................................................................................ 16 THE EXTREME VALUE THEORY (EVT) ............................................................................ 16 3.1 Introduction ......................................................................................................................... 16 3.2 The Data .............................................................................................................................. 18 3.3 Domain of Attraction .......................................................................................................... 18 3.4 Extreme Value Distribution (EVD); Extremal Type Theorem (Maxima) .......................... 19 3.5 The Generalized Extreme Value (GEV) Distribution ......................................................... 20 3.6 Asymptotic Models for Minima .......................................................................................... 23 3.7 The Generalized Pareto Distribution (GPD) ....................................................................... 24 3.8 Parameter Estimation .......................................................................................................... 27 3.8.1 Block Maxima (or Minima) Approach ......................................................................... 27 3.8.2 Maximum Likelihood (ML) Estimation Method.......................................................... 28 3.8.3 Extreme quantiles and other parameters ....................................................................... 29 3.8. 4 Confidence Interval Estimations ................................................................................. 30 3. 8.5 Model Verification for the Generalized Extreme Value Distribution ......................... 32 3.8.6 Peak-over threshold approach: threshold selection ...................................................... 33 3.8.7 Maximum likelihood Estimation of the GP model ....................................................... 36 3.8.8 Extreme Quantiles and Other Parameters ..................................................................... 37 3.8.9 Confidence Interval Estimation for the GP distribution Estimates .............................. 38 3.8.10 Model Verification for the Generalized Pareto Distribution ...................................... 40 3.9 Extremes of Dependent (Stationary Series) Sequence ........................................................ 41 3.9.1 Maxima (or Minima) of Stationary Series .................................................................... 42 3.9.2 Models for Block Maximum (or Minimum) ................................................................ 44 3.9.3 Modeling threshold Exceedances ................................................................................. 45 3.9.4 Estimation of Extreme Quantile for a Stationary GP model ........................................ 46 University of Ghana http://ugspace.ug.edu.gh vi 3.10 Extremes of Non-stationary Sequences............................................................................. 47 3.10.1 The Structure of the GEV Model under non-stationarity ........................................... 47 3.10.2 The Structure GP Model under non-stationarity ........................................................ 49 3.10.3 Model Choice ............................................................................................................. 50 3.11 Parameter Estimation for Non-stationary model ............................................................... 51 3.11.2 Extreme quantiles Estimation for the non-stationary GEV distribution ..................... 52 3.11.3 Peak-over threshold approach: threshold selection .................................................... 52 3.11.4 Quantile Regression .................................................................................................... 53 3.11.5 Maximum likelihood Estimation for the GP distribution ........................................... 54 3.11.6 Extreme Quantiles Estimation for the non-stationary GP distribution ....................... 54 3.12 Model Verification ............................................................................................................ 55 CHAPTER FOUR ......................................................................................................................... 58 ESTIMATION AND INTERPRETATION OF THE EXTREME VALUE MODELS ............... 58 4.1 Introduction ......................................................................................................................... 58 4.2 Data and Preliminary Analysis ............................................................................................ 58 4.3 Fitting the Generalized Extreme Value Model to the Monthly extremes ........................... 60 4. 3. 1 Test of Stationarity ..................................................................................................... 60 4.3.2 The Blocking Method; Generalized Extreme Value Model ......................................... 61 4.4 Fitting the Generalized Pareto Model to the daily water levels of the Akosombo dam ..... 68 4.4.1 Test of stationarity for the daily water levels ............................................................... 69 4.4.2 The Peak-Over Threshold Method; Threshold Selection ............................................. 69 CHAPTER FIVE .......................................................................................................................... 81 DISCUSSION, CONCLUSION AND RECCOMENDATION ................................................... 81 5.1 Discussion and Conclusions ................................................................................................ 81 5.2 Recommendations ............................................................................................................... 83 University of Ghana http://ugspace.ug.edu.gh vii REFERENCES ............................................................................................................................. 84 APPENDICES .............................................................................................................................. 89 University of Ghana http://ugspace.ug.edu.gh viii LIST OF FIGURES Figure 4.1: Akosombo dam Water Level ...................................................................................... 59 Figure 4.2: Monthly Maximum Water Levels, left panel; Monthly Minimum Water levels, right panel. ............................................................................................................................................. 61 Figure 4.3: Exponential QQ-plot for Maximum water levels, left panel; Exponential QQ-plot for Minimum water levels, right panel. .............................................................................................. 62 Figure 4.4: Diagnostic plots for the fitted GEV model for the monthly minimum water levels, left panel; monthly maximum water levels, right panel. ..................................................................... 65 Figure 4.5: Mean residual life plot for daily minimum water level (left panel) and daily maximum water level (right panel) of the Akosombo dam .......................................................... 70 Figure4.6a: Parameter estimates against threshold for daily maximum water levels ................... 71 Figure 4.7: Plot of exceedances for daily minimum water levels, left panel, and daily maximum water levels, right panel. ............................................................................................................... 72 Figure 4.8: Diagnostic plots for fitted GP model using cluster maxima, left tail (left panel) and right tail (right panel). ................................................................................................................... 74 Figure 4.9: Diagnostic plots for the fitted GP model, right tail (left panel) and left tail (right panel)............................................................................................................................................. 78 Figure 1-A:Diagnostic plots for the simulated GP distribution; right tail .................................... 91 Figure 1-B: A diagnostic plot for the simulated GP distribution; left tail .................................... 92 University of Ghana http://ugspace.ug.edu.gh ix LIST OF TABLES Table 4.1: Descriptive Statistics of the Akosombo dam ............................................................... 58 Table 4.2a: Parameter estimates of the GEV distribution for the right tail of the Akosombo dam water levels and their corresponding standard errors (in parentheses). ........................................ 63 Table 4.3: Profile log-likelihood of the parameters of the fitted GEV ......................................... 64 Table 4.4: Exceedance Probabilities estimates using the GEV model ......................................... 67 Table 4.5: Return Periods estimates for the GEV model .............................................................. 68 Table 4.6: Parameter Estimates of GP model for right tail (cluster maxima) and their corresponding standard errors (in parentheses). ........................................................................... 73 Table 4.7: Return period estimates and their corresponding return levels for the fitted GP model (cluster maxima) ........................................................................................................................... 75 Table 4.8: Parameter Estimates for the GP model for all exceedances ignoring dependence, right tail. ................................................................................................................................................ 76 Table 4.9: Parameter Estimates for the GP model for all exceedances ignoring dependence, left tail. ................................................................................................................................................ 77 Table 4.10: Profile log-likelihood of the GP distribution parameters .......................................... 77 Table 4.11: Estimated exceedance probabilities for the fitted GP distribution ............................ 79 Table 4.12: Return period estimates for the fitted GP distribution ............................................... 