![International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 4 Issue 1 || January. 2016 || PP-21-24
www.ijmsi.org 21 | Page
Mathematical Model of Affinity Predictive Model for Multi-Class
Prediction
Hamdan O. Alanazi1, 2
, Abdul Hanan Abdullah 1
, Moussa Larbani3
1
Department Computer Science, Faculty of Computing, Universiti Teknologi Malaysia, 81310 Johor,Malaysia
2
Faculty of Applied Medical Science, Al-Majmaah University, Kingdom of Saudi Arabia
3
Faculty of Economics, IIUM University, Jalan Gombak, Kuala Lumpur, 53100, Malaysia
Abstract: The notion of affinity which is one of the predictive models can bedefined as the distance or closeness
between two objects.Unlike the fuzzy Set and Rough Set, the affinity can deal with third objects and deals with
time dimension. In addition, it could deal with entities or abstract side by side with real objects. However, the
existing model of affinity is developed for binary classification or prediction. In this paper, Affinity Predictive
Modelhas been proposed in order to provide a multi-classprediction. This developed method can be used in
many applications when multi-classpredictions are needed.
Keywords: Affinity Set, Predictive Model, Multi –Class Prediction
I. INTRODUCTION
owadays, machine learning relates to computational models to improve the performance through automating
the attainment of knowledge from past experience. Based on [1], the term predictive analytics might be used
swapped with the term predictive model. In addition, the term pattern recognition, predictive model, machine
learning and predictive analytics are used interchangeably [1-3]. It is important to predictive models to predict
future outcomes [4]. Predictive models have many uses, including guiding healthcare policy; determining study
eligibility of patients for new treatments; selecting appropriate tests and therapies in individual patient
management including supporting decisions. Chen and Larbani initiated the theory of the affinity set and
defined this set as the distance between two objects, where the distance measurement could be real or abstract
[5]. This means the affinity set is a natural liking for or attraction to objects or abstracts. In order for this to
happen, the affinity needs two elements namely the subjects between whom the affinity takes place and the
affinity itself. Unlike the fuzzy Set and Rough Set, the affinity can deal with third objects and deals with time
dimension. A notable feature is that it could deal with entities or abstract side by side with real objects.Huang et
al. have used Affinity Set for measuring the performance of non-profit organization [6]. In the study of [7], they
have provided a topology concept of Affinity Set as the data mining tool to classify and focus on the key
attributes causing delayed diagnosis. Study of [8]provided a topology concept of Affinity Set as the data mining
tool to classify and focus on the key attributes causing delayed diagnosis.Yuhet al.introduced the first
classification model by using affinity Set [9]. The existing models of Affinity Predictive Models would not be
able to provide a multi-class prediction. Thus, theAffinity Predictive Models should be improved to provide a
multi-class prediction. The objective of this paper is to develop a mathematical model of affinity predictive
model for multi-class prediction.
II. Mathematical Model Of AffinityPredictive Model For Multi- Class Prediction
Abstractly, Multi-class Affinity is a predictive model: given a problem instance to be classifiedusing affinity
between entities, represented as following:
i. Definition of Affinity
Consider x and y are two entities. Then, the Affinity degree between entity (x) and entity (y) can be defined as
follow:
𝐴𝐹𝑥
𝑦
When 𝐴𝐹𝑥
𝑦
= 1: Means the entity (x) has a very strong relationship with the entity (y).
When 𝐴𝐹𝑥
𝑦
= 0: Means the entity (x) has not a relationship with the entity y.
