Search results for: Forecast models

Authors: Wardah, T., Kamil, A.A., Sahol Hamid, A.B., Maisarah, W.W.I

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Quantitative precipitation forecast (QPF) from atmospheric model as input to hydrological model in an integrated hydro-meteorological flood forecasting system has been operational in many countries worldwide. High-resolution numerical weather prediction (NWP) models with grid cell sizes between 2 and 14 km have great potential in contributing towards reasonably accurate QPF. In this study the potential of two NWP models to forecast precipitation for a flood-prone area in a tropical region is examined. The precipitation forecasts produced from the Fifth Generation Penn State/NCAR Mesoscale (MM5) and Weather Research and Forecasting (WRF) models are statistically verified with the observed rain in Kelantan River Basin, Malaysia. The statistical verification indicates that the models have performed quite satisfactorily for low and moderate rainfall but not very satisfactory for heavy rainfall.

Keywords: MM5, Numerical weather prediction (NWP), quantitative precipitation forecast (QPF), WRF

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Authors: J. Vilgerts, L. Timma, D. Blumberga

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The objective of the paper is to develop the forecast model for the HW flows. The methodology of the research included 6 modules: historical data, assumptions, choose of indicators, data processing, and data analysis with STATGRAPHICS, and forecast models. The proposed methodology was validated for the case study for Latvia. Hypothesis on the changes in HW for time period of 2010-2020 have been developed and mathematically described with confidence level of 95.0% and 50.0%. Sensitivity analysis for the analyzed scenarios was done. The results show that the growth of GDP affects the total amount of HW in the country. The total amount of the HW is projected to be within the corridor of – 27.7% in the optimistic scenario up to +87.8% in the pessimistic scenario with confidence level of 50.0% for period of 2010-2020. The optimistic scenario has shown to be the least flexible to the changes in the GDP growth.

Keywords: Forecast models, hazardous waste management, sustainable development, waste management indicators.

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Authors: N. Sopipan, A. Intarasit, K. Chuarkham

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 In this paper, we forecast the volatility of Baht/USDs using Markov Regime Switching GARCH (MRS-GARCH) models. These models allow volatility to have different dynamics according to unobserved regime variables. The main purpose of this paper is to find out whether MRS-GARCH models are an improvement on the GARCH type models in terms of modeling and forecasting Baht/USD volatility. The MRS-GARCH is the best performance model for Baht/USD volatility in short term but the GARCH model is best perform for long term.

Keywords: Volatility, Markov Regime Switching, Forecasting.

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Authors: Josip Vasilj, Petar Sarajcev, Damir Jakus

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One of the major difficulties introduced with wind power penetration is the inherent uncertainty in production originating from uncertain wind conditions. This uncertainty impacts many different aspects of power system operation, especially the balancing power requirements. For this reason, in power system development planing, it is necessary to evaluate the potential uncertainty in future wind power generation. For this purpose, simulation models are required, reproducing the performance of wind power forecasts. This paper presents a wind power forecast error simulation models which are based on the stochastic process simulation. Proposed models capture the most important statistical parameters recognized in wind power forecast error time series. Furthermore, two distinct models are presented based on data availability. First model uses wind speed measurements on potential or existing wind power plant locations, while the seconds model uses statistical distribution of wind speeds.

Keywords: Wind power, Uncertainty, Stochastic process, Monte Carlo simulation.

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Authors: João Paulo Teixeira, Paula Odete Fernandes

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The authors have been developing several models based on artificial neural networks, linear regression models, Box- Jenkins methodology and ARIMA models to predict the time series of tourism. The time series consist in the “Monthly Number of Guest Nights in the Hotels" of one region. Several comparisons between the different type models have been experimented as well as the features used at the entrance of the models. The Artificial Neural Network (ANN) models have always had their performance at the top of the best models. Usually the feed-forward architecture was used due to their huge application and results. In this paper the author made a comparison between different architectures of the ANNs using simply the same input. Therefore, the traditional feed-forward architecture, the cascade forwards, a recurrent Elman architecture and a radial based architecture were discussed and compared based on the task of predicting the mentioned time series.

Keywords: Artificial Neural Network Architectures, time series forecast, tourism.

