Search results for: Time Series Forecasting.

Authors: Neha Asthana, Vishal Krishna Prasad

Abstract:

Collaborative Planning and Forecasting is an innovative and systematic approach towards productive integration and assimilation of data synergized into information. The changing and variable market dynamics have persuaded global business chains to incorporate Collaborative Planning and Forecasting as an imperative tool. Thus, it is essential for the supply chains to constantly improvise, update its nature, and mould as per changing global environment.

Keywords: Information transfer, Forecasting, Optimization.

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Authors: Defu Zhang, Qingshan Jiang, Xin Li

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This paper deals with the application of a well-known neural network technique, multilayer back-propagation (BP) neural network, in financial data mining. A modified neural network forecasting model is presented, and an intelligent mining system is developed. The system can forecast the buying and selling signs according to the prediction of future trends to stock market, and provide decision-making for stock investors. The simulation result of seven years to Shanghai Composite Index shows that the return achieved by this mining system is about three times as large as that achieved by the buy and hold strategy, so it is advantageous to apply neural networks to forecast financial time series, the different investors could benefit from it.

Keywords: Data mining, neural network, stock forecasting.

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Authors: Mohammad H. Fattahi

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Chaotic analysis has been performed on the river flow time series before and after applying the wavelet based de-noising techniques in order to investigate the noise content effects on chaotic nature of flow series. In this study, 38 years of monthly runoff data of three gauging stations were used. Gauging stations were located in Ghar-e-Aghaj river basin, Fars province, Iran. Noise level of time series was estimated with the aid of Gaussian kernel algorithm. This step was found to be crucial in preventing removal of the vital data such as memory, correlation and trend from the time series in addition to the noise during de-noising process.

Keywords: Chaotic behavior, wavelet, noise reduction, river flow.

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Authors: Fazıl Gökgöz, Fahrettin Filiz

Abstract:

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: 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: A. Kablan

Abstract:

The prediction of financial time series is a very complicated process. If the efficient market hypothesis holds, then the predictability of most financial time series would be a rather controversial issue, due to the fact that the current price contains already all available information in the market. This paper extends the Adaptive Neuro Fuzzy Inference System for High Frequency Trading which is an expert system that is capable of using fuzzy reasoning combined with the pattern recognition capability of neural networks to be used in financial forecasting and trading in high frequency. However, in order to eliminate unnecessary input in the training phase a new event based volatility model was proposed. Taking volatility and the scaling laws of financial time series into consideration has brought about the development of the Intraday Seasonality Observation Model. This new model allows the observation of specific events and seasonalities in data and subsequently removes any unnecessary data. This new event based volatility model provides the ANFIS system with more accurate input and has increased the overall performance of the system.

Keywords: Adaptive Neuro-fuzzy Inference system, High Frequency Trading, Intraday Seasonality Observation Model.

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Authors: Zhiyong Li, Zhigang Chen, Chao Fu, Shipeng Zhang

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Load forecasting has always been the essential part of an efficient power system operation and planning. A novel approach based on support vector machines is proposed in this paper for annual power load forecasting. Different kernel functions are selected to construct a combinatorial algorithm. The performance of the new model is evaluated with a real-world dataset, and compared with two neural networks and some traditional forecasting techniques. The results show that the proposed method exhibits superior performance.

Keywords: combinatorial algorithm, data mining, load forecasting, support vector machines

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Authors: B.P. Bezruchko, D.A. Smirnov

Abstract:

Quantitative characterization of nonlinear directional couplings between stochastic oscillators from data is considered. We suggest coupling characteristics readily interpreted from a physical viewpoint and their estimators. An expression for a statistical significance level is derived analytically that allows reliable coupling detection from a relatively short time series. Performance of the technique is demonstrated in numerical experiments.

Keywords: Nonlinear time series analysis, directional couplings, coupled oscillators.