80 Table 1-A: Parameter estimates of the simulated GP distribution; right tail ................................ 91 Table 1-B: Parameter estimates of the simulated GP distribution; left tail .................................. 92 University of Ghana http://ugspace.ug.edu.gh 1 CHAPTER ONE GENERAL INTRODUCTION 1.1 Introduction The classical statistical theory is often concerned with the behavior of the mean, which is described through the expected value ( ) of the underlying distribution. On the basis of the law of large numbers, the sample mean is used as a consistent estimator of ( ). The central limit theorem yields the asymptotic behavior of the sample mean. This result can be used to provide a confidence interval for ( ) in case the sample size is sufficiently large, a necessary condition when invoking the central limit theorem. Now what happens if the second moment, ( ) or even the mean, ( ) is not finite? Then the central limit theorem cannot be applied and so is the classical statistical theory, or what if one wants to estimate ( ) for where is the largest order statistic in sample and the estimate ̂ defined above yields the value 0? We cannot simply assume that such values are impossible. However, the traditional technique based on the empirical distribution function, does not yield any useful information concerning this type of questions. This shows that there is the need to develop special techniques that focuses on the extreme values of a sample, on extremely high quantiles or on small tail probabilities. In real life situations, these extreme values are often of key interest (Extracted from Beirlant et al. (2004)). Extreme Value Theory (EVT) is an area of statistics, which deals with the statistical techniques for modeling and the estimation of extreme (rare) events. By definition, Extreme Values are scarce, meaning that estimates are often required for levels of a process that are much greater (or less) than have already been observed, which implies an extrapolation from observed levels to University of Ghana http://ugspace.ug.edu.gh 2 unobserved levels and EVT provides a class of models to enable such extrapolation, (Coles, 2001). EVT is unique as a Statistical discipline in that, it develops techniques and models for describing the unusual rather than the usual since it provides a framework in which an estimate of anticipated forces could be made using historical data, (Coles, 2001). The distinguishing feature of an extreme value analysis is the objective to quantify the stochastic behavior of a process at unusually large or small levels, which usually requires estimation of the probability of events that are more (or less) extreme than any that have already been observed, (Coles, 2001). In this thesis, we shall focus on extreme high and low levels of the water levels of a dam. 1.2 Background of the Akosombo Dam The construction of the Akosombo dam started in 1961 and was commissioned into operation by Dr. Kwame Nkrumah, Ghana‟s First president in January 1965. The Akosombo dam, also known as the Akosombo Hydroelectric Project, is a hydroelectric dam on the Volta River in the Southeastern Ghana in the Akosombo gorge. The construction of the dam flooded part of the Volta River Basin and the subsequent creation of the Lake Volta. Lake Volta is the World‟s largest man-made Lake, covering 8,502 square kilometers, which is 3.6% of Ghana‟s land area. On the east side of the dam are two adjacent spillways that can discharge approximately 34,000 square meters per second (1,200,000 ) of water. The dam power plant contains six 170MW Francis turbines. Each turbine is supplied with water via 112 – 116 meters (367 – 381ft) University of Ghana http://ugspace.ug.edu.gh 3 long and 7.2 meters (24ft) diameter penstock with a maximum of 68.8 meters (226ft) of hydraulic afforded. The dam provides electricity to Ghana and some neighboring West African Countries including Togo and Benin. Power demands, along with unforeseen environmental trends, have resulted in rolling blackouts and major power outages. An overall trend of lower lake levels has been observed, sometimes below the minimum requirement for the operation of the dam. There have also been records of lake levels very close to the maximum requirement for the operation of the dam. Lower water levels are accompanied with load shedding of electricity and the high water levels result in the opening of the floodgates to protect the dam from collapsing, which result in flooding in the Communities downstream. The dam operates between a minimum level of 240ft and a maximum level of 278ft. The lowest ever water level was recorded for the period under study is 234.00ft., this was recorded in 1966 and the highest (277.54ft) ever water level was recorded in 2010. The critical level of the dam is 226ft and at this level, all the turbines must be shut down. 1.3 Problem Statement The Ghana Energy Commission, in their 2014 energy supply and demand outlook for Ghana reported that, as at December 2013, 1,580MW (53.80%) of Ghana‟s energy security depends on hydro power plants and out of this, the Akosombo dam alone contribute 1,020MW (34.70%). For hydro power plants, the water level at a particular time is a major factor in determining the power output from the plant. Water levels below the requirement for operation result in low power output from the plant since not all turbines can function at such minimal level. This may University of Ghana http://ugspace.ug.edu.gh 4 result in major power outages leading to power rationing in many parts of the country, which affect households, business, economic and social activities. In the 2014 energy supply and demand report by Ghana Energy Commission, it was reported that, the Nation‟s real Gross Domestic Product (GDP) growth in 2013 reduced from 8.8% in 2012 to 7.1%. This was attributed to negative growth in the manufacturing Subsector and the services sector, which was largely due to the inadequate grid power supplied which led to nationwide load shedding for most part of the year. During the early part of 2007, there was a reduction in electricity supply from the Akosombo dam due to low water levels of the dam and this had an adverse effect on the country‟s economy. However, when there are abundant rain and increased inflow of water, the dam approaches the maximum operation head level of 278ft., which leads to water discharge to safeguard the integrity of the dam structure. In the later part of 2010 for instance, the dam approached its maximum operation head level and due to that, water was discharged for several weeks which resulted in flooding downstream, (VRA, 2010). This led to loss of properties and evacuation of communities which also has an adverse effect on the Country‟s economy. Therefore, an extreme water level of the dam at any giving time directly affects power supply, lives and properties of people along the Volta Lake. This is the reason why this study sort to study the extremes of the dam. University of Ghana http://ugspace.ug.edu.gh 5 1.4 Objectives of the Study This study seeks to use EVT to analyze the water levels of the dam to enable us determine: (i) The lowest water level and the highest water level that the Akosombo dam can attain. (ii) The exceedance probabilities for very low water levels and very high water levels of the Akosombo dam. (iii)The return periods (recurrence interval) and their corresponding return levels for the Akosombo dam. 1.5 Rationale of the Study This study can help the Authorities in charge of the dam, have a fair idea as to when, the water level can fall below the minimum operating level and also when it can rise above the maximum operating level and the chances of these events happening, so that they can put in place contingency plans. This study will also add to the existing literature on the application of EVT in hydrology, specifically to dams, since not much work has been done to this effect. This will enrich the knowledge in this field of statistics. 1.6 Scope of the study The data consist of daily water readings of the Akosombo dam between the periods; January 1, 1966 and December 31, 2013. University of Ghana http://ugspace.ug.edu.gh 6 1.7 Limitations of the study The data at hand did not enable us to consider other factors which may influence the water levels of the Akosombo dam. Factors such as rainfall, inflow of water from other sources, water spillage from the dam and evaporation. This study considered only the water levels of the dam. University of Ghana http://ugspace.ug.edu.gh 7 CHAPTER TWO LITERATURE REVIEW 2.1 Historical Review (the Birth of Extreme Value Theory) Historically, work on extreme value problems may be traced back to as early as 1709, when Nicolas Bernoulli discussed the mean largest distance from the origin given n points lying randomly on a straight line of a fixed length t (Kotz and Saralees, 2000). Systematic development of the general theory may be regarded as having started with the paper by Von Bortkiewicz (1922), which dealt with the distribution of range in random samples from a normal distribution. The importance of the paper by Bortkiewicz is inherent in the fact that the concept of distribution of largest value was clearly introduced in it for the first time (Kotz and Saralees, 2000). The key result obtained by Fisher and Tippet in 1928 on the possible limit laws of the sample maximum seemingly created the idea that EVT was something rather special, very different from classical central limit theory (Beirlant et al., 2004). In 1943, Gnedenko presented a rigorous foundation for the Extreme Value Theory (EVT) and provided necessary and sufficient conditions for the weak convergence of the extreme order statistics (Kotz and Saralees, 2000). Gnedenko (1943) unified and formalized the ideas into the fundamental assumption in EVT known as the Extreme Value Condition (Kotz and Saralees, 2000). The Extreme Value Condition provides a Semi-parametric model solely for the tails of the distribution function. University of Ghana http://ugspace.ug.edu.gh 8 The doctoral dissertation by Haan in 1970, on Regular Variation and its Application to the Weak Convergence of Sample Extremes, is an important contribution to the theoretical development of EVT (Beirlant et al., 2004). Theoretical developments of the 1920s and 1930s were followed in the late 1930s and 1940s by a number of papers dealing with practical applications of extreme value statistics in distributions of human lifetimes, Radioactive emission (Gumbel, 1937a, b), strength of materials (Weibull, 1939), flood analysis (Gumbel, 1941, 1944, 1945, 1949a) and rainfall analysis (Potter, 1949), as summarized by Kotz and Saralees (2000). 