Statistical Modeling of Temperature Extremes Behaviour in Ghana

The Generalized Pareto (GP) and Generalized extreme value (GEV) distributions play an important role in extreme value analyses, as models for threshold excesses and block maxima respectively. For each of these distributions we consider Bayesian inference using "reference" prior distributions (in the general sense of priors constructed using formal rules) for the model parameters, specifically a Jeffreys prior, the maximal data information (MDI) prior and independent uniform priors on separate model parameters. We investigate the important issue of whether these improper priors lead to proper posterior distributions. We show that, in the GP and GEV cases, the MDI prior, unless modified, never yields a proper posterior and that in the GEV case this also applies to the Jeffreys prior. We also show that a sample size of three (four) is sufficient for independent uniform priors to yield a proper posterior distribution in the GP (GEV) case.

Extreme value theory (EVT) is useful for modeling the impact of crashes or situations of extreme stress on investor portfolios. EVT is mostly utilized in financial modeling, risk management, insurance, and hydrology. The price of gold fluctuates considerably over time, and this introduces a risk on its own. The goal of this study is to analyze the risk of gold investment by applying the EVT to historical daily data for extreme daily losses and gains in the price of gold. We used daily gold prices in the Pakistan Bullion Market from August 1, 2011 to July 30, 2021. This paper covers two methods such as Block Maxima (BM) and Peak Over Threshold (POT) modeling. The risk measures which are adopted in this paper are Value at Risk (VaR) and Expected Shortfall (ES). The point and interval estimates of VaR and ES are obtained by fitting the Generalized Pareto (GPA) distribution. Moreover, in this paper, return-level forecasting is also included for the next 5 and 10 years by analyzing the Generalized Extreme Value (GEV) distribution.

Cryptocurrency markets have emerged as a dynamic and intriguing domain, with Bitcoin at the forefront, captivating the attention of investors, researchers, and enthusiasts alike. The volatile nature of Bitcoin prices presents both opportunities and challenges for market participants seeking to understand and anticipate its movements. In this study, we delve into the realm of time series analysis to explore the feasibility of predicting The research journey begins with meticulous data preprocessing steps to ensure the quality and integrity of the input data. Leveraging Python libraries such as pandas and NumPy, we cleanse and format the historical Bitcoin price data, laying the foundation for subsequent analysis. Key preprocessing tasks include handling missing values, normalization, and addressing any anomalies or outliers that may distort the underlying patterns. With the data prepared, our attention turns to assessing the stationarity of the Bitcoin price time series—a fundamental prerequisite for applying classical time series models. Through visual inspection and statistical tests such as the Augmented Dickey-Fuller (ADF) test, we ascertain the presence of trends or seasonality that could influence the modelling process. To mitigate such effects, we employ techniques such as differencing and transformations, including the Box-Cox transformation, to stabilize the variance of the data. Armed with a stationary time series, we embark on the core of our analysis: modelling Bitcoin prices using Autoregressive Integrated Moving Average (ARIMA) and Seasonal Autoregressive Integrated Moving Average with Exogenous Variables (SARIMAX) models. These models, renowned for their versatility and effectiveness in capturing temporal dependencies, offer a sophisticated framework for forecasting time series data. Guided by the principles of parsimony and model selection criteria such as the Akaike Information Criterion (AIC), we systematically explore the parameter space to identify the optimal specifications for our models. The efficacy of the chosen models is rigorously evaluated through diagnostic checks, encompassing residual analysis, model fit statistics, and out-of-sample validation. Insights gleaned from these assessments inform our confidence in the models' predictive capabilities and guide our interpretation of the forecasted outcomes. Finally, armed with a validated model, we turn our gaze to the future, employing it to generate forecasts of Bitcoin prices for forthcoming periods. Visualizations juxtaposing predicted prices against observed values provide a compelling narrative of the model's performance and offer stakeholders valuable insights into potential market trends and dynamics. In summary, this research contributes to the burgeoning field of cryptocurrency analytics by showcasing the application of time-tested statistical methodologies to forecast Bitcoin prices. By leveraging the power of ARIMA and SARIMAX models, we illuminate the intricate patterns underlying Bitcoin's price dynamics, empowering market participants with actionable intelligence for informed decision-making in an ever-evolving landscape. Keywords Cryptocurrency Markets, Bitcoin Price Prediction, Time Series Analysis, Python Programming, Data Preprocessing, Pandas, NumPy, Stationarity Testing, Augmented Dickey-Fuller (ADF) Test, Box-Cox Transformation, ARIMA Model, SARIMAX Model, Model Selection, Akaike Information Criterion (AIC), Diagnostic Checks, Residual Analysis, Out-of-Sample Validation, Forecasting, Visualization, Market Trends, Decision-Making, Cryptocurrency Analytics