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Authors: Xiaolei Hu, Enrico Ferrera, Riccardo Tomasi, Claudio Pastrone

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Load Forecasting plays a key role in making today's and future's Smart Energy Grids sustainable and reliable. Accurate power consumption prediction allows utilities to organize in advance their resources or to execute Demand Response strategies more effectively, which enables several features such as higher sustainability, better quality of service, and affordable electricity tariffs. It is easy yet effective to apply Load Forecasting at larger geographic scale, i.e. Smart Micro Grids, wherein the lower available grid flexibility makes accurate prediction more critical in Demand Response applications. This paper analyses the application of short-term load forecasting in a concrete scenario, proposed within the EU-funded GreenCom project, which collect load data from single loads and households belonging to a Smart Micro Grid. Three short-term load forecasting techniques, i.e. linear regression, artificial neural networks, and radial basis function network, are considered, compared, and evaluated through absolute forecast errors and training time. The influence of weather conditions in Load Forecasting is also evaluated. A new definition of Gain is introduced in this paper, which innovatively serves as an indicator of short-term prediction capabilities of time spam consistency. Two models, 24- and 1-hour-ahead forecasting, are built to comprehensively compare these three techniques.

Keywords: Short-term load forecasting, smart micro grid, linear regression, artificial neural networks, radial basis function network, Gain.

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Authors: Moustafa S. El-Sammany

Abstract:

Flash floods are considered natural disasters that can cause casualties and demolishing of infra structures. The problem is that flash floods, particularly in arid and semi arid zones, take place in very short time. So, it is important to forecast flash floods earlier to its events with a lead time up to 48 hours to give early warning alert to avoid or minimize disasters. The flash flood took place over Wadi Watier - Sinai Peninsula, in October 24th, 2008, has been simulated, investigated and analyzed using the state of the art regional weather model. The Weather Research and Forecast (WRF) model, which is a reliable short term forecasting tool for precipitation events, has been utilized over the study area. The model results have been calibrated with the real data, for the same date and time, of the rainfall measurements recorded at Sorah gauging station. The WRF model forecasted total rainfall of 11.6 mm while the real measured one was 10.8 mm. The calibration shows significant consistency between WRF model and real measurements results.

Keywords: Early warning system, Flash floods forecasting, WadiWatier, WRF 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: Daekook Kang, Wooseok Jang, Hyeonjeong Lee, Hyun Joung No

Abstract:

Technology changes have been acknowledged as a critical factor in determining competitiveness of organization. Under such environment, the right anticipation of technology change has been of huge importance in strategic planning. To monitor technology change, technology forecasting (TF) is frequently utilized. In academic perspective, TF has received great attention for a long time. However, few researches have been conducted to provide overview of the TF literature. Even though some studies deals with review of TF research, they generally focused on type and characteristics of various TF, so hardly provides information about patterns of TF research and which TF method is used in certain technology industry. Accordingly, this study profile developments in and patterns of scholarly research in TF over time. Also, this study investigates which technology industries have used certain TF method and identifies their relationships. This study will help in understanding TF research trend and their application area.

Keywords: Technology forecasting, technology industry, TF trend, technology trajectory.

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Authors: Shigeaki Sakurai, Ken Ueno, Ryohei Orihara

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This paper proposes a method that discovers time series event patterns from textual data with time information. The patterns are composed of sequences of events and each event is extracted from the textual data, where an event is characteristic content included in the textual data such as a company name, an action, and an impression of a customer. The method introduces 7 types of time constraints based on the analysis of the textual data. The method also evaluates these constraints when the frequency of a time series event pattern is calculated. We can flexibly define the time constraints for interesting combinations of events and can discover valid time series event patterns which satisfy these conditions. The paper applies the method to daily business reports collected by a sales force automation system and verifies its effectiveness through numerical experiments.

Keywords: Text mining, sequential mining, time constraints, daily business reports.

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Authors: Yang Zhang, Yuncai Liu

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Accurately predicting non-peak traffic is crucial to daily traffic for all forecasting models. In the paper, least squares support vector machines (LS-SVMs) are investigated to solve such a practical problem. It is the first time to apply the approach and analyze the forecast performance in the domain. For comparison purpose, two parametric and two non-parametric techniques are selected because of their effectiveness proved in past research. Having good generalization ability and guaranteeing global minima, LS-SVMs perform better than the others. Providing sufficient improvement in stability and robustness reveals that the approach is practically promising.

