[2023 Huashu Cup National College Student Mathematical Contest in Modeling] C-question The impact of mother's physical and mental health on infant growth 45-page paper and python code

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1 topic

A mother is one of the most important people in a baby's life, providing not only nutrition and physical protection but also emotional support and a sense of security. Adverse conditions in the mother's mental health, such as depression, anxiety, and stress, may have negative effects on the baby's cognition, emotion, and social behavior. A stressed mother can negatively affect a baby's physical and psychological development, such as affecting sleep.

The appendix presents data on 390 infants aged 3 to 12 months and their mothers. These data cover a variety of subjects, the mother's physical indicators include age, marital status, education level, gestational duration, mode of delivery, and maternal psychological indicators CBTS (Baby-related Post Traumatic Stress Disorder Questionnaire), EPDS (Edinburgh Postpartum Depression Scale) , HADS (Hospital Anxiety and Depression Scale) and indicators of infant sleep quality including duration of sleep throughout the night, number of awakenings and falling asleep patterns.

Please refer to the relevant literature, understand the professional background, build a mathematical model based on the topic data, and answer the following questions.

Many studies have shown that the mother's physical and psychological indicators have an impact on the baby's behavioral characteristics and sleep quality. I would like to ask whether there is such a rule and conduct research based on the data in the attachment.
The Infant Behavior Questionnaire is a scale used to assess infant behavioral characteristics that includes questions about the infant's emotions and responses. We divide the behavioral characteristics of infants into three types: quiet, moderate, and ambivalent. Please establish a relationship model between the baby's behavioral characteristics and the mother's physical and psychological indicators. In the last 20 groups of infants (No. 391-410) in the data table, the behavior characteristics information has been deleted. Please judge what type they belong to.
Intervention on maternal anxiety can help improve the mother's mental health, improve the quality of mother-infant interactions, and promote the cognitive, emotional, and social development of infants. The change rate of the treatment cost of CBTS, EPDS, and HADS relative to the degree of illness is proportional to the treatment cost. After investigation, the treatment costs corresponding to the two scores are given, as shown in Table 1. There is an infant whose behavioral characteristics are ambivalent, numbered 238. Please build a model and analyze how much treatment costs are needed at least to change the behavioral characteristics of the baby from the contradictory type to the moderate type?

How would the treatment plan need to be adjusted in order to change his behavioral profile to a quieter type?

Table 1. Disease score and treatment cost

CBTS	EPDS	HADS
Score	Treatment cost (yuan)	Score	Treatment cost (yuan)	Score	Treatment cost (yuan)
0	200	0	500	0	300
3	2812	2	1890	5	12500

The baby's sleep quality indicators include the duration of the whole night's sleep, the number of times of waking up, and the way of falling asleep. Please make a comprehensive evaluation of the baby's sleep quality in four categories: excellent, good, medium, and poor, and establish a correlation model between the baby's comprehensive sleep quality and the mother's physical and psychological indicators, and predict the last 20 groups (No. 391-410) of babies comprehensive sleep quality.
On the basis of Question 3, if it is necessary to make the sleep quality of baby No. 238 rated as excellent, does the treatment strategy for Question 3 need to be adjusted? How to adjust?

2 Problem Analysis

2.1 Question 1

This is a regression analysis modeling problem for the effect of mother's physical and psychological indicators on infant's behavioral characteristics and sleep quality. Infant behavioral characteristics and infant sleep duration throughout the night were used as dependent variables, while mother's physical and psychological indicators (maternal age, marital status, education level, gestational duration, mode of delivery, CBTS, EPDS, and HADS) were analyzed as independent variables . Generally, multiple linear regression models are used for modeling. The goal of a multiple linear regression model is to find a set of linear relationships that connect the independent variables to the dependent variable. The form of the regression model can be expressed as:

$Y = b 0 + b 1 * X 1 + b 2 * X2_+ ... + β n * X n + ε$
where Y represents the dependent variable (i.e. infant behavioral characteristics and infant sleep quality), X1 to Xn represent independent variables (i.e. mother’s physical and psychological indicators), β0 to βn represent regression coefficients, and ε represents the error term.