When 0 ≤ 𝐴𝐹𝑥
𝑦
≤ 1, Means that (x) has a relationship with the entity (y)
N](https://image.slidesharecdn.com/d041021024-160912064849/85/Mathematical-Model-of-Affinity-Predictive-Model-for-Multi-Class-Prediction-1-320.jpg)
![Mathematical Model of AffinityPredictiveModelfor Multi…
www.ijmsi.org 22 | Page
ii. Definition of rule set
Rules in our rule set can be defined as follows:
𝑅 𝑘 : 𝑥1 𝑖𝑠 𝐹1
𝑘
… . . 𝑥 𝑛 𝑖𝑠 𝐹𝑛
𝑘
𝑇ℎ𝑒𝑛 𝐶𝑙𝑎𝑠𝑠 𝑖𝑠 𝑂1 𝑤𝑖𝑡ℎ 𝐴𝐹1
𝑘
… 𝑎𝑛𝑑 𝑂𝑐 𝑤𝑖𝑡ℎ 𝐴𝐹𝑐
𝑘
Where 𝑅 𝑘 is the k-th rule1 ≤ 𝑘 ≤ 𝐿 and𝑥1, . . . , 𝑥 𝑛 are the input variables (Features), 𝐿 is number of rules and
𝐹1
𝑘
, … , 𝐹𝑛
𝑘
are the discrete values of the input variables, 𝑂𝑗 is the class label, where j = {1,…,c} where c is
number of classes. 𝐴𝐹𝑗
𝑘
is the affinity degree of the rule 𝑅 𝑘 with the class 𝑂𝑗 .
iii. Generating all possible rules
Generate possible rules where the number of rules 𝑁𝑝 = 𝑛1 ∗ … ∗ 𝑛 𝑛 , where 𝑛1, … , 𝑛 𝑛 represent the number of
discrete values of the input variables 𝑥1, . . . , 𝑥 𝑛 , respectively.
iv. Affinity between rules and classes through training set
Finding the Affinity between rules and classes through training set can be illustrated as following:
1. Calculate the frequency 𝐹𝑅𝑗
𝑘
which is the number of 𝑅 𝑘 occurrences in the training data set forclass 𝑂𝑗 .
2. Finding the Affinity Degree 𝐴𝐹𝑘𝑐𝑗
𝑘
between each rule 𝑅 𝑘 and the class pattern 𝑂𝑗 using the following
formula:
𝐴𝐹𝑘𝑐𝑗
𝑘
=
𝑊𝑘𝑗
𝑘
∗ 𝐹𝑅𝑗
𝑘
𝑊𝑘𝑗
𝑘
× 𝐹𝑅𝑗
𝑘𝑐
𝑗=1
, 𝑗 = 1, … , 𝑐
This can be formulated as:
𝐴𝐹𝑘𝑐𝑗
𝑘
=
𝑊𝑘𝑗
𝑘
∗ 𝐹𝑅𝑗
𝑘
𝐹𝑅1
𝑘
× 𝑊1
𝑘
+ ⋯ + 𝐹𝑅 𝑐
𝑘 × 𝑊𝑐
𝑘
, 𝑗 = 1, … , 𝑐
Where the weights of rule 𝑅 𝑘 with the class pattern 𝑂𝑗 , where Wkj
k
whereWkj
k
∈ [0,1], 𝑊𝑘𝑗 = 1c
j=1
v. Affinity between rules and classes through other rules within 𝒄𝒐𝒓𝒆 − 𝒓
Finding the Affinity between rules and classes through other rules 𝐴𝐹klc within 𝑐𝑜𝑟𝑒 − 𝑟 can be illustrated as
following:
1. Finding the Affinity between Rules
Affinity between two rules 𝑘 and 𝑙 can be calculated with this equation
𝐴𝐹𝑘𝑙
𝐹𝑚
𝑛
𝑚=1
𝑛
Where 𝐹𝑚 = 1 𝑤ℎ𝑒𝑛 𝐹𝑖
𝑘
= 𝐹𝑖
𝑙
otherwise 𝐹𝑚 = 0 , Where 𝐹𝑖
𝑘
, 𝐹𝑖
𝑙
are the discrete values of the i-th input
variable for the rules 𝑘 and 𝑙, respectively where 𝑖 ∈ [1, n]
2. Finding the rule list within assigned core
Finding the core-set of the k-th rule 𝐶𝑆 𝑅 𝑘
among the total number of rules NR where the affinity between