Keywords: Parametric and Nonparametric Techniques, Non-peak Traffic Forecasting

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Authors: Jiangyong Liu, Xiangxiang Xu, Bote Luo, Xiaoxue Luo, Jiang Zhu, Lingzhi Yi

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To address the problems of non-linearity and high randomness of the original power load sequence causing the degradation of power load forecasting accuracy, a short-term load forecasting method is proposed. The method is based on the least square support vector machine (LSSVM) optimized by an improved sparrow search algorithm combined with the variational mode decomposition proposed in this paper. The application of the variational mode decomposition technique decomposes the raw power load data into a series of intrinsic mode functions components, which can reduce the complexity and instability of the raw data while overcoming modal confounding; the proposed improved sparrow search algorithm can solve the problem of difficult selection of learning parameters in the LSSVM. Finally, through comparison experiments, the results show that the method can effectively improve prediction accuracy.

Keywords: Load forecasting, variational mode decomposition, improved sparrow search algorithm, least square support vector machine.

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Authors: S. V. Yendiyarov

Abstract:

Pattern discovery from time series is of fundamental importance. Particularly, when information about the structure of a pattern is not complete, an algorithm to discover specific patterns or shapes automatically from the time series data is necessary. The dynamic time warping is a technique that allows local flexibility in aligning time series. Because of this, it is widely used in many fields such as science, medicine, industry, finance and others. However, a major problem of the dynamic time warping is that it is not able to work with structural changes of a pattern. This problem arises when the structure is influenced by noise, which is a common thing in practice for almost every application. This paper addresses this problem by means of developing a novel technique called adaptive dynamic time warping.

Keywords: Pattern recognition, optimal control, quadratic programming, dynamic programming, dynamic time warping, sintering control.

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Authors: Ajay Shekhar Pandey, D. Singh, S. K. Sinha

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A clustering based technique has been developed and implemented for Short Term Load Forecasting, in this article. Formulation has been done using Mean Absolute Percentage Error (MAPE) as an objective function. Data Matrix and cluster size are optimization variables. Model designed, uses two temperature variables. This is compared with six input Radial Basis Function Neural Network (RBFNN) and Fuzzy Inference Neural Network (FINN) for the data of the same system, for same time period. The fuzzy inference system has the network structure and the training procedure of a neural network which initially creates a rule base from existing historical load data. It is observed that the proposed clustering based model is giving better forecasting accuracy as compared to the other two methods. Test results also indicate that the RBFNN can forecast future loads with accuracy comparable to that of proposed method, where as the training time required in the case of FINN is much less.

Keywords: Load forecasting, clustering, fuzzy inference.

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Authors: João Paulo Teixeira, Paula Odete Fernandes

Abstract:

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: Shumin Hou, Yourong Li, Sanxing Zhao

Abstract:

Much time series data is generally from continuous dynamic system. Firstly, this paper studies the detection of the nonlinearity of time series from continuous dynamics systems by applying the Phase-randomized surrogate algorithm. Then, the Delay Vector Variance (DVV) method is introduced into nonlinearity test. The results show that under the different sampling conditions, the opposite detection of nonlinearity is obtained via using traditional test statistics methods, which include the third-order autocovariance and the asymmetry due to time reversal. Whereas the DVV method can perform well on determining nonlinear of Lorenz signal. It indicates that the proposed method can describe the continuous dynamics signal effectively.

Keywords: Nonlinearity, Time series, continuous dynamics system, DVV method

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Authors: H. Anıl, G. Kar

Abstract:

In addition to being rich in renewable energy resources, Turkey is one of the countries that promise potential in geothermal energy production with its high installed power, cheapness, and sustainability. Increasing imbalance penalties become an economic burden for organizations, since the geothermal generation plants cannot maintain the balance of supply and demand due to the inadequacy of the production forecasts given in the day-ahead market. A better production forecast reduces the imbalance penalties of market participants and provides a better imbalance in the day ahead market. In this study, using machine learning, deep learning and time series methods, the total generation of the power plants belonging to Zorlu Doğal Electricity Generation, which has a high installed capacity in terms of geothermal, was predicted for the first one-week and first two-weeks of March, then the imbalance penalties were calculated with these estimates and compared with the real values. These modeling operations were carried out on two datasets, the basic dataset and the dataset created by extracting new features from this dataset with the feature engineering method. According to the results, Support Vector Regression from traditional machine learning models outperformed other models and exhibited the best performance. In addition, the estimation results in the feature engineering dataset showed lower error rates than the basic dataset. It has been concluded that the estimated imbalance penalty calculated for the selected organization is lower than the actual imbalance penalty, optimum and profitable accounts.