2.2 Some Application of Extreme Value Theory Application of Extreme Value Theory has become one of the most important statistical discipline for the applied Sciences over the years. The statistical analysis of extreme values is employed in numerous disciplines; including hydrology, meteorology, engineering, finance, economics, reinsurance, and telecommunication, sport science and demography. The environmental sciences are a rich source of extreme value problems; for example, in predicting unusually strong wind, high waves and the excessive river levels. Gilli and Kellezi (2006) used Extreme Value Theory to compute tail risk measures and their related confidence intervals and also applied it to several major stock market indices. Jockovic (2012) used Extreme Value Theory (the Generalized Pareto Distributions) in finance, to estimate Value-at-Risk (VaR). Marimoutou et al.(2009) modeled VaR for long and short trading position in oil market by applying Extreme Value Theory models to forecast the Value-at-Risk (VaR). University of Ghana http://ugspace.ug.edu.gh 9 Gencay and Faruk (2004) employed Extreme Value Theory to generate Value-at-Risk estimates; their results indicated that, EVT-based Value –at – Risk estimates are more accurate at higher quantiles. Cotter (2005) measured Extreme risk in futures of contracts traded on the London International Financial Futures Exchange via Extreme Value Theory. Coles (2001) applied EVT in sport science to study the fastest annual race time for the women‟s 1500 meter event over the period 1972 – 1992. Adam and Tawn (2012) employed models of extreme value theory to model the times of gold medalist‟s Swimmers in the Olympic 400m freestyle from 1924 - 2004. Chan and Nadarajah (2015) explained the temporal changes in the extremes of the returns of electricity demand in the UK with the generalized Pareto distribution. Blanchet et al. (2009) analyzed snowfall data from the Swiss Alpine region, using EVT based on a “peaks-over-threshold” approach and a Poisson point process representation, in which they found out that, a significant number of the stations do not follow the Gumbel distribution. They also found out that low altitude stations in the Swiss plateau are heavy tailed because of rare extraordinary snowfall events. Watts et al. (2013) used extreme value models to quantify the upper tail of the distribution of human lifespans. University of Ghana http://ugspace.ug.edu.gh 10 2.3 Extreme Value Analysis Models The Classical extreme value theory assumes independence and constant distribution through time. The Classical theorem applies only to sequences of Independent and Identically Distributed (IID) random variables, however, it also holds when the hypotheses are relaxed moderately; for instance, when there exists a relatively weak statistical dependence between the random variables (Méndez and Menéndez, 2006). According to Méndez and Menéndez (2006), the theorem can be employed in practice, if every year can be conceived to be composed of many small “sampling” intervals (e.g. hours), such that successive values of the process are appropriately identically distributed and show “relatively weak” statistical dependence. Hawkes et al. (2008) stated that, standard methodologies employed in extreme value analysis should consist of adopting asymptotic models to describe the stochastic variations of the process. D‟Onofrio et al. (1999) employed the Generalized Extreme Value (GEV) distribution to estimate the return periods of extreme water levels. Haung et al. (2008) employed the GEV model to analyze annual maximum water levels on the coast of the United States. Xu and Haung (2010) employed the GEV model to estimate extreme water level at Wusong station near Shanghai. What all these studies have in common is that, all the authors assumed independence and constant distribution through time, but in the context of environmental processes, the assumption of constant distribution through time restrict the findings in the form of the data used in the study. The moment there is a change in the characteristics of the data over time; such findings will no longer be valid. Environmental processes are usually non-stationary; non-stationary University of Ghana http://ugspace.ug.edu.gh 11 processes have characteristics that change systematically through time. Non-stationarity is often apparent in environmental processes because of seasonal effects, perhaps due to different climate patterns in different months or in a form of trends, possibly due to long-term climate changes (Coles, 2001). In modeling extreme non-stationary processes, the general assumptions of independence and constant distribution through time cannot be applied. For the non-stationary approach, the parameters of the distribution functions are replaced with time-dependent parameters, so that the results of the extreme value analysis also vary with time (Coles (2001); Katz et al. (2002); El Adlouni et al. (2007); Hundecha et al. (2008); Mudersbach and Jensen (2010); Khaliq et al. (2006)). According to Mudersbach and Jensen (2010), the non-stationary problem can be solved by de- trending the data, in order to apply the classical extreme value analysis, however, the non- stationary approach has the benefit of enabling the extrapolation of the results up to future time horizons. Northrop and Jonathan (2010) mentioned that, in the application of EVT to environmental data, it is common for extremes of variables of interest to be non-stationary, which changes systematically in space, time or with the values of covariates. Coles (2001) studied the annual maximum sea-level data at Fremantle where the location parameter was modeled as a linear time-dependent model. Jensen and Mudersbach (2010), applied the non-stationary statistical model to annual maximum water levels from 1849 - 2007 at the German North sea gauge at Cuxharen using linear and exponential time models for the location and scale parameters of the GEV. The results were University of Ghana http://ugspace.ug.edu.gh 12 compared with stationary methods and it was found that, the non-stationary GEV approach is suitable for determining coastal design water levels. Katz et al. (2002) employed a non-stationary GEV model and recommended a linear model for the location and a log-transformed model for the scale parameter but with fixed shape parameter. El Adlouni et al. (2007) studied the conditional distribution of the annual maximal precipitation according to the SOI index at the Randsburg station in California using the classical model, GEV0, linear trend model, GEV1 and quadratic dependence structure model, GEV2. The comparison of these models by the deviance statistic indicated that, the GEV2 model more adequately represents the data variance. El Adlouni et al. (2007) also compared GEV1 and GEV11 and concluded that, GEV11 is more adequate to represent the data. El Adlouni et al. (2007) stated that, empirical studies of non-stationary models indicate that, it‟s preferable to represent the non-stationarity in both the location and the scale parameters. Hundecha et al. (2008) implemented a non-stationary extreme value analysis to estimate quantiles of extreme wind speed and their changes against time. It was applied to 10-m wind speed data from the North American Regional Reanalysis (NARR) data set and data from some selected stations of environment of Canada. Zhang et al. (2004b) compared different methods of detecting trends in extreme values and concluded that methods based on modeling trends in the parameter of the distribution of the extremes are powerful methods of detecting statistically significant trends in the extremes (as cited by Hundecha et al. (2008)). University of Ghana http://ugspace.ug.edu.gh 13 Tramblay et al.(2013) analyzed heavy rainfall events from the Southern France using both the Non-stationary model and the classical stationary model and concluded that the non-stationary model is better. Watts et al. (2013) employed the non-stationary Generalized Extreme Value and Generalized Pareto distributions in modeling life span data. Bezak et al. (2014) compared the Peak-Over-Threshold method and the Annual maxima method and concluded that, the Peak-Over-Threshold method yields better results than the Annual maxima method. 2.4 Parameter Estimation In every statistical modeling, parameter estimation is of great importance, since a wrong choice in the estimation procedure weakens the inference drawn from the study, thus a wrong choice affects the quality of the work. Several estimation techniques have been exploited and their strengths and weaknesses discussed in details by different authors, such as; Coles (2001), Hawkes et al. (2008), Beirlant et al. (2004), Katz et al. (2002), Ribereau et al. (2008), Mendez and Menendez (2006) and Bezak et al. (2014). Ribereau et al. (2008) compared to the maximum likelihood (ML) method, the Generalized Probability Weighted Moment (GPWM) method and found out that the GPWM method performs better and is computationally easy but cannot obtain confidence interval. The GPWM can only have a non-stationary location parameter, whereas, with the ML method the three parameters of the GEV distribution can be non-stationary. Ribereau et al. (2008) also found that, when the University of Ghana http://ugspace.ug.edu.gh 14 Independent and Identically Distributed (IID) assumption is not satisfied, the ML method offers an elegant way to handle non-stationarities by letting the GEV parameters be time dependent. Katz et al. (2002) stated that the probability weighted moments (PWM) or L-moment due to their computational simplicity and also their better performance for small samples, are more common in the applications of hydrologic extremes than the ML estimation method. According to Katz et al. (2002) and Scarf and Laycock (1996) due to the ML estimation‟s adaptability to model changes, it makes it easy to handle more complex situations that involve covariate but the PWM cannot easily incorporate covariates. Bezak et al. (2014) compared the annual maxima approach and the POT approach using L- moment, the conventional moments and the ML estimation techniques and concluded