𝑅 𝑘 and these rules are greater or equal than 𝑐𝑜𝑟𝑒 − 𝑟where 𝑐𝑜𝑟𝑒 − 𝑟 ∈ 0,1 . These rules are selected from the
all rules𝑇𝑅_𝑙𝑖𝑠𝑡 as follows:
𝐶𝑆 𝑅 𝑘
= {0};
𝑖𝑛𝑑𝑒𝑥_𝑟1 = 1;
For 𝑖𝑛𝑑𝑒𝑥_𝑟 =1: NR
𝐼𝑓 𝐴𝐹𝑘𝑙 ≥ 𝑐𝑜𝑟𝑒 − 𝑟
𝐶𝑆 𝑅 𝑘
{𝑖𝑛𝑑𝑒𝑥_𝑟1} = 𝑇𝑅_𝑙𝑖𝑠𝑡 𝑖𝑛𝑑𝑒𝑥_𝑟 + 𝐶𝑆 𝑅 𝑘
𝑖𝑛𝑑𝑒𝑥_𝑟1 = 𝑖𝑛𝑑𝑒𝑥_𝑟1 + 1;](https://image.slidesharecdn.com/d041021024-160912064849/85/Mathematical-Model-of-Affinity-Predictive-Model-for-Multi-Class-Prediction-2-320.jpg)
![Mathematical Model of AffinityPredictiveModelfor Multi…
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𝐴𝐹𝑘𝑠𝑐𝑗
𝑘
=
𝑊𝑠𝑗
𝑖𝑛𝑑𝑒𝑥 _𝑠
∗ 𝐹𝑆𝑗
𝑖𝑛𝑑𝑒𝑥 _𝑠
𝑊𝑠𝑗
𝑖𝑛𝑑𝑒𝑥 _𝑠
× 𝐹𝑆𝑗
𝑖𝑛𝑑𝑒𝑥 _𝑠𝑐
𝑗=1
+ 𝐴𝐹𝑘𝑠𝑐𝑗
𝑘
End
Where the weights of rule 𝑅 𝑘 with the class pattern 𝑂𝑗 , where S_list is the number of rules in 𝐶𝑆𝑆 𝑅 𝑠
and Wsj
s
whereWkj
s
∈ [0,1], 𝑊𝑠𝑗 = 1c
j=1
x. Affinity between rules and classes through Frequentist probability
Finding the Affinity between rules and classes through Frequentist probability can be found with this equation:
𝐴𝐹𝑘𝑓𝑐 𝑘
𝑗
=
𝑁𝑗
𝑁
Where 𝑁𝑗 is the number of training patterns of class 𝑂𝑗 and 𝑁is the total number of training patterns
xi. Final Affinity between rules and classes through all affinity relationships
Finding Final Affinity 𝐴𝐹𝐹 between each rule 𝑅 𝑘 and class 𝑂𝑗 through all affinity relationships can be found
with this equation:
𝐴𝐹𝐹𝑗
𝑘
=
𝐴𝐹𝑘𝑗
𝑘
+ 𝐴𝐹𝑘𝑙𝑐𝑗
𝑘
+ 𝐴𝐹𝑘𝑠𝑐𝑗
𝑘
+ 𝐴𝐹𝑘𝑓𝑐𝑗
𝑘
𝑁_𝐴𝑓𝑓
Where 𝐴𝐹k: Affinity between rules and classes through training set, 𝐴𝐹klc: Affinity between rules and classes
through other rules within𝑐𝑜𝑟𝑒 − r , 𝐴𝐹𝑘𝑠𝑐: Affinity between rules and classes through Super rules within
core − s, 𝐴𝐹kfc: Affinity between rules and classes through Frequentist probability, 𝑁_𝐴𝑓𝑓 : Number of
Affinity relationships which equals to 4 and 1 ≤ k ≤ 𝐿 and 1 ≤ j ≤ 𝑐
III. CONCLUSION
Predictive models are using widely for providing aclassification and prediction. The affinity predictive model
can be used for classification and prediction problems. A review of related literature reveals that existing models
of Affinity Predictive provide a binary classification. In multi-class classification, the input is to be classified
into one, and only one, of non-overlapping outcomes. Indeed, the multi- class prediction is needed in many
applications. Therefore, a mathematical model of Affinity have been presented in this paper.
