2020年数学建模B题小美赛论文.pdf

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Symbol Definition 6

I. Introduction 7

1.1 Problem Description 7

1.2 Terminology and Definitions 7

1.2.1What is the oxygen saturation? 7

1.2.2What is the Body Mass Index (BMI)? 8

1.2.3 Clinical application of oxygen saturation monitoring technology. 8

II. The Description of the Problem 9

2.1 The impact of age on oxygen saturation 9

2.2 The impact of BMI on oxygen saturation 9

2.3 The impact of gender on oxygen saturation 10

2.4 The impact of Smoking on oxygen saturation 10

2.5 The essay construction 11

III. Assumption 11

3.1 Assumptions 11

IV. Models 12

4.1 ModelsⅠ：Regression analysis model 12

4.1.1 The principle of binary Logistic regression 12

4.1.2 The principle of multiple linear regression 13

4.1.3 Analysis of binary logistics regression model 14

4.1.4 The foundation of Multiple linear regression model 16

4.1.5 Analysis of the Result 18

4.1.6 Strength and Weakness 19

4.2 ModelⅡ：Back Propagation Algorithm (BP Algorithm) 20

4.2.1 The principle of BP Algorithm 20

4.2.2 Description of BP Algorithm in Mathematics 20

4.2.3 Modification of Weight Value 21

4.2.4 The foundation of BP neural network model 22

4.2.5 Analysis of the Result 25

4.2.6 Strength and Weakness 28

IV. Conclusions 29

4.1 Conclusions of the problem 29

4.2 Methods used in our models 29

4.3 Applications of our models 30

V. Future Work 30

5.1 General overview of our models 30

5.2 Limitations and the way to improved it 30

VI. References 31

VII. Appendix 32

Symbol Definition

I. Introduction

1.1 Problem Description

1.2 Terminology and Definitions

1.2.1What is the oxygen saturation?

1.2.2What is the Body Mass Index (BMI)?

1.2.3 Clinical application of oxygen saturation monitoring technology.

II. The Description of the Problem

2.1 The impact of age on oxygen saturation

2.2 The impact of BMI on oxygen saturation

2.3 The impact of gender on oxygen saturation

2.4 The impact of Smoking on oxygen saturation

2.5 The essay construction

III. Assumption

3.1 Assumptions

IV. Models

4.1 ModelsⅠ：Regression analysis model

4.1.1 The principle of binary Logistic regression

4.1.2 The principle of multiple linear regression

4.1.3 Analysis of binary logistics regression model

4.1.4 The foundation of Multiple linear regression model

4.1.5 Analysis of the Result

4.1.6 Strength and Weakness

4.2 ModelⅡ：Back Propagation Algorithm (BP Algorithm)

4.2.1 The principle of BP Algorithm

4.2.2 Description of BP Algorithm in Mathematics

4.2.3 Modification of Weight Value

4.2.4 The foundation of BP neural network model

4.2.5 Analysis of the Result

4.2.6 Strength and Weakness

IV. Conclusions

4.1 Conclusions of the problem

4.2 Methods used in our models

4.3 Applications of our models

V. Future Work

5.1 General overview of our models

5.2 Limitations and the way to improved it

VI. References

VII. Appendix

第九届“认证杯”数学中国数学建模国际赛承诺书我们仔细阅读了第九届“认证杯”数学中国数学建模国际赛的竞赛规则。我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。我们知道，抄袭别人的成果是违反竞赛规则的, 如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。如有违反竞赛规则的行为，我们将受到严肃处理。我们允许数学中国网站(www.madio.net)公布论文，以供网友之间学习交流，数学中国网站以非商业目的的论文交流不需要提前取得我们的同意。我们的参赛队号为：2216 我们选择的题目是： B 参赛队员 (签名) ：队员 1：王嘉琪队员 2：于子航队员 3：胡柄宇参赛队教练员 (签名)：陈万忠

Team # 2216 Page 2 of 33 第九届“认证杯”数学中国数学建模国际赛编号专用页参赛队伍的参赛队号：（请各个参赛队提前填写好）： 2216 竞赛统一编号（由竞赛组委会送至评委团前编号）：竞赛评阅编号（由竞赛评委团评阅前进行编号）：

Team # 2216 Page 3 of 33 The Variability of Oxygen Saturation Abstract: The variability of oxygen saturation plays a significant role in mon- itoring various physiological indicators of the human body. In this paper, we establish two models to describe the patterns of oxygen saturation. In the process of analyzing this problem, we find there are four pa- rameters which can impact the variability of oxygen saturation more or less. Therefore, we find that there are two problems that we need to solve. Question 1: we should find out the influence of gender, age, smok- ing status and BMI on the change of blood oxygen saturation. Question 2: we need to find one or more models to help us predict and analyze the change of blood oxygen saturation, that is, when we get a group of subject’s information about gender, age, smoking status and BMI, we can get the general oxygen saturation of people who meet these physical conditions. Though we ran into several obstacles while trying to solve the prob- lems, we all solved them successfully. In view of the above two problems, we propose two solutions. Solution1: We use SPSS analysis tools and Binary Logistic Re- gression Analysis to analyze the OR values of four parameters. We can obtain some reliable scientific basis. We use these bases to transform text information into numerical information and input the transformed infor- mation into multiple linear regression model for correlation analysis to obtain the influence coefficient. Finally, a mathematical expression which can represent blood oxygen saturation is obtained. After a large number of data testing, the error between the predicted and actual SaO2 values is controlled within 1%. Solution2: We used the BP prediction model to divide 36 subjects into 5:1 group. The age, gender, smoking status and BMI of 30 subjects were used as the training set input, and the parameter values of 6 subjects were used as the testing set input. The experiment shows that the BP pre- diction model which we constructed can well predict the oxygen satura- tion value.

