Türkiye Rüzgar Verilerinin Bayesyen Maksimum Entropi Yaklaşımıyla Uzay-Zaman Modeli

Abstract

ÖZET 19. yüzyılın başlarında bağlı olduğu prensibin temeli Sadi Carnot tarafından atılan, ilk olarak 1854 yılında Rudolf Julius Emanuel Clausius tarafından kullanılan entropi kavramı, Temodinamiğin II. Kanunu'nun da temelini oluşturur. Daha sonra, Boltzmann bağıntısıyla, entropi, termodinamik olasılık ile ifade edilmiştir. Entropi, kısaca, bir sistemdeki belirsizliğin ölçüsü olarak tanımlanabilir. Entropi artışı, sistemdeki düzensizliğin artışına neden olacaktır. Claude Elwood Shannon, “İletişimin Matematiksel Teorisi” adlı makalesiyle, bilgi ile entropi arasındaki ilişkiyi ortaya koymuş ve bir olasılık dağılımının belirsizliğinin ölçüsünü, entropi olarak adlandırmıştır.Hakkında kesin bilgi bulunan olayların gerçekleşme olasılığı yüksek olduğundan, sisteme dair taşıdığı bilgi azdır. Bu yönüyle de, entropi, belirsizliğin bir ölçüsüdür. Jaynes, Bilgi Teorisi’yle, olası bütün dağılımlar arasından, maksimum entropiye sahip olan dağılımın seçilmesi gerekliliğini öne sürmüştür. Bu ilkeye ise; Maksimum Entropi İlkesi denir. Bayesyen Maksimum Entropi (BME), Bayesyen koşullandırma ile entropi maksimizasyonunu aynı süreç içerisinde kullanan, uzay-zamansal haritalama ve analiz yapan bir metoddur. Ayrıca BME, uzay-zamansal haritalamada, sadece veriyi değil aynı zamanda sisteme ait bilgiyi de kullanan tek yaklaşımdır. BME ile verinin işlenmesi sırasında, fizik kanunları, hipotezler, deneyimler, bilimsel teoriler, yüksek dereceli uzay/zaman momentleri, çeşitli formdaki belirsiz bilgi, model çıktıları vb. sürece dahil edilir. Edinilen bilginin maksimizasyonu, en doğru analiz ve tahmin için taban oluşturur. Klasik metodlar, muhakemede tümevarımı kullanırken, BME tümdengelim yaklaşımını kullanır. Bu çalışmada, BME ile Türkiye rüzgar verilerinin tahmini yapılarak, uzay-zamansal bir Türkiye rüzgar haritasının çıkarılmıştır. Elde edilen tüm tahmin sonuçları, güven aralığı sınırları içerisinde kalmıştır ve tahminlerin hata varyansları oldukça düşüktür. Bu haritanın güncel Türkiye rüzgar atlası olabileceği düşünülmektedir. Ayrıca, Türkiye rüzgar verilerinin, entropi ile analizi yapılarak, verinin taşıdığı bilgiyi temsil edebilecek bir indeks olarak Normalize Marjinal Entropi (NME) önerilmiştir. Önerilen bu indeksin, veri kalitesinin bir ölçütü olarak da kullanılabileceği öngörülmektedir. Entropi analizinin sonucunda, en düşük ve en yüksek entropi değerlerine karşılık gelen temsili birkaç meteoroloji gözlem istasyonu verisinin, Destek Vektör Regresyonu (DVR) ile tahmin sonuçlarının entropi ile ilişkisi irdelenmiştir. Entropi kavramını destekleyecek şekilde, entropisi yüksek istasyonların tahmin edilebilirliğinin daha yüksek olduğu, düşük entropiye sahip istasyonların verilerinin tahmininin daha yüksek tahmin hatalarına neden olduğu görülmüştür. Bu çalışmayı özgün kılan temel unsurlar; Türkiye’de BME kullanılarak yapılan ilk ve tek çalışma olması ve dünyada da rüzgar verisine BME yaklaşımının uygulandığı tek çalışma olması, tahmin edilebilirlik ile entropi arasındaki ilişkiyi sayısal olarak ifade eden ve veri kalitesi için bir ölçüt olabileceği düşünülen yeni bir indeks (NME) önerilmesidir.

