Table of contents
2. Gray comprehensive evaluation method (gray correlation analysis)
3. Fuzzy comprehensive evaluation method
4. BP neural network comprehensive evaluation method
2. Regression prediction method
3. Time series analysis method
●Differential equation model steps:
What is an optimization problem?
What are the three categories? ? ?
Evaluation class, prediction class and optimization class
1. Evaluation algorithm
1. Analytic hierarchy process
●Basic idea:
●It is a multi-criteria decision-making and evaluation method that combines qualitative and quantitative methods. Decompose the relevant elements of decision-making into the target layer, criterion layer and program layer , and sort the pros and cons of decision-making programs through people's judgment, and conduct qualitative and quantitative analysis on this basis . It hierarchizes and quantifies people's thinking process, and uses mathematics to provide quantitative basis for analysis, decision-making, evaluation, forecasting and control.
●Basic steps:
Construct a hierarchical structure model; construct a pairwise comparison matrix; hierarchical single sorting and consistency test (that is, judge whether the subjectively constructed pairwise comparison matrix has good consistency overall); hierarchical overall sorting and consistency test (test hierarchical consistency between).
●Advantages:
It relies entirely on subjective evaluation to rank the pros and cons of solutions, requires a small amount of data, and takes a short time to make decisions. On the whole, AHP introduces quantitative analysis into the complex decision-making process, and makes full use of the preference information given by decision-makers in pairwise comparisons for analysis and decision support. It not only effectively absorbs the results of qualitative analysis, but also exerts quantitative influence. The advantage of analysis makes the decision-making process highly organized and scientific, and is especially suitable for use in decision-making analysis of socio-economic systems.
●Disadvantages
Decision-making with AHP is highly subjective. When the decision-maker's judgment is too much affected by his subjective preferences, resulting in some distortion of objective laws, the results of AHP are obviously unreliable.
●Scope of application:
●Especially suitable for situations where people's qualitative judgment plays an important role and it is difficult to directly and accurately measure the decision-making results. To make the decision-making conclusions of AHP conform to objective laws as much as possible, decision-makers must have a relatively in-depth and comprehensive understanding of the problems they face. In addition, when encountering evaluation problems with many factors and large scale, this model is prone to problems. It requires the evaluator to have a thorough grasp of the nature of the problem, the elements involved and the logical relationships between them. Otherwise, the evaluation The results are unreliable and accurate.
●Improvement methods:
(1) The pairwise comparison matrix can be obtained using the Delphi method (expert opinion method).
(2) If there are too many evaluation indicators (generally more than 9), the weight obtained by using the analytic hierarchy process will have a certain deviation, and then the results of the combined evaluation model will no longer be reliable. According to the actual situation and characteristics of the evaluation object, certain methods can be used to layer and classify the original indicators so that the number of indicators in each category is less than 9.
2. Gray comprehensive evaluation method (gray correlation analysis)
●Basic idea:
●The essence of gray correlation analysis is that the evaluation objects can be compared and sorted by using the degree of correlation between each plan and the optimal plan. The greater the correlation, the more consistent the changing trends of the comparison sequence and the reference sequence are, on the contrary, the changing trends are inconsistent. From this the evaluation results can be obtained.
●Basic steps:
●Establish the original indicator matrix; determine the optimal indicator sequence; perform indicator standardization or dimensionless processing; find the difference sequence, maximum difference and minimum difference; calculate the correlation coefficient; calculate the degree of correlation.
●Advantages:
●It is an effective model for evaluating systems with a large amount of unknown information. It is a comprehensive evaluation model that combines qualitative analysis and quantitative analysis. This model can better solve the problem that evaluation indicators are difficult to accurately quantify and count, and can eliminate the effects of human factors. influence, making the evaluation results more objective and accurate. The entire calculation process is simple, easy to understand, and easy for people to master; the data does not need to be normalized, and the original data can be used for direct calculation, which is highly reliable; the evaluation index system can be increased or decreased according to specific circumstances; there is no need for a large number of samples, as long as there are A representative small sample is sufficient.