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Authors: Anastasia K. Eleftheriadou, Athanasios Sfetsos, Nikolaos Gounaris

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The current paper presents an extensive bottom-up framework for assessing building sector-specific vulnerability to climate change: energy supply and demand. The research focuses on the application of downscaled seasonal models for estimating energy performance of buildings in Greece. The ARW-WRF model has been set-up and suitably parameterized to produce downscaled climatological fields for Greece, forced by the output of the CFSv2 model. The outer domain, D01/Europe, included 345 x 345 cells of horizontal resolution 20 x 20 km2 and the inner domain, D02/Greece, comprised 180 x 180 cells of 5 x 5 km2 horizontal resolution. The model run has been setup for a period with a forecast horizon of 6 months, storing outputs on a six hourly basis.

Keywords: Urban environment, vulnerability, climate change, energy performance, seasonal forecast models.

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Authors: Shamsuddin Ahmed, M. G. M. Khan, Biman Prasad, Avlin Prasad

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The focus of this paper is to construct daily time series exchange rate forecast models of Samoan Tala/USD and Tala/AUD during the year 2008 to 2012 with neural network The performance of the models was measured by using varies error functions such as Root Square mean error (RSME), Mean absolute error (MAE), and Mean absolute percentage error (MAPE). Our empirical findings suggest that AR (1) model is an effective tool to forecast the Tala/USD and Tala/AUD.

Keywords: Neural Network Forecasting Model, Autoregressive time series, Exchange rate, Tala/AUD, winters model.

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Authors: Snezhana G. Gocheva-Ilieva, Maya P. Stoimenova

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Ambient air pollution with fine particulate matter (PM10) is a systematic permanent problem in many countries around the world. The accumulation of a large number of measurements of both the PM10 concentrations and the accompanying atmospheric factors allow for their statistical modeling to detect dependencies and forecast future pollution. This study applies the classification and regression trees (CART) method for building and analyzing PM10 models. In the empirical study, average daily air data for the city of Pleven, Bulgaria for a period of 5 years are used. Predictors in the models are seven meteorological variables, time variables, as well as lagged PM10 variables and some lagged meteorological variables, delayed by 1 or 2 days with respect to the initial time series, respectively. The degree of influence of the predictors in the models is determined. The selected best CART models are used to forecast future PM10 concentrations for two days ahead after the last date in the modeling procedure and show very accurate results.

Keywords: Cross-validation, decision tree, lagged variables, short-term forecasting.

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Authors: Nabilah Filzah Mohd Radzuan, Andi Putra, Zalinda Othman, Azuraliza Abu Bakar, Abdul Razak Hamdan

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Precipitation forecast is important in avoid incident of natural disaster which can cause loss in involved area. This review paper involves three techniques from artificial intelligence namely logistic regression, decisions tree, and random forest which used in making precipitation forecast. These combination techniques through VAR model in finding advantages and strength for every technique in forecast process. Data contains variables from rain domain. Adaptation of artificial intelligence techniques involved on rain domain enables the process to be easier and systematic for precipitation forecast.

Keywords: Logistic regression, decisions tree, random forest, VAR model.

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Authors: Tahseen A. Jilani, Aqil S. Burney, C. Ardil

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Time series models have been used to make predictions of academic enrollments, weather, road accident, casualties and stock prices, etc. Based on the concepts of quartile regression models, we have developed a simple time variant quantile based fuzzy time series forecasting method. The proposed method bases the forecast using prediction of future trend of the data. In place of actual quantiles of the data at each point, we have converted the statistical concept into fuzzy concept by using fuzzy quantiles using fuzzy membership function ensemble. We have given a fuzzy metric to use the trend forecast and calculate the future value. The proposed model is applied for TAIFEX forecasting. It is shown that proposed method work best as compared to other models when compared with respect to model complexity and forecasting accuracy.

Keywords: Quantile Regression, Fuzzy time series, fuzzy logicalrelationship groups, heuristic trend prediction.

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Authors: Fatemeh Mosalman Yazdi

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One of the most important requirements for the operation and planning activities of an electrical utility is the prediction of load for the next hour to several days out, known as short term load forecasting. This paper presents the development of an artificial neural network based short-term load forecasting model. The model can forecast daily load profiles with a load time of one day for next 24 hours. In this method can divide days of year with using average temperature. Groups make according linearity rate of curve. Ultimate forecast for each group obtain with considering weekday and weekend. This paper investigates effects of temperature and humidity on consuming curve. For forecasting load curve of holidays at first forecast pick and valley and then the neural network forecast is re-shaped with the new data. The ANN-based load models are trained using hourly historical. Load data and daily historical max/min temperature and humidity data. The results of testing the system on data from Yazd utility are reported.