REFERENCES
[1] Burke, Jason (2013). Health analytics: Gaining the insights to transform health care. Vol. 71. John Wiley & Sons,
[2] Omer A., Dominique L.(2015). Predictive Marketing: Easy Ways Every Marketer Can Use Customer Analytics and Big Data,
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[3] Hildebrandt M. and Katja D.(2013). Privacy, Due Process and the Computational Turn: The Philosophy of Law Meets the
Philosophy of Technology. Routledge.
[4] Vogenberg, F. Randy(2009). "Predictive and prognostic models: implications for healthcare decision-making in a modern
recession." American health & drug benefits 2.6: 218.
[5] Chen Y, Larbani M (2006) Developing the affinity set and its applications. In: Proceeding of the Distinguished Scholar
Workshop by National Science Council, July, pp. 14-18.
[6] Huang, Wen-Tsung, and Yuh-Wen Chen(2012). "Qualitative data envelopment analysis by affinity set: a survey of subjective
opinions for NPOs." Quality & Quantity: 1-15.
[7] Chen, Y.W., Larbani, M., Shen, C.M. and Chen, C.W.(2009) "Using Affinity Set on Finding the Key Attributes of Delayed
Diagnosis." Applied Mathematical Sciences 3.7: 297-316.
[8] Larbani, Moussa, and Yuh-WehChen (2009). "A Fuzzy Set Based Framework for Concept of Affinity." Applied Mathematical
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[9] Chen YW, Larbani M, Hsieh CY, Chen CW (2009) Introduction of affinity set and its application in data-mining example of
delayed diagnosis. Expert SystAppl 36:10883-10889.](https://image.slidesharecdn.com/d041021024-160912064849/85/Mathematical-Model-of-Affinity-Predictive-Model-for-Multi-Class-Prediction-4-320.jpg)
Mathematical Model of Affinity Predictive Model for Multi-Class Prediction
- 1. International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 4 Issue 1 || January. 2016 || PP-21-24 www.ijmsi.org 21 | Page Mathematical Model of Affinity Predictive Model for Multi-Class Prediction Hamdan O. Alanazi1, 2 , Abdul Hanan Abdullah 1 , Moussa Larbani3 1 Department Computer Science, Faculty of Computing, Universiti Teknologi Malaysia, 81310 Johor,Malaysia 2 Faculty of Applied Medical Science, Al-Majmaah University, Kingdom of Saudi Arabia 3 Faculty of Economics, IIUM University, Jalan Gombak, Kuala Lumpur, 53100, Malaysia Abstract: The notion of affinity which is one of the predictive models can bedefined as the distance or closeness between two objects.Unlike the fuzzy Set and Rough Set, the affinity can deal with third objects and deals with time dimension. In addition, it could deal with entities or abstract side by side with real objects. However, the existing model of affinity is developed