2019年美国大学生数学建模比赛E题.pdf

美国大学生数学建模比赛

数学建模

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Contents 
Introduction ............................................................................................................... 1 
1 Background .......................................................................................................... 1 
2 Problem description ........................................................................................... 1 
3 Our work .............................................................................................................. 1 
Assumptions and Justifications ............................................................................. 2 
Symbols and Descriptions ...................................................................................... 2 
    Model I : STARMA-network model .................................................................... 3 
1 Flow chart .............................................................................................................. 3 
2 Spatio-temporal prediction model of opioid use................................................... 3 
2.1 Space-time data analysis and processing ................................................... 3 
2.2 Spatial weight matrix and adjacent hierarchy division ............................... 4 
2.3 Modeling ..................................................................................................... 5 
2.4 Model Solution ........................................................................................... 6 
2.5 Result analysis and conclusion .................................................................... 7 
Model II: Projection pursuit model ................................................................... 10 
1 Data Processing .................................................................................................... 10 
2 Establish of Projection pursuit model’s ............................................................... 11 
2.1 Modeling ................................................................................................... 11 
2.3 Result analysis and conclusion .................................................................. 13 
3 Adjustment of model I ......................................................................................... 14 
  How to fight against the opioid crisis .................................................................. 15 
1 Governance Strategy ............................................................................................ 15 
2 Validity of the test model strategy ....................................................................... 16 
  Sensitivity Analysis ................................................................................................. 18 
Strengths and Weaknesses ..................................................................................... 19 
1 Strength .............................................................................................................. 19 
2 Weaknesses ........................................................................................................ 19 
MEMORANDUM .......................................................................................................... 20 
 
 
 
 
 
 
 

Team # 1921203 Page 1 of 22

1 Introduction

1.1 Background

According to the New York Times’ estimation in June 2017 after integrated data of

American states, the number of opioid overdose deaths in America is likely to exceed

59,000, which is more than the death toll of shootings and traffic accidents across the

country[1]. Opioid abuse also affects the quantity and quality of the American workforce

and the prospects for U.S. economic growth. Goldman Sachs’ recent report showed that

the opioid abuse had become the key factor of why workers of prime age, mainly male,

unable or unwilling to find employment. President Trump told the media in 2017 that the

opioid crisis is one of the biggest challenges America met. The media called the northwest

corner of West Virginia, which borders Kentucky, Huntington, as the epicenter of the

opioid crisis. The five states around the epicenter are the “disaster zone” of drug abuse:

Ohio, Kentucky, West Virginia, Virginia, and Pennsylvania.

1.2 Problem description

According to the requirements of the problem，We need to solve the following

problems:

➢ Use the data provided by NFLIS to model the data over time.

➢ Describe the evolutionary characteristics of synthetic opioid and heroin events

between five states and their counties based on our models.

➢ Use the established models to predict the origins of opioid abuse in five states.

➢ Identify the threshold at which the drug abuse has a significant impact on society

based on the patterns of spread and characteristics the established models obtained.

➢ Use the established models to predict which state will reach the threshold we

determined at what time.

➢ Use data from the U.S. census to position the characteristics of opioid abuse

populations.

➢ Identify what factors caused the use of opioids and the growth of drug addiction, and

why many people still use opioids despite knowing their perniciousness.

➢ Modify the first part of the models based on important factors in the data set.

➢ Integrate previous conclusions to identify possible strategies to combat the opioid

crisis and model to test the effectiveness of the proposed strategies and identify any

significant parameters that result in the success (or failure) of the strategies.

1.3 Our work

In this paper, we break our work into sections as follows:

➢ Process the primary data .

➢ Establish a spatio-temporal hybrid model of neural network and STARMA (time-

space autocorrelation moving average model), and solve the synthesis of opioid and

heroin events over time.

➢ Establish a projection pursuit evaluation model, and analyze the characteristics of

opioid abusers.

➢ 4. Establish a probability model in order to verify the rationality of the model.

Team # 1921203 Page 4 of 22

In order to dealing with nonstationary sequences，we decided to use neural network

to extract the large-scale trend term. The neural network sets a hidden layer. The input

layer includes 4 variables: latitude of the state, longitude of the state, time, and quantity of

Drug Reports. The hidden layer includes 3 nodes, and each node is a nonlinear function：

sigmoid function[2]. The activation function is a linear regression function. The final

output of the predicted number of crimes is the trend term of the space-time sequence.

The parameters of the neural network model are shown in table 1. The test result of the

neural network extraction of global space-time trend term shows that the root mean square

error (RMSE) was 0.222, and the relative square root error (RSE) is 0.049. Both errors are

small, so it can be considered that the predicted values of the neural network are the trend

terms of the space-time sequence.

Table 1 The parameters of neural network model

Node1(sigmoid)

Node2(sigmoid)

Node3(sigmoid)

Regression(linear)

X: 1.105

X: 4.318

X: -18.60

Node1: -1.077

Y: 2.834

Y: -5.110

Y: 3.586

Node2: -1.370

t: 0.055

t: -0.076

t: -0.488

Node3: -1.062

Bias :-1.355

Bias: -8.475

Bias: -2.876

Threshold: 0.446

4.2.2 Spatial weight matrix and adjacent hierarchy division

According to the topology of spatial unit, there are three principles of spatial

proximity determination mentioned in the literature: Rooks、Bishops and Queens. No

matter which principle is used, the nearest neighbor is called the direct nearest neighbor

of the central unit X (n-order nearest neighbor), and on this basis, the direct nearest

neighbor forms the second order nearest neighbor of X, which can also be extended to the

n-order nearest neighbor.

Through spatial weight matrix W, we can quantify the spatial proximity between n

spatial units.

is as follows:

11 12 1

21 22

31 32

w w w







（1）

The element

in W shows the neighborhood relationship between spatial unit

and spatial unit

. If the principle of spatial proximity determination is true, the value is

1/d

; otherwise, the value is 0, that is:

1/ ,

ij ij

ot e















（2）

In response to the n-order nearest neighbor, there are also partitions of n-order

neighbor (n-order spatial weight matrix). We made following definitions and extended

them: the spatial unit

itself is defined as zero-order, and the zero-order spatial weight

matrix is denoted as

(0)

; the first-order refers to the direct nearest neighbor region of

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