Keywords: Artificial neural network, Holiday forecasting, pickand valley load forecasting, Short-term load-forecasting.

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Authors: Bin Mu, Site Li, Shijin Yuan

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Under the circumstance of environment deterioration, people are increasingly concerned about the quality of the environment, especially air quality. As a result, it is of great value to give accurate and timely forecast of AQI (air quality index). In order to simplify influencing factors of air quality in a city, and forecast the city’s AQI tomorrow, this study used MATLAB software and adopted the method of constructing a mathematic model of PCA-GABP to provide a solution. To be specific, this study firstly made principal component analysis (PCA) of influencing factors of AQI tomorrow including aspects of weather, industry waste gas and IAQI data today. Then, we used the back propagation neural network model (BP), which is optimized by genetic algorithm (GA), to give forecast of AQI tomorrow. In order to verify validity and accuracy of PCA-GABP model’s forecast capability. The study uses two statistical indices to evaluate AQI forecast results (normalized mean square error and fractional bias). Eventually, this study reduces mean square error by optimizing individual gene structure in genetic algorithm and adjusting the parameters of back propagation model. To conclude, the performance of the model to forecast AQI is comparatively convincing and the model is expected to take positive effect in AQI forecast in the future.

Keywords: AQI forecast, principal component analysis, genetic algorithm, back propagation neural network model.

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Authors: Irwin Ma, Tony Wong, Thiagas Sankar

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By systematically applying different engineering methods, difficult financial problems become approachable. Using a combination of theory and techniques such as wavelet transform, time series data mining, Markov chain based discrete stochastic optimization, and evolutionary algorithms, this work formulated a strategy to characterize and forecast non-linear time series. It attempted to extract typical features from the volatility data sets of S&P100 and S&P500 indices that include abrupt drops, jumps and other non-linearity. As a result, accuracy of forecasting has reached an average of over 75% surpassing any other publicly available results on the forecast of any financial index.

Keywords: Discrete stochastic optimization, genetic algorithms, genetic programming, volatility forecast

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Authors: Hamid R. S. Mojaveri, Seyed S. Mousavi, Mojtaba Heydar, Ahmad Aminian

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The aim of this paper is to present a methodology in three steps to forecast supply chain demand. In first step, various data mining techniques are applied in order to prepare data for entering into forecasting models. In second step, the modeling step, an artificial neural network and support vector machine is presented after defining Mean Absolute Percentage Error index for measuring error. The structure of artificial neural network is selected based on previous researchers' results and in this article the accuracy of network is increased by using sensitivity analysis. The best forecast for classical forecasting methods (Moving Average, Exponential Smoothing, and Exponential Smoothing with Trend) is resulted based on prepared data and this forecast is compared with result of support vector machine and proposed artificial neural network. The results show that artificial neural network can forecast more precisely in comparison with other methods. Finally, forecasting methods' stability is analyzed by using raw data and even the effectiveness of clustering analysis is measured.

Keywords: Artificial Neural Networks (ANN), bullwhip effect, demand forecasting, Support Vector Machine (SVM).

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Authors: Indra Budi, Rizal Fathoni Aji, Agus Widodo

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Prediction of future research topics by using time series analysis either statistical or machine learning has been conducted previously by several researchers. Several methods have been proposed to combine the forecasting results into single forecast. These methods use fixed combination of individual forecast to get the final forecast result. In this paper, quite different approach is employed to select the forecasting methods, in which every point to forecast is calculated by using the best methods used by similar validation dataset. The dataset used in the experiment is time series derived from research report in Garuda, which is an online sites belongs to the Ministry of Education in Indonesia, over the past 20 years. The experimental result demonstrates that the proposed method may perform better compared to the fix combination of predictors. In addition, based on the prediction result, we can forecast emerging research topics for the next few years.

Keywords: Combination, emerging topics, ensemble, forecasting, machine learning, prediction, research topics, similarity measure, time series.

