Article directory
0 Ideas for the competition
(Share on CSDN as soon as the competition questions come out)
https://blog.csdn.net/dc_sinor?type=blog
1 Competition Information
The National Undergraduate Mathematical Contest in Modeling (hereinafter referred to as the competition) is a mass scientific and technological activity for college students across the country sponsored by the Chinese Society of Industrial and Applied Mathematics. To improve comprehensive ability, encourage students to actively participate in extracurricular science and technology activities, broaden their knowledge, cultivate creativity and cooperation awareness, and promote the reform of university mathematics teaching system, teaching content and methods.
The competition topics generally come from practical problems that have been properly simplified and processed in the fields of science and engineering technology, humanities and social sciences (including economic management). Participants are not required to master in-depth specialized knowledge in advance, and only need to have studied basic mathematics courses in colleges and universities . There is greater flexibility in the topic for contestants to develop their creativity. Participants should complete a paper (answer sheet) including model assumption, establishment and solution, calculation method design and computer implementation, result analysis and testing, model improvement, etc. according to the topic requirements. The main criteria for the competition awards are the rationality of assumptions, the creativity of modeling, the correctness of results and the clarity of written expressions.
The competition is divided into undergraduate group and specialist group. Undergraduate students can only participate in the undergraduate group competition, not the specialist group competition. College (higher vocational college) students generally participate in the competition of the specialist group, and can also participate in the competition of the undergraduate group. No matter which group of competition to participate in, it must be determined at the time of registration, and the registration group cannot be changed after the registration deadline. Students on the same team must be from the same school.
2 race time
Registration end time: September 4, 2023 at 20:00
Competition start time: September 7, 2023 (Thursday) 18:00
Competition end time: September 10, 2023 (Sunday) 20:00
3 Types of Common Problems in Modeling
While the competition questions have not been updated yet, Mr. A will summarize the mathematical models that are often used in mathematical modeling in the national competition. The questions basically belong to the following four types of problems, and the corresponding solutions are also given by Mr. A.
They are:
classification model
optimization model
predictive model
evaluation model
3.1 Classification problem
Discriminant analysis:
Also known as the "discrimination method", it is a multivariate statistical analysis method to distinguish the type of a research object according to various characteristic values of a certain research object under the condition of a certain classification.
The basic principle is to establish one or more discriminant functions according to certain discriminant criteria; use a large amount of data of the research object to determine the undetermined coefficients in the discriminant function, and calculate the discriminant index; based on this, it can be determined which category a certain sample belongs to. When a new sample data is obtained, it is necessary to determine which of the known types the sample belongs to. This type of problem belongs to the discriminant analysis problem.
Cluster analysis:
Clustering analysis or clustering is to divide similar objects into different groups or more subsets through static classification, so that the member objects in the same subset have similar attributes, which are commonly included in the coordinate system In the shorter space distance and so on.
Cluster analysis itself is not a specific algorithm, but a general task to be solved. It can be achieved with different algorithms that vary greatly in terms of understanding what constitutes clusters and how to find them efficiently.
Neural Network Classification:
BP neural network is a neural network learning algorithm. It is a hierarchical neural network composed of an input layer, an intermediate layer, and an output layer, and the intermediate layer can be extended to multiple layers. RBF (Radial Basis Function) neural network: The radial basis function (RBF-Radial Basis Function) neural network is a three-layer feed-forward network with a single hidden layer. It simulates the neural network structure in the human brain with locally adjusted, mutually overlapping receptive fields. Perceptron neural network: It is a neural network with a single layer of computational neurons, and the transfer function of the network is a linear threshold unit. It is mainly used to simulate the perceptual characteristics of the human brain. Linear neural network: It is a relatively simple neural network consisting of one or more linear neurons. A linear function is used as the transfer function, so the output can be any value. Self-organizing neural network: self-organizing neural network includes self-organizing competition network, self-organizing feature map network, learning vector quantization and other network structure forms. K-nearest neighbor algorithm: K-nearest neighbor classification algorithm is a relatively mature method in theory and one of the simplest machine learning algorithms.
3.2 Optimization problem
Linear programming:
Mathematical theories and methods for studying the extremum problems of linear objective functions under linear constraints. English abbreviation LP. It is an important branch of operations research, widely used in production planning, logistics and transportation, resource allocation, financial investment and other fields. Modeling method: List the constraints and objective function; draw the feasible region represented by the constraints; find the optimal solution and optimal value of the objective function in the feasible region.