the L- moment performance better than the other two techniques. ML estimation is widely used in extreme value modeling since it provides asymptotically unbiased parameter estimators and also from all unbiased estimators, it has the least mean square error Hawkes et al. (2008). The authors also said that the ML estimation approach is the best if we have large sample. El Adlouni et al. (2007) extended the Generalized Maximum Likelihood (GML) method introduced by Martins and Stedinger (2001), for the estimation of parameters to the non- stationary case. They compared the ML and GML estimation methods, with a simulation experiment. It was shown that GML method produces the best results with respect to bias and root-mean-square error (RMSE). The GML method is a particular case of the Bayesian approach based on the maximum likelihood method with additional prior information. University of Ghana http://ugspace.ug.edu.gh 15 According to Castillo and Hadi (1997), the Method of moments and the Probability-Weighted Moments can yield estimates which are inconsistent with the observed data. They also stated that, for large sample, ML estimation is to be preferred due to its efficiency in the case of large samples. Castillo and Hadi further indicated that, the PWM provides better results for small samples. Northrop and Jonathan (2010) proposed a method for setting a covariate-dependent threshold in which they employed quantile regression to select the appropriate threshold to model non- stationary exceedances of high threshold. University of Ghana http://ugspace.ug.edu.gh 16 CHAPTHER THREE THE EXTREME VALUE THEORY (EVT) METHODOLOGY 3.1 Introduction Consider a random sample { } from a distribution F, where is a random value of the daily flow at a dam site for day with a distribution function F. The behaviour of the mean has often been the focus of classical statistical theory. The discussion of the classical statistical theory of the mean is based on the Central Limit Theorem (CLT) and it often approximates to the normal distribution as a basis for statistical inference. The classical CLT states that the distribution of 1 ( ) var(X) n i i n x E X T n         (3.1.1) approximates to the standard normal for sufficiently large n (i.e. as ) In general, the Central limit problem considers the sum 1 n n i i s x   and manipulate it to find constants and such that n n n n s b Y a   (3.1.2)   converges in distribution to a non-degenerate distribution. Then from the limiting distribution of the quantity , the distribution of can be determined. The normal distribution is attained as a limit for the sum (or average) of independent and identically distributed random variables, except when the underlying distribution possesses a University of Ghana http://ugspace.ug.edu.gh 17 heavy tail. The extremes produced by such sample will distort the averages so that an asymptotic behaviour, which is different from the normal behaviour is obtained. Extreme Value Theory (EVT) is concerned with probabilistic and statistical questions related to very high or very low values in sequences of random variables and in stochastic processes. Extreme Value Theory is the theory of modeling the behaviour of sample extremes and measuring events which occur with very small probability. The behaviour of such order statistics may be achieved through their exact distribution function or through their limiting distribution function. Let * + denote the Sample maximum, where is a sequence of independent random variables having a common distribution function . Theoretically, the distribution of can be derived exactly for all values of :        , 1 2 1 P P , ,..., P ( ) n n n n n i i X x X x X x X x X x F x          (3.1.3) But this approach has two flaws associated with it; the first is ( ) , so ( ) and the other is the distribution of may not be known. The purpose of EVT is finding the behaviour of the Sample extremes for sufficiently large University of Ghana http://ugspace.ug.edu.gh 18 3.2 The Data The Volta River Authority (VRA) measures the daily elevation (in feet) of the Akosombo dam and keeps records on these elevations. That data was obtained from the Engineering Services Department of the VRA, the Akuse office. The data covers 1965 to 2013, however the 1965 data has some irregularities so it will be excluded from the analysis. The highest ever water level recorded from 1966 to 2013 is 278.54ft., which was recorded on the 8 th and 9 th of November, 2010. The lowest ever is 234.00ft recorded on 28 th June, 1966. Two approaches would be considered for the analysis. One approach would make use of monthly maximum and minimum water levels. Therefore, these extrema will be extracted and use in the analysis. The other approach will set two thresholds; one for the selection of maximum water levels and the other for the selection of minimum water levels. Any water level which fall above the threshold for the maximum is considered an extreme and those that fall below the threshold for the minimum is also considered an extreme. 3.3 Domain of Attraction Definition 3.3.1 Let be independent and identically distributed random variables with common distribution function . We say F belong to the domain of attraction of a distribution V, if there exists norming constants and centering constants such that as lim ( )n n n n s b P V x a       University of Ghana http://ugspace.ug.edu.gh 19 at all continuity points of . Definition 3.3.2 The random variable belongs to the maximum domain of attraction of a non-degenerate distribution H, if for a suitable Sequence { + and * + such that at all continuity point of the distribution function, H   , lim lim ( ) nn n n n n n n n X b P x F a x b H x a           (3.3.1) 3.4 Extreme Value Distribution (EVD); Extremal Type Theorem (Maxima) Theorem 3.4.1 If there exist sequences of constants { } and { } such that , ( ) n n n n X b P x H x a        as n  where is a non-degenerate distribution function, then ( ) belongs to one of the following families: I: ( ) exp exp x b H x a                x    II: 0, ( ) exp , x b H x x b x b a                  University of Ghana http://ugspace.ug.edu.gh 20 III: exp ( ) , 1, x b H x x ba x b                       for parameters and in the case of families II and III . In simple language, Theorem 3.4.1 states that the rescaled Sample maxima  , /n n n nX b a converges in distribution to a variable having a distribution within one of the families labeled I, II and III, with types I, II and III known as Gumbel, Fréchet and Weibull families respectively. Each family has a location and scale parameter and respectively; but the Fréchet and Weibull families have a shape parameter α in addition. These three types of extreme value distributions (EVD) are the only possible limits for the distribution of irrespective of the distribution of F for the population (Coles, 2001; Beirlant et al., 2004) Theorem 3.4.1 provides an extreme value analog of the Central Limit Theorem. The proof of the Theorem can be found in Beirlant et al. (2004). 3.5 The Generalized Extreme Value (GEV) Distribution The Gumbel, Fréchet and the Weibull distributions have distinct forms of behaviour, corresponding to different forms of tail behaviour for the distribution function F of the . This can be made precise by considering the behaviour of the limit distribution H at its upper endpoint. For the Weibull the upper endpoint is finite, while the Fréchet and the Gumbel have an infinite upper endpoint. The density of H decays exponentially for the Gumbel distribution and polynomially for the Fréchet distribution. The Fréchet family contains distributions that are University of Ghana http://ugspace.ug.edu.gh 21 heavy tailed. The infinite endpoint of the distribution function implies some moment do not exist for this class of distribution. The Weibull family contains distributions that have a lighter tail than the exponential. The three different families give different representations of extreme value behaviour in application. In the early applications of EVT, one of the families was usually adopted and then used to estimate the relevant parameters of the distribution, but two weaknesses are associated with this; a. A technique is required to choose the most appropriate distribution from the three families for the data in question and; b. The moment a choice is made; all inferences assume the choice to be correct and do not allow for uncertainty of the choice made, even though this uncertainty may be substantial. A better approach was offered by a reformulation of the models, which was independently proposed by Von Mises (1954) and Jenkinson (1955) (Coles, 2001). The Gumbel, Fréchet and Weibul families can be combined into a single family of models having the distribution function of the form (Coles, 2001); 1 exp 1 , 0 ( ) exp exp , 0 x H x x                                               (3.5.1) University of Ghana http://ugspace.ug.edu.gh 22 defined on the set :1 x x              , where the parameters satisfy . This is known as the Generalized Extreme Value family of distributions. The model has three parameters: a location parameter, ; a scale parameter, ; and a shape parameter, . The shape parameter, if the underlying distribution has a finite upper endpoint, if the underlying distribution decreases exponentially and if the tail decreases polynomially. Thus, corresponds to the Gumbel distribution, corresponds to the Fréchet distribution with ⁄ and corresponds to the Weibul distribution with ⁄ . The unification of the original three families of the extreme value distribution into a single family was a breakthrough for statistical implementation. From inference on , the data at hand determines the most appropriate type of tail behaviour and therefore, it is no more necessary to make subjective priori judgments about which individual extreme value distribution would be appropriate for the data (Coles, 2001). The uncertainty in the inferred value of measures the lack of certainty as to which of the original three types is most appropriate for a given data set (Coles, 2001). The GEV provides a model for the distribution of block maxima. Its application consists of blocking the data into blocks of equal length and fitting the GEV to the set of block maxima. But in implementing this model for any particular data sets, the choice of block size can be critical. The choice amounts to a trade-off between bias