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Authors: Ramin Vafadary, Maryam Khanbaghi

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Forecasting electricity load is important for various purposes like planning, operation and control. Forecasts can save operating and maintenance costs, increase the reliability of power supply and delivery systems, and correct decisions for future development. This paper compares various time series methods to forecast 24 hours ahead of electricity load. The methods considered are the Holt-Winters smoothing, SARIMA Modeling, LSTM Network, Fbprophet and Tensorflow probability. The performance of each method is evaluated by using the forecasting accuracy criteria namely, the Mean Absolute Error and Root Mean Square Error. The National Renewable Energy Laboratory (NREL) residential energy consumption data are used to train the models. The results of this study show that SARIMA model is superior to the others for 24 hours ahead forecasts. Furthermore, a Bagging technique is used to make the predictions more robust. The obtained results show that by Bagging multiple time-series forecasts we can improve the robustness of the models for 24 hour ahead electricity load forecasting.

Keywords: Bagging, Fbprophet, Holt-Winters, LSTM, Load Forecast, SARIMA, tensorflow probability, time series.

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Authors: Ubaidillah Zuhdi, Shunsuke Mori, Kazuhisa Kamegai

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The purpose of this study is to forecast the influences of information and communication technology (ICT) on the structural changes of Japanese economies. In this study, input-output (IO) and statistical approaches are used as analysis instruments. More specifically, this study employs Leontief IO coefficients and constrained multivariate regression (CMR) model in order to achieve the purpose. The periods of initial and forecast in this study are 2005 and 2015, respectively. In this study, ICT is represented by ICT capital stocks. This study conducts two levels of analysis, namely macro and micro. The results of macro level analysis show that the dynamics of Japanese economies on the forecast period, relative to the initial period, are not so high. We focus on (1) commerce, (2) business services and office supplies, and (3) personal services sectors when conducting the analysis of the micro level. Further, we analyze its specific IO coefficients when doing this analysis. The results of the analysis explain that ICT gives a strong influence on the changes of these coefficients from initial to forecast periods.

Keywords: Forecast, ICT, Structural changes, Japanese economies.

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Authors: Jagadish H. Pujar

Abstract:

Fuzzy Load forecasting plays a paramount role in the operation and management of power systems. Accurate estimation of future power demands for various lead times facilitates the task of generating power reliably and economically. The forecasting of future loads for a relatively large lead time (months to few years) is studied here (long term load forecasting). Among the various techniques used in forecasting load, artificial intelligence techniques provide greater accuracy to the forecasts as compared to conventional techniques. Fuzzy Logic, a very robust artificial intelligent technique, is described in this paper to forecast load on long term basis. The paper gives a general algorithm to forecast long term load. The algorithm is an Extension of Short term load forecasting method to Long term load forecasting and concentrates not only on the forecast values of load but also on the errors incorporated into the forecast. Hence, by correcting the errors in the forecast, forecasts with very high accuracy have been achieved. The algorithm, in the paper, is demonstrated with the help of data collected for residential sector (LT2 (a) type load: Domestic consumers). Load, is determined for three consecutive years (from April-06 to March-09) in order to demonstrate the efficiency of the algorithm and to forecast for the next two years (from April-09 to March-11).

Keywords: Fuzzy Logic Control (FLC), Data DependantFactors(DDF), Model Dependent Factors(MDF), StatisticalError(SE), Short Term Load Forecasting (STLF), MiscellaneousError(ME).

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Authors: M. Ionascu, L. Olimid

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In the last few years, several steps were taken in order to improve the quality of corporate governance for Romanian listed companies. Higher standards of corporate governance is documented in the literature to lead to a better information environment, and, consequently, to increase analysts forecast accuracy. Accordingly, the purpose of this paper is to investigate the extent to which corporate governance policies affect analysts forecasts for companies listed on Bucharest Stock Exchange. The results showed that there is indeed a negative correlation between a corporate governance index – used as a proxy for the quality of corporate governance practices - and analysts forecast errors.

Keywords: corporate governance, aanalysts' forecasts, information environment

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Authors: Ε. Giovanis

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In this paper we present, propose and examine additional membership functions for the Smoothing Transition Autoregressive (STAR) models. More specifically, we present the tangent hyperbolic, Gaussian and Generalized bell functions. Because Smoothing Transition Autoregressive (STAR) models follow fuzzy logic approach, more fuzzy membership functions should be tested. Furthermore, fuzzy rules can be incorporated or other training or computational methods can be applied as the error backpropagation or genetic algorithm instead to nonlinear squares. We examine two macroeconomic variables of US economy, the inflation rate and the 6-monthly treasury bills interest rates.

