One of the reasons for writing this article is to summarize the algorithms I learned before to prepare for the American tournament, but to list the algorithms and explain them in a few words so that I can look them up when I need them in the future.

Mathematical modeling problems are always divided into four categories:

1. Classification problem 2. Optimization problem 3. Evaluation question 4. Prediction problem

I wrote this book based on mathematical modeling algorithm and application.

Optimization problem

Linear programming and nonlinear programming methods are the most basic classics: the idea of objective function and constraint function.

Modern optimization algorithm: tabu search; Simulated annealing; Genetic algorithm; artificial neural network

Simulated annealing algorithm;

Introduction: Research achievements of statistical mechanics of materials. Statistical mechanics shows that different structures in materials correspond to different particle energy levels. At high temperature, particles have higher energy and can move freely and rearrange. At low temperature, the particle energy is low. If the temperature drops very slowly from high temperature (this process is called annealing), the particles can reach thermal equilibrium at each temperature. When the system is completely cooled, crystals in low energy state are finally formed.

Ideas can be used to solve mathematical problems. In the process of solving, the new solution is transformed one way at a time, and then the annealing process of the state (new solution) is accepted by probability: probability transformation, and the probability is the energy /KT power of the natural base.

Genetic algorithm: Genetic algorithm is a search algorithm based on the principle of natural selection and natural genetic mechanism. Simulate the evolution mechanism of life in nature and realize the optimization of specific goals in artificial systems.

The essence of genetic algorithm is through group search technology (? ), according to the principle of survival of the fittest, finally get the optimal solution or quasi-optimal solution.

Specific implementation process (P329~33 1)

* coding

* determine the fitness function (that is, the objective function)

* Determine the evolution parameters: population size m, crossover probability Pc, mutation probability Pm and evolution termination conditions.

* coding

* Determine the initial population and use the classical improved circle algorithm.

* objective function

* cross operation

* mutation operation

* choose

Improved genetic algorithm

There are two improvements: the crossover operation is changed to the pairing based on the principle of "door-to-door matching", and the mutation operation of determining the position of bad points with chaotic sequences is separated from the crossover operation.

Two classification problems (and some multivariate analysis methods)

* support vector machine SVM

* Cluster analysis

* Principal component analysis

* discriminant analysis

* Canonical correlation analysis

Support Vector Machine (SVM): The main idea is to find a hyperplane, so that it can separate two kinds of data points as correctly as possible, and at the same time make the separated two kinds of data points farthest from the classification plane.

Cluster analysis (an extremely classic algorithm): the classification of samples is called Q- cluster analysis, and the classification of indicators is called R- cluster analysis.

Basis: the measurement of sample similarity-quantification, distance-such as the distance of Min.

Principal component analysis: its main purpose is to explain most of the variations in the original data with fewer variables, and transform many highly correlated variables into independent or irrelevant variables. Usually, several new variables and principal components are selected, which are less than the original variables and can explain the variation in most data. Essentially, it is a dimension reduction method.

Discriminant analysis: it is a statistical method to infer individual types according to the observation indexes of the individuals studied. This criterion is optimal in a sense, such as the minimum probability of misjudgment or the minimum loss of misjudgment. This method is like a general term for classification methods. Such as distance discrimination, Bayesian discrimination and Fisher discrimination.

Canonical correlation analysis: study the correlation between two groups of variables relative to calculating all correlation coefficients, find out some linear combination of two groups of variables by using the idea of similar principal components, and discuss the correlation between linear combinations.

Third, evaluation and decision-making issues.

Evaluation methods can be divided into two categories, the difference is to determine the weight: one is subjective weighting; the comprehensive information evaluation determines the weight; The other is objective weighting: the weight is determined according to the correlation of each index or the change degree of each index value.

* Ideal solution

* Fuzzy comprehensive evaluation method

* Data Envelopment Analysis Method

* Grey relational analysis method

* Principal component analysis (omitted)

* Ideal solution of rank sum ratio comprehensive evaluation method

Thought: Take the distance from the optimal solution (ideal solution) as the standard for evaluating samples.

Fuzzy comprehensive evaluation method is used in personnel assessment and other fuzzy problems. There are multi-level fuzzy comprehensive evaluation methods.

Data Envelopment Analysis (DEA) is an effective method to evaluate multi-index input and multi-index output systems. Is based on the concept of relative efficiency.

The idea of grey correlation analysis: calculate the grey weighted correlation degree between the object to be evaluated and the ideal object, similar to TOPSIS method.

Principal component analysis (omitted)

The concept of sample rank in the comprehensive evaluation method of rank sum ratio: the benefit indicators are sorted from small to large, and the cost indicators are sorted from large to small, and then the rank sum ratio is calculated, and finally statistical regression is carried out.

Four forecasting problems

* Differential equation model

* Grey prediction model

* Markov prediction

* Time series (omitted)

* interpolation and fitting (omitted)

* Neural network

Lanchester War Prediction Model. .

The main feature of the grey prediction model is that the original data series is not used, but the generated data series is used. Advantages: A large amount of data is not needed, and the essence of the system can be fully explored by using differential equations with high accuracy. Irregular raw data can be generated, and a regular generation sequence can be obtained. Disadvantages: only suitable for short-term and medium-term forecasts, only suitable for exponential growth forecasts.

Markov's prediction of the future of a system is only related to the present state and has nothing to do with the past.

Markov chain

Time-homogeneous Markov chain

Time series (omitted)

Interpolation and fitting (omitted)

Neural network (omitted)