Some events are difficult to avoid and gives considerable influence to humans and the environment is extreme weather and climate change. Many of the problems that require knowledge about the behavior of extreme values and one of the methods used are the Extreme Value Theory (EVT). EVT used to draw up reliable systems in a variety of conditions, so as to minimize the risk of a major disaster. There are two methods for identifying extreme value, Block Maxima with Generalized Extreme Value (GEV) distribution approach and Peaks over Threshold (POT) with Generalized Pareto Distribution (GPD) approach. This research in Indramayu with January 1961-December 2003 period, the method used is Block Maxima with GEV distribution approach. The result showed that there is no climate change in Indramayu with January 1961-December 2003 period.

Accurate estimation of extreme (i.e., 1-day maximum) rainfall is essential for water resources management, flood forecasting, agricultural planning, and climate change impact studies. This can be achieved through fitting the extreme value family of probability distributions (EVD) that consists of Extreme Value Type-1 (EV1), Extreme Value Type-2 (EV2), Generalized Extreme Value (GEV), and Generalized Pareto (GPA) to the series of observed annual 1-day maximum rainfall (AMR), whereas the parameters are determined by the Method of L-Moments (LMO). This paper presents a study on the comparison of LMO estimators of EVD for the determination of design rainfall depth at Akkalkuwa, Kamrej, Navapur, Sakri, Shahada, and Taloda rain gauge sites. For this purpose, the AMR series was generated from the daily rainfall data observed at the sites during the period 1960 to 2022 and used. The adequacy of fitting LMO of EVD to the AMR series was examined through Goodness-of-Fit (GoF) tests, viz., Chi-Square and Kolmogorov-Smirnov (KS), while the selection of the best-fit distribution was made through model performance analysis with various indicators, viz., correlation coefficient (CC), Nash- Sutcliffe model efficiency (NSE), root mean squared error (RMSE), and cross-correlation matrix analysis (CCMA). The Chi- Square test results uniformly supported the use of EV1 and GEV for modelling the AMR data of six sites, whereas KS test results supported all four distributions of EVD for all six sites. The results indicated that the CC values obtained from four distributions vary between 0.960 and 0.994. The study showed that the NSE computed by EV1, GEV, and GPA varies from 91.7% to 98.7%. The outcomes of CCMA showed that there is a perfect correlation between the estimated rainfall by EV1 and GEV, and also nearer to 1.000. On the basis of evaluation of the results with quantitative (viz., CC, NSE, and RMSE) and qualitative assessments, it was found that GEV is the best choice for rainfall estimation for Akkalkuwa, Kamrej, Navapur, Sakri, Shahada, and Taloda. The estimated extreme rainfall by GEV distribution could be considered as a design rainfall depth while planning water resources management projects and their related activities in the respective sites.

Estimating financial risk is a critical issue for banks and insurance companies. Recently, quantile estimation based on extreme value theory (EVT) has found a successful domain of application in such a context, outperforming other methods. Given a parametric model provided by EVT, a natural approach is maximum likelihood estimation. Although the resulting estimator is asymptotically efficient, often the number of observations available to estimate the parameters of the EVT models is too small to make the large sample property trustworthy. In this paper, we study a new estimator of the parameters, the maximum Lq-likelihood estimator (MLqE), introduced by Ferrari and Yang (Estimation of tail probability via the maximum Lq-likelihood method, Technical Report 659, School of Statistics, University of Minnesota, 2007 http//:www.stat.umn.edu/~dferrari/research/techrep659.pdf ). We show that the MLqE outperforms the standard MLE, when estimating tail probabilities and quantiles of the generalized extreme value (GEV) and the generalized Pareto (GP) distributions. First, we assess the relative efficiency between the MLqE and the MLE for various sample sizes, using Monte Carlo simulations. Second, we analyze the performance of the MLqE for extreme quantile estimation using real-world financial data. The MLqE is characterized by a distortion parameter q and extends the traditional log-likelihood maximization procedure. When q→1, the new estimator approaches the traditional maximum likelihood estimator (MLE), recovering its desirable asymptotic properties; when q ≠ 1 and the sample size is moderate or small, the MLqE successfully trades bias for variance, resulting in an overall gain in terms of accuracy (mean squared error).