SUMMARY The term "entropy", the basis of the cherished principle of which was set by the Sadi Carnot at the beginning of the 19th century and firstly used in 1954 by Emanuel Clausius, also underlies the basis of the 2nd Law of Thermodynamics. Then, entropy was described with thermodynamics probability in Boltzmann formula. Entropy can be identified as a measure of uncertainty of a system whereas probability indicates uncertainty about the realization of a case. An increase of the entropy causes an increase of the disorder. In the article, “A Mathematical Theory of Information”, Claude Elwood Shannon presented the relationships between information and entropy and stated that the entropy called as Shannon entropy or marginal entropy is a measure of uncertainty of a probability distribution. Because the probability of the realization of the cases which there is a certain information about them is high, the cases have less information. The cases which they have less probability of the realization carry more information because they have a factor of the surprise.From this aspect, it can be said that entropy is a measure of uncertainty. Edwin Thompson Jaynes showed how a typical probability distribution is attained merging Shannon entropy that it is a measure of the uncertainty of a probability distribution with testable information in Information Theory. According to Information Theory, it is obvious that there are infinitely many probability distributions. At this point, Jaynes asserted that the distribution which it has maximum entropy should be chosen among probable distributions. This rule and this distribution are called Maximum Entropy Principle and maximum entropy distribution, respectively. The distribution which it has a maximum entropy is an unbiased forecast when considered the loss information. Statistical methods can be separated to two classes as Bayesian methods and classical methods. Bayesian methods allow the use of subsidiary information and use subjective probabilities whereas classical methods utilize relative probabilities. Moreover classical and Bayesian methods use induction and deduction in reasoning, respectively. Bayesian Maximum Entropy (BME), a nonlinear geostatistical approach, is the method of spatiotemporal analysis and mapping in a spatiotemporal domain. BME method can be employed for the forecast and simulation of geological variables. The goal of the method is that it comprises multivariable probability densities which they optimize the use of information. Generally, it can be said that BME is the method which uses Bayesian conditionalizition and entropy maximization in the same process and realizes a spatiotemporal mapping and an analysis. It fulfills epistemic targets and incorporates physical knowledge bases in a rigorous and systematic manner. Non-Gaussian laws are automatically included because it does not necessitate any assumption regarding the shape of the underlying probability law. BME can be applied to either spatial or spatiotemporal domains. It can model both nonhomogeneous and nonstationary data. BME leads to nonlinear estimators and can obtain well-known kriging estimators as its limiting cases. It provides multipoint mappings while most traditional approaches do not offer. BME has global forecast features since it incorporates physical laws into spatiotemporal mapping. Moreover, BME is the only approach which uses not only data on the process but also information of the data in a spatiotemporal mapping. During processing the data with BME, physical laws, hypotheses, experiences, scientific theories, high order space/time moments, various types of uncertain information, outputs of models etc. are incorporated to the process. In the application of BME, there are three stages which they provide gaining, interpreting and processing of information, respectively. In prior stage, the probability density function is calculated with using of given general knowledge and applying maximum entropy theory. Then, specificatory knowledge which it is added to BME formulation is collected and arranged in meta-prior stage. Finally, by using general and specificatory knowledge and Bayesian conditionalization, the prior probability density function upgrades to posterior probability density function and this probability density function is maximized in posterior or integration stage. Maximization of the gained information generates a base for the most accurate analysis and forecast. This study have more than one goals. First of all, an entropy map is generated for whole of Turkey. In consequence of entropy analysis, the relationship between the forecast results of some representative meteorological observation stations’ data which have the lowest and highest entropy values using Support Vector Regression (SVR) and entropy values is scrutinized. SVR is a subcategory of Support Vector Machines (SVMs) which they are learning machines and have good generalization capability applying a structural risk minimization on a limited number of learning patterns. The idea of SVR is based on the computation of a lineer regression function in a high dimensional feature space. SVR attemps to minimize the generalization error bound so as to achieve generalized performance. Moreover, analysing of Turkey wind speed data via entropy, Normalised Marginal Entropy (NME) is proposed as the index that it represents the information carried from data. NME can be stated as the proportion between a marginal entropy and maximum entropy. In short, the data with high entropy have high NME and higher NME means higher information about data thanks to suprise factor inside the data. Therefore the data can be forecasted more accurately if data have higher NME. From this point, NME can be considered as a forecastability index and also it can be projected that it may be used as a criterion of the data quality. In other words, if the data is measured properly and faultlessly, namely; quality of the data is fine, the forecastability of the data may be high. Ultimately, it is generated a spatiotemporal Turkey mean wind speed map forecasting of mean wind speed values with BME. Generally, BME provides smaller confidence level than other kriging methods and performance criteria of forecast results are given with error variances. In this study, mean wind speed data of