for binary classification or prediction. In this paper, Affinity Predictive Modelhas been proposed in order to provide a multi-classprediction. This developed method can be used in many applications when multi-classpredictions are needed. Keywords: Affinity Set, Predictive Model, Multi –Class Prediction I. INTRODUCTION owadays, machine learning relates to computational models to improve the performance through automating the attainment of knowledge from past experience. Based on [1], the term predictive analytics might be used swapped with the term predictive model. In addition, the term pattern recognition, predictive model, machine learning and predictive analytics are used interchangeably [1-3]. It is important to predictive models to predict future outcomes [4]. Predictive models have many uses, including guiding healthcare policy; determining study eligibility of patients for new treatments; selecting appropriate tests and therapies in individual patient management including supporting decisions. Chen and Larbani initiated the theory of the affinity set and defined this set as the distance between two objects, where the distance measurement could be real or abstract [5]. This means the affinity set is a natural liking for or attraction to objects or abstracts. In order for this to happen, the affinity needs two elements namely the subjects between whom the affinity takes place and the affinity itself. Unlike the fuzzy Set and Rough Set, the affinity can deal with third objects and deals with time dimension. A notable feature is that it could deal with entities or abstract side by side with real objects.Huang et al. have used Affinity Set for measuring the performance of non-profit organization [6]. In the study of [7], they have provided a topology concept of Affinity Set as the data mining tool to classify and focus on the key attributes causing delayed diagnosis. Study of [8]provided a topology concept of Affinity Set as the data mining tool to classify and focus on the key attributes causing delayed diagnosis.Yuhet al.introduced the first classification model by using affinity Set [9]. The existing models of Affinity Predictive Models would not be able to provide a multi-class prediction. Thus, theAffinity Predictive Models should be improved to provide a multi-class prediction. The objective of this paper is to develop a mathematical model of affinity predictive model for multi-class prediction. II. Mathematical Model Of AffinityPredictive Model For Multi- Class Prediction Abstractly, Multi-class Affinity is a predictive model: given a problem instance to be classifiedusing affinity between