The evaluation indicators for the fitting program of the regression model include adjusted R square, AIC, BIC and so on. Note that in regression analysis, it is necessary to put forward assumptions and analyze the regression results, taking into account the coefficient estimates of each indicator and the results of its significance test. Such as the following three aspects.

Estimated value of coefficient: Indicates the influence strength of the independent variable on the dependent variable, for example, β1 represents the influence strength of the independent variable x1 on the dependent variable y, β2 represents the influence strength of the independent variable x2 on the dependent variable y, and so on.
T-value and p-value: Used to test the significance degree of each coefficient estimate. The t value can represent the ratio between the standard error of the variable and the coefficient, and the p value can represent the range of the confidence interval of the t value at the significance level. Generally speaking, when the p value is less than 0.05 or 0.01, it means that the estimated value of the coefficient is statistically significant at the significance level, that is, the independent variable has a significant impact on the dependent variable.
The fitting degree of regression equation: It can be measured by R square and adjusted R square. The R-square indicates how well the model fits the data and can be interpreted as the percentage of the variance of the independent variable to the dependent variable. Adjusting the R square takes into account the influence of the number of independent variables and has a more robust fitting effect. In addition, since the data set contains many correlation coefficients, we can also consider using principal component analysis (PCA) to reduce the dimension of independent variables, extract main features, and establish a principal component regression model.

In addition, in multiple regression analysis, it is necessary to consider whether there is multicollinearity among the various features. Before regression analysis, principal component analysis (PCA) can be used to reduce the dimension of independent variables, extract main features, and then establish a principal component regression model.

2.2 Question 2

This is a classification problem and needs to predict the class of 20 examples. The steps to build the classification model are as follows:

Data preprocessing: Perform preprocessing steps such as missing value processing, outlier processing, and standardization on the collected data.
Feature engineering: perform operations such as feature transformation, feature combination, or selection to extract more discriminative features.
Model selection: According to the characteristics of the data and the requirements of the model, select a suitable multi-category classification algorithm, such as logistic regression, support vector machine, decision tree, random forest, neural network, etc.
Model evaluation: Use cross-validation, confusion matrix, accuracy, recall, F1 score and other indicators to evaluate and optimize the model.
Model application: use the trained model to predict and classify new samples, and then the baby's behavioral feature classification results can be obtained.

2.3 Question 3

Linear regression models were first used to establish the relationship between CBTS, EPDS, and HADS scores and treatment costs. The scores of CBTS, EPDS, and HADS can be used as independent variables, and treatment costs can be used as dependent variables, and the linear relationship between scores and treatment costs can be estimated by fitting a linear regression model.

The fitted linear regression model can be used to predict the minimum cost of treatment for infants whose behavioral characteristics become intermediate. For the infant numbered 238, the corresponding treatment cost can be obtained by inputting its current CBTS, EPDS and HADS scores into the linear regression model. Then, the CBTS, EPDS, and HADS scores can be adjusted one by one, and the corresponding treatment costs can be calculated until the infant's behavioral characteristics change from ambivalent to moderate. Record the treatment cost of each adjustment, and finally choose the smallest treatment cost as the answer.

To change the infant's behavioral characteristics into a quiet type, further adjustments to the CBTS, EPDS, and HADS scores are required. You can use some library functions with optimization methods to make adjustments. With the goal of minimizing the total treatment cost, the constrained optimization method can be used to set the scores of CBTS, EPDS and HADS as variables, and set constraints to make the infant's behavior characteristics reach the quiet type. Then use the optimization algorithm to solve the problem of minimizing the total treatment cost, and get the corresponding CBTS, EPDS and HADS scores and the minimum treatment cost.

2.4 Question 4

This is a clustering problem, using cluster analysis to classify infant sleep quality into four categories: excellent, good, fair and poor. According to the indicators of the baby's sleep time throughout the night, the number of times of waking up, and the way of falling asleep, the baby's sleep quality can be classified by clustering algorithms with a specified K value such as K-Means and Birch. All samples in the dataset are divided into four groups using four classes as the number of clusters. Each group represents a category of sleep quality.