Objective: This study examines the application of Extreme Value Theory (EVT) in economics to understand and manage rare, high-impact events that affect financial markets and economic stability. Theoretical framework: EVT provides a statistical foundation for modeling extreme variations in financial data, making it an essential tool for risk assessment and economic decision-making. By estimating the probability of extreme fluctuations, EVT enhances the understanding of financial risk exposure. Method: We analyze daily Eurodollar exchange rate data from January 1, 1999, to March 22, 2024, obtained from Yahoo Finance, totaling approximately 6,580 observations. The monthly maximum values were extracted and modeled using the Generalized Extreme Value (GEV) distribution. Results and Discussion: The"Return Level Plot" derived from the GEV model highlights potential changes in extreme values of the Eurodollar exchange rate over time. The findings provide valuable insights into financial risk exposure and potential economic fluctuations. The estimated parameters for the GEV distribution are: location parameter (1.15), scale parameter (0.15), and shape parameter (-0.27). To ensure the robustness of results, the Kolmogorov-Smirnov test was applied, confirming the goodness-of-fit of the GEV model. Research Implications: This study contributes to financial risk management by offering a methodological approach to modeling extreme exchange rate fluctuations. It assists policymakers and financial analysts in preparing for and mitigating risks associated with extreme currency movements. Originality/value: This research provides empirical evidence supporting the use of EVT in financial market analysis. It emphasizes the relevance of EVT in risk management and economic decision-making by demonstrating its effectiveness in modeling extreme variations in exchange rates.

The SARIMA time series model is fitted to the monthly average maximum and minimum temperature data sets collected at Giridih, India for the years 1990-2011. From the time-series plots, we observe that the patterns of both the series are quite different; maximum temperature series contain sharp peaks in almost all the years while it is not true for the minimum temperature series and hence both the series are modeled separately (also for the sake of simplicity). SARIMA models are selected based on observing autocorrelation function (ACF) and partial autocorrelation function (PACF) of the monthly temperature series. The model parameters are obtained by using maximum likelihood method with the help of three tests [i.e., standard error, ACF and PACF of residuals and Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC) and corrected Akaike Information Criteria (AICc)]. Adequacy of the selected models is determined using diagnostic checking with the standardized residuals, ACF of residuals, normal Q-Q plot of the standardized residuals and p-values of the Ljung-Box statistic. The models ARIMA (1; 0; 2) × (0; 1; 1)12 and ARIMA (0; 1; 1) × (1; 1; 1)12 are finally selected for forecasting of monthly average maximum and minimum temperature values respectively for the eastern plateau region of India.

The quantiles of annual maximum wind speed (AMWS) can be estimated for different meteorological stations of interest by using at‐site frequency analysis and extreme value theory. These estimates are of immense importance for the codification of wind speed. However, the historical data of wind speed at the number of meteorological stations are sometimes unavailable and often insufficient due to the shorter length, especially in developing countries like Pakistan. The scarcity of the data increases the uncertainty of the quantiles estimates regarding policy implications. To cope with the problem, an approach of Regional Frequency Analysis (RFA) is opted here. In this study, RFA of AMWS using linear‐moments (L‐moments) is carried out by considering wind speed data of nine meteorological stations of province Punjab, Pakistan. No station is found to be discordant. A single homogenous region is constituted from these nine stations using a subjective approach based on their geographical locations. Heterogeneity measures justify that these nine stations of Punjab form a single homogeneous region. Regional quantiles estimates are found through the most appropriate probability distribution among generalized normal (GNO), generalized logistic (GLO), Pearson Type 3 (P3), generalized Pareto (GPA), Weibull (WEI), log Pearson Type 3 (LP3) and generalized extreme value (GEV) distributions. Z‐statistic and L‐moment ratio diagram suggest that GLO and GNO distributions are better choices than others. Robustness of both distributions is evaluated through relative bias (RB) and relative root mean square error (RRMSE). Findings indicate that overall, GLO distribution is better than GNO. Further, we also find at‐site quantiles from dimensionless quantities (regional quantiles) using the sample mean and median as scaling factors. Quantiles' estimates calculated from this study can be used in codified structural designs for policy implications.

