Several common models in mathematical modeling:
(1) Forecasting and Forecasting
1. Gray prediction model (must master)
It can be used if two conditions are met:
① The number of data sample points is small, 6-15
② The data is in the form of an exponential or curve.
For example: the time point when the next stable point and extreme point will appear can be predicted through extreme points and stable points.
2. Prediction of differential equations (high-end, backup)
The relationship between the original data cannot be found directly, but the relationship between the changing speed of the original data can be found, and converted into a relationship between the original data through formula derivation. The relationship between differential equations is relatively complex, so if you are not very good at mathematics, you will generally not choose to use it. For example, I am the editor.
3. Regression analysis prediction (must master)
Find the relationship between a dependent variable and several independent variables. If the independent variable changes, find how the dependent variable changes;
the number of sample points has requirements:
① The covariance between independent variables is relatively small, preferably close to 0 , the correlation between independent variables is small;
②The number of sample points n>3k+1, k is the number of independent variables;
③The dependent variable must conform to the normal distribution
4. Markov prediction (standby)
There is no information transfer between a sequence, and there is no connection before and after. The data are highly random and do not affect each other. Today's temperature has no direct connection with yesterday or the background. To predict the probability of high, medium, or low temperature the day after tomorrow, we can only get probability
5. Time series forecasting (must master)
Complementary to Markov chain prediction, at least two points need to transmit information, AR model, MA model, ARMA model, period model, seasonal model, etc.
6. Wavelet analysis and prediction (higher level)
The data is irregular and massive. The waves are separated to separate periodic data and regular data. It can produce data that cannot be produced by time series and has a wide range of applications.
7. Neural network prediction (standby)
A large amount of data does not require a model. It only requires input and output. Black box processing is recommended as a verification method.
8. Chaos sequence prediction (higher level)
It is difficult to master and requires high mathematical skills.
(2) Evaluation and decision-making
1. Fuzzy comprehensive evaluation (frequently used, needs to be mastered)
Evaluate an object at levels of excellent, medium, or poor, evaluate a school, etc., and cannot be sorted.
2. Principal component analysis (frequently used, needs to be mastered)
Evaluate and sort the levels of multiple objects, with strong correlation between indicators
3. Analytical Hierarchy Process (AHP) (frequently used, needs to be mastered)
Make decisions, where to travel, make decisions based on indicators and comprehensive considerations
4. Data Envelopment (DEA) Analysis Method
Optimize issues and evaluate the development status of each province
5. Rank sum ratio comprehensive evaluation method (often used, needs to be mastered)
Evaluate and sort each object, the correlation between indicators is not strong
6. The distance method between superior and inferior solutions (TOPSIS method)
7. Projection pursuit comprehensive evaluation method
Combining multiple algorithms, such as genetic algorithms, optimization theory, etc.
8. Analysis of variance, analysis of covariance, etc. (frequently used, need to be mastered)
Analysis of variance: Check whether there are differences and differential effects between several types of data, for example: whether elements have an impact on wheat yield, and the amount of difference; (1992, the problem of fertilization effects on crop growth) Analysis of covariance: how
many Factors, we only consider the impact of one factor on the problem, ignore other factors, but pay attention to the dimensions of the initial data and the initial situation. (2006, Issues in the evaluation and prediction of AIDS therapies)
(3) Classification and discrimination
1. Distance clustering (system clustering) (commonly used, needs to be mastered)
2. Correlation clustering (commonly used, needs to be mastered)
3. Hierarchical clustering
4. Density clustering
5. Other clusters
6. Bayesian discrimination (statistical discrimination method, need to master)
7. Fisher’s discrimination (there are many training samples and need to be mastered)
8. Fuzzy recognition (there are relatively few data points that can be classified into good categories)
(4) Correlation and causation
1. Gray correlation analysis method (the number of sample points is relatively small)
2. Sperman or Kendall grade correlation analysis
3. Person related (the number of sample points is relatively large)
4. Copula related (more difficult, financial mathematics, probability mathematics)
5. Canonical correlation analysis (the dependent variable group Y1234, the independent variable group X1234, the correlation between the respective variable groups is relatively strong, ask which dependent variable has a closer relationship with which independent variable?)
6. Standardized regression analysis
Several independent variables and one dependent variable. Which independent variable has a closer relationship with the dependent variable?
7. Survival analysis (event history analysis) is difficult
There are missing data in the data. What factors affect the dependent variable?
8. Granger causality test
Econometrics, does last year’s x have any impact on this year’s y?
(5) Optimization and control
1. Current planning, integer planning, 0-1 planning (with constraints, determined goals)
2. Nonlinear programming and intelligent optimization algorithms
3. Multi-objective programming and goal programming (flexible constraints, objective functions, exceed)
4. Dynamic programming
5. Network optimization (multiple factors are intertwined and complex)
6. Queuing theory and computer simulation
7. Fuzzy planning (scope constraints)
8. Gray planning (difficult)
◆Mathematical modeling methods involved:
Geometric theory, current algebra, calculus, combinatorial probability, statistical (regression) analysis, optimization method (planning), graph theory and network optimization, comprehensive evaluation, interpolation and fitting, difference calculation, differential equations, queuing theory, fuzzy mathematics, Stochastic decision-making, multi-objective decision-making, stochastic simulation, gray system theory, neural network, time series, mechanism analysis and other methods.
