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Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model
BBE 186 1–9 biocybernetics and biomedical engineering xxx (2017) xxx–xxx
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/bbe 1 2 3
Original Research Article
Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model
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Narges Shafaei Bajestani a,*, Ali Vahidian Kamyad a, Ensieh Nasli Esfahani b, Assef Zare c a
Department of Electrical Engineering, Tehran Science and Research Branch, Islamic Azad University, Tehran, Iran Diabetes Research Center, Endocrinology and Metabolism Clinical Sciences Institute, Tehran University of Medical Sciences, Tehran, Iran c Department of Electrical Engineering, Gonabad Branch, Islamic Azad University, Tehran, Iran b
article info
abstract
Article history:
Choosing a proper method to predict and timely prevent the complications of diabetes could
Received 9 May 2016
be considered a significant step toward optimally controlling the disease. Since in medical
Received in revised form
research only small sample sizes of data are available and medical data always includes high
17 January 2017
levels of uncertainty and ambiguity, a type-2 fuzzy regression model seems to be an
Accepted 19 January 2017
appropriate procedure for finding the relationship between outcome and explanatory
Available online xxx
variables in medical decision-making. In this paper, a new type-2 fuzzy regression model based on type-2 fuzzy time series concepts is used to forecast nephropathy in diabetic
Keywords:
patients. Results in two examples show model efficiency. The use of such models in diabetes
Type-2 fuzzy logic
clinics is proposed.
Fuzzy time series
© 2017 Published by Elsevier B.V. on behalf of Nałęcz Institute of Biocybernetics and
Fuzzy regression
Biomedical Engineering of the Polish Academy of Sciences.
Nephropathy Forecasting
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1.
Introduction
Diabetes is among the most common and dangerous diseases of the modern world, causing huge loss of life and financial resources in many societies every year. It is a difficult and incurable, but controllable, disease. Properly controlling this disease prevents or postpones further complications.
Therefore, choosing a proper method to predict and timely prevent diabetes complications could be considered a significant step toward optimally controlling the disease. Among patients starting renal replacement therapy, diabetic nephropathy is the most prevalent cause of kidney disease, affecting 40% of type 1 and type 2 diabetic patients [1]. Due to the prevalence of nephropathy among diabetic patients, they are usually advised to go for check-ups several
* Corresponding author at: Department of Electrical Engineering, Tehran Science and Research Branch, Islamic Azad University, Tehran, Iran. E-mail addresses: [email protected], [email protected] (N.S. Bajestani). http://dx.doi.org/10.1016/j.bbe.2017.01.003 0208-5216/© 2017 Published by Elsevier B.V. on behalf of Nałęcz Institute of Biocybernetics and Biomedical Engineering of the Polish Academy of Sciences. Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003
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times a year, yet some of these check-ups are not necessary. Unnecessary check-ups are omitted from the treatment procedure for cases in which kidney complications are predicted, while more check-ups are prescribed for patients who need them. The existing research implicates the above omissions and additions as well. To detect and follow impairment of renal function, a knowledge of glomerular filtration rate (GFR) is required to allow the correct dosage of drugs cleared by the kidneys and to use the potentially nephrotoxic radiographic contrast media [2]. In clinical research, to evaluate the severity of disease in patients, such linguistic terms like high, medium, low, etc. are used. These terms can be modeled as fuzzy sets. Moreover, the borderline between these fuzzy sets is not crisp, although they are measured in numerical scale. As an example, to define high blood glucose in diagnosing diabetic patients, a cut-off point of 140 mg/dl for two-hour plasma glucose throughout an oral glucose tolerance test is not the precise borderline. In other words, cases within the neighborhood of the borderline indicate a vague status regarding the disease [3]. For these reasons, fuzzy models have been used in medical and especially diabetes research [4–10]. Some researchers have focused on diabetic complications with fuzzy models. For instance, the study of Bin Mansour et al. presented an algorithm based on fuzzy morphology for the computer-assisted improvement of exudates in fundus images of the human retina for the diagnosis of diabetic retinopathy [11]. The study by Narasimhan et al. researched the risk classification of diabetic nephropathy using fuzzy logic [12]. Rama Devi et al. used the design methodology of a fuzzy knowledge-based system to predict the risk of diabetic nephropathy. In their paper, the manageable risk factors like hyperglycemia, insulin, ketones, lipids, obesity, blood pressure and protein/creatinine ratio were considered as independent parameters, and the stages of renal disorder became the output parameter [13]. Since in medical research only small sample sizes of data are available, and as previously mentioned, medical data always includes some levels of uncertainty and ambiguity, some researchers have used a kind of fuzzy regression model to find the relationship between outcome and explanatory variables in medical decision-making. In 2005, Bolotin modeled two examples with fuzzy regression: one associated the quality of life with BMI categories, and the other modeled the analysis of high hemoglobin HbA1c levels among diabetic patients [14]. Pourahmad et al. predicted the existence of diabetes with respect to participants' sex, age, BMI, family history, and two-hour plasma glucose [3]. There are high levels of uncertainty in diabetes data (uncertainty in measurement device, doctor decision, patient body, patient life style, etc.) which encourage the use of type-2 fuzzy logic which can handle high levels of uncertainty. Some researchers have used type-2 fuzzy regressions as the modeling structure. Yicheng Wei and Junzo Watada built a type-2 fuzzy qualitative regression model. They implied that they used a general type-2 fuzzy number, while it seems that they used an interval type-2 fuzzy regression model. Poleshchuk and Komarov presented a regression model for
interval type-2 fuzzy sets based on the least squares estimation technique [15]. Hosseinzadeh et al. presented a weighted goal programming approach to fuzzy linear regression with crisp inputs and type-2 fuzzy outputs (WGP) [16]. This model, however, only tried to close the membership functions of observed and estimated responses by closing some of their parameters. It seems that none of the studies mentioned above could adequately model the type-2 fuzzy regression; instead, they reduced their models to only some points of type-2 fuzzy numbers. Fuzzy time series models have been applied to real life phenomena. Different fuzzy methods have been proposed to solve fuzzy time series problems. Watada applied fuzzy regression to solve the problems of fuzzy time series [17]. Song and Chissom proposed novel definitions for fuzzy time series [18]. Chen improved Song and Chissom's model [19]. Some other researchers also improved fuzzy time series models [20–22]. Fuzzy time series models have been proposed for various applications, such as enrollment, stock indexes, load forecasting, tourism demand forecasting, etc. [23–27]. Huarng and Yu proposed a framework for a type-2 fuzzy time series model to improve forecasting results [28]. Shafaei Bajestani et al. have optimized Huarng and Yu's model to forecast the Taiwan Stock Index based on optimized highorder type-2 fuzzy time series [29,30]. The GA algorithm was also used in fuzzy regression and fuzzy time series demands for optimizing models and their results [22,31–33]. The incorporation of fuzzy regression models and type-2 fuzzy time series models is named T2FRFTS, and using the benefits of both allowed the presentation in this research of a different viewpoint to type-2 fuzzy regression models that could predict GFR efficiently in two diabetic patients. In this study, our aim is to predict nephropathy due to priori of diabetic patient. It is expected that past data of GFR models could predict future GFR with good accuracy. In fact, the condition of kidneys in future referrals could be predicted using the proposed model. GFR (t) and GFR (t + 1) are the inputs and outputs of the model, respectively. Uncertainties of measurement devices make the data recorded in the clinics have a high level of ambiguity and uncertainty. These uncertainties confirm the use of type-2 fuzzy for modeling this data. GFR has three common formulas. These three formulas are another reason for using type-2 fuzzy sets to consider the effects of all three formulas in proper ways. It is expected that the proposed model, considering the effect of uncertainty, will be able to present a good prediction of GFR to predict nephropathy in a diabetic patient. Our motivation and final objective is to offer software which can predict diabetic complications to specialists and diabetes clinics as a large number of the world population suffers from diabetes. This prediction can help both the patients and the doctors to prevent unnecessary experiments and check-ups. Moreover, doctors can make better decisions for the type and dosage of medication they prescribe to postpone retinopathy among diabetic patients. The rest of current paper is organized as follows. The studies relevant to this research are reviewed in Section 2. In Section 3, the proposed model is described using two examples. Finally, Section 4 presents the conclusion.
Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003
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2.
are minimized. This can be achieved by using the following linear program:
Review
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As mentioned in the previous section, the proposed model was constructed based on some type-2 fuzzy time series and fuzzy regression concepts. Therefore, in this section type-1 fuzzy regression and type-2 fuzzy time series concepts are presented.
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2.1.
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At first, a formulation for type-1 fuzzy regression that proposed by Tanka is presented in this section. This model is the basis of most of type-1 fuzzy regression model. The model is assumed to be as below:
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~ 1 xi1 þ A ~ 2 xi2 þ þ A ~ n xin ~i ¼ A Y
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~ j ð j ¼ 1; 2; :::; nÞ are the fuzzy coefficients in the form of where A (aj, cj) where aj is the middle and cj is the spread. The membership functions are assumed triangular membership as: 8 xaj < 1 ; aj cj xaj þ cj Ai ðxÞ ¼ (2) cj : 0 otherwise
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i¼1
j¼0 k X ðajð1HÞ: cj Þxij þdþ iL þ diL ¼ yi ð1HÞei þ dþ iU ; diU ; diL ; diL 0 i ¼ 1; :::; n
(3)
~ j that minimize the Fuzzy regression model seeks to find A spread of the fuzzy estimated output for all data sets subject to a H-cut of observed output yi ¼ ðyi ; ei Þ is included in estimated output. ~ j which are the More specially, problem is to find out A solution of following linear programming problem: m X n n X X c0 þ cj xij min J ¼ subject to :
i¼1 j¼1
j¼1
n X
X
j¼1
j
aj xij þ ð1HÞ
cj jxij j yi þ ð1HÞei
n X X aj xij þ ð1HÞ cj jxij j yi þ ð1HÞei j¼1
i ¼ 1; :::; n
j¼0
(1)
From (1) and (2), we get: P 8 n > aj xij y > > < 1 P j¼1 x 6¼ 0 n mY ðyÞ ¼ j¼1 cj xij j > > x ¼ 0; y ¼ 0 > : 1 0 x ¼ 0; y 6¼ 0
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n X þ ðdþ iU þ diU þ diL þ diL Þ
k X subject to : ðaj þð1HÞ: cj Þxij þdþ iU þdiU ¼ yi þð1HÞei i ¼ 1; :::; n
Type-1 fuzzy regression
a;c
(4)
j
c0 i ¼ 1; :::; m: 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197
Minimize
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One of the proper models in type-1 fuzzy regression that shows its ability is Hojati's model used in this research. Hojati's model is briefly explained here. For more details please read the reference [34] which contains all necessary definition and literature review. ~ j ¼ ðaj ; cj Þ in In Hojati's model to find the fuzzy coefficients A ~ ~ ~ 2 xi2 þ þ the fuzzy linear regression model, Y i ¼ A1 xi1 þ A ~ n xin , the following simple goal programming-like approach is A proposed. The observed value is assumed to be a symmetric triangular fuzzy number with center yi and half-width ei. In this model the fuzzy regression coefficients are chosen so that the total deviation of the upper points of H-certain predicted and associated observed intervals and the deviation of the lower points of H-certain predicted and associated observed intervals
aj ¼ free;
cj 0 j ¼ 0; :::; k (5)
where jdþ d iU j refers the distance between the upper point of iU the H-certain predicted interval and the upper point of the H certain observed interval, and jdþ iL diL j presents the distance between the lower point of H-certain predicted interval and the lower point of the H-certain observed interval. Minimizing the summation of these two distances is objective function [34].
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2.2.
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Required concepts
To understand the proposed model, it is necessary to know some definitions that are presented here.
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Definition 1. (Type-2 fuzzy set) type-2 fuzzy set is an extension of type-1 fuzzy set that there can be a fuzzy set for any degree of membership. For example, there is a triangular fuzzy set (0.4, 0.5, 0.6) as a degree of membership for x = 1 in Fig. 1.
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Definition 2. (Fuzzy time series) Y(t) (t = . . ., 0, 1, 2,. . .), is a subset of R. Let Y(t) be the universe of discourse defined by fuzzy set fi(t). If FðtÞ consists of fi(t) (i = 1,2,. . .), FðtÞ is defined as a fuzzy time series on Y(t).
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Definition 3. Suppose F(t) is caused by F(t 1) only. The relationship between F(t 1) = Ai and F(t) = Aj can be represented by Ai ! Aj, if F(t 1) = Ai (a fuzzy set) and F(t) = Aj, where Ai and Aj are the LHS (left-hand side) and the RHS (right-hand side) of the fuzzy logic relationships (FLR), respectively [29].
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Definition 4. Assume there are the subsequent FLRs:
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Ai ! Aj1 Ai ! Aj1 .. . Ai ! Ajl
(6)
Fig. 1 – Type-2 fuzzy set.
Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003
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Following Chen's model [19], these FLRs can be set into a fuzzy logic relationship group (FLRG) as: Ai ! Aj1, Aj2, . . ., Ajl
in the population that are selected for deletion. For more details of the GA algorithm please read reference [36].
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Definition 5. (Type-2 fuzzy time series) in definition 1, if fi(t) is type-2 fuzzy set, F(t) is defined as type-2 fuzzy time series.
3.
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The proposed type-2 fuzzy regression model utilizes the fuzzy relationships established by a type-1 fuzzy regression model based in type-1 and type-2 observations. Operators are used to include or screen out fuzzy relationships obtained from types 1 and 2 observations. Type-2 forecasts are then calculated from these fuzzy relationships. Two operators were defined. The first involves including and the other screening out fuzzy relationships. Hence, we propose union and intersection operators in type 2 model accordingly.
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Definition 6. The relationships between two FLR or FLRGs are obtained by defining operators of union (_) and intersection (^):
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_ ðLHSd ; LHSe Þ ¼ RHSd [ RHSe ^ ðLHSd ; LHSe Þ ¼ RHSd \ RHSe
(7)
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where [ and \ refer the union the intersection operator for set theory, respectively. The LHSd indicates the LHS of an FLRG and RHSd indicates RHS of an FLRG.
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Definition7. The definition of operators of union and intersection for multiple FLRGs, _m and ^m are as follows: _ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ _ . . .ð _ ð _ ðLHSc ; LHSd Þ; LHSe Þ; . . .Þ ^ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ ^ . . .ð ^ ð ^ ðLHSc ; LHSd Þ; LHSe Þ; . . .Þ (8)
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_ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ ðRHSc [ RHSd [ RHSe [ . . .Þ ^ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ ðRHSc \ RHSd \ RHSe \ . . .Þ
(9)
_ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ ;, then Definition 8. (a) If _ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ LHSx , where LHSx is acquired from the FLRG resulted through observations of type-1. (b) If ^ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ ;, then ^ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ LHSx , where LHSx, is obtained from the FLRG resulted through observations of type-1. For more details please read Refs. [28,29,35] that are type-2 fuzzy time series model. The GA algorithm, which is a powerful random global search technique to handle optimization problems and can help find the globally optimal solution over a domain, was used for optimizing in the proposed model. GA consists of gradient free parallel-optimization algorithms that use a performance criterion for evaluation and a population of possible solutions to search for a global optimum. These structured random search techniques are capable of handling complex and irregular solution spaces. GAs is inspired by the biological process of Darwinian evolution where selection, mutation, and crossover play a major role. Good solutions are selected and manipulated to achieve new and possibly better solutions. The manipulation is done by the genetic operators that work on the chromosomes in which the parameters of possible solutions are encoded. In each generation of the GA, the new solutions replace the solutions
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Proposed type-2 fuzzy regression model
In this section, first the proposed model steps have been listed, and then the model is explained in two examples. Fig. 2 shows a flowchart of this model. The proposed algorithm for the type2 fuzzy regression models is listed below. 1. Selecting a model of type-1 fuzzy regression (Hojati's model). 2. Choosing observations of type-1. 3. Selecting observations of type-2. 4. Applying the model of type-1 fuzzy regression to observations of type-1 and type-2. 5. Using operators to the FLRGs for each of observations. 6. Defuzzifying the predictions. 7. Computing predictions for model of type-2. 8. Evaluating the model efficiency. Two examples are used to describe this model. GFR(t) is the input of model and GFR(t + 1) is the output of model.
3.1.
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GFR forecasting
In this research about 500 files of diabetic patients were studied and checked. 106 patients that had attained some level of nephropathy were chosen for this research. The clinical records of these patients were followed for about 5 years. Of the 106 patients, 47 who regularly went to the diabetic clinic were used for this research as GFR values must be recorded orderly. The proposed model in more than 90% of patients presents the best results compared to other models. Two patients that have the best results with the proposed model were reported in this paper as examples. In this study, our aim is to predict nephropathy due to priori of diabetic patient. It is expected that using past GFR data models could predict future GFR with good accuracy. In fact, the condition of kidney could be predicted using proposed model. GFR(t) and GFR(t + 1) is input and output of model respectively. In the proposed model, 70% of the data was used as train data and 30% as the test value. GFR has three common formulas. As mentioned previously, these three formulas are one reason for using type-2 fuzzy sets to consider the effects of all three formulas.
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Weight GFR1 ¼ ð140AgeÞ 72 pcr
(10)
GFR2 ¼ 175ðpcrÞ1:154 ðAgeÞ:203 ð:742 if femaleÞ
(11)
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(12)
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Chronic kidney disease (CKD) stages based on GFR level are shown in Table 1. In proposed model, type-1 and type-2 observations were required. The mean of three formulas in every t (every referral
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GFR3 ¼
ð0:41HeightÞ pcr
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Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003
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Fig. 2 – Flowchart of T2FRFTS model.
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of patients) was used as a type-1 observation because the mean of three formulas is a good choice for GFR prediction if type-1 fuzzy regression is used. For the type-2 fuzzy model, type-2 observations were required. Minimum and maximum of GFR1, GFR2, GFR3 were chosen as the type-2 observation. These type-1 and type-2 observations make type 2 fuzzy model in steps that are explained in the following section. Proposed model steps have been applied for GFR forecasting. Step 1: Choosing a model of type-1 fuzzy regression (Hojati's model). In this study, Hojati's model was chosen. It provides relatively simpler calculations and better forecasting results. Step 2: Selecting a variable as observation of type-1. The mean of three GFR equation results has been chosen as the main value in the type-1 fuzzy model. This value was selected as type-1 observation.
Table 1 – CKD Stage based on GFR level. CKD Stage Stage Stage Stage Stage Stage
1 2 3 4 5
GFR level (mL/min/1.73 m2) ≥90 60–89 30–59 15–29 <15
Step 3. Selecting observations of type-2 In three equations, the minimum and maximum outcomes are selected as observations of type-2. For example, in the fourth value, GFR1 = 56.07, GFR2 = 71.65, GFR3 = 82.00; the mean of them is 69.91, and that has been chosen as the type-1 observation. Low value is 56.07 and high value is 82. Step 4: Applying the model of type-1 fuzzy regression to the observations of type-1 and type-2. The type-1 fuzzy regression model can then be applied to type-1 and type-2 observations and the forecasts can be obtained. For this goal, these steps should be followed: Step 4-1: Defining fuzzy sets for observations With reference to the authors' prior model [29], triangular fuzzy sets with indeterminate supports were used. This means that triangular legs were not fixed, but were determined after the solution of an optimization problem. The GA algorithm, a powerful random global search technique to handle optimization problems and which can help find the globally optimal solution over a domain, was used for optimizing fuzzy sets. The GA tool of MATLAB was used to find the optimized fuzzy sets support and parameter / that is defined in the following steps. Initial fuzzy sets supports, [0 10], [10 20],. . ., [90 100], and /=0.5 as initial population, uniform mutation function, scattered crossover function, and stochastic uniform selection function are used in the GA tool setting. Stopping criteria was number of generations 100. Other parameters
Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003
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Table 2 – Parameters of GA algorithm.