●Disadvantages:
●Requires sample data and has time series characteristics; it only identifies the merits of the evaluation object and does not reflect the absolute level. Therefore, the
comprehensive evaluation based on gray correlation analysis has all the shortcomings of "relative evaluation".
●Scope of application:
●There are no strict requirements on the sample size, and it is not required to obey any distribution. It is suitable for problems with only a small amount of observation data; when applying this method
for indicator system and weight distribution are a key issue, and the appropriateness of the choice directly affects the final result. Evaluation results.
●Improvement methods:
●(1) Use the combined weighting method: The weight coefficient is obtained based on the combination of the objective weighting method and the subjective weighting method.
●(2) Combined with the TOPSIS method : not only pay attention to the correlation between the sequence and the positive ideal sequence, but also pay attention to the correlation between the sequence and the negative ideal sequence, and
calculate the final correlation according to the formula.
3. Fuzzy comprehensive evaluation method
●Basic idea:
●Based on fuzzy mathematics, it applies the principle of fuzzy relationship synthesis to quantify some factors with unclear boundaries and difficult to quantify, and comprehensively evaluates the status of the subordinate level (or set of comments) of the evaluated things from multiple factors. a method. Comprehensive evaluation assigns a non-negative real number evaluation index to each object based on the given conditions for all evaluation objects, and then sorts and selects the best ones accordingly.
●Basic steps:
●Determine factor sets and comment sets; construct a fuzzy relationship matrix; determine index weights for fuzzy synthesis and evaluation.
●Advantages:
●The mathematical model is simple, easy to master, and has good evaluation effect on complex problems with multiple factors and levels. The fuzzy evaluation model can not only evaluate and sort the evaluation objects according to the comprehensive score, but also evaluate the level of the object according to the value on the fuzzy evaluation set according to the principle of maximum membership. The result contains a rich amount of information. The evaluation is carried out pair by pair, and the evaluated object has a unique evaluation value, which is not affected by the object collection in which the evaluated object is located. It is close to the thinking habits and description methods of Eastern people, so it is more suitable for evaluating social and economic system problems.
●Disadvantages:
●It cannot solve the problem of duplication of evaluation information caused by the correlation between evaluation indicators. There is no systematic method to determine the membership function, and the
synthesis algorithm also needs to be further explored. The evaluation process makes extensive use of human subjective judgment. Since the determination of the weight of each factor has
a certain degree of subjectivity, in general, fuzzy comprehensive evaluation is a comprehensive evaluation method based on subjective information.
●Application scope:
●Widely used in economic management and other fields. The reliability and accuracy of the comprehensive evaluation results depend on the index selection factors, the weight
distribution of factors and the synthesis operator of the comprehensive evaluation, etc.
●Improvement methods:
●Adopt combined weighting method: The weight coefficient is obtained based on the combination of objective weighting method and subjective weighting method.
4. BP neural network comprehensive evaluation method
●Basic idea:
●It is an interactive evaluation method that can continuously modify the weight of the indicator according to the output expected by the user until the user is satisfied. Therefore, generally speaking, the results obtained by the artificial neural network evaluation method will be more consistent with the actual situation.
●Advantages:
●Neural networks have adaptive capabilities and can give an objective evaluation of multi-index comprehensive evaluation problems, which is very beneficial for weakening human factors in weight determination. In the previous evaluation methods, the traditional weight design has great ambiguity, and the human factors in the weight determination also have a great influence. With the passage of time and space, the degree of influence of each indicator on its corresponding questions may also change, and the determined initial weight may not necessarily conform to the actual situation. Furthermore, considering that the entire analysis and evaluation is a complex nonlinear large system, a weight learning mechanism must be established. These aspects are the advantages of artificial neural networks. In view of the limitations of the variable selection method in the comprehensive evaluation modeling process, the neural network principle can be used to analyze the contribution of variables, and then eliminate factors with insignificant and unimportant effects to establish a simplified model, which can avoid the influence of subjective factors on variable selection. interference.