Keywords: Machine learning, deep learning, time series models, feature engineering, geothermal energy production forecasting.

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Authors: S. M. Anowarul Haque, Md. Asiful Islam

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Electrical load forecasting is considered to be one of the most indispensable parts of a modern-day electrical power system. To ensure a reliable and efficient supply of electric energy, special emphasis should have been put on the predictive feature of electricity supply. Artificial Neural Network-based approaches have emerged to be a significant area of interest for electric load forecasting research. This paper proposed an Artificial Neural Network model based on the particle swarm optimization algorithm for improved electric load forecasting for Mymensingh, Bangladesh. The forecasting model is developed and simulated on the MATLAB environment with a large number of training datasets. The model is trained based on eight input parameters including historical load and weather data. The predicted load data are then compared with an available dataset for validation. The proposed neural network model is proved to be more reliable in terms of day-wise load forecasting for Mymensingh, Bangladesh.

Keywords: Load forecasting, artificial neural network, particle swarm optimization.

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Authors: Karim Hamidi Machekposhti, Hossein Sedghi, Abdolrasoul Telvari, Hossein Babazadeh

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Floods have huge environmental and economic impact. Therefore, flood prediction is given a lot of attention due to its importance. This study analysed the annual maximum streamflow (discharge) (AMS or AMD) of Karkheh River in Karkheh River Basin for flood predicting using ARIMA model. For this purpose, we use the Box-Jenkins approach, which contains four-stage method model identification, parameter estimation, diagnostic checking and forecasting (predicting). The main tool used in ARIMA modelling was the SAS and SPSS software. Model identification was done by visual inspection on the ACF and PACF. SAS software computed the model parameters using the ML, CLS and ULS methods. The diagnostic checking tests, AIC criterion, RACF graph and RPACF graphs, were used for selected model verification. In this study, the best ARIMA models for Annual Maximum Discharge (AMD) time series was (4,1,1) with their AIC value of 88.87. The RACF and RPACF showed residuals’ independence. To forecast AMD for 10 future years, this model showed the ability of the model to predict floods of the river under study in the Karkheh River Basin. Model accuracy was checked by comparing the predicted and observation series by using coefficient of determination (R²).

Keywords: Time series modelling, stochastic processes, ARIMA model, Karkheh River.

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Authors: M. Cizmesija, N. Erjavec, V. Bahovec

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While the European Union (EU) harmonized methodology is a benchmark of worldwide used business survey (BS) methodology, the choice of variables that are components of the confidence indicators, as the leading indicators, is not strictly determined and unique. Therefore, the aim of this paper is to investigate and to quantify the relationship between all business survey variables in manufacturing industry and industrial production as a reference macroeconomic series in Croatia. The assumption is that there are variables in the business survey, that are not components of Industrial Confidence Indicator (ICI) and which can accurately (and sometimes better then ICI) predict changes in Croatian industrial production. Empirical analyses are conducted using quarterly data of BS variables in manufacturing industry and Croatian industrial production over the period from the first quarter 2005 to the first quarter 2013. Research results confirmed the assumption: three BS variables which is not components of ICI (competitive position, demand and liquidity) are the best leading indicator then ICI, in forecasting changes in Croatian industrial production instantaneously, with one, two or three quarter ahead.

Keywords: Balance, Business Survey, Confidence Indicators, Industrial Production, Forecasting.

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Authors: Ren-Jieh Kuo, Chun-Shou Huang

Abstract:

The research of juice flavor forecasting has become more important in China. Due to the fast economic growth in China, many different kinds of juices have been introduced to the market. If a beverage company can understand their customers’ preference well, the juice can be served more attractive. Thus, this study intends to introducing the basic theory and computing process of grapes juice flavor forecasting based on support vector regression (SVR). Applying SVR, BPN, and LR to forecast the flavor of grapes juice in real data shows that SVR is more suitable and effective at predicting performance.

Keywords: Flavor forecasting, artificial neural networks, support vector regression, grape juice flavor.