entities, represented as following: i. Definition of Affinity Consider x and y are two entities. Then, the Affinity degree between entity (x) and entity (y) can be defined as follow: 𝐴𝐹𝑥 𝑦 When 𝐴𝐹𝑥 𝑦 = 1: Means the entity (x) has a very strong relationship with the entity (y). When 𝐴𝐹𝑥 𝑦 = 0: Means the entity (x) has not a relationship with the entity y. When 0 ≤ 𝐴𝐹𝑥 𝑦 ≤ 1, Means that (x) has a relationship with the entity (y) N
- 2. Mathematical Model of AffinityPredictiveModelfor Multi… www.ijmsi.org 22 | Page ii. Definition of rule set Rules in our rule set can be defined as follows: 𝑅 𝑘 : 𝑥1 𝑖𝑠 𝐹1 𝑘 … . . 𝑥 𝑛 𝑖𝑠 𝐹𝑛 𝑘 𝑇ℎ𝑒𝑛 𝐶𝑙𝑎𝑠𝑠 𝑖𝑠 𝑂1 𝑤𝑖𝑡ℎ 𝐴𝐹1 𝑘 … 𝑎𝑛𝑑 𝑂𝑐 𝑤𝑖𝑡ℎ 𝐴𝐹𝑐 𝑘 Where 𝑅 𝑘 is the k-th rule1 ≤ 𝑘 ≤ 𝐿 and𝑥1, . . . , 𝑥 𝑛 are the input variables (Features), 𝐿 is number of rules and 𝐹1 𝑘 , … , 𝐹𝑛 𝑘 are the discrete values of the input variables, 𝑂𝑗 is the class label, where j = {1,…,c} where c is number of classes. 𝐴𝐹𝑗 𝑘 is the affinity degree of the rule 𝑅 𝑘 with the class 𝑂𝑗 . iii. Generating all possible rules Generate possible rules where the number of rules 𝑁𝑝 = 𝑛1 ∗ … ∗ 𝑛 𝑛 , where 𝑛1, … , 𝑛 𝑛 represent the number of discrete values of the input variables 𝑥1, . . . , 𝑥 𝑛 , respectively. iv. Affinity between rules and classes through training set Finding the Affinity between rules and classes through training set can be illustrated as following: 1. Calculate the frequency 𝐹𝑅𝑗 𝑘 which is the number of 𝑅 𝑘 occurrences in the training data set forclass 𝑂𝑗 . 2. Finding the Affinity Degree 𝐴𝐹𝑘𝑐𝑗 𝑘 between each rule 𝑅 𝑘 and the class pattern 𝑂𝑗 using the following formula: 𝐴𝐹𝑘𝑐𝑗 𝑘 = 𝑊𝑘𝑗 𝑘 ∗ 𝐹𝑅𝑗 𝑘 𝑊𝑘𝑗 𝑘 × 𝐹𝑅𝑗 𝑘𝑐 𝑗=1 , 𝑗 = 1, … , 𝑐 This can be formulated as: 𝐴𝐹𝑘𝑐𝑗 𝑘 = 𝑊𝑘𝑗 𝑘 ∗ 𝐹𝑅𝑗 𝑘 𝐹𝑅1 𝑘 × 𝑊1 𝑘 + ⋯ + 𝐹𝑅 𝑐 𝑘 × 𝑊𝑐 𝑘 , 𝑗 = 1, … , 𝑐 Where the weights of rule 𝑅 𝑘 with the class pattern 𝑂𝑗 , where Wkj k whereWkj k ∈ [0,1], 𝑊𝑘𝑗 = 1c j=1 v. Affinity between rules and classes through other rules within 𝒄𝒐𝒓𝒆 − 𝒓 Finding the Affinity between rules and classes through other rules 𝐴𝐹klc within 𝑐𝑜𝑟𝑒 − 𝑟 can be illustrated as following: 1. Finding the Affinity between Rules Affinity between two rules 𝑘 and 𝑙 can be calculated with this equation 𝐴𝐹𝑘𝑙 𝐹𝑚 𝑛 𝑚=1 𝑛 Where 𝐹𝑚 = 1 𝑤ℎ𝑒𝑛 𝐹𝑖 𝑘 = 𝐹𝑖 𝑙 otherwise 𝐹𝑚 = 0 , Where 𝐹𝑖 𝑘 , 𝐹𝑖 𝑙 are the discrete values of the i-th input variable for the rules 𝑘 and 𝑙, respectively where 𝑖 ∈ [1, n] 2. Finding the rule list within assigned core Finding the core-set of the k-th rule 𝐶𝑆 𝑅 𝑘 among the total number of rules NR where the affinity between 𝑅 𝑘 and these rules are greater or equal than 𝑐𝑜𝑟𝑒 − 𝑟where 𝑐𝑜𝑟𝑒 − 𝑟 ∈ 0,1 . These rules are selected from the all rules𝑇𝑅_𝑙𝑖𝑠𝑡 as follows: 𝐶𝑆 𝑅 𝑘 = {0}; 𝑖𝑛𝑑𝑒𝑥_𝑟1 = 1; For 𝑖𝑛𝑑𝑒𝑥_𝑟 =1: NR 𝐼𝑓 𝐴𝐹𝑘𝑙 ≥ 𝑐𝑜𝑟𝑒 − 𝑟 𝐶𝑆 𝑅 𝑘 {𝑖𝑛𝑑𝑒𝑥_𝑟1} = 𝑇𝑅_𝑙𝑖𝑠𝑡 𝑖𝑛𝑑𝑒𝑥_𝑟 + 𝐶𝑆 𝑅 𝑘 𝑖𝑛𝑑𝑒𝑥_𝑟1 = 𝑖𝑛𝑑𝑒𝑥_𝑟1 + 1;