●Disadvantages:
●The biggest problem encountered in the application of ANN is that it cannot provide analytical expressions. The weight cannot be interpreted as a regression coefficient, nor can it be used to analyze causal relationships. At present, it is not possible to explain ANN theoretically or practically. The meaning of weight. It requires a large number of training samples, the accuracy is not high, and the scope of application is limited. The biggest application obstacle is the complexity of the evaluation algorithm. People can only use computers to process it, and commercial software in this area is not mature enough.
●Scope of application:
●The neural network evaluation model has adaptive capabilities, fault tolerance, and can handle nonlinear and non-local large-scale complex systems. In the training of learning samples, there is no need to consider the weight coefficients between input factors. ANN automatically adjusts and adapts along the original connection weight through the error comparison between the input value and the expected value. Therefore, this method reflects the interaction between factors. effect.
●Improvement methods:
●Adopt combined evaluation method: For the results obtained by other evaluation methods, select part as training samples and part as test samples for testing. In this way, the neural network is trained until the requirements are met, and better results can be obtained.
2. Prediction algorithms
1. Gray prediction model
●Basic idea:
●Gray prediction is a method of predicting systems containing uncertain factors. Gray prediction identifies the degree of dissimilarity in development trends between system factors, that is, performs correlation analysis, and generates and processes original data to find the rules of system changes, generates data sequences with strong regularities, and then establishes corresponding differential equations model to predict the future development trend of things. It constructs a gray prediction model using a series of quantitative values observed at equal intervals that reflect the characteristics of the prediction object, and predicts the characteristic quantity at a certain moment in the future, or the time to reach a certain characteristic quantity.
●Basic steps:
●1) Data inspection and processing to determine whether the level ratio of the data column falls within the capacity coverage, thereby determining whether the known data column can be used for gray prediction; 2) Establish a gray model based on the prediction algorithm to obtain the predicted value; 3) Test the predicted value----residual error test, grade ratio deviation value test; 4) Give the prediction, that is, the conclusion.
●Advantages:
●When processing less eigenvalue data, the sample space of the data does not need to be large enough to solve the
problems of small historical data, sequence integrity and low reliability, and can generate irregular original data. Generate sequences with strong regularity.
●Disadvantages:
●It is only suitable for short- and medium-term forecasts and is only suitable for forecasts that approximate exponential growth.
●Scope of application:
●This model uses not the sequence of original data, but the generated data sequence. The core system is Gray Model. It is
a method of cumulative generation (or other processing generation) of original data to obtain an approximate exponential law and then modeling.
2. Regression prediction method
●The regression prediction method predicts based on the correlation between independent variables and dependent variables. The number of independent variables can be one or more. According to the number of independent variables, it can be divided into single regression prediction and multiple regression prediction. At the same time, according to the correlation between independent variables and dependent variables, it is divided into linear regression prediction method and nonlinear regression method. Learning a regression problem is equivalent to function fitting: choosing a function curve that fits the known data well and predicts the unknown data well.
●Advantages:
●1) The regression analysis method is simpler and more convenient when analyzing multi-factor models; 2) Using the regression model, as long as the model and data used are the same, a unique result can be calculated through standard statistical methods, but in the figures and tables In the form, the interpretation of the relationship between data often varies from person to person, and the fitting curves drawn by different analysts are likely to be different; 3) Regression analysis can accurately measure the degree of correlation between various factors and regression fitting. The degree of improvement improves the effect of the prediction equation;
●Disadvantages:
Sometimes in regression analysis, which factor to use and what expression the factor uses is just a guess. This affects the diversity of factors and the unmeasurability of some factors, making regression analysis limited in some cases.
●Scope of application:
●Regression analysis is suitable for a certain correlation between independent variables and dependent variables, and the relationship can be fitted by linear and nonlinear functions.
3. Time series analysis method
●The full name of the ARIMA model is called the autoregressive moving average model, and the full name is (AR|MA, Autoregressive Integrated Moving Average Model). i is also recorded as AR |MA(p, d, q), which is the most common model among statistical models used for time series prediction.