Table 3 – GFR Forecast after ^m and _m.
Parameter
Value
Population size Mutation rate Crossover rate generations
40 0.01 0.8 100
No 4
GFR1 GFR2 GFR3
Forecast after ^m
Forecast after _m
A8
A6, A8
A6 ! A6 A8 ! A8 A9 ! A8
Then, ^m and _m are both used to the predictions of all date. In Table 3, ^m and _m are applied to all predictions, including those of observation of type-1 and type-2. Step 6. Defuzzifying the predictions The defuzzyfied prediction for ^m is equivalent to:
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deffuzzificationintersection ðno:4Þ ¼ 74:79 Correspondingly, the defuzzyfied prediction for _m is as follows: Fig. 3 – Fuzzy sets after optimization with GA.
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were used show in Table 2. After no changes in parameters, minimization was robust between about 30 runs. Min, max, mean and variance of optimal values were 3.9, 5.1, 4.38 and 0.2 respectively for patient 1 and 1.33, 2.35, 1.77 and 0.14 for patient 2. Optimized fuzzy set supports found with GA in patient 1 are (Fig. 3): [0 10.10], [0.04 20.00], [19.89 32.16], [27.96 40.00], [39.98 51.75], [46.85 60.00], [59.99 70.30], [68.56 80.01], [79.62 90.07], [90.00 100.00]. Step 4-2: Fuzzifying the observations Different GFR values can be fuzzified into the corresponding fuzzy sets. All observation of type-1 and type-2 are fuzzified in this step. For example, the GFR for the fourth value was 71.65, which was fuzzified to A8. Step 4-3: Applying model of type-1 fuzzy regression to observations of type-1 and type-2 and acquiring predictions. Hojati's model was applied to the main, high, and low values that were fuzzified before. Fig. 4 shows the results of the type-1 fuzzy regression model for type-1 and type-2 observations (second column of Table 3). Step 5: Using operators to the FLRGs for each of observations.
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defuzzificationunion ð11=8Þ ¼ ð53:42 þ 74:79Þ=2 ¼ 64:11 Step 7. Obtaining predictions for model of type-2 / was used as a coefficient in calculating type-2 forecasting [29]:
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defuzzificationðtÞ ¼ a ðdefuzzificationintersection ðtÞ þ ð1aÞðdefuzzificationunion ðtÞÞ
(13)
where a can be achieved through solving the optimization problem. This means that there is another parameter in the optimization problem. The best value of a is 0.126. Thus, it is used for calculating type-2 forecasting:
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defuzzificationðtÞ ¼ 0:874ðdefuzzificationintersection ðtÞÞ þ 0:126ðdefuzzificationunion ðtÞÞ
(14)
Step 8: Evaluating the model efficiency. The assessment of the efficiency of the model is performed by applying the RMSE:
RMSE ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX n u u ðactual valueðtÞforecasted valueðtÞÞ2 t i¼1
n
444 445 446 447 448 449
(15)
Fig. 4 – Apply type-1 fuzzy regression model to type-1 and type-2 values in patient 1. Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003
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Fig. 5 – GFR forecasting results in patient 1. Fig. 7 – GFR forecasting results in patient 2.
Table 4 – Forecasts of the GFR in patient 1. No.
Actual
Huarng and Yu
WGP
Shafaei
T2FRFTS
.. . 37.28 11.28 9.46 8.16 7.48 .. .
.. . 33.3 20 15 15 15 .. .
.. . 31.5 18.3 11.2 10.3 10.3 .. .
.. . 35 15 7.87 7.87 7.87
.. . 33.98 11.02 10.02 10.02 10.02
.. . 8 9 10 11 12 .. .
Table 6 – Comparison of RMSE of GFR forecasting in patient 2. Model Train-RMSE Test-RMSE
Hojati
Huarng and Yu
WGP
Shafaei
T2FRFTS
7.32 8.88
4.12 5.25
2.88 5.17
2.50 4.53
1.33 2.65
Table 5 – Comparison of RMSE of GFR forecasting in patient 1. Model Train-RMSE Test-RMSE
452 450 451 453 454 455
Hojati
Huarng and Yu
WGP
Shafaei
T2FRFTS
15.10 24.42
13.29 23.50
13.01 23.11
8.97 15.57
3.91 6.85
Figs. 4 and 5 and Tables 3 and 4 show the results for patient 1. Figs. 6 and 7 show the results for patient 2. Tables 5 and 6 show a comparison of the RMSE of several models. The results indicate that the suggested model was able to perform
predictions relatively more accurately in two patients. The RMSE in both train and test values is better than other models. Such an accurate prediction is very helpful for diabetic patients and diabetes clinics. As mentioned before, 47 patients were used in this research. Min, max, mean and variance of RMSEs between 47 patients were 1.33, 20.20, 8.67, and 19.86 respectively. In more than 90% of patient proposed model has better performance and minimum RMSE. The two patients that were reported had the best results with proposed model.
Fig. 6 – Apply type-1 fuzzy regression model to type-1 and type-2 values in patient 1. Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003
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This study presents a GFR forecasting with different perspectives to type-2 fuzzy regression models based on type-2 fuzzy time series concepts (T2FRFTS) to forecast nephropathy in diabetic patients. GFR forecasting clinically helps patients and society decrease the cost of control and treatment by optimizing the number of check-ups. Unnecessary regular check-ups worry patients, even though they may not suffer from the illness for years to come. In addition, GFR forecasting can help doctors make better decisions regarding the type and dosage of medication needed to postpone retinopathy among diabetic patients. The proposed type-2 fuzzy regression model avoids additional complexity and helps model more uncertainty in diabetic values. Therefore, the authors propose the use of such models in diabetes clinics. From the empirical analysis, it can be inferred that, in addition to its intelligible steps, this model can adequately forecast GFR.