In addition, on the basis of the above classification, label the data, and use methods such as linear regression to establish a relationship model between the baby's comprehensive sleep quality and the mother's physical and psychological indicators. The independent variables of the regression model include the mother's physical indicators (such as age, pregnancy time, etc.) and psychological indicators (such as CBTS, EPDS, HADS score), and the dependent variable is the baby's comprehensive sleep quality score. The regression model obtained by training can be used to predict the comprehensive sleep quality score of infants in the last 20 groups (No. 391-410).

2.5 Question Five

On the basis of question 3, predict the number 238 after adjustment, which category is the current sleep quality under the characteristics of the quiet type after the adjustment of the diagnosis and treatment plan. If it belongs to excellent, there is no need to adjust it, and if it belongs to others, it needs to be adjusted.

3 Introduction to the paper

Research on infant sleep quality based on machine learning combination model

Summary

This article aims to explore the influence of mother's mental health status on infant development, especially the association with infant sleep quality. As one of the most important people in a baby's life, the mother not only provides nutrients and physical protection, but also provides emotional support and a sense of security for the baby. However, poor maternal mental health, such as depression, anxiety, and stress, may negatively affect infants' cognition, emotion, and social behavior, and even affect their sleep quality. Therefore, we collected data on 390 infants aged 3 to 12 months and their mothers, including maternal physical indicators (age, marital status, education, gestational duration, mode of delivery) and maternal psychological indicators (CBTS, EPDS, HADS ). In addition, we recorded sleep quality metrics for infants, including how long they slept throughout the night, how many times they woke up, and how they fell asleep.

By analyzing these data, we hope to reveal the relationship between maternal mental health and infant sleep quality, and to learn more about other potential associated factors. The research results are of great significance for in-depth understanding of mother-infant relationship and formulating corresponding solutions.

This paper attempts to establish various combination models such as structural equation model, decision tree and random forest to explore the influence of mother's physical and mental health on infant growth, and proposes solutions to improve the quality of mother-infant interaction and infant sleep quality.

For question one, we used structural equation modeling and Spearman correlation analysis to explore whether the mother's physical and psychological indicators have an impact on the infant's behavioral characteristics and sleep quality. In this question, the structural equation model consists of two parts, the measurement model and the structural model, which are used to describe the relationship between observed variables and latent variables. Then, we use Spearman correlation analysis to test whether there is a statistically significant relationship between the variables, and analyze the positive and negative correlation coefficient and the degree of correlation, and finally draw a conclusion: the mother's physical indicators and psychological indicators have a significant impact on Infants' behavioral characteristics and sleep quality are affected.

For problem 2, we use a variety of machine learning models to judge the baby's behavior characteristics information, such as: logistic regression, random forest, XGBOOST, neural network, etc., and finally select the model with the highest performance index as the random forest, and use the TPE algorithm Carry out global optimization, create Seaborn heat map and histogram for visualization, and finally get the behavioral characteristic information types of the last 20 groups (No. 391-410) of babies.

For question three, we divide the model into two parts. In the first part, we build a prediction model that can predict the behavioral characteristics of infants based on the three variables of CBTS, EPDS, and HADS; in the second part, we configure CBTS, EPDS , HADS treatment combination so as to realize the transformation of the infant's behavioral characteristics. In the selection of models, we choose XGBoost model, decision tree model, random forest model, LGBM model, Adaboost model and GBOT model. Finally, the decision tree model with the highest model performance was selected, and it was finally concluded that it would cost at least 1445 yuan for treatment, which could change the behavioral characteristics of the baby from contradictory to moderate. When CBTS, EPDS, and HADStt are 13, 21, and 18 respectively, the treatment cost is 3119 yuan, and the characteristics can be changed into quiet type.

For questions 4 and 5, we also divide the model into two parts. The first model is used to evaluate the baby's sleep quality. It is mainly constructed based on the baby's sleep quality indicators, and then the weight is determined by the AHP and the entropy weight method. Then, a comprehensive evaluation model was constructed by the rsr rank sum ratio method and topsis, and finally the sleep quality score was obtained. The second model is constructed based on comprehensive sleep quality scores, mother's physical indicators, and psychological indicators, and finally predicts the comprehensive sleep quality of the last 20 groups of babies.

Key words: mother mental health infant development structural equation model correlation analysis random forest decision tree
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