Keywords: Forecast , Fuzzy membership functions, Smoothingtransition, Time-series

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Authors: Paul Lajbcygier, Seng Lee

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Co-integration models the long-term, equilibrium relationship of two or more related financial variables. Even if cointegration is found, in the short run, there may be deviations from the long run equilibrium relationship. The aim of this work is to forecast these deviations using neural networks and create a trading strategy based on them. A case study is used: co-integration residuals from Australian Bank Bill futures are forecast and traded using various exogenous input variables combined with neural networks. The choice of the optimal exogenous input variables chosen for each neural network, undertaken in previous work [1], is validated by comparing the forecasts and corresponding profitability of each, using a trading strategy.

Keywords: Artificial neural networks, co-integration, forecasting, trading rule.

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Authors: Shih-Ching Lo

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Discrete choice model is the most used methodology for studying traveler-s mode choice and demand. However, to calibrate the discrete choice model needs to have plenty of questionnaire survey. In this study, an aggregative model is proposed. The historical data of passenger volumes for high speed rail and domestic civil aviation are employed to calibrate and validate the model. In this study, different models are compared so as to propose the best one. From the results, systematic equations forecast better than single equation do. Models with the external variable, which is oil price, are better than models based on closed system assumption.

Keywords: forecasting, passenger volume, dynamic competition model, external variable, oil price

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Authors: Karim Hamidi Machekposhti, Hossein Sedghi

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This study was designed to find the best stochastic model (using of time series analysis) for annual extreme streamflow (peak and maximum streamflow) of Karkheh River at Iran. The Auto-regressive Integrated Moving Average (ARIMA) model used to simulate these series and forecast those in future. For the analysis, annual extreme streamflow data of Jelogir Majin station (above of Karkheh dam reservoir) for the years 1958–2005 were used. A visual inspection of the time plot gives a little increasing trend; therefore, series is not stationary. The stationarity observed in Auto-Correlation Function (ACF) and Partial Auto-Correlation Function (PACF) plots of annual extreme streamflow was removed using first order differencing (d=1) in order to the development of the ARIMA model. Interestingly, the ARIMA(4,1,1) model developed was found to be most suitable for simulating annual extreme streamflow for Karkheh River. The model was found to be appropriate to forecast ten years of annual extreme streamflow and assist decision makers to establish priorities for water demand. The Statistical Analysis System (SAS) and Statistical Package for the Social Sciences (SPSS) codes were used to determinate of the best model for this series.

Keywords: Stochastic models, ARIMA, extreme streamflow, Karkheh River.

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Authors: T. K. Sriram, Joydip Dhar

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Models based on stage structure have found varied applications in population models. This paper proposes a stage structured model to study the trends in the computer and video game playing population of US. The game paying population is divided into three compartments based on their age group. After simulating the mathematical model, a forecast of the number of game players in each stage as well as an approximation of the average age of game players in future has been made.

Keywords: Age structure, Forecasting, Mathematical modeling, Stage structure.

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Authors: Petr Seďa

Abstract:

This paper deals with heterogeneous autoregressive models of realized volatility (HAR-RV models) on high-frequency data of stock indices in the USA. Its aim is to capture the behavior of three groups of market participants trading on a daily, weekly and monthly basis and assess their role in predicting the daily realized volatility. The benefits of this work lies mainly in the application of heterogeneous autoregressive models of realized volatility on stock indices in the USA with a special aim to analyze an impact of the global financial crisis on applied models forecasting performance. We use three data sets, the first one from the period before the global financial crisis occurred in the years 2006-2007, the second one from the period when the global financial crisis fully hit the U.S. financial market in 2008-2009 years, and the last period was defined over 2010-2011 years. The model output indicates that estimated realized volatility in the market is very much determined by daily traders and in some cases excludes the impact of those market participants who trade on monthly basis.

Keywords: Global financial crisis, heterogeneous autoregressive model, in-sample forecast, realized volatility, U.S. stock market.