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Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/bbe 1 2 3
Original Research Article
Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model
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Q1
Narges Shafaei Bajestani a,*, Ali Vahidian Kamyad a, Ensieh Nasli Esfahani b, Assef Zare c a
Department of Electrical Engineering, Tehran Science and Research Branch, Islamic Azad University, Tehran, Iran Diabetes Research Center, Endocrinology and Metabolism Clinical Sciences Institute, Tehran University of Medical Sciences, Tehran, Iran c Department of Electrical Engineering, Gonabad Branch, Islamic Azad University, Tehran, Iran b
article info
abstract
Article history:
Choosing a proper method to predict and timely prevent the complications of diabetes could
Received 9 May 2016
be considered a significant step toward optimally controlling the disease. Since in medical
Received in revised form
research only small sample sizes of data are available and medical data always includes high
17 January 2017
levels of uncertainty and ambiguity, a type-2 fuzzy regression model seems to be an
Accepted 19 January 2017
appropriate procedure for finding the relationship between outcome and explanatory
Available online xxx
variables in medical decision-making. In this paper, a new type-2 fuzzy regression model based on type-2 fuzzy time series concepts is used to forecast nephropathy in diabetic
Keywords:
patients. Results in two examples show model efficiency. The use of such models in diabetes
Type-2 fuzzy logic
clinics is proposed.
Fuzzy time series
© 2017 Published by Elsevier B.V. on behalf of Nałęcz Institute of Biocybernetics and
Fuzzy regression
Biomedical Engineering of the Polish Academy of Sciences.
Nephropathy Forecasting
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1.
Introduction
Diabetes is among the most common and dangerous diseases of the modern world, causing huge loss of life and financial resources in many societies every year. It is a difficult and incurable, but controllable, disease. Properly controlling this disease prevents or postpones further complications.
Therefore, choosing a proper method to predict and timely prevent diabetes complications could be considered a significant step toward optimally controlling the disease. Among patients starting renal replacement therapy, diabetic nephropathy is the most prevalent cause of kidney disease, affecting 40% of type 1 and type 2 diabetic patients [1]. Due to the prevalence of nephropathy among diabetic patients, they are usually advised to go for check-ups several
* Corresponding author at: Department of Electrical Engineering, Tehran Science and Research Branch, Islamic Azad University, Tehran, Iran. E-mail addresses: [email protected], [email protected] (N.S. Bajestani). http://dx.doi.org/10.1016/j.bbe.2017.01.003 0208-5216/© 2017 Published by Elsevier B.V. on behalf of Nałęcz Institute of Biocybernetics and Biomedical Engineering of the Polish Academy of Sciences. Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003
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times a year, yet some of these check-ups are not necessary. Unnecessary check-ups are omitted from the treatment procedure for cases in which kidney complications are predicted, while more check-ups are prescribed for patients who need them. The existing research implicates the above omissions and additions as well. To detect and follow impairment of renal function, a knowledge of glomerular filtration rate (GFR) is required to allow the correct dosage of drugs cleared by the kidneys and to use the potentially nephrotoxic radiographic contrast media [2]. In clinical research, to evaluate the severity of disease in patients, such linguistic terms like high, medium, low, etc. are used. These terms can be modeled as fuzzy sets. Moreover, the borderline between these fuzzy sets is not crisp, although they are measured in numerical scale. As an example, to define high blood glucose in diagnosing diabetic patients, a cut-off point of 140 mg/dl for two-hour plasma glucose throughout an oral glucose tolerance test is not the precise borderline. In other words, cases within the neighborhood of the borderline indicate a vague status regarding the disease [3]. For these reasons, fuzzy models have been used in medical and especially diabetes research [4–10]. Some researchers have focused on diabetic complications with fuzzy models. For instance, the study of Bin Mansour et al. presented an algorithm based on fuzzy morphology for the computer-assisted improvement of exudates in fundus images of the human retina for the diagnosis of diabetic retinopathy [11]. The study by Narasimhan et al. researched the risk classification of diabetic nephropathy using fuzzy logic [12]. Rama Devi et al. used the design methodology of a fuzzy knowledge-based system to predict the risk of diabetic nephropathy. In their paper, the manageable risk factors like hyperglycemia, insulin, ketones, lipids, obesity, blood pressure and protein/creatinine ratio were considered as independent parameters, and the stages of renal disorder became the output parameter [13]. Since in medical research only small sample sizes of data are available, and as previously mentioned, medical data always includes some levels of uncertainty and ambiguity, some researchers have used a kind of fuzzy regression model to find the relationship between outcome and explanatory variables in medical decision-making. In 2005, Bolotin modeled two examples with fuzzy regression: one associated the quality of life with BMI categories, and the other modeled the analysis of high hemoglobin HbA1c levels among diabetic patients [14]. Pourahmad et al. predicted the existence of diabetes with respect to participants' sex, age, BMI, family history, and two-hour plasma glucose [3]. There are high levels of uncertainty in diabetes data (uncertainty in measurement device, doctor decision, patient body, patient life style, etc.) which encourage the use of type-2 fuzzy logic which can handle high levels of uncertainty. Some researchers have used type-2 fuzzy regressions as the modeling structure. Yicheng Wei and Junzo Watada built a type-2 fuzzy qualitative regression model. They implied that they used a general type-2 fuzzy number, while it seems that they used an interval type-2 fuzzy regression model. Poleshchuk and Komarov presented a regression model for
interval type-2 fuzzy sets based on the least squares estimation technique [15]. Hosseinzadeh et al. presented a weighted goal programming approach to fuzzy linear regression with crisp inputs and type-2 fuzzy outputs (WGP) [16]. This model, however, only tried to close the membership functions of observed and estimated responses by closing some of their parameters. It seems that none of the studies mentioned above could adequately model the type-2 fuzzy regression; instead, they reduced their models to only some points of type-2 fuzzy numbers. Fuzzy time series models have been applied to real life phenomena. Different fuzzy methods have been proposed to solve fuzzy time series problems. Watada applied fuzzy regression to solve the problems of fuzzy time series [17]. Song and Chissom proposed novel definitions for fuzzy time series [18]. Chen improved Song and Chissom's model [19]. Some other researchers also improved fuzzy time series models [20–22]. Fuzzy time series models have been proposed for various applications, such as enrollment, stock indexes, load forecasting, tourism demand forecasting, etc. [23–27]. Huarng and Yu proposed a framework for a type-2 fuzzy time series model to improve forecasting results [28]. Shafaei Bajestani et al. have optimized Huarng and Yu's model to forecast the Taiwan Stock Index based on optimized highorder type-2 fuzzy time series [29,30]. The GA algorithm was also used in fuzzy regression and fuzzy time series demands for optimizing models and their results [22,31–33]. The incorporation of fuzzy regression models and type-2 fuzzy time series models is named T2FRFTS, and using the benefits of both allowed the presentation in this research of a different viewpoint to type-2 fuzzy regression models that could predict GFR efficiently in two diabetic patients. In this study, our aim is to predict nephropathy due to priori of diabetic patient. It is expected that past data of GFR models could predict future GFR with good accuracy. In fact, the condition of kidneys in future referrals could be predicted using the proposed model. GFR (t) and GFR (t + 1) are the inputs and outputs of the model, respectively. Uncertainties of measurement devices make the data recorded in the clinics have a high level of ambiguity and uncertainty. These uncertainties confirm the use of type-2 fuzzy for modeling this data. GFR has three common formulas. These three formulas are another reason for using type-2 fuzzy sets to consider the effects of all three formulas in proper ways. It is expected that the proposed model, considering the effect of uncertainty, will be able to present a good prediction of GFR to predict nephropathy in a diabetic patient. Our motivation and final objective is to offer software which can predict diabetic complications to specialists and diabetes clinics as a large number of the world population suffers from diabetes. This prediction can help both the patients and the doctors to prevent unnecessary experiments and check-ups. Moreover, doctors can make better decisions for the type and dosage of medication they prescribe to postpone retinopathy among diabetic patients. The rest of current paper is organized as follows. The studies relevant to this research are reviewed in Section 2. In Section 3, the proposed model is described using two examples. Finally, Section 4 presents the conclusion.
Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003
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2.
are minimized. This can be achieved by using the following linear program:
Review
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As mentioned in the previous section, the proposed model was constructed based on some type-2 fuzzy time series and fuzzy regression concepts. Therefore, in this section type-1 fuzzy regression and type-2 fuzzy time series concepts are presented.
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At first, a formulation for type-1 fuzzy regression that proposed by Tanka is presented in this section. This model is the basis of most of type-1 fuzzy regression model. The model is assumed to be as below:
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~ 1 xi1 þ A ~ 2 xi2 þ þ A ~ n xin ~i ¼ A Y
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~ j ð j ¼ 1; 2; :::; nÞ are the fuzzy coefficients in the form of where A (aj, cj) where aj is the middle and cj is the spread. The membership functions are assumed triangular membership as: 8 xaj < 1 ; aj cj xaj þ cj Ai ðxÞ ¼ (2) cj : 0 otherwise
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i¼1
j¼0 k X ðajð1HÞ: cj Þxij þdþ iL þ diL ¼ yi ð1HÞei þ dþ iU ; diU ; diL ; diL 0 i ¼ 1; :::; n
(3)
~ j that minimize the Fuzzy regression model seeks to find A spread of the fuzzy estimated output for all data sets subject to a H-cut of observed output yi ¼ ðyi ; ei Þ is included in estimated output. ~ j which are the More specially, problem is to find out A solution of following linear programming problem: m X n n X X c0 þ cj xij min J ¼ subject to :
i¼1 j¼1
j¼1
n X
X
j¼1
j
aj xij þ ð1HÞ
cj jxij j yi þ ð1HÞei
n X X aj xij þ ð1HÞ cj jxij j yi þ ð1HÞei j¼1
i ¼ 1; :::; n
j¼0
(1)
From (1) and (2), we get: P 8 n > aj xij y > > < 1 P j¼1 x 6¼ 0 n mY ðyÞ ¼ j¼1 cj xij j > > x ¼ 0; y ¼ 0 > : 1 0 x ¼ 0; y 6¼ 0
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n X þ ðdþ iU þ diU þ diL þ diL Þ
k X subject to : ðaj þð1HÞ: cj Þxij þdþ iU þdiU ¼ yi þð1HÞei i ¼ 1; :::; n
Type-1 fuzzy regression
a;c
(4)
j
c0 i ¼ 1; :::; m: 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197
Minimize
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One of the proper models in type-1 fuzzy regression that shows its ability is Hojati's model used in this research. Hojati's model is briefly explained here. For more details please read the reference [34] which contains all necessary definition and literature review. ~ j ¼ ðaj ; cj Þ in In Hojati's model to find the fuzzy coefficients A ~ ~ ~ 2 xi2 þ þ the fuzzy linear regression model, Y i ¼ A1 xi1 þ A ~ n xin , the following simple goal programming-like approach is A proposed. The observed value is assumed to be a symmetric triangular fuzzy number with center yi and half-width ei. In this model the fuzzy regression coefficients are chosen so that the total deviation of the upper points of H-certain predicted and associated observed intervals and the deviation of the lower points of H-certain predicted and associated observed intervals
aj ¼ free;
cj 0 j ¼ 0; :::; k (5)
where jdþ d iU j refers the distance between the upper point of iU the H-certain predicted interval and the upper point of the H certain observed interval, and jdþ iL diL j presents the distance between the lower point of H-certain predicted interval and the lower point of the H-certain observed interval. Minimizing the summation of these two distances is objective function [34].
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2.2.
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Required concepts
To understand the proposed model, it is necessary to know some definitions that are presented here.
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Definition 1. (Type-2 fuzzy set) type-2 fuzzy set is an extension of type-1 fuzzy set that there can be a fuzzy set for any degree of membership. For example, there is a triangular fuzzy set (0.4, 0.5, 0.6) as a degree of membership for x = 1 in Fig. 1.
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Definition 2. (Fuzzy time series) Y(t) (t = . . ., 0, 1, 2,. . .), is a subset of R. Let Y(t) be the universe of discourse defined by fuzzy set fi(t). If FðtÞ consists of fi(t) (i = 1,2,. . .), FðtÞ is defined as a fuzzy time series on Y(t).
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Definition 3. Suppose F(t) is caused by F(t 1) only. The relationship between F(t 1) = Ai and F(t) = Aj can be represented by Ai ! Aj, if F(t 1) = Ai (a fuzzy set) and F(t) = Aj, where Ai and Aj are the LHS (left-hand side) and the RHS (right-hand side) of the fuzzy logic relationships (FLR), respectively [29].
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Definition 4. Assume there are the subsequent FLRs:
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Ai ! Aj1 Ai ! Aj1 .. . Ai ! Ajl
(6)
Fig. 1 – Type-2 fuzzy set.
Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003
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Following Chen's model [19], these FLRs can be set into a fuzzy logic relationship group (FLRG) as: Ai ! Aj1, Aj2, . . ., Ajl
in the population that are selected for deletion. For more details of the GA algorithm please read reference [36].
