Grey theory is an effective theory that deals with systems that lack weak information and / or information. With this theory, effective and very accurate estimates can be created for future times by utilizing a small number of data. The grey modeling method is a sub-branch of the theory of grey systems and the modeling process is carried out with the help of differences equations and differential equations. The least squares approach plays a role in the precise results of the method. Using the GM (1,1) modeling method, which is the basis of grey prediction models with its accuracy and usefulness, health expenditures in Turkey that will be achieved in the coming years were estimated.These estimates are particularly useful for health and economic policies.Gri teorisi, zayıf bilgi ve / veya bilgi içermeyen sistemlerle ilgilenen etkili bir teoridir. Bu teoriyle, az sayıda veri kullanılarak gelecek zamanlar için etkili ve çok doğru tahminler oluşturulabilir. Gri modelleme yöntemi, gri sistem teorisinin bir alt dalıdır ve modelleme işlemi, denklemler ve diferansiyel denklemler yardımıyla gerçekleştirilir. En küçük kareler yaklaşımı, yöntemin kesin sonuçlarında rol oynar. Gri tahmin modellerinin temeli olan GM (1,1) modelleme yönteminin doğruluğu ve kullanışlılığı ile Türkiye’de önümüzdeki yıllarda sağlanacak sağlık harcamaları tahmin edilmiştir. Bu tahminler özellikle sağlık ve ekonomi politikaları için faydalıdır

Bilgil, Halis

Öztürk, Zafer

Aksaray University Institutional Repository

 52 ErciyesUniversity Journal of Institue Of Scienceand Technology Volume 35, Issue 3, 2019    Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi Cilt 35 Sayı 3 ,2019    Mathematical Estimation of Expenditures in The Health Sector in TURKEY with Grey Modeling   Zafer ÖZTÜRK*1, Halis BİLGİL2,    1* Aksaray University, Graduate School of Natural and Applied Sciences Department of Mathematics, AKSARAY 2Aksaray University, Department of Mathematics, AKSARAY  (Alınış / Received: 22.07.2019, Kabul / Accepted: 28.12.2019, Online Yayınlanma/ Published Online: 31.12.2019)                                                        Keywords Grey Estimation Model,    Grey System Theory, Forecast Accuracy,           Least Squares Method,  Abstract: Grey theory is an effective theory that deals with systems that lack weak information  and / or  information. With this theory, effective and very accurate estimates can be created for future  times  by utilizing  a small number of data. The grey modeling method is a sub-branch of the theory of grey systems and the modeling process is carried out with the help of differences equations and differential equations. The least squares approach plays a role in the precise results of the method. Using the GM (1,1) modeling method, which is the basis of grey prediction models with its accuracy and usefulness, health expenditures in Turkey that will be achieved in the coming years were estimated.These estimates are particularly useful for health and economic policies. Gri Modelleme ile Türkiye Sağlık Sektöründe Harcamaların Matematiksel Tahmini                                                  Anahtar Kelimeler Gri Tahmin Modeli,               Gri Sistem Teorisi,       Tahmini Doğruluk,                En Küçük Kareler Yöntemi, Öz: Gri teorisi, zayıf bilgi ve / veya bilgi içermeyen sistemlerle ilgilenen etkili bir teoridir. Bu teoriyle, az sayıda veri kullanılarak gelecek zamanlar için etkili ve çok doğru tahminler oluşturulabilir. Gri modelleme yöntemi, gri sistem teorisinin bir alt dalıdır ve modelleme işlemi, denklemler ve diferansiyel denklemler yardımıyla gerçekleştirilir. En küçük kareler yaklaşımı, yöntemin kesin sonuçlarında rol oynar. Gri tahmin modellerinin temeli olan GM (1,1) modelleme yönteminin doğruluğu ve kullanışlılığı ile Türkiye’de önümüzdeki  yıllarda  sağlanacak  sağlık  harcamaları  tahmin edilmiştir. Bu tahminler özellikle sağlık ve ekonomi politikaları için faydalıdır.  