and variance; blocks that are too small mean that approximation by the GEV is likely to be poor, leading to bias in estimation and extrapolation. University of Ghana http://ugspace.ug.edu.gh 23 3.6 Asymptotic Models for Minima There are some applications of EVT which require models for extremely small observations. For instance, in the study of the water levels of a dam, it will be of interest to study the tail behaviour of the minimum water levels since the level of the water determines how much hydro-electric power is generated. Let * +, where the denote the individual daily water levels. Assume the to be independent and identically distributed, similar arguments apply to as was applied to , leading to a limiting distribution of a suitable re-scaled variable. Let , for , this implies that small values of corresponds to large values of . So if * + then * + * + * + This implies Hence for large        1, 1 1 1 1 ( ) 1 exp 1 ˆ 1 exp 1 n n n nP X x P Y x P Y x P Y x H x x x                                                              Therefore, the GEV distribution for minima is defined as University of Ghana http://ugspace.ug.edu.gh 24 1 ˆ 1 exp 1 , 0 H*( ) ˆ 1 exp exp , 0 x x x                                           (3.5.1) { ( ̂) } and where ̂ , ̂ , and . In situations where a model for the block minima is required, the GEV distribution for the minima can directly be applied. Another method is to make use of the duality between the distribution for maxima and minima. Given the data that are realization from the GEV distribution for minima, with parameters( ̂ ), implies fitting the GEV distribution for maxima to the data . The maximum likelihood estimate of the parameters of this distribution corresponds exactly to that of the required GEV distribution for minima apart from the sign correction ̂ . 3.7 The Generalized Pareto Distribution (GPD) The GEV method is very useful when only block maxima (or minima) data are available, but when other extremes (higher or lower water levels) data are available it‟s wasteful. For instance, two or three very high water levels may occur within one year (or month), yet only one of these levels is included in the estimation. This may possibly lead to underestimation of the frequency of occurrence of such levels. Therefore, if hourly, daily, weekly or monthly observations are available, the data can be put to better use by avoiding the blocking method. University of Ghana http://ugspace.ug.edu.gh 25 Let be a sequence of independent and identically distributed random variables, with common distribution function F. is considered as extreme event if it exceed some threshold u. If denote; an arbitrary term in , then, the stochastic behaviour of extreme events is given by the conditional probability (Coles, 2001),   1 ( ) , 0 1 ( ) F u y P X u y X u y F u         (3.7.1) With the parent distribution F known, the distribution of threshold exceedances in equation (3.7.1) can be derived, however, in practice the parent distribution is usually not known, hence we sought to approximation that is applicable for high values of threshold. Theorem 3.7.1 Let be a sequence of independent and identically distributed random variables with distribution function F and let * +. Let denote an arbitrary term in the sequence and assume that F satisfies the GEV conditions, so that for sufficiently large,  ,n ( )nP X x H x  , where H is as defined in equation (3.3.1). Then, for sufficiently large , the distribution function of ( ) conditional on is approximately (Coles, 2001); 1 ( ) 1 1 y H y            (3.7.2) University of Ghana http://ugspace.ug.edu.gh 26 defined on { ( ) }, and where The family of distributions defined by equation (3.7.2) is known as the Generalized Pareto Distribution (GPD). Theorem 3.7.1 implies that, if block maxima have approximating distribution , then threshold excesses have corresponding approximate within the generalized Pareto family. The rigorous connection of the Pareto distribution with classical extreme value theory was established by Pickands (1975) (Coles, 2001). The parameters of the GP distribution of threshold excesses are uniquely determined by that of the associated GEV distribution of the block maxima. The shape parameter, of the GEV distribution is the same as the corresponding GP distribution parameter. There is a close parallel between limit results for sample maxima and limit results of exceedance over thresholds. According to Coles (2001), also for the GP distribution, if , the distribution of excesses have an upper bound of ; if the distribution has no upper limit and if , the distribution is unbounded, which we may take the limit as to get ( ) 1 exp , 0 y H y y           (3.7.3) which is exponential distribution with mean . University of Ghana http://ugspace.ug.edu.gh 27 3.8 Parameter Estimation The estimation of the Extreme Value Index (EVI) is of key interest in extreme value analysis since it determines the tail behaviour of the underlying distribution F. Once the EVIs are known; other equally important estimators for statistical inferences can then be obtained. Some of these important estimators are the extreme quantiles, exceedance probabilities and the return periods. 3.8.1 Block Maxima (or Minima) Approach The block maxima (or minima) method consist of dividing the sample into, say non-over lapping blocks. The blocks are usually equal in length and the block sizes are usually selected naturally. The maximum (or minimum) of each block is selected as independent sample maxima (or minima) from the observed data and then fitted to the GEV distribution. The choice of block size is very important and it is a trade-off between bias and variance. Blocks of small size are likely to yield poor approximation by the GEV distribution and blocks of larger size produce few block maxima (or minima), which might lead to large variance. Though, block size may not refer to months, in this study, the block size would be monthly. Thus monthly maxima and monthly minima water levels of the Akosombo dam would be used, instead of the usual annual extremes. Though, the block sizes in this case may differ because the months do not have the same number of days, but we do not expect this to affect the result significantly. We use monthly extremes due to the scarcity of data and also, for us to consider more than one datum per year. According to Mendez and Menendez (2006), to reduce the problem of data scarcity, it is necessary to employ monthly maxima method, peak-over-threshold method or r-largest order statistics in order to consider more than one extreme datum per year. According to them, University of Ghana http://ugspace.ug.edu.gh 28 monthly random variables of an extreme event can be expected to be more homogeneous than within years. 3.8.2 Maximum Likelihood (ML) Estimation Method The parameters of the GEV distribution may be estimated by Maximum Likelihood (ML), Probability Weighted Moments (PWM) or the L-moments method. The advantage of the ML over the other techniques of parameter estimation is its adaptability to changes in model structure. That is, although the estimating equations change if a model is modified, the underlying methodology is essentially unchanged. The ML also has a convenient set of “off-the- shelf” large sample inference properties (Coles, 2001). Considering the assumption that are independent, then the likelihood function of the GEV is given as;     1 ( ) ; , , , , , n i i L g x           , where ( ) denotes the GEV density function with parameters evaluated at . If , the log-likelihood is;   1 1 1 ( ) log 1 1 log 1 1 n n i i i i x x l n                                            provided that ( ) . If , the log-likelihood reduces to; University of Ghana http://ugspace.ug.edu.gh 29 1 1 ( ) log exp n n i i i i x x l n                               The estimates of the parameters ( ) are obtained by maximizing the log-likelihood function. Beirlant et al. (2004) and Coles (2001) stated that, since the support of the GEV depends on unknown parameter values, the usual regularity conditions underlying the asymptotic properties of ML estimators are not automatically applicable. Smith (1985) studied this problem in detailed and his findings are summarized in Coles (2001) as:  When , maximum likelihood estimators are regular, and hence have the usual asymptotic properties.  When , maximum likelihood estimators do not have the usual asymptotic properties, but it is obtainable.  When , maximum likelihood estimators are not likely obtainable. 3.8.3 Extreme quantiles and other parameters When the estimates of of the GEV distribution have been obtained, other equally important extreme events of any extreme value analysis are extreme quantiles, exceedance probabilities, and return levels and their corresponding periods. An important objective of extreme value modeling is the estimation of an extreme quantile which correspond to a given return period, q, or the maximum value observed in q years (Jonathan and Ewans, 2013) The extreme quantile of the GEV can be estimated by inverting the GEV distribution function in (3.5.1); University of Ghana http://ugspace.ug.edu.gh 30      ˆ , ˆ ˆˆ 1 log 1 , 0 ˆ ˆˆ ˆ log log 1 , 0 x p p q p                         (3.8.1) where ( ) The parameters ̂ ̂ ̂ are corresponding ML estimates of respectively. In simple language, is the return level associated with the return period . Usually long return periods which correspond to small value of p, are of key interest in practice. For , an inference can be made on the upper endpoint of the distribution. This is the return period for the infinite observation which corresponds to with and the ML estimate is ̂ ̂ ̂⁄ . 3.8. 