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Definition 5. (Type-2 fuzzy time series) in definition 1, if fi(t) is type-2 fuzzy set, F(t) is defined as type-2 fuzzy time series.
3.
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The proposed type-2 fuzzy regression model utilizes the fuzzy relationships established by a type-1 fuzzy regression model based in type-1 and type-2 observations. Operators are used to include or screen out fuzzy relationships obtained from types 1 and 2 observations. Type-2 forecasts are then calculated from these fuzzy relationships. Two operators were defined. The first involves including and the other screening out fuzzy relationships. Hence, we propose union and intersection operators in type 2 model accordingly.
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Definition 6. The relationships between two FLR or FLRGs are obtained by defining operators of union (_) and intersection (^):
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_ ðLHSd ; LHSe Þ ¼ RHSd [ RHSe ^ ðLHSd ; LHSe Þ ¼ RHSd \ RHSe
(7)
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where [ and \ refer the union the intersection operator for set theory, respectively. The LHSd indicates the LHS of an FLRG and RHSd indicates RHS of an FLRG.
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Definition7. The definition of operators of union and intersection for multiple FLRGs, _m and ^m are as follows: _ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ _ . . .ð _ ð _ ðLHSc ; LHSd Þ; LHSe Þ; . . .Þ ^ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ ^ . . .ð ^ ð ^ ðLHSc ; LHSd Þ; LHSe Þ; . . .Þ (8)
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_ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ ðRHSc [ RHSd [ RHSe [ . . .Þ ^ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ ðRHSc \ RHSd \ RHSe \ . . .Þ
(9)
_ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ ;, then Definition 8. (a) If _ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ LHSx , where LHSx is acquired from the FLRG resulted through observations of type-1. (b) If ^ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ ;, then ^ m ðLHSc ; LHSd ; LHSe ; . . .Þ ¼ LHSx , where LHSx, is obtained from the FLRG resulted through observations of type-1. For more details please read Refs. [28,29,35] that are type-2 fuzzy time series model. The GA algorithm, which is a powerful random global search technique to handle optimization problems and can help find the globally optimal solution over a domain, was used for optimizing in the proposed model. GA consists of gradient free parallel-optimization algorithms that use a performance criterion for evaluation and a population of possible solutions to search for a global optimum. These structured random search techniques are capable of handling complex and irregular solution spaces. GAs is inspired by the biological process of Darwinian evolution where selection, mutation, and crossover play a major role. Good solutions are selected and manipulated to achieve new and possibly better solutions. The manipulation is done by the genetic operators that work on the chromosomes in which the parameters of possible solutions are encoded. In each generation of the GA, the new solutions replace the solutions
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Proposed type-2 fuzzy regression model
In this section, first the proposed model steps have been listed, and then the model is explained in two examples. Fig. 2 shows a flowchart of this model. The proposed algorithm for the type2 fuzzy regression models is listed below. 1. Selecting a model of type-1 fuzzy regression (Hojati's model). 2. Choosing observations of type-1. 3. Selecting observations of type-2. 4. Applying the model of type-1 fuzzy regression to observations of type-1 and type-2. 5. Using operators to the FLRGs for each of observations. 6. Defuzzifying the predictions. 7. Computing predictions for model of type-2. 8. Evaluating the model efficiency. Two examples are used to describe this model. GFR(t) is the input of model and GFR(t + 1) is the output of model.
3.1.
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GFR forecasting
In this research about 500 files of diabetic patients were studied and checked. 106 patients that had attained some level of nephropathy were chosen for this research. The clinical records of these patients were followed for about 5 years. Of the 106 patients, 47 who regularly went to the diabetic clinic were used for this research as GFR values must be recorded orderly. The proposed model in more than 90% of patients presents the best results compared to other models. Two patients that have the best results with the proposed model were reported in this paper as examples. In this study, our aim is to predict nephropathy due to priori of diabetic patient. It is expected that using past GFR data models could predict future GFR with good accuracy. In fact, the condition of kidney could be predicted using proposed model. GFR(t) and GFR(t + 1) is input and output of model respectively. In the proposed model, 70% of the data was used as train data and 30% as the test value. GFR has three common formulas. As mentioned previously, these three formulas are one reason for using type-2 fuzzy sets to consider the effects of all three formulas.
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Weight GFR1 ¼ ð140AgeÞ 72 pcr
(10)
GFR2 ¼ 175ðpcrÞ1:154 ðAgeÞ:203 ð:742 if femaleÞ
(11)
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(12)
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Chronic kidney disease (CKD) stages based on GFR level are shown in Table 1. In proposed model, type-1 and type-2 observations were required. The mean of three formulas in every t (every referral
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GFR3 ¼
ð0:41HeightÞ pcr
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Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003
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Fig. 2 – Flowchart of T2FRFTS model.
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of patients) was used as a type-1 observation because the mean of three formulas is a good choice for GFR prediction if type-1 fuzzy regression is used. For the type-2 fuzzy model, type-2 observations were required. Minimum and maximum of GFR1, GFR2, GFR3 were chosen as the type-2 observation. These type-1 and type-2 observations make type 2 fuzzy model in steps that are explained in the following section. Proposed model steps have been applied for GFR forecasting. Step 1: Choosing a model of type-1 fuzzy regression (Hojati's model). In this study, Hojati's model was chosen. It provides relatively simpler calculations and better forecasting results. Step 2: Selecting a variable as observation of type-1. The mean of three GFR equation results has been chosen as the main value in the type-1 fuzzy model. This value was selected as type-1 observation.
Table 1 – CKD Stage based on GFR level. CKD Stage Stage Stage Stage Stage Stage
1 2 3 4 5
GFR level (mL/min/1.73 m2) ≥90 60–89 30–59 15–29 <15
Step 3. Selecting observations of type-2 In three equations, the minimum and maximum outcomes are selected as observations of type-2. For example, in the fourth value, GFR1 = 56.07, GFR2 = 71.65, GFR3 = 82.00; the mean of them is 69.91, and that has been chosen as the type-1 observation. Low value is 56.07 and high value is 82. Step 4: Applying the model of type-1 fuzzy regression to the observations of type-1 and type-2. The type-1 fuzzy regression model can then be applied to type-1 and type-2 observations and the forecasts can be obtained. For this goal, these steps should be followed: Step 4-1: Defining fuzzy sets for observations With reference to the authors' prior model [29], triangular fuzzy sets with indeterminate supports were used. This means that triangular legs were not fixed, but were determined after the solution of an optimization problem. The GA algorithm, a powerful random global search technique to handle optimization problems and which can help find the globally optimal solution over a domain, was used for optimizing fuzzy sets. The GA tool of MATLAB was used to find the optimized fuzzy sets support and parameter / that is defined in the following steps. Initial fuzzy sets supports, [0 10], [10 20],. . ., [90 100], and /=0.5 as initial population, uniform mutation function, scattered crossover function, and stochastic uniform selection function are used in the GA tool setting. Stopping criteria was number of generations 100. Other parameters
Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003
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Table 2 – Parameters of GA algorithm.