Department of Mathematics, Aksaray University, AKSARAY Department of Mathematics, Aksaray University, AKSARAY *Corresponding author,  email: zaferozturk@aksaray.edu.tr  1.Introduction        Grey modeling method plays an important role in grey systems theory and was first introduced by Deng (1982).This model is expressed in the form of GM(1,1) and is used to produce high precision estimates based on a small number of data [1]. The grey prediction modeling method has been used successfully in many fields such as industry, science and technology, economy, energy consumption [2-8]. In recent years, improved or modified versions of GM(1.1) have been studied by researchers [9-13]. Grey modeling methods provide effective results in estimating exponential number sequences. In this study, with the help of GM(1,1) modeling method, expenditures on the health sector in Turkey between 2004-2017 were collected and next expenditures were estimated by 2025. As a result of the predictions made with the same model, the error rate was very low, and this situation encouraged the certainty of subsequent predictions. This article is composed of 4 parts. The Grey modeling GM(1,1) method is given in the section 2. The implementation of the GM(1,1) method on health expenditures derived from the health sector in Turkey was carried out in the third section. In the fourth and final section, results based on the findings are given.   Mathematical Estimation of Expenditures in The Health Sector  in TURKEY with Grey Modeling 53  2. Material and Method  2.1  Grey Modelling Theory  Step 1:  The initial data is created with an array of 𝑋(0) data. 𝑋(0) = (𝑥(0)(1), 𝑥(0)(2), … , 𝑥(0)(𝑛)).                                                                                                                                               (1) Step 2:      𝑋(1) accumulated generating sequence is created.  𝑋(1) = (𝑥(1)(1), 𝑥(1)(2), … , 𝑥(1)(𝑛)).                                                                                                                                              (2)   𝑥(1)(𝑘) =∑𝑥(0)(𝑖).𝑘𝑖=1 Step 3: The first-order average value operator 𝑍(1) is created.   𝑍(1)(𝑘) =𝑥(1)(𝑘) + 𝑥(1)(𝑘 − 1)2 where k = 2, 3, … , n  𝑍(1) = (𝑧(1)(2), 𝑧(1)(3), … , 𝑧(1)(𝑛)).                                                                                                                                            (3) Step 4: The equation    𝑥(0)(𝑘) + 𝑎𝑧(1)(𝑘) = 𝑏                                                                                                                                                                       (4) is called the basic form of the GM (1,1) model. 𝑎 and 𝑏 coefficients of equality are determined by the least squares method. Where k is the coefficient of development and progress of a time point 𝑎 and 𝑏 respectively [13]. The equation (4) can be expanded as,    𝑥(0)(2) + 𝑎𝑧(1)(2) = 𝑏              𝑥(0)(3) + 𝑎𝑧(1)(3) = 𝑏                                                                                                                                                                      (5)                    ⋮     𝑥(0)(𝑛) + 𝑎𝑧(1)(𝑛) = 𝑏  from the system (5) can be written as follow,     𝑌 = 𝐵?̂?                                                                                                                                                                                                  (6) where, 𝐵 =( −𝑧(1)(2) 1−𝑧(1)(3) 1⋮−𝑧(1)(𝑛) 1)  Mathematical Estimation of Expenditures in The Health Sector  in TURKEY with Grey Modeling 54           𝑌 =( 𝑥(0)(2)𝑥(0)(3)⋮𝑥(0)(𝑛))      ?