- 3. Mathematical Model of AffinityPredictiveModelfor Multi… www.ijmsi.org 23 | Page End End 3. Finding the Affinity between rules and classes through other rules within 𝑐𝑜𝑟𝑒 − 𝑟 4. Calculate the 𝐴𝐹𝑘𝑙𝑐 between 𝑅 𝑘 and class 𝑂𝑗 through the other 𝐶𝑆 𝑅 𝑘 as follows: 𝐴𝐹𝑘𝑙𝑐 = 0; For 𝑖𝑛𝑑𝑒𝑥_𝑟=1: N_list % N_list is the number of rules in 𝐶𝑆 𝑅 𝑘 𝐴𝐹𝑘𝑙𝑐 𝑘 𝑗 = 𝑊𝑘𝑗 𝑖𝑛𝑑𝑒𝑥 _𝑟 ∗ 𝐹𝑅𝑗 𝑖𝑛𝑑𝑒𝑥 _𝑟 𝑊𝑘𝑗 𝑖𝑛𝑑𝑒𝑥 _𝑟 × 𝐹𝑅𝑗 𝑖𝑛𝑑𝑒𝑥 _𝑟𝑐 𝑗=1 + 𝐴𝐹𝑘𝑙𝑐 𝑘 𝑗 End vi. Affinity between rules and classes through Super rules within 𝐜𝐨𝐫𝐞– 𝐬 Finding the Affinity between rules and classes through Super rules within core –s (𝐴𝐹𝑘𝑠𝑐) can 1. Finding super rules To find the super rules these following steps need to be followed: Find the super rules 𝑆𝑅_𝑙𝑖𝑠𝑡 through rules which occur at least one time in the training set. Calculate the frequency 𝐹𝑆𝑗 𝑘 which is the number of 𝑆𝑅_𝑙𝑖𝑠𝑡occurrences in the training data set forclass 𝑂𝑗 . vii. Finding the Affinity between rules and super rules Finding the Affinity between rules 𝑘 and super rules 𝑠 can be calculated with this equation 𝐴𝐹𝑘𝑠 𝐹𝑚 𝑛 𝑚=1 𝑛 Where 𝐹𝑚 = 1 𝑤ℎ𝑒𝑛 𝐹𝑡 𝑘 = 𝐹𝑖 𝑠 otherwise 𝐹𝑚 = 0 , Where 𝐹𝑡 𝑘 , 𝐹𝑡 𝑠 are the discrete values of the t-th input variable for the rules 𝑘 and 𝑠, respectively, where 1 ≤ t ≤ 𝑛 viii. Finding the rule list of super rules within assigned core Finding the Core-Super-Set of the k-th rule 𝐶𝑆𝑆 𝑅 𝑠 among the super rules where the affinity between 𝑅 𝑘 and these rules are greater or equal than 𝑐𝑜𝑟𝑒 − 𝑠 where 𝑐𝑜𝑟𝑒 − 𝑠 ∈ 0,1 . These rules are selected from the all super rules 𝑆𝑅_𝑙𝑖𝑠𝑡 as follows: 𝐶𝑆 𝑅 𝑘 = {0}; 𝐶𝑆𝑆 𝑅 𝑠 = 0 ; 𝑖𝑛𝑑𝑒𝑥_𝑠1 = 1; For 𝑖𝑛𝑑𝑒𝑥_𝑠 = 1 : SR 𝐼𝑓 𝐴𝐹𝑘𝑠 ≥ 𝑐𝑜𝑟𝑒 − 𝑠 𝐶𝑆𝑆 𝑅 𝑠 {𝑖𝑛𝑑𝑒𝑥_𝑠1} = 𝑆𝑅_𝑙𝑖𝑠𝑡 𝑖𝑛𝑑𝑒𝑥_𝑠 + 𝐶𝑆𝑆 𝑅 𝑠 𝑖𝑛𝑑𝑒𝑥_𝑠1 = 𝑖𝑛𝑑𝑒𝑥_𝑠1 + 1; End End ix. Finding the Affinity between rules and classes through super rules within 𝒄𝒐𝒓𝒆 − 𝒔 Calculating the 𝐴𝐹𝑘𝑠𝑐𝑗 𝑘 between 𝑅 𝑘 and class 𝑂𝑗 through the super rules 𝐶𝑆𝑆 𝑅 𝑠 as follows: 𝐴𝐹𝑘𝑠𝑐 = 0; For 𝑖𝑛𝑑𝑒𝑥_𝑠 =1: S_list
- 4. Mathematical Model of AffinityPredictiveModelfor Multi… www.ijmsi.org 24 | Page 𝐴𝐹𝑘𝑠𝑐𝑗 𝑘 = 𝑊𝑠𝑗 𝑖𝑛𝑑𝑒𝑥 _𝑠 ∗ 𝐹𝑆𝑗 𝑖𝑛𝑑𝑒𝑥 _𝑠 𝑊𝑠𝑗 𝑖𝑛𝑑𝑒𝑥 _𝑠 × 𝐹𝑆𝑗 𝑖𝑛𝑑𝑒𝑥 _𝑠𝑐 𝑗=1 + 𝐴𝐹𝑘𝑠𝑐𝑗 𝑘 End Where the weights of rule 𝑅 𝑘 with the class pattern 𝑂𝑗 , where S_list is the number of rules in 𝐶𝑆𝑆 𝑅 𝑠 and Wsj s whereWkj s ∈ [0,1], 𝑊𝑠𝑗 = 1c j=1 x. Affinity between rules and classes through Frequentist probability Finding the Affinity between rules and classes through Frequentist probability can be found with this equation: 𝐴𝐹𝑘𝑓𝑐 𝑘 𝑗 = 𝑁𝑗 𝑁 Where 𝑁𝑗 is the number of training patterns of class 𝑂𝑗 and 𝑁is the total number of training patterns xi. Final