4 Confidence Interval Estimations The estimator ML is asymptotically normal and hence normal confidence interval can be derived for the GEV parameters ( ). If is any one of the parameters, then the ( ) confidence interval for is (Beirlant et al., 2004);  1 ˆ ˆ 1 / 2 iiV m     , where ̂ is the ML estimates of and ̂ denotes the diagonal element of the variance covariance matrix of ML estimator and m is the number of sample maxima (or minima) available. University of Ghana http://ugspace.ug.edu.gh 31 Similarly, inference concerning the GEV quantiles can be based on the normal limiting behaviour. A straightforward application of the delta method produces    , , ˆ 0, ' ,D x p x pm q q N V   as m  , where ̂ denotes the estimator for which is obtained by substituting the ML estimator into (3.8.1) and where is a covariance matrix of ML estimator and , , , ' , , x p x p x pq q q                According to Beirlant et al. (2004), inference based on the normal limit results may be misleading as the normal approximation to the true sampling distribution of the respective estimators may be poor; however, the profile likelihood function provides better confidence intervals than the normal interval. The profile likelihood function of is given by (Beirlant et al., 2004)   , / ( ) max , ,pL L        Therefore, the profile likelihood ratio statistic 0( ) ˆ( ) p p L L     equals the classical likelihood ratio statistic for testing the hypothesis 0 0:H   vrs 1 0:H   and hence for . University of Ghana http://ugspace.ug.edu.gh 32 At significance level ( ) the profile likelihood-based ( ) confidence interval for is given by    2 12 1 ( ) 1 ˆ: 2log 1 : log ( ) log ( ) ˆ( ) 2 p p p p L CI L L L                                   .. Similarly, profile likelihood- based confidence intervals for the other GEV parameters can be constructed. 3. 8.5 Model Verification for the Generalized Extreme Value Distribution Coles (2001) stated that, it is impossible to verify the validity of an extrapolation from a GEV model, but some assessment can be made with reference to the observed data. The probability plot, quantile plot and a density plot are good graphical tools for assessing the goodness-of-fit of a GEV model (Coles, 2001). A probability plot and quantile plot involves the comparison of the empirical and a fitted distribution. Let ( ) ( ) ( ), then the empirical distribution function evaluated at ( ) is defined as ( )( ) 1i iF x k   (Coles, 2001). When the estimated parameters is substituted into (3.5.1), the resulting model-based estimates are 1 ˆ ( ) ( ) ˆ ˆ ˆ( ) exp 1 ˆ i i x F x                     University of Ghana http://ugspace.ug.edu.gh 33 If the GEV model is appropriate, ( ) ( ) ˆ( ) ( ); 1,2,...,i iF x F x i k  (Coles, 2001). Hence a probability plot will consist of the points (Coles, 2001)   ( ) ( ) ˆ( ), ( ) ; 1,2,...,i iF x F x i k and should yield a straight one-to-one line of points, any deviation from linearity is an indication of failure of the GEV model. Additionally, the quantile plot consist of points (Coles, 2001)   1 ( ) ˆ ( 1), ; 1, 2,..., kiF i k x i   Hence from (3.8.1) we have (Coles, 2001)   ˆ1 1 1 1 1 ˆˆ ˆ( ) 1 log( ) ˆk k G              The quantile plot should also yield a straight one-to-one line of points. Any deviation from linearity in the plot indicates failure of the model. 3.8.6 Peak-over threshold approach: threshold selection An event is considered extreme if it exceeds a high threshold u, that is, if it is defined as * +. Where the are a random sample of size n from the random variable X having a distribution function F. We however have to select an appropriate threshold such that it is enough to justify the GP distribution assumption. The selection of threshold is similar to the choice of block size in the block maxima approach; it is a balance between bias and variance. In University of Ghana http://ugspace.ug.edu.gh 34 this case, too low a threshold is likely to violate the asymptotic basis for the model, leading to bias and too high a threshold will increase the variance since it will generate few excesses. Two procedures will be used to aid in the selection of an appropriate threshold for the GP model. One of the procedures consists of an exploratory technique which is carried out prior to the estimation of the model and the other procedure involves an assessment of the stability of parameter estimates, which is based on fitting the model across a range of different thresholds (Coles, 2001). The Mean Residual Life Plot This procedure depends on the mean of the GP distribution, let Y have a GP distribution with parameters , thus scale and shape respectively, then ( ) 1 E Y     , for (3.8.2) Suppose the GP distribution is a valid model for the excess of a threshold produced by a series , then by (3.8.2), we have   0 0/ 1 u E X u X u       , for where X is an arbitrary term of the series and denotes the scale parameter corresponding to excesses of the threshold . Now if the GP distribution is valid for the excesses of , then it should equally be for all thresholds . This is subject to an appropriate change of the scale parameter to . Now for and from equation (3.7.3) University of Ghana http://ugspace.ug.edu.gh 35   0/ 1 1 uu u E X u X u            (3.8.3) For  0 , /u u E X u X u   is a linear function of the threshold u. The estimates by (3.8.3) are expected to change linearly with u and for values of u for which the GP model is appropriate. This leads to the locus of points  ( ) max 1 1 , : un i iu u x u u x n               where ( ) ( ) consist of the observations that exceeds is the maximum of the , this is known as the mean residual life plot or mean excess plot. For values above the threshold , for which the GP distribution yields a valid approximation for the excess distribution, the mean residual life plot should be approximately linear. The mean residual life plot can also be used to assess the domain of attraction of the underlying distribution (Beirlant et al., 2004). According to Beirlant et al. (2004), when the distribution of a random variable, say X, has a tail heavier than the exponential (HTE), then the mean residual life function increases and decreases for tails lighter than the exponential distribution (LTE). Therefore, the shape of the mean residual life function provides important information on the tails of the distribution in question. Gosh and Resnick (2010), the mean excess plot is a graphical tool which is widely used in the study of extreme values for the verification of a GP model for the excess distribution. University of Ghana http://ugspace.ug.edu.gh 36 The Threshold Stability Plot According to Coles (2001), if a GP distribution is an appropriate model for excess of a threshold , then excesses of a higher threshold u should also follow a GP distribution and the shape parameters of the two distributions are identical. Let denote the value of the GP scale parameter for a threshold of , then from equation (3.7.1)   0 0u u u u     (3.8.4) When , the scale parameter is unchanged. This problem is corrected by reparametering the GP scale parameter as * u u    , which is fixed with respect to u by (3.8.4) Now if is a valid threshold for excesses to follow the GP distribution, then the estimates of both should be stable above . The estimates of these quantities will not exactly be constant, due to sampling variability but they should be stable after allowance for their sampling errors (Coles, 2001). 3.8.7 Maximum likelihood Estimation of the GP model After a suitable threshold has been determined, the parameters of the GP distribution can be estimated by maximum likelihood. Let be the excesses over a threshold . The log-likelihood function if is given by University of Ghana http://ugspace.ug.edu.gh 37   1 ( , ) log 1 1 log 1 m i i l m y                     , provided If , the log-likelihood function reduces to 1 1 ( ) log m i i l m y        3.8.8 Extreme Quantiles and Other Parameters Extreme quantiles of the GP distribution can be obtained by inverting the GP distribution function to yield (Beirlant et al., 2004)  ˆˆ ˆ1 , 0 ˆ ˆ ˆlog , 0 r p x p             (3.8.5) where the parameters ( ̂ ̂) are ML estimates of ( ) respectively. If , the right endpoint of the GP distribution is finite and is given by ˆ ˆ x     , in (3.8.5) If the data are not exact GP distributed, (3.7.1) implies that 1 ( ) ( ) 1 ( ) u F u y F y y F u            Where ̅( ) ( ) ̅( ) ( ) University of Ghana http://ugspace.ug.edu.gh 38 When 1 ( ) ( ) 1F x F u          Estimating ̅( ) by the proportion of exceedances over the threshold u, and replacing by their corresponding estimates yields   1 ˆˆ ( ) 1 ˆ m F u x u n            (3.8.6) The POT estimator for extreme quantiles can now be obtained from inverting the right-hand side of (3.8.6). In the case where , we obtain the extreme quantile estimator as ˆ ˆ 1 ˆ r np x u m                (3.8.7) ̂ is the r observation return level and yield an upper end-point ˆ ˆ ˆ x u      , as 0p  in (3.8.7) 3.8.9 Confidence Interval Estimation for the GP distribution Estimates As in the case of the GEV distribution, ( ) confidence interval for the parameters of the GP distribution can be derived. Let denote any of the parameters of the GP distribution, then a ( ) confidence interval of is given by, Beirlant et al. (2004) University of Ghana http://ugspace.ug.edu.gh 39  1 ˆ ˆ 1 2 iiV m     Where ̂ is the ML estimator of any of the parameters ( ) and ̂ denotes the diagonal elements of the variance covariance matrix of the ML estimates. Inference about return levels can be drawn in a similar way. Straightforward application of the delta method gives    ˆ 0, 'D r rm x x N w Vw  Where V is a covariance matrix of ML estimator and ' ,r rx x w            Therefore, a ( ) confidence interval for is given by  1 ' ˆ 1 2r w Vw x m    Beirlant et al. (2004) again proposed that, better confidence intervals can be constructed on the basis of the profile likelihood ratio test statistic. By similar arguments as in the case of the GEV distribution, the ( ) profile likelihood confidence interval for the parameter can be obtained as      2 1 1 ˆ ˆ: log log 2 p pCI L L                University of Ghana http://ugspace.ug.edu.gh 40 The profile likelihood confidence intervals for the other GP parameters can be obtained through similar arguments. 3.8.10 Model Verification for the Generalized Pareto Distribution Quantile plots, probability plots and density plots are very good graphical tools for assessing the validity of a fitted GP model (Coles, 2001). Suppose there exist, a threshold u, threshold excesses ( ) ( ) ( ) and an estimated model ̂. Then its probability plot consists of the pairs (Coles, 2001)   ( ) ˆ(r 1), ( ) : 1,..., rii F y i  Where 1 ˆˆ ˆ1 1 , 0 ˆˆ ( ) ˆ1 exp , 0 ˆ y F y y                         And the quantile plot is defined as (Coles, 2001)   1 ( ) ˆ ( ( 1), : i 1,..., riF i r y   Where  ˆ1 ˆˆ ( ) 1 ˆ F y u y       , for ˆ 0  If the GP distribution is a valid model for the excesses of u, then both the probability plot and the quantile plot are expected to be approximately linear. University of Ghana http://ugspace.ug.edu.gh 41 A histogram of the threshold exceedances is overlay with the density of the GP model and also compared to see if the GP is a valid model for the data of interest. 