●Modeling process:
●1) Import experimental data. 2) Determine the ARMA model order. 3) Residual test. 4) Give the result
●Scope of application:
●Based on the continuous regularity of the development of objective things, we use past historical data and statistical analysis to further speculate on the future development trend of the market. Time series is at the core of time series analysis and forecasting methods.
●Advantages:
●Generally, the ARMA model is used to fit a time series and predict the future value of the time series. DanieI tests for stationarity. Auto regressive AR (Auto regressive) and moving average MA (MovingAverage) prediction models have relatively high prediction accuracy and are suitable for medium and long-term prediction problems.
●Disadvantages:
●When encountering major changes in the external world, there will often be large deviations. The time series prediction method is more effective for short- and medium-term predictions than long-term predictions.
4. Differential equations
●The differential equation model is a common and important model in our daily life. We often encounter this type of questions in our daily courses. For example, we often encounter Newton's second law. to related questions.
●Differential equation model steps:
●1) Determine the actual quantities (all required independent variables, unknown functions, necessary parameters) and determine the coordinate system.
●2) Find out the basic relationships between these quantities (physics, chemistry, biology, geometry, etc.).
●3) Use these relationships to list equations and definite solution conditions.
●Advantages:
●It is suitable for short, medium and long-term forecasts. Such as the prediction model of infectious diseases, the prediction model of economic growth (or population), and the Lanchester war prediction model.
●Disadvantages:
●Reflects the internal laws and internal relationships of things, but because the establishment of the equation is based on the assumption of independence of local laws, when used as a long-term prediction, the error is large, and the solution of the differential equation is difficult to obtain.
●Scope of application:
●Causal prediction models suitable for basic correlation principles are mostly typical problems in physics or geometry. Assumptions are made, mathematical symbols are used to express rules, and equations are listed. The solution result is the answer to the problem.
3. Optimization algorithms
What is an optimization problem?
●The so-called optimization is the discipline of selecting the most reasonable one from all possible solutions to achieve the optimal goal. In real life, whatever people do, whether they are analyzing problems or making decisions, they must use a standard to measure whether they have reached the optimal level (such as fund investment). In various scientific problems, engineering problems, production management, and social and economic problems, people always hope to obtain the maximum gain (such as insurance) at the lowest possible cost under limited resource conditions.
●In mathematics, an optimization problem has three elements: decision variables, objective function, and constraints. The optimization problem refers to adjusting the decision variables to minimize (or maximize) the value of the objective function while satisfying the constraints.
●The particle swarm algorithm is mostly used for optimization problems where the decision variables are continuous variables (its convergence speed is fast, but its ability to jump out of the local optimal solution is
relatively weak.
●Genetic algorithm (GeneticAIgorithm) is a computational model that simulates the biological evolution process of natural selection and genetic mechanisms of biological evolution
. It is a method of searching for optimal solutions by simulating the natural evolution process. It is mainly used for optimization problems where the decision variables are discrete variables
, such as integer programming, 0-1 programming, etc. Its convergence speed is relatively slow, but its ability to jump out of the local optimal solution is strong.
●The simulated annealing algorithm was born out of the physical process of nature and derived from the principle of solid annealing. It is a probability-based algorithm that
heats the solid to a sufficiently high temperature and then allows it to cool slowly. When it is heated, the internal particles of the solid rise with the temperature. It becomes disordered and the internal energy increases. As it slowly
cools, the particles gradually become more ordered and reach an equilibrium state at each temperature. Finally, they reach the ground state at normal temperature and the internal energy is reduced to a minimum.
●Relatively speaking, the simulated annealing algorithm has no constraints on the type of decision variables. It can be
solved good ability to jump out of the local optimal solution and can easily find the global optimal solution. Its disadvantage is that it only It can operate in a single thread, but cannot conduct
a large-scale search. When the dimension of the decision variable is high, the algorithm converges very slowly.