Integer programming:
The variables in a program are restricted (in whole or in part) to integers and are called integer programs. When the variables in a linear model are restricted to integers, it is called integer linear programming. The currently popular methods for solving integer programming are often only applicable to integer linear programming. A class of mathematical programming that requires all or some of the variables in the solution of a problem to be integers. From the composition of constraints, it can be subdivided into linear, quadratic and nonlinear integer programming.
Nonlinear programming:
Nonlinear programming is a mathematical programming with nonlinear constraints or objective functions, and is an important branch of operations research. Nonlinear programming studies the extremum problem of an n-ary real function under a set of constraints of equality or inequality, and at least one of the objective function and the constraints is a nonlinear function of unknown quantity. The case where both the objective function and the constraints are linear functions is called linear programming.
Dynamic programming:
Including knapsack problem, production and operation problem, fund management problem, resource allocation problem, shortest path problem and complex system reliability problem, etc.
Dynamic programming is mainly used to solve the optimization problem of the dynamic process divided into stages by time, but some static programming (such as linear programming and nonlinear programming) that have nothing to do with time can be regarded as a multi-stage decision-making process as long as the time factor is artificially introduced , can also be solved conveniently by dynamic programming method.
Multi-objective programming:
Multi-objective programming is a branch of mathematical programming. Study the optimization of more than one objective function over a given domain. Any multi-objective programming problem consists of two basic parts:
(1) More than two objective functions;
(2) Several constraints. There are n decision variables, k objective functions, and m constraint equations, then:
Z=F(X) is a k-dimensional function vector, Φ(X) is an m-dimensional function vector; G is an m-dimensional constant vector;
3.3 Prediction problem
Regression Fit Forecast
Fitting forecasting is the process of building a model to approximate the actual data series, suitable for developmental systems. When building a model, it is usually necessary to specify a time origin and a time unit with clear meaning. Also, the model should still make sense as t tends to infinity. The significance of taking the fitting prediction as a kind of system research is to emphasize its only "symbol" nature. The establishment of a prediction model should conform to the actual system as much as possible, which is the principle of fitting. The degree of fit can be measured by least squares, maximum likelihood, and minimum absolute deviation.
gray forecast
Gray predictions are predictions made on gray systems. It is a method for predicting systems with uncertain factors. Gray prediction is to identify the degree of difference between the development trends of system factors, that is, to conduct correlation analysis, and generate and process the original data to find the law of system changes, generate a data sequence with strong regularity, and then establish a corresponding differential equation Models to predict the future development trend of things. It uses a series of quantitative values that reflect the characteristics of the predicted object observed at equal time intervals to construct a gray prediction model to predict the characteristic quantity at a certain moment in the future, or the time to reach a certain characteristic quantity.
Markov forecast: It is a method that can be used to predict the supply of internal human resources in an organization. Its basic idea is to find out the law of personnel changes in the past, so as to speculate on the trend of future personnel changes. The conversion matrix is actually The transition probability matrix, which describes the overall form of employee inflow, outflow and internal mobility in the organization, can be used as the basis for predicting internal labor supply.
BP Neural Network Prediction
The BP network (Back-ProPagation Network), also known as the back-propagation neural network, continuously corrects the network weights and thresholds through the training of sample data so that the error function decreases along the negative gradient direction and approaches the desired output. It is a widely used neural network model, which is mostly used for function approximation, model recognition and classification, data compression and time series prediction.
support vector machine method
Support Vector Machine (SVM), also known as Support Vector Network [1], is a supervised learning model and related learning algorithm that uses classification and regression analysis to analyze data. Given a set of training samples, each training sample is labeled as belonging to one or the other of the two categories. The training algorithm of a support vector machine (SVM) creates a model that assigns new samples to one of two classes, making it a non-probabilistic binary linear classifier (although in the probabilistic classification setting, there are corrections like Prato's Such methods use support vector machines). The support vector machine model represents samples as points in a map in space, so that samples with a single class can be separated as clearly as possible. All such new samples map to the same space, and it is possible to predict which class they belong to based on which side of the interval they fall on.