Affinity between rules and classes through all affinity relationships Finding Final Affinity 𝐴𝐹𝐹 between each rule 𝑅 𝑘 and class 𝑂𝑗 through all affinity relationships can be found with this equation: 𝐴𝐹𝐹𝑗 𝑘 = 𝐴𝐹𝑘𝑗 𝑘 + 𝐴𝐹𝑘𝑙𝑐𝑗 𝑘 + 𝐴𝐹𝑘𝑠𝑐𝑗 𝑘 + 𝐴𝐹𝑘𝑓𝑐𝑗 𝑘 𝑁_𝐴𝑓𝑓 Where 𝐴𝐹k: Affinity between rules and classes through training set, 𝐴𝐹klc: Affinity between rules and classes through other rules within𝑐𝑜𝑟𝑒 − r , 𝐴𝐹𝑘𝑠𝑐: Affinity between rules and classes through Super rules within core − s, 𝐴𝐹kfc: Affinity between rules and classes through Frequentist probability, 𝑁_𝐴𝑓𝑓 : Number of Affinity relationships which equals to 4 and 1 ≤ k ≤ 𝐿 and 1 ≤ j ≤ 𝑐 III. CONCLUSION Predictive models are using widely for providing aclassification and prediction. The affinity predictive model can be used for classification and prediction problems. A review of related literature reveals that existing models of Affinity Predictive provide a binary classification. In multi-class classification, the input is to be classified into one, and only one, of non-overlapping outcomes. Indeed, the multi- class prediction is needed in many applications. Therefore, a mathematical model of Affinity have been presented in this paper. REFERENCES [1] Burke, Jason (2013). Health analytics: Gaining the insights to transform health care. Vol. 71. John Wiley & Sons, [2] Omer A., Dominique L.(2015). Predictive Marketing: Easy Ways Every Marketer Can Use Customer Analytics and Big Data, book. DOI: 10.1002/9781119175803 [3] Hildebrandt M. and Katja D.(2013). Privacy, Due Process and the Computational Turn: The Philosophy of Law Meets the Philosophy of Technology. Routledge. [4] Vogenberg, F. Randy(2009). "Predictive and prognostic models: implications for healthcare decision-making in a modern recession." American health & drug benefits 2.6: 218. [5] Chen Y, Larbani M (2006) Developing the affinity set and its applications. In: Proceeding of the Distinguished Scholar Workshop by National Science Council, July, pp. 14-18. [6] Huang, Wen-Tsung, and Yuh-Wen Chen(2012). "Qualitative data envelopment analysis by affinity set: a survey of subjective opinions for NPOs." Quality & Quantity: 1-15. [7] Chen, Y.W., Larbani, M., Shen, C.M. and Chen, C.W.(2009) "Using Affinity Set on Finding the Key Attributes of Delayed Diagnosis." Applied Mathematical Sciences 3.7: 297-316. [8] Larbani, Moussa, and Yuh-WehChen (2009). "A Fuzzy Set Based Framework for Concept of Affinity." Applied Mathematical Sciences 3.7: 317-332. [9] Chen YW, Larbani M, Hsieh CY, Chen CW (2009) Introduction of affinity set and its application in data-mining example of delayed diagnosis. Expert SystAppl 36:10883-10889.