̂? = (𝑎𝑏) The goal is to determine a and b coefficients. The least squares are multiplied by 𝐵𝑇  on both sides of the equation  𝑌 = 𝐵?̂?  according to the least squares method, 𝐵𝑇𝑌 = 𝐵𝑇𝐵?̂? and thence, ?̂? = (𝐵𝑇𝐵)−1𝐵𝑇𝑌 can be found. The matrix multiplication algorithm and the least squares method are used to calculate the parameters of this model. Proof: Substitute all data values into the grey differential equation 𝑥(1)(𝑘 + 1) = 1𝑥(1) + 2 gives that 𝑥(1)(2) = 1𝑥(1)(1) + 2, 𝑥(1)(3) = 1𝑥(1)(2) + 2, …       …       …      …  ,  𝑥(1)(𝑛) = 1𝑥(1)(𝑛 − 1) + 2, That is, Y = B?̂? , for the evaluated values of  1 and 2, substitute  𝑥(1)(𝑘 + 1), (k=1,2,…, n-1) with  1𝑥(1)(𝑘) +2 , which gives the error sequence  𝜀 = 𝑌 − 𝐵?̂?. Hence, the S operator is can be defined as, S = 𝜀𝑇𝜀 = (𝑌 − 𝐵?̂?)𝑇(𝑌 − 𝐵?̂?) = ∑ (𝑥(1)(𝑘 + 1) − 1𝑥(1)(𝑘) − 2)2𝑛−1𝑘=1                    For minimum values of S, 1 and 2 should satisfy following equations, 𝜕𝑆𝜕1= −2∑ ((𝑥(1)(𝑘 + 1) − 1𝑥(1)(𝑘) − 2)𝑛−1𝑘=1 𝑥(1)(𝑘)) = 0  𝜕𝑆𝜕2= −2∑ ((𝑥(1)(𝑘 + 1) − 1𝑥(1)(𝑘) − 2)𝑛−1𝑘=1          = 0 These equations can be solved easly.Therefore 1 and 2 are found as, 1=∑ 𝑥(1)(𝑘 + 1)𝑥(1)(𝑘) −1𝑛−1𝑛−1𝑘=1 ∑ 𝑥(1)(𝑘 + 1)∑ 𝑥(1)(𝑘)𝑛−1𝑘=1𝑛−1𝑘=1∑ (𝑥(1)(𝑘))2 −1𝑛−1(∑ 𝑥(1)(𝑘))𝑛−1𝑘=12𝑛−1𝑘=1 2=1𝑛 − 1∑𝑥(1)(𝑘 + 1) −𝑛−1𝑘=11∑𝑥(1)(𝑘)𝑛−1𝑘=1 Mathematical Estimation of Expenditures in The Health Sector  in TURKEY with Grey Modeling 55  From  Y = B?̂? , it follows that 𝐵𝑇𝐵?̂? = 𝐵𝑇𝑌, ?̂? = (𝐵𝑇𝐵)−1𝐵𝑇𝑌 It is clear that, 𝐵𝑇𝐵 =( 𝑥(1)(1)       1𝑥(1)(2)       1⋮𝑥(1)(𝑛 − 1) 1) 𝑇( 𝑥(1)(1)       1𝑥(1)(2)       1⋮𝑥(1)(𝑛 − 1) 1)  = (∑ 𝑥(1)(𝑘)𝑥(1)(𝑘 + 1)𝑛−1𝑘=1∑ 𝑥(1)(𝑘 + 1)𝑛−1𝑘=1)                                                                           Therefore,  ?̂?  = (𝐵𝑇𝐵)−1𝐵𝑇𝑌 =1(𝑛 − 1)∑ (𝑥(1)(𝑘))2− 1𝑛−1(∑ 𝑥(1)(𝑘))𝑛−1𝑘=12𝑛−1𝑘=1 x(   (𝑛 − 1)∑𝑥(1)(𝑘)𝑥(1)(𝑘 + 1) −∑𝑥(1)(𝑘 + 1)∑𝑥(1)(𝑘)𝑛−1𝑘=1𝑛−1𝑘=1𝑛−1𝑘=1−∑𝑥(1)(𝑘)∑𝑥(1)(𝑘)𝑥(1)(𝑘 + 1) +𝑛−1𝑘=1𝑛−1𝑘=1∑𝑥(1)(𝑘 + 1)∑(𝑥(1)(𝑘))2𝑛−1𝑘=1𝑛−1𝑘=1 )    = (∑ 𝑥(1)(𝑘)𝑥(1)(𝑘+1)𝑛−1𝑘=1 −1𝑛−1∑ 𝑥(1)(𝑘+1)∑ 𝑥(1)(𝑘)𝑛−1𝑘=1𝑛−1𝑘=1∑ (𝑥(1)(𝑘))2𝑛−1𝑘=1 − 1𝑛−1∑ (𝑥(1)(𝑘))2𝑛−1𝑘=11𝑛−1(∑ 𝑥(1)(𝑘 + 1) −𝑛−1𝑘=1 1∑ 𝑥(1)(𝑘)𝑛−1𝑘=1) = [1, 2]𝑇 Step 5: The Grey differential equation is defined follow as, 𝑑𝑥(1)(𝑘)𝑑𝑘+ 𝑎𝑥(1)(𝑘) = 𝑏 where the grey developmental coefficient 𝑎 and grey control parameter 𝑏 are the model parameters to be estimated. The grey derivative for the first-order grey differential equation with (1) data as the intermediate information is conventionally represented as 𝑑𝑥(1)(𝑘)𝑑𝑘= lim𝑘→0𝑥(1)(𝑘 + 𝑘) − 𝑥(1)(𝑘)𝑘 and  𝑑𝑥(1)(𝑘)𝑑𝑘=𝑥(1)(𝑘)𝑘= 𝑥(1)(𝑘 + 1) − 𝑥(1)(𝑘)  = 𝑥(0)(𝑘 + 1) when 𝑘→1 roughly. The background value of  𝑑𝑥(1)(𝑘)𝑑𝑘, 𝑥(1)(𝑘) is taken as the mean of 𝑥(1)(𝑘) 𝑎𝑛𝑑 𝑥(1)(𝑘 + 1). The solution to equation    𝑑𝑥(1)(𝑘)𝑑𝑘+ 𝑎𝑥(1)(𝑘) = 𝑏 Mathematical Estimation of Expenditures in The Health Sector  in TURKEY with Grey Modeling 56  with system parameters determined by least-squares method and initial condition      𝑥(1)(1) = 𝑥(0)(1)  is  ?̂?(1)(𝑘 + 1) = (𝑥(0)(1) −𝑏𝑎) 𝑒−𝑎𝑘 +𝑏𝑎                                                                                                                                             (7) is obtained in the form of  (k=1,2,…,n-1). The GM(1,1) model is a special modeling approach based on the exponential functions of the solution.Therefore the prediction values can be generated by,  ?̂?(0)(1) = 𝑥(0)(1)                                                                                                                                                                                  (8)  ?̂?(0)(𝑘) = ?̂?(1)(𝑘) − ?̂?