Turkey between the years 2010-2015 which they are measured at 10 m. Considered regulations about windpower plants, the data of meteorological stations which they have fill rate of 90%, 95% and 100% are tried in each years but the data of the stations with fill rate of 90% are chosen as more data is necessary for more knowledge. Because the number of stations with fill rate of 90% belong to 2010, the data of the year of 2010 is preferred in the entropy analysis. When observed entropy map in 2010, areas with medium and high entropies are Çanakkale, Balıkesir, Tekirdağ, Afyonkarahisar, Bursa, İzmir, Ankara, Manisa, İstanbul, Kocaeli, Tokat, Sivas, Kayseri, Elazığ, Şanlıurfa, Mardin, Artvin and Bartın. The entropy map is in accordance with Wind Energy Potential Atlas prepared from Ministry of Energy and Natural Resources in 2006. Furthermore, it is seen that there is a medium level increase in Black Sea Region, the east of Central Anatolia Region and the neighbourhood of Van Lake in 2012. Similarly, areas with high entropy have high NME values. The most important point of the part of the study is that NME provides a scale between 0% and 100% while entropy values do not give a range for interpretation. The highest NME value is found as 44% in the study. When taken into account that NME represents amount of information in data, this amount is low. It can be thought that the reason of the lowness is mistakes in process of data entry workloads or collectors. In other part of entropy analysis of the study, ten meteorological stations’ data which they have the highest and the smallest entropy values are chosen and 3-month forecast is realized via SVR. Inasmuch as SVR requires a special input matix, Chaotic Approach is implemented. To prepare SVR input matrix, a phase space is reconstructed determining embedding parameters, embedding dimensions and time delays. An algorithm based on False Nearest Neighbour Method is implemented in order to find optimum embedding dimensions. Autocorrelation function and Mutual Information function are employed so as to determine appropriate time delays. According to optimum embedding parameters, the phase space is reconstructed. Because the SVR transforms input space which is formed from the observations into high dimensional feature space by way of a kernel function and performs a linear regression in this space, the SVR application throughout this study, Radial Basis Function (RBF) has been chosen as the kernel function.The RBF is the most commonly used kernel function because of its flexibility in applications. Besides, it has a strong learning ability and is able to reduce computational complexity of the training process and improve the generalization performance of the SVR. Then, by means of a fortran code written, the most appropriate SVR parameters are determined. For all stations chosen, forecasts are realized and mean absolute errors and correlation coefficients are calculated as performance criteria. It is seen that forecast accuracies of the stations which they have high entropy is more than the others. Summarily, in furtherance of the entropy concept, it is seen that the more entropy the data have, the more forecastability the data show or vice versa. In the part of the forecast using with BME, the kind of input data is Gaussian data and one-dimension BME analysis is realized. Theoretical covariance models fitted to experimenteal covariance models are nested models and consist four separable exponential functions. In the study, hard, soft, detrended soft, logaritmic hard, logaritmic soft and logaritmic detrended soft data are used but the studies which they use any kind of logaritmic data show unreasonable results. So the results of these studies do not added the thesis. This situation shows clearly that logaritmic transformation is not applicable for wind speed data. In yearly forecast, 0.050, 0.10, 0.30, 0.50, 10- spatial range and one day, one week, one month-temporal range are tried in order to determine the most appropriate spatial and temporal range. Different kinds of local means are used so as to find proper kriging method. Moreover, BME forecast is employed for each month. BME forecasts with hard data give the best results with taking local mean as constant (ordinary kriging). All forecast results observed stays the interval of confidence levelsand error variances of forecast results are low. When considering in respect to wind energy, the wind speed at 50 m should be 7 m/s at least. With using Hellman equation, it is found that the wind speed at 10 m should be 4.1 m/s. So, the stations which they have wind speeds more than 4.1 m/s are determined. BME forecasts with soft data give the best results with taking local mean as constant, too. All forecast results observed stays the interval of confidence levels again. When compared the study with hard data, it is seen that error variance values reduce by half. From this point, it can be said that the usage of the soft data increase the accuracy of the forecasts. In addition, wind speed values of the areas which they are not determined with hard data is calculated and wind speed values over the seas are identified. It is thougth that the spatiotemporal wind map for Turkey may be considered as a current wind atlas. BME forecasts with detrended soft data give the best results with taking local mean as constant, too. When seen the error variance maps, the forecast with detrendend soft data is not appropriate for wind speed data. The last part of the study, monthly forecasts is realized. When observed monthly statistics, the highest mean wind speed value belongs to July with 2.41 m/s whereas the lowest mean wind speed value belongs to November with 1.89 m/s. In contrast to the yearly forecast with soft data, detrended soft data gives the best results in monthly forecasts. Results show that high wind speed values in January, February and March spread all over of Turkey and the lowest wind speed values is generally seen in May. The reasons of that the study is orginal; there is the first and only study which it uses BME in Turkey and BME application to wind speed data in the world. Besides, the new index that it numerically expresses the relationship between predictability and entropy is proposed. It is thought that this index may be used as a criterion of data quality.