Table 3 – GFR Forecast after ^m and _m.
Parameter
Value
Population size Mutation rate Crossover rate generations
40 0.01 0.8 100
No 4
GFR1 GFR2 GFR3
Forecast after ^m
Forecast after _m
A8
A6, A8
A6 ! A6 A8 ! A8 A9 ! A8
Then, ^m and _m are both used to the predictions of all date. In Table 3, ^m and _m are applied to all predictions, including those of observation of type-1 and type-2. Step 6. Defuzzifying the predictions The defuzzyfied prediction for ^m is equivalent to:
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deffuzzificationintersection ðno:4Þ ¼ 74:79 Correspondingly, the defuzzyfied prediction for _m is as follows: Fig. 3 – Fuzzy sets after optimization with GA.
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were used show in Table 2. After no changes in parameters, minimization was robust between about 30 runs. Min, max, mean and variance of optimal values were 3.9, 5.1, 4.38 and 0.2 respectively for patient 1 and 1.33, 2.35, 1.77 and 0.14 for patient 2. Optimized fuzzy set supports found with GA in patient 1 are (Fig. 3): [0 10.10], [0.04 20.00], [19.89 32.16], [27.96 40.00], [39.98 51.75], [46.85 60.00], [59.99 70.30], [68.56 80.01], [79.62 90.07], [90.00 100.00]. Step 4-2: Fuzzifying the observations Different GFR values can be fuzzified into the corresponding fuzzy sets. All observation of type-1 and type-2 are fuzzified in this step. For example, the GFR for the fourth value was 71.65, which was fuzzified to A8. Step 4-3: Applying model of type-1 fuzzy regression to observations of type-1 and type-2 and acquiring predictions. Hojati's model was applied to the main, high, and low values that were fuzzified before. Fig. 4 shows the results of the type-1 fuzzy regression model for type-1 and type-2 observations (second column of Table 3). Step 5: Using operators to the FLRGs for each of observations.
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defuzzificationunion ð11=8Þ ¼ ð53:42 þ 74:79Þ=2 ¼ 64:11 Step 7. Obtaining predictions for model of type-2 / was used as a coefficient in calculating type-2 forecasting [29]:
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defuzzificationðtÞ ¼ a ðdefuzzificationintersection ðtÞ þ ð1aÞðdefuzzificationunion ðtÞÞ
(13)
where a can be achieved through solving the optimization problem. This means that there is another parameter in the optimization problem. The best value of a is 0.126. Thus, it is used for calculating type-2 forecasting:
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defuzzificationðtÞ ¼ 0:874ðdefuzzificationintersection ðtÞÞ þ 0:126ðdefuzzificationunion ðtÞÞ
(14)
Step 8: Evaluating the model efficiency. The assessment of the efficiency of the model is performed by applying the RMSE:
RMSE ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX n u u ðactual valueðtÞforecasted valueðtÞÞ2 t i¼1
n
444 445 446 447 448 449
(15)
Fig. 4 – Apply type-1 fuzzy regression model to type-1 and type-2 values in patient 1. Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003
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Fig. 5 – GFR forecasting results in patient 1. Fig. 7 – GFR forecasting results in patient 2.
Table 4 – Forecasts of the GFR in patient 1. No.
Actual
Huarng and Yu
WGP
Shafaei
T2FRFTS
.. . 37.28 11.28 9.46 8.16 7.48 .. .
.. . 33.3 20 15 15 15 .. .
.. . 31.5 18.3 11.2 10.3 10.3 .. .
.. . 35 15 7.87 7.87 7.87
.. . 33.98 11.02 10.02 10.02 10.02
.. . 8 9 10 11 12 .. .
Table 6 – Comparison of RMSE of GFR forecasting in patient 2. Model Train-RMSE Test-RMSE
Hojati
Huarng and Yu
WGP
Shafaei
T2FRFTS
7.32 8.88
4.12 5.25
2.88 5.17
2.50 4.53
1.33 2.65
Table 5 – Comparison of RMSE of GFR forecasting in patient 1. Model Train-RMSE Test-RMSE
452 450 451 453 454 455
Hojati
Huarng and Yu
WGP
Shafaei
T2FRFTS
15.10 24.42
13.29 23.50
13.01 23.11
8.97 15.57
3.91 6.85
Figs. 4 and 5 and Tables 3 and 4 show the results for patient 1. Figs. 6 and 7 show the results for patient 2. Tables 5 and 6 show a comparison of the RMSE of several models. The results indicate that the suggested model was able to perform
predictions relatively more accurately in two patients. The RMSE in both train and test values is better than other models. Such an accurate prediction is very helpful for diabetic patients and diabetes clinics. As mentioned before, 47 patients were used in this research. Min, max, mean and variance of RMSEs between 47 patients were 1.33, 20.20, 8.67, and 19.86 respectively. In more than 90% of patient proposed model has better performance and minimum RMSE. The two patients that were reported had the best results with proposed model.
Fig. 6 – Apply type-1 fuzzy regression model to type-1 and type-2 values in patient 1. Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003
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4.
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This study presents a GFR forecasting with different perspectives to type-2 fuzzy regression models based on type-2 fuzzy time series concepts (T2FRFTS) to forecast nephropathy in diabetic patients. GFR forecasting clinically helps patients and society decrease the cost of control and treatment by optimizing the number of check-ups. Unnecessary regular check-ups worry patients, even though they may not suffer from the illness for years to come. In addition, GFR forecasting can help doctors make better decisions regarding the type and dosage of medication needed to postpone retinopathy among diabetic patients. The proposed type-2 fuzzy regression model avoids additional complexity and helps model more uncertainty in diabetic values. Therefore, the authors propose the use of such models in diabetes clinics. From the empirical analysis, it can be inferred that, in addition to its intelligible steps, this model can adequately forecast GFR.
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Conflict of interest
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None declared.
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references
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Calculation
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Please cite this article in press as: Bajestani NS, et al. Nephropathy forecasting in diabetic patients using a GA-based type-2 fuzzy regression model. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.01.003