In this work, we introduce a method of estimating stochastic freeway capacity using elements of both extreme value theory and survival analysis. First, we define capacity data, or estimates of the capacity of the roadway, as the daily maximum flow values. Then, under a survival analysis premise, we introduce censoring into our definition. That is, on days when flows are sufficiently high and congestion occurs, corresponding flow maxima are considered true estimates of capacity; otherwise, for those days that do not observe high flows or congestion, flow maxima are deemed censored observations and capacities must be higher than the observations. By extreme value theory, the collection of flow maxima (block maxima) can be appropriately approximated with a generalized extreme value (GEV) distribution. Because of small sample sizes and the presence of censoring, a Bayesian framework is pursued for model fitting and parameter estimation. To lend credence to our proposed methodology, the procedure is applied to real-world traffic stream data collected by the New Hampshire Department of Transportation (NHDOT) at a busy location on Interstate I-93 near Salem, New Hampshire. Data were collected over a period of 11 months and raw data were aggregated into 15-minute intervals. To assess our procedure, and to provide proof of concept, several validation procedures are presented. First, using distinct training and validation subsets of our data, the procedure yields accurate predictions of highway capacity. Next, our procedure is applied to a training set to yield random capacities which are then used to predict breakdown in the validation set. The frequency of these predicted breakdowns is found to be statistically similar to observed breakdowns observed in our validation set. Lastly, after comparing our methodology to other methods of stochastic capacity estimation, we find our procedure to be highly successful.

India gets three fourths of its annual rainfall during the south-west monsoon season (June–September). The study of extreme events is significant in the stochastic behaviour of rainfall pattern. The aim of the present work is to compare different methods; and find a suitable method to study extreme rainfall trend analysis. In this study, frequency distribution method, generalized extreme value (GEV) distribution, Mann–Kendall and Sen’s slope estimator are used for rainfall trend analysis over the Nethravathi basin located in the southern part of India. The rainfall data during the monsoon months (June–September) were analysed for a period of 1971–2010. The comparison of all the methods had been carried out and it has been observed that there is an increasing trend of frequency in class-1 and decreasing trend in class-2 and class-3, respectively. The interpretation of the results is carried by using the GEV distribution and non-parametric trend analysis (Mann–Kendall and Sen’s slope estimator test). It turns out the best results to identify the extreme rainfall trend are obtained by the statistical techniques – Block Maxima (GEV) distribution, Mann–Kendall and Sen’s slope estimator test as compared to frequency-based method. The above results which help to study climate change will contribute towards sustainable development of the Nethravathi River basin.

Extreme value modeling for extreme rainfall is one of the most important processes in the field of hydrology. For the improvement of extreme value modeling and its physical meaning, large-scale climate modes have been widely used as covariates of distribution parameters, as they can physically account for climate variability. This study proposes a novel procedure for extreme value modeling of rainfall based on the significant relationship between the long-term trend of the annual maximum (AM) daily rainfall and large-scale climate indices. This procedure is characterized by two main steps: (a) identifying significant seasonal climate indices (SCIs), which impact the long-term trend of AM daily rainfall using statistical approaches, such as ensemble empirical mode decomposition, and (b) selecting an appropriate generalized extreme value (GEV) distribution among the stationary GEV and nonstationary GEV (NS-GEV) using time and SCIs as covariates by comparing their model fit and uncertainty. Our findings showed significant relationships between the long-term trend of AM daily rainfall over South Korea and SCIs (i.e., the Atlantic Meridional Mode, Atlantic Multidecadal Oscillation in the fall season, and North Atlantic Oscillation in the summer season). In addition, we proposed a model selection procedure considering both the Akaike information criterion and residual bootstrap method to select an appropriate GEV distribution among a total of 59 GEV candidates. As a result, the NS-GEV with SCI covariates generally showed the best performance over South Korea. We expect that this study can contribute to estimating more reliable extreme rainfall quantiles using climate covariates.