3.9 Extremes of Dependent (Stationary Series) Sequence The extreme value models described in the previous sections have been derived under the assumption that, the underlying processes consist of sequences of independent and identically distributed random variables. But in practice, the types of data to which extreme value models are commonly applied to are usually not independent in nature. For example, considering the Akosombo water levels, which consist of daily records. Intuitively, we expect dependence from one day to the next and such dependence is likely to be present in the extremes of the process. In threshold exceedances analysis we are likely to include all daily observations and these extreme water levels are also very likely to depend on one another from one day to the next. This observation is common to the extremes of most environmental time series and such temporal dependence violates the basic assumption of classical extreme value modeling; the assumption of independence of the sequence of extremes. The most natural generalization of a sequence of independent random variables is to a stationary series, which is a more realistic assumption for many physical processes (Coles, 2001). Stationarity implies that, given any subset of variables, the joint distribution of the same subset viewed n times point later remains unchanged. Thus, if is a stationary process, then must have the same distribution as and the joint distribution of ( ) must be identical to that of ( ). Stationarity does not prevent being dependent on previous values, rather, trends, seasonality and other deterministic cycles are excluded by an assumption of stationarity. University of Ghana http://ugspace.ug.edu.gh 42 It worth noting that, under some conditions on the dependence structure, the asymptotic distributions of the maximum of a stationary stochastic process can be related with the maximum of an associated independent sequence of random variables with the same distribution function of the dependent one (Freitas and Freitas, 2008). 3.9.1 Maxima (or Minima) of Stationary Series A very important result mostly used to formulate a condition that makes the notion of extreme events near independent is “Ledbetter‟s ( ) condition”, which makes sure long-range dependence is sufficiently weak so as not to affect the asymptotics of an extreme value analysis. Definition 3.9.2 (Ledbetter’s ( ) condition), A stationary series is said to satisfy the ( ) condition if, for all 1 1... i ... jp qi j     with       1 1 1 1 ,..., , ,..., ,..., ,..., ( , ) p q p qi n i n j n j n i n i n j n j nP X u X u X u X u P X u X u P X u X u n l          (3.9.1) Where ( ) for some sequence such that ⁄ . For sequence of independent variables the difference in probabilities expressed in (3.9.1) is exactly zero for any sequence . More generally, we will require that the ( ) condition holds only for a specific sequence of threshold that increases with . For such a sequence, the ( ) condition ensures that, for sets of variables that are far enough apart, the difference in University of Ghana http://ugspace.ug.edu.gh 43 probabilities expressed in (3.9.1) if not zero, is sufficiently close to zero to have no effect on the limit laws of extremes. Theorem 3.9.2 Let be a stationary process and let * +. Then if * + and * + are sequences of constant such that *( ) ⁄ + ( ), where H is a non- degenerate distribution function and the ( ) condition is satisfied with , then H is a member of the generalized extreme value family of distributions. Theorem 3.9.3 (Extreme of dependent Sequences) Let be a stationary series satisfying Leadbetter‟s ( ) condition and let * +. Let be an independent series with having the same distribution as and let * +. Then if has a non-degenerate limit law given by   , ( )n n n nP X b a x H x   , it follows that;   * , ( )n n n nP X b a x H x   (3.9.2) for some constant such that . The parameter is known as the Extremal index and quantifies the extent of Extremal dependence; for a complete independent process and with increasing level of Extremal dependence. Since H in Theorem 3.8.3 is necessarily an extreme value distribution and due to the max-stability property, then the distribution of maxima in processes displaying short- range temporal dependence (characterized by Extremal index ) is also a GEV distribution; the University of Ghana http://ugspace.ug.edu.gh 44 powering of the limit distribution by only affects the location and the scale parameters of the distribution. Theorem 3.9.3 implies that if maxima of a stationary series converge, which by theorem 3.9.2, they will do, then provided an appropriate ( ) condition is satisfied, then its limiting distribution is related to the limiting distribution of an independent series. The effect of the dependence as seen in expression (3.9.2) is just a replacement of H as the limit distribution with . More precisely, if H corresponds to the GEV distribution with parameters ( ) and then, Coles (2001) 11 * ( ) exp 1 exp 1 x x H x                                                 Where ( ) Thus if the distribution of is GEV with parameters ( ), then that of is also GEV with parameters ( ). Note that the EVI remains unchanged. 3.9.2 Models for Block Maximum (or Minimum) If long-range dependence at extreme levels is weak, so that the data can be reasonably assumed as a process satisfying the ( ) condition, the distribution of the block maximum falls within the same family of distribution as would be appropriate if the series were truly independent. Although the parameters would be different from those that would suppose the series is actually independent but since the parameters needs to be estimated, this is not that important (Coles, University of Ghana http://ugspace.ug.edu.gh 45 2001). Therefore, dependence in data can be ignored when modeling block maxima (Coles, 2001). 3.9.3 Modeling threshold Exceedances Though the modeling procedures for fitting the GEV for the set of block maxima remains unchanged for stationary series, but for the threshold exceedance method, some modification is needed, since extremes may have the tendency to cluster in a stationary series (Coles, 2001). For a stationary series, the usual asymptotic argument means that marginal distribution of the excesses of a high threshold is a Generalized Pareto, but does not specify the joint distribution of neighbouring excesses. There is no general theory to provide an alternative likelihood that incorporates the dependence in observations (Coles, 2001). According Coles (2001), there are various suggestions, which offer a way of dealing with the problem of dependent exceedances in the threshold exceedance approach. These include: (i) The Declustering method (ii) Fitting the GP distribution to all exceedances, ignoring dependence, but then appropriately adjusting the inference (usually an inflation of the standard error) to account for reduction in information (iii)Explicitly modeling the temporal dependence in the process (but this approach makes use of multivariate EVT) The most widely-used method is the declustering, which corresponds to a filtering of the dependent observations to obtain a set of threshold excesses that are approximately independent. This works by going through the following steps: University of Ghana http://ugspace.ug.edu.gh 46 (i) Choose an auxiliary declustering parameter, say k. (ii) A cluster of threshold excesses is then considered to have ended as soon as at least k consecutive observations fall below the threshold. (iii) Go through the entire series identifying clusters in this manner. (iv) The maximum (or peak) observation from each cluster is then extracted and the GP distribution is fitted to the set of cluster peak excesses. This method is simple, but has its limitations. Results can be sensitive to the arbitrary choices made in cluster determination and there is arguably wastage of information in discarding all data except the cluster maxima. 3.9.4 Estimation of Extreme Quantile for a Stationary GP model The rate at which clusters occur, rather than the rate of individual exceedances must be considered when finding an m-observation return level (Coles, 2001). Therefore, the m- observation return level is defined by   ˆˆ ˆˆ ˆ 1 ˆ mx u m           where are the GP distribution parameters, is the probability of an exceedance of the threshold u, is the Extremal index. Letting denote the number of exceedances of the threshold u, the number of clusters obtained above u and n the sample size, ̂ ̂ University of Ghana http://ugspace.ug.edu.gh 47 According to Coles (2001), inference on return levels is robust to the choice of the threshold u and the declustering parameter k. Also instability in the Extremal index estimate does not impact on the estimation of return levels excessively (Coles, 2001). 3.10 Extremes of Non-stationary Sequences Non-stationary processes have characteristics that change systematically through time. Non- stationarity is common in environmental processes and this may be due to seasonal effects and climate changes for different periods of time. In modeling extremes of non-stationary processes, the general assumption of independence and constant distribution through time cannot be applied. In the case of non-stationarity, the parameters of the distribution function are replaced by time-dependent parameters, so that the result of the extreme value analysis also varies with time. For the non-stationary approach, the parameters of the distribution are modelled as a covariate of time. Based on the observed trend in the time series model, the parameters can be modeled accordingly, thus as a linear function, a quadratic function, cubic function or an exponential function. 