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Authors: Rampal Singh, S. Balasundaram

Abstract:

In this paper, we study the application of Extreme Learning Machine (ELM) algorithm for single layered feedforward neural networks to non-linear chaotic time series problems. In this algorithm the input weights and the hidden layer bias are randomly chosen. The ELM formulation leads to solving a system of linear equations in terms of the unknown weights connecting the hidden layer to the output layer. The solution of this general system of linear equations will be obtained using Moore-Penrose generalized pseudo inverse. For the study of the application of the method we consider the time series generated by the Mackey Glass delay differential equation with different time delays, Santa Fe A and UCR heart beat rate ECG time series. For the choice of sigmoid, sin and hardlim activation functions the optimal values for the memory order and the number of hidden neurons which give the best prediction performance in terms of root mean square error are determined. It is observed that the results obtained are in close agreement with the exact solution of the problems considered which clearly shows that ELM is a very promising alternative method for time series prediction.

Keywords: Chaotic time series, Extreme learning machine, Generalization performance.

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Authors: Guoyuan Qi , Yskandar Hamam, Barend Jacobus van Wyk, Shengzhi Du

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The majority of existing predictors for time series are model-dependent and therefore require some prior knowledge for the identification of complex systems, usually involving system identification, extensive training, or online adaptation in the case of time-varying systems. Additionally, since a time series is usually generated by complex processes such as the stock market or other chaotic systems, identification, modeling or the online updating of parameters can be problematic. In this paper a model-free predictor (MFP) for a time series produced by an unknown nonlinear system or process is derived using tracking theory. An identical derivation of the MFP using the property of the Newton form of the interpolating polynomial is also presented. The MFP is able to accurately predict future values of a time series, is stable, has few tuning parameters and is desirable for engineering applications due to its simplicity, fast prediction speed and extremely low computational load. The performance of the proposed MFP is demonstrated using the prediction of the Dow Jones Industrial Average stock index.

Keywords: Forecast, model-free predictor, prediction, time series

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Authors: Mohsen Hayati, Zahra Mohebi

Abstract:

In this paper, the application of neural networks to study the design of short-term temperature forecasting (STTF) Systems for Kermanshah city, west of Iran was explored. One important architecture of neural networks named Multi-Layer Perceptron (MLP) to model STTF systems is used. Our study based on MLP was trained and tested using ten years (1996-2006) meteorological data. The results show that MLP network has the minimum forecasting error and can be considered as a good method to model the STTF systems.

Keywords: Artificial neural networks, Forecasting, Weather, Multi-layer perceptron.

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Authors: Tushnik Sarkar, Mofazzal H. Khondekar, Subrata Banerjee

Abstract:

This paper investigates the nature of the fluctuation of the daily average Solar wind speed time series collected over a period of 2492 days, from 1^st January, 1997 to 28^th October, 2003. The degree of self-similarity and scalability of the Solar Wind Speed signal has been explored to characterise the signal fluctuation. Multi-fractal Detrended Fluctuation Analysis (MFDFA) method has been implemented on the signal which is under investigation to perform this task. Furthermore, the singularity spectra of the signals have been also obtained to gauge the extent of the multifractality of the time series signal.

Keywords: Detrended fluctuation analysis, generalized Hurst exponent, holder exponents, multifractal exponent, multifractal spectrum, singularity spectrum, time series analysis.

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Authors: Steffen C. Eickemeyer, Tim Borcherding, Peter Nyhuis, Hannover

Abstract:

Planning capacities when regenerating complex investment goods involves particular challenges in that the planning is subject to a large degree of uncertainty regarding load information. Using information fusion – by applying Bayesian Networks – a method is being developed for forecasting the anticipated expenditures (human labor, tool and machinery utilization, time etc.) for regenerating a good. The generated forecasts then later serve as a tool for planning capacities and ensure a greater stability in the planning processes.

Keywords: Bayesian networks, capacity planning, complex investment goods, damages library, forecasting, information fusion, regeneration.

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Authors: Khaled M. EL-Naggar, Khaled A. AL-Rumaih

Abstract:

This paper presents performance comparison of three estimation techniques used for peak load forecasting in power systems. The three optimum estimation techniques are, genetic algorithms (GA), least error squares (LS) and, least absolute value filtering (LAVF). The problem is formulated as an estimation problem. Different forecasting models are considered. Actual recorded data is used to perform the study. The performance of the above three optimal estimation techniques is examined. Advantages of each algorithms are reported and discussed.

Keywords: Forecasting, Least error squares, Least absolute Value, Genetic algorithms

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