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Authors: Fazıl Gökgöz, Fahrettin Filiz

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Electricity prices have sophisticated features such as high volatility, nonlinearity and high frequency that make forecasting quite difficult. Electricity price has a volatile and non-random character so that, it is possible to identify the patterns based on the historical data. Intelligent decision-making requires accurate price forecasting for market traders, retailers, and generation companies. So far, many shallow-ANN (artificial neural networks) models have been published in the literature and showed adequate forecasting results. During the last years, neural networks with many hidden layers, which are referred to as DNN (deep neural networks) have been using in the machine learning community. The goal of this study is to investigate electricity price forecasting performance of the shallow-ANN and DNN models for the Turkish day-ahead electricity market. The forecasting accuracy of the models has been evaluated with publicly available data from the Turkish day-ahead electricity market. Both shallow-ANN and DNN approach would give successful result in forecasting problems. Historical load, price and weather temperature data are used as the input variables for the models. The data set includes power consumption measurements gathered between January 2016 and December 2017 with one-hour resolution. In this regard, forecasting studies have been carried out comparatively with shallow-ANN and DNN models for Turkish electricity markets in the related time period. The main contribution of this study is the investigation of different shallow-ANN and DNN models in the field of electricity price forecast. All models are compared regarding their MAE (Mean Absolute Error) and MSE (Mean Square) results. DNN models give better forecasting performance compare to shallow-ANN. Best five MAE results for DNN models are 0.346, 0.372, 0.392, 0,402 and 0.409.

Keywords: Deep learning, artificial neural networks, energy price forecasting, Turkey.

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Authors: S. Esfandeh, M. Sedighizadeh

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Weather systems use enormously complex combinations of numerical tools for study and forecasting. Unfortunately, due to phenomena in the world climate, such as the greenhouse effect, classical models may become insufficient mostly because they lack adaptation. Therefore, the weather forecast problem is matched for heuristic approaches, such as Evolutionary Algorithms. Experimentation with heuristic methods like Particle Swarm Optimization (PSO) algorithm can lead to the development of new insights or promising models that can be fine tuned with more focused techniques. This paper describes a PSO approach for analysis and prediction of data and provides experimental results of the aforementioned method on realworld meteorological time series.

Keywords: Weather, Climate, PSO, Prediction, Meteorological

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Authors: Nkongho Ayuketang Arreyndip, Ebobenow Joseph

Abstract:

In this paper, temperature extremes are forecast by employing the block maxima method of the Generalized extreme value(GEV) distribution to analyse temperature data from the Cameroon Development Corporation (C.D.C). By considering two sets of data (Raw data and simulated data) and two (stationary and non-stationary) models of the GEV distribution, return levels analysis is carried out and it was found that in the stationary model, the return values are constant over time with the raw data while in the simulated data, the return values show an increasing trend but with an upper bound. In the non-stationary model, the return levels of both the raw data and simulated data show an increasing trend but with an upper bound. This clearly shows that temperatures in the tropics even-though show a sign of increasing in the future, there is a maximum temperature at which there is no exceedence. The results of this paper are very vital in Agricultural and Environmental research.

Keywords: Return level, Generalized extreme value (GEV), Meteorology, Forecasting.

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Authors: Tomohiro Hachino, Hitoshi Takata, Seiji Fukushima, Yasutaka Igarashi

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This paper presents a Gaussian process model-based short-term electric load forecasting. The Gaussian process model is a nonparametric model and the output of the model has Gaussian distribution with mean and variance. The multiple Gaussian process models as every hour ahead predictors are used to forecast future electric load demands up to 24 hours ahead in accordance with the direct forecasting approach. The separable least-squares approach that combines the linear least-squares method and genetic algorithm is applied to train these Gaussian process models. Simulation results are shown to demonstrate the effectiveness of the proposed electric load forecasting.

Keywords: Direct method, electric load forecasting, Gaussian process model, genetic algorithm, separable least-squares method.

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Authors: Panudet Saengseedam, Nanthachai Kantanantha

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This paper proposes a linear mixed model (LMM) with spatial effects to forecast rice and cassava yields in Thailand at the same time. A multivariate conditional autoregressive (MCAR) model is assumed to present the spatial effects. A Bayesian method is used for parameter estimation via Gibbs sampling Markov Chain Monte Carlo (MCMC). The model is applied to the rice and cassava yields monthly data which have been extracted from the Office of Agricultural Economics, Ministry of Agriculture and Cooperatives of Thailand. The results show that the proposed model has better performance in most provinces in both fitting part and validation part compared to the simple exponential smoothing and conditional auto regressive models (CAR) from our previous study.

Keywords: Bayesian method, Linear mixed model, Multivariate conditional autoregressive model, Spatial time series.

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