3.10.1 The Structure of the GEV Model under non-stationarity Let GEV( ) denote the GEV distribution with parameters , a suitable model for , the annual (or monthly) maximum water level at time , becomes ( ( ) ( ) ( )). This implies that, the location parameter, the scale parameter and shape parameter are time dependent. Due to the difficulty in estimating the shape parameter with precision, the shape parameter is usually kept constant (Coles, 2001). Also, in order to ensure the University of Ghana http://ugspace.ug.edu.gh 48 positivity of the scale parameter for all values of , an exponential model is usually used for the scale parameter (Coles, 2001). The variations through time in the observed process can be modeled as a polynomial or an exponential trend in the location parameter, based on the kind of trend observed in the process. Therefore, the models for the parameters may be; ( ) ( ) ( ) ( ) Or ( ) ( ) ( ) ( ) The parameters are estimated from the observed data. It should be noted that, more complex models than what has been stated above can be obtained. The GEV distribution then becomes; 1 ( ) exp 1 , 0 ( ) ( , ) ( ) exp exp , 0 ( ) x t t G x t x t t                                               (3.10.1) University of Ghana http://ugspace.ug.edu.gh 49 where is the independent water level, ( ) the time-dependent location parameter, ( ) the time-dependent scale parameter and the constant shape parameter. 3.10.2 The Structure of GP Model under non-stationarity Using the notation ( ) to denote the GP distribution with parameters , the time- dependent model for threshold exceedances can be expressed as ( ( ) ( )) ( ( ) )⁄ , where ( ( ) ) are the time-dependent GP distribution parameters and ( ) is a time- dependent threshold. Just as in the GEV case, the shape parameter, would be kept constant and the scale parameter modeled as in the case of the GEV distribution. The GP distribution is now defined as; 1 1 1 , 0 ( ) ( ) 1 exp , 0 ( ) t t y t H y y t                         (3.10.2) Where ( ) and are the time series of daily observation of the Akosombo water levels at a given time t. University of Ghana http://ugspace.ug.edu.gh 50 3.10.3 Model Choice Due to the possibility of modeling any combination of the extreme value model parameters as a function of time, there is a large catalogue of models to select from and selecting an appropriate model becomes an important issue. The basic principle is parsimony, obtaining the simplest model possible, which explains as much variation in the data as possible (Coles, 2001). The model is needed as a description of the process that generated the data and not for the data themselves, so it is necessary to assess the strength of evidence for the more complex model structure (Coles, 2001). If the evidence is not particularly strong, the simpler model should be selected in preference. Maximum likelihood estimation of nested models leads to a simple test procedure of one model against the other. With models , the deviance statistic is defined as * ( ) ( )+ Where ( ) ( ) are the maximized log-likelihood under models respectively. Large values of D indicates that model explains substantially more of the variation in the data than while small values of D suggest that the increase in model size does not bring any improvements in the model‟s capacity to explain the data. The asymptotic distribution of the deviance function is used to determine how large D should be before model is preferred to model . is rejected by a test at the of significance if , where is the ( ) quantile of the distribution and k is the difference in the dimensionality of and . University of Ghana http://ugspace.ug.edu.gh 51 3.11 Parameter Estimation for Non-stationary model Now we consider how to estimate the Extreme Value Index, which is very key in extreme value analysis and other equally important extreme events. 3.11.1 Maximum Likelihood (ML) Estimation Method for the GEV distribution The parameters of the GEV distribution may be estimated by Maximum Likelihood (ML) estimation technique. Let ( ( ) ( ) ), a non-stationary GEV model to describe the distribution of , where each of ( ) ( ) have an expression in terms of a parameter vector. The likelihood function of the non-stationary GEV is then given; 1 ( ) ( , ( ), ( ), ) m t t L g x t t      , ( ( ) ( ) ) Where ( ( ) ( ) ) denotes the GEV density function with parameters ( ) ( ) evaluated at . If , the log-likelihood is;   1 1 ( ) ( ) ( ) log ( ) 1 1 log 1 1 ( ) ( ) m t t t x t x t l t t t                                              provided that ( ( ) ( ) ) . If , the log-likelihood reduces to; 1 ( ) ( ) ( ) log (t) exp ( ) ( ) m t t t x t x t l t t                                 University of Ghana http://ugspace.ug.edu.gh 52 The estimates of the parameters ( ( ) ( ) ) are obtained by maximizing the log-likelihood function. 3.11.2 Extreme quantiles Estimation for the non-stationary GEV distribution When the estimators of ( ) ( ) of the GEV distribution have been obtained, the other most important parameters of any extreme value analysis is to be able to estimate extreme quantiles, exceedance probabilities and return periods. The extreme quantile of the GEV can be estimated by inverting the GEV distribution function in (3.10.1);      , ( ) ( ) 1 log 1 , 0 ( ) ( ) log log(1 p) , 0 tx p t t p q t t                       (3.11.1) for . The parameters ( ) ( ) are replaced with their corresponding ML estimates. 3.11.3 Peak-over threshold approach: threshold selection In extremes of non-stationary sequences, the assumption of constant threshold of exceedances is not the best, since the characteristics of the process change systematically through time. Hence a temporally changing threshold is required, one way of determining such a threshold is by the use quantile regression. However, an appropriate quantile should be chosen such that, the value of the threshold is enough to justify the GPD assumption. The choice of a threshold is a balance between bias and variance as described in the previous sections. University of Ghana http://ugspace.ug.edu.gh 53 3.11.4 Quantile Regression Since we wish to set a non-constant threshold, we employ quantile regression which is used to quantify how a given conditional quantile (or quantiles) of a response variable Y depends on the observed values of covariates (Koenker and Hallock, 2001). The quantile regression model condition quantiles as function of predictors. The quantile regression model is an extension of the linear regression model. While the linear regression model specifies the changes in the conditional mean of the dependent variable associated with a change in the covariates, the quantile regression model specifies changes in the conditional quantile. Any predetermined position of the distribution can be modeled since quantiles are used. Quantile regression is used to quantify how a conditional quantile of a response variable depends on the observed values of covariates Let denote the conditional quantile of Y, satisfying ( ) for ( ) . According to Koenker and Bassett (1978), the regression parameters are estimated by minimizing         |, | ˆ ( ) arg min 1 i i n Y i i Yy i i i iy Y y Q Y y Y y                             With respect to ( ) , where . The R package quantreg (Koenker, 2015) can be employed to estimate . If the probability of threshold exceedance is quadratic (or cubic), then the form of the GP model fitted implies that particular form of the quantile regression model be used to set the threshold. Therefore, for a given GP model, we know which quantile model to fit. University of Ghana http://ugspace.ug.edu.gh 54 3.11.5 Maximum likelihood Estimation for the GP distribution After a suitable threshold has been determined, the parameters of the GP distribution can be estimated by maximum likelihood. Let be the excesses over a threshold ( ). The log-likelihood function if is given by   1 ( ( ), ) log ( ) 1 1 log 1 ( ) m t t l t t y t                     provided ( ) If , the log-likelihood function reduces to 1 ( ( )) log ( ) ( ) m t t y l t t t             3.11.6 Extreme Quantiles Estimation for the non-stationary GP distribution Extreme quantiles of the GP distribution can be obtained by inverting the GP distribution function in (3.9.2) to yield   , ( ) 1 , 0 ( ) log , 0 ty p t p x t p             (3.11.2) where the parameters ( ( ) ) are substituted with their corresponding ML estimates. University of Ghana http://ugspace.ug.edu.gh 55 3.12 Model Verification After the estimates of various parameters of possible model have been determined, we need to verify the appropriateness of the adopted model. For the non-stationary case, the lack of homogeneity in the distributional assumptions for each observation calls for some modification in the diagnostic plots discussed for the independent and identical distribution. It is generally only possible to use such diagnostic plots on standardized version of the data, given the fitted parameter values (Coles, 2001). For instance, based on the estimated model ( ̂( ) ̂( ) ̂), the standardized variables ̃ , is defined by (Coles, 2001) ˆ1 ( ) ˆlog 1 ˆ ˆ ( ) t t t X x t                (3.12.1) and each of the standardized variables ̃ have the standard Gumbel distribution, with probability density function    exp exp( )tP X x x    (3.12.2) The probability and the quantile plots of the observed ̃ are constructed with reference to the Gumbel distribution (Coles, 2001). Letting ̃( ) ̃( ) denote the ordered values of ̃ , the probability plots consists of the pairs (Coles, 2001);    ( )( 1),exp exp( ) : 1,...,ii m x i m    and the quantile plot consists of the pair (Coles, 2001); University of Ghana http://ugspace.ug.edu.gh 56    ( ) , log ( 1) : 1,...,ix i m i m   According to Coles (2001), the probability plot is invariant to the Gumbel distribution as a reference distribution but the quantile plot is not and so any choice other than the Gumbel would produce a different plot. Similar techniques can be adopted for the GP distribution. For the GP distribution, a set of temporally varying thresholds ( ), which yield threshold excesses is considered. The estimated model is then defined as ( ̂( ) ̂). In this case, since the exponential distribution is a special case of the GP family with , the exponential distribution would be used as the reference distribution (Coles, 2001).   1 ˆ ˆ t t t Y u Y t          is the standardized variables for the GP distribution. Letting ̃( ) ̃( ) denote the ordered values of the observed ̃ , the probability plot consist of the pairs (Coles, 2001),       / 1 ,1  exp : 1,  ,  i i k y i k     and the quantile plot consist of pairs (Coles, 2001),     , log 1 ; 1,  ,  1i iy i k k                  University of Ghana http://ugspace.ug.edu.gh 57 Just as in the case the GEV distribution, the probability plot is invariant to the choice of reference distribution but the quantile plot is specific to the choice of exponentia