(1)(𝑘 − 1), k= 2, 3, … , n 3. Results  3.1 An Application of The GM (1,1) Model on Health Expenditures in TURKEY  Turkish Statiscal Institute (TUIK) which reports complete, correct and official data. The data includes the amount of total health expenditures between 2004 and 2017 (Million TL). This section presents a systematic predicting methodology which includes data collection, parameter estimation, result analysis and future health expenditures for Turkey.  Table 1: Health expenditures and forecasts in health sector in Turkey  Years Expenditures Amount (Million TL)  GM(1,1) Error Rate (𝜀) 2004 30020,8457 31732,0974 5,70 2005 35358,9075 35636,1038 0,78 2006 44068,6811 40020,4209 9,19 2007 50904,3009 44944,1415 11,71 2008 57740,0000 50473,6283 12,58 2009 57910,7320 56683,4091 2,12 2010 61677,5979 63657,1805 3,21 2011 68607,4094 71488,9364 4,20 2012 74188,7119 80284,2348 8,22 2013 84390,0912 90161,6205 6,84 2014 94749,5074 101254,2231 6,87 2015 104567,5392 113711,5507 8,74 2016 119755,7799 127701,5060 6,63 2017 140647,3488 143412,6483 1,97 Average Error Rate   6,34 Health Expenditures Estimates (Million TL) 2018  161056,7356  2019  180871,5786  2020  203124,2457  2021  228114,6630  2022  256179,6565  2023  287697,4918  2024  323092,9727  2025  362843,1671   4. Discussion and Conclusions  In the study, expenditures in the health sector in Turkey between 2004-2017 were analyzed and forecasts were made for the following  years by using the grey modelling theory. The data at hand were compared with the generated model and the mean error was calculated as 6.34%. In the literature, the results for models with an error rate below 10% are acceptable [9]. The size of a country's health economy represents the development of the country.It is clear that,major investments in the health sector are inevitable to create quality health care.To ensure success in the field of health, the health economy needs to be managed properly.Increased population and following of emerging technologies shows that the inevitable increase in health spending in Turkey. Mathematical Estimation of Expenditures in The Health Sector  in TURKEY with Grey Modeling 57    (a)  (b) Figure 1: Actual values and forecasting values of health expenditures   Error calculation, 𝜀(𝑘) = |𝑥(0)(𝑘)−𝑥(0)(𝑘)𝑥(0)(𝑘)| ∗ 100                                                                                                                                                               (9) and average error,      Average Error =1𝑛∑ 𝜀𝑛𝑘=1 (𝑘)                                                                                                                                                     (10) calculated in the form of. The results show that the increase in health expenditures is mainly due to increased population and other factors. 01000002000003000004000002000 2020 2040Real ValueForecastingValue050000100000150000200000250000300000350000400000Million TLYearsTotal health expenditureYearsReal ValueForecastingValue2004200820122010201420162018202020222024Mathematical Estimation of Expenditures in The Health Sector  in TURKEY with Grey Modeling 58  References [1]  J.L.Deng 1982. Control problem of grey system, Syst. Control Lett. 5 (1982) 288-294. [2] Yong-Huang Lin, Pin-Chan Lee, Ta-Peng Chang 2009. Adaptive and high-precision grey forecasting model, Expert Syst. Appl. 36 (2009) 9658-9662. [3] S.F. Liu, Y.G. Dang, Z.G. Fang 2004. Grey System Theory and its Application, third ed., Science Press, Beijing, 2004 (Chapter 5).  [4] W. Zhou, J.-M. He 2013. Generalized gm (1,1) model and its application in forecasting of fuel production, Applied Mathematical Modelling 37 (9) (2013) 6234-6243.  [5] S. Javed, S. 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Gri modelleme ile Türkiye sağlık sektöründe harcamaların matematiksel tahmini

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