3.4 Evaluation Questions
AHP
It refers to taking a complex multi-objective decision-making problem as a system, decomposing the goal into multiple goals or criteria, and then decomposing it into several levels of multiple indicators (or criteria, constraints), and calculating the hierarchical single ranking ( Weights) and total ranking as a systematic method for objective (multi-indicator) and multi-scheme optimization decision-making.
Pros and cons solution distance method
Also known as the ideal solution, it is an effective multi-index evaluation method. This method constructs the positive ideal solution and negative ideal solution of the evaluation problem, that is, the maximum and minimum values of each index, and calculates the relative closeness of each scheme to the ideal scheme, that is, the distance close to the positive ideal solution and far away from the negative ideal solution. degree, to sort the schemes, so as to select the optimal scheme.
fuzzy comprehensive evaluation method
It is a comprehensive bid evaluation method based on fuzzy mathematics. The comprehensive evaluation method converts qualitative evaluation into quantitative evaluation according to the membership degree theory of fuzzy mathematics, that is, uses fuzzy mathematics to make an overall evaluation of things or objects restricted by various factors. It has the characteristics of clear results and strong system. It can solve fuzzy and difficult-to-quantify problems well, and is suitable for solving various non-deterministic problems.
Gray relational analysis method (gray comprehensive evaluation method)
For the factors between two systems, the measure of the magnitude of their correlation with time or different objects is called the degree of correlation. In the process of system development, if the changing trends of the two factors are consistent, that is, the degree of synchronous change is high, it can be said that the degree of correlation between the two is high; otherwise, it is low. Therefore, the gray relational analysis method is based on the degree of similarity or dissimilarity in the development trend between factors, that is, the "gray relational degree", as a method to measure the degree of correlation between factors.
Canonical correlation analysis method: It is an understanding of the cross-covariance matrix, and it is a multivariate statistical analysis method that uses the correlation relationship between comprehensive variable pairs to reflect the overall correlation between two groups of indicators. Its basic principle is: in order to grasp the correlation between the two groups of indicators as a whole, two representative comprehensive variables U1 and V1 are respectively extracted from the two groups of variables (respectively, the linear variables of each variable in the two variable groups Combination), using the correlation between these two comprehensive variables to reflect the overall correlation between the two sets of indicators.
Principal Component Analysis (Dimensionality Reduction)
is a statistical method. Through orthogonal transformation, a group of variables that may be correlated is converted into a group of linearly uncorrelated variables, and the converted group of variables is called the principal component. When using statistical analysis methods to study multi-variable topics, too many variables will increase the complexity of the topic. People naturally hope that the number of variables will be less and more information will be obtained. In many cases, there is a certain correlation between variables. When there is a certain correlation between two variables, it can be explained that the two variables reflect a certain overlap in the information of this topic. Principal component analysis is to delete redundant variables (closely related variables) for all the variables originally proposed, and establish as few new variables as possible, so that these new variables are irrelevant in pairs, and these new variables are reflecting The information aspect of the subject should be kept as original as possible. Trying to recombine the original variables into a new group of several comprehensive variables that are irrelevant to each other, and at the same time, according to actual needs, a few less comprehensive variables can be extracted from them to reflect as much information as possible on the original variables. The statistical method is called principal component analysis or Principal component analysis is also a method used in mathematics for dimensionality reduction.
Factor Analysis (Dimensionality Reduction)
Factor analysis refers to the study of statistical techniques to extract common factors from variable groups. It was first proposed by the British psychologist CE Spearman. He found that there is a certain correlation between the scores of students in various subjects. Students with good scores in one subject often have better scores in other subjects, so he speculates whether there are some potential common factors, or some general intelligence conditions. affect students' academic performance. Factor analysis can find hidden representative factors among many variables. Classifying variables of the same nature into one factor can reduce the number of variables and test the hypothesis of the relationship between variables.
BP Neural Network Comprehensive Evaluation Method
It is a multi-layer feed-forward network trained by the error back propagation algorithm, and it is one of the most widely used neural network models. The BP network can learn and store a large number of input-output pattern mapping relationships without revealing the mathematical equations describing the mapping relationship in advance. Its learning rule is to use the steepest descent method to continuously adjust the weights and thresholds of the network through backpropagation to minimize the sum of squared errors of the network. The topological structure of BP neural network model includes input layer (input), hidden layer (hide layer) and output layer (output layer).
4 Modeling data
Data Sharing: The strongest modeling data