A great challenge has been appeared on if the assumption of data stationary for flood frequency analysis is justifiable. Results for frequency analysis (FA) could be substantially different if non-stationarity is incorporated in the data analysis. In this study, extreme water levels (annual maximum and daily instantaneous maximum) in a coastal part of New York City were considered for FA. Annual maximum series (AMS) and peak-over threshold (POT) approaches were applied to build data timeseries. The resulted timeseries were checked for potential trend and stationarity using statistical tests including Man-Kendall, Augmented Dickey–Fuller (ADF) and Kwiatkowski–Phillips–Schmidt–Shin (KPSS). Akaike information criterion (AIC) was utilized to select the most appropriate probability distribution models. Generalized Extreme Value (GEV) distribution and Generalized Pareto Distribution (GPD) were then applied as the probability distribution functions on the selected data based on AMS and POT methods under non-stationary assumption. Two methods of maximum likelihood and penalized maximum likelihood were applied and compared for the estimation of the distributions’ parameters. Results showed that by incorporating non-stationarity in FA, design values of extreme water levels were significantly different from those obtained under the assumption of stationarity. Moreover, in the non-stationary FA, consideration of time-dependency for the distribution parameters resulted in a range of variation for design floods. The findings of this study emphasize on the importance of FA under the assumptions of data stationarity and non-stationarity, and taking into account the worst case flooding scenarios for future planning of the watershed against the probable flood events. There is a need to update models developed for stationary flood risk assessment for more robust and resilient hydrologic predictions. Applying non-stationary FA provides an advanced method to extrapolate return levels up to the desired future time perspectives.

Abstract. In this study we detect and quantify changes in the distribution of the annual maximum daily maximum temperature (TXx) in a large observation-based gridded data set of European daily temperature during the years 1950–2018. Several statistical models are considered, each of which analyses TXx using a generalized extreme-value (GEV) distribution with the GEV parameters varying smoothly over space. In contrast to several previous studies which fit independent GEV models at the grid-box level, our models pull information from neighbouring grid boxes for more efficient parameter estimation. The GEV location and scale parameters are allowed to vary in time using the log of atmospheric CO2 as a covariate. Changes are detected most strongly in the GEV location parameter, with the TXx distributions generally shifting towards hotter temperatures. Averaged across our spatial domain, the 100-year return level of TXx based on the 2018 climate is approximately 2 ∘C (95 % confidence interval of [2.03,2.12] ∘C) hotter than that based on the 1950 climate. Moreover, averaged across our spatial domain, the 100-year return level of TXx based on the 1950 climate corresponds approximately to a 6-year return level in the 2018 climate.

Determination of the extent of peak rainfall for different return periods is an essential ingredient for the accurate design of hydraulic structures such as drains, dams and culverts as well as detection of flood risk areas. The focus of this study is to analyze annual maximum daily rainfall series in some selected sites within the coastal region of Nigeria using three parameter probability distribution models, namely, Generalized Logistics (GLO), Generalized Extreme Value (GEV) and Generalized Pareto (GPA) with the view of identifying the best fit probability distribution model per station that can be employed to estimate the rainfall magnitude for selected return periods. Specific time series analysis test, namely, detection of outlier and homogeneity test were performed to certify that the data utilized are adequate and suitable. Descriptive statistics such as sample mean, variance, standard deviation, kurtosis, skewness, and coefficient of variation were computed using basic statistical equations. The probability weighted moment parameters (b0, b1, b2 and b3), L-moment values (λ1, λ2, λ3 and λ4) and ratios (τ2, τ 3 and τ4) including the distribution parameters, namely, shape (k), scale (α) and location (ξ) parameters were computed based on L-moments procedures. To select the best-fit probability distribution model per station, carefully chosen goodness-of-fit statistics, namely, root mean square error, relative root mean square error, maximum absolute deviation index, maximum absolute error and probability plot correlation coefficient were employed since they can adequately assess the fitted distribution at a site. Results obtained indicate that the GLO is the best fit distribution for analyzing annual maximum daily rainfall series from Warri and Calabar while GPA for Port Harcourt and Uyo.