Team # 2216 Page 4 of 33 Key words: SPSS, BP Prediction Model, Logistic regression model, , Multiple linear regression, MATLAB

Team # 2216 Page 5 of 33 Contents Symbol Definition ................................................................................................................ 6 I. Introduction .......................................................................................................................... 7 1.1 Problem Description ..................................................................................................... 7 1.2 Terminology and Definitions ........................................................................................ 7 1.2.1What is the oxygen saturation? ............................................................................... 7 1.2.2What is the Body Mass Index (BMI)? .................................................................... 8 1.2.3 Clinical application of oxygen saturation monitoring technology. ...................... 8 II. The Description of the Problem ........................................................................................ 9 2.1 The impact of age on oxygen saturation ...................................................................... 9 2.2 The impact of BMI on oxygen saturation ................................................................... 9 2.3 The impact of gender on oxygen saturation .............................................................. 10 2.4 The impact of Smoking on oxygen saturation .......................................................... 10 2.5 The essay construction ................................................................................................ 11 III. Assumption ...................................................................................................................... 11 3.1 Assumptions ................................................................................................................. 11 IV. Models .............................................................................................................................. 12 4.1 ModelsⅠ：Regression analysis model ........................................................................ 12 4.1.1 The principle of binary Logistic regression ......................................................... 12 4.1.2 The principle of multiple linear regression ......................................................... 13 4.1.3 Analysis of binary logistics regression model...................................................... 14 4.1.4 The foundation of Multiple linear regression model .......................................... 16 4.1.5 Analysis of the Result ........................................................................................... 18 4.1.6 Strength and Weakness ........................................................................................ 19 4.2 ModelⅡ：Back Propagation Algorithm (BP Algorithm) ........................................ 20 4.2.1 The principle of BP Algorithm ............................................................................ 20 4.2.2 Description of BP Algorithm in Mathematics ..................................................... 20 4.2.3 Modification of Weight Value .............................................................................. 21 4.2.4 The foundation of BP neural network model ...................................................... 22 4.2.5 Analysis of the Result ........................................................................................... 25 4.2.6 Strength and Weakness ........................................................................................ 28 IV. Conclusions ...................................................................................................................... 29 4.1 Conclusions of the problem ........................................................................................ 29 4.2 Methods used in our models ....................................................................................... 29 4.3 Applications of our models ......................................................................................... 30 V. Future Work ..................................................................................................................... 30 5.1 General overview of our models ................................................................................ 30 5.2 Limitations and the way to improved it .................................................................... 30 VI. References ........................................................................................................................ 31 VII. Appendix ........................................................................................................................ 32

Team # 2216 Page 6 of 33 Symbol Definition variable Definition xk gk ρ R )(nV j )(nd C jiw η ix iw je the matrix of current weight value and threshold value the gradient of current function Significant Goodness of fit Local induction domain Expected response vector All neurons in the output layer Synaptic weights Reverse learning rate influencing factors weight The homeopathic energy of the neuron in the hidden layer

Team # 2216 Page 7 of 33 I. Introduction 1.1 Problem Description Humans obtain most of the energy for various life activities through aerobic respiration. Thus, in the medical field, measuring the variability of oxygen saturation plays a significant role in monitoring various physi- ological indicators of the human body. It is believed that many factors affect our body’s oxygen saturation, such as age, BMI, gender, Smoking history and/or current smoking status, and any signiﬁcant medical conditions. In order to indicate the problems about how these conditions affect oxygen saturation, the fol- lowing background is worth mentioning. 1.2 Terminology and Definitions 1.2.1What is the oxygen saturation? Blood oxygen saturation is the concentration of oxygen in the blood. In the field of clinical medicine, measuring the oxygen carrying capacity of blood and judging the level of oxygen content in individual blood is reflected by the index of blood oxygen saturation, because it represents the ratio of the actual oxygen content in a certain unit of blood to the maximum oxygen content carried by hemoglobin in blood. In order to unify the detection site, the arterial oxygen saturation ( SaO ) is usually taken as the evaluation standard, that is, the percentage of hemoglobin bound oxygen capacity in arterial blood to the maximum oxygen capacity that hemoglobin can combine [1]: C 2 (1) SaO 2 = C HbO 2 HbO 2 + C Hb × 100% The results show that the oxygen saturation of arterial blood of healthy people is 98% and the oxygen saturation of venous blood is 75% [2]. There are 8 parts of foot wrist, finger, toe, back of hand, instep, auricle, nose and forehead, where are no significant difference between pulse ox- ygen saturation and arterial oxygen saturation.

Team # 2216 Page 8 of 33 1.2.2What is the Body Mass Index (BMI)? Body mass index (BMI) is a formula used to assess a person’s body weight relative to height. This information is crucial in evaluating a per- son’s risk for heart disease [3]. Use either the table provided or calculate utilizing formula below. Fig 1.1 Massive data of BMI 1.2.3 Clinical application of oxygen saturation monitoring technology. Pulse oximetry is routinely used for monitoring patients’ oxygen saturation levels. In medical institutions, any patient with hypoxemia risk can use pulse oximeter for monitoring. Therefore, pulse oximeter has been widely used in emergency room, operating room, clinical anesthesia, monitoring treatment, emergency transportation and other environment. Continuous pulse oxygen saturation monitoring can help doctors to un- derstand the degree of hypoxia and provide effective treatment for pa- tients with pulmonary diseases and respiratory disorders.

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