Table of contents
2. Gray comprehensive evaluation method (gray correlation degree 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?
Which three categories? ? ?
Evaluation class, prediction class and optimization class
1. Evaluation Algorithms
1. Analytic Hierarchy Process
●Basic idea:
●It is a multi-criteria decision-making and evaluation method combining qualitative and quantitative. Decompose the relevant elements of decision-making into target layer, criterion layer and program layer , and rank the pros and cons of decision-making programs through people's judgment, and conduct qualitative and quantitative analysis on this basis . It stratifies and quantifies people's thinking process, and uses mathematics to provide quantitative basis for analysis, decision-making, evaluation, forecasting and control.
●Basic steps:
Construction of hierarchical structure model; construction of pairwise comparison matrix; hierarchical single sorting and consistency test (that is, judging whether the subjectively constructed pairwise comparison matrix has good consistency on the whole); hierarchical total sorting and consistency test (test consistency among them).
●Advantages:
It completely relies on subjective evaluation to rank the pros and cons of the schemes, requires a small amount of data, and takes a very short time for decision-making. On the whole, AHP introduces quantitative analysis in the complex decision-making process, and makes full use of the preference information given by decision-makers in pairwise comparisons for analysis and decision-making support. It not only effectively absorbs the results of qualitative analysis, but also exerts quantitative Therefore, the decision-making process is highly organized and scientific, and it is especially suitable for use in the decision-making analysis of social and economic systems.
●Disadvantages
Decision making with AHP is highly subjective. When the judgment of decision makers is too much affected by their subjective preferences, resulting in some kind of distortion of objective laws, the results of AHP are obviously unreliable.
●Scope of application:
●Especially suitable for occasions where people's qualitative judgment plays an important role and it is difficult to directly and accurately measure the decision-making results. To make AHP's decision-making conclusions conform to objective laws as much as possible, decision-makers must have a relatively in-depth and comprehensive understanding of the problems they are facing. In addition, when encountering a large-scale evaluation problem with many factors, the 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 relationship between them, otherwise the evaluation The results are not reliable and accurate.
●Improvement method:
(1) The pairwise comparison matrix can be obtained by Delphi method (expert opinion method).
(2) If there are too many evaluation indicators (generally more than 9), the weight obtained by using the AHP will have a certain deviation, and then the result of the combined evaluation model will no longer be reliable. According to the actual situation and characteristics of the evaluation objects, certain methods can be used to stratify 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 degree analysis)
●Basic idea:
●The essence of gray relational 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 degree, the more consistent the change trend of the comparison sequence and the reference sequence is, otherwise, the change trend is contrary. From this, evaluation results can be obtained.
●Basic steps:
●Establish the original index matrix; determine the optimal index sequence; carry out index standardization or dimensionless processing; find the difference sequence, the maximum difference and the minimum difference; calculate the correlation coefficient; calculate the correlation degree.
●Advantages:
●It is an effective model for evaluating a system with a large amount of unknown information. It is a comprehensive evaluation model combining qualitative analysis and quantitative analysis. To make the evaluation result more objective and accurate. The entire calculation process is simple, easy to understand, and easy for people to grasp; the data does not need to be normalized, and the original data can be used for direct calculation, which has strong reliability; the evaluation index system can be increased or decreased according to the specific situation; there is no need for a large number of samples, as long as there are A small representative sample will suffice.
●Disadvantages:
●Requires sample data and has time series characteristics; it only distinguishes the pros and cons of the evaluation object, and does not reflect the absolute level, so the
comprehensive evaluation based on gray relational analysis has all the shortcomings of "relative evaluation".
●Scope of application:
●There is no strict requirement 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 this method is used
for index system and weight distribution are a key issue, and whether the selection is appropriate or not will directly affect the final result. Evaluation results.
●Improvement method:
●(1) Combination weighting method is adopted: the weighting coefficient is obtained based on the combination of objective weighting method and subjective weighting method.
●(2) Combined with the TOPSIS method : not only pay attention to the correlation degree between the sequence and the positive ideal sequence, but also pay attention to the correlation degree between the sequence and the negative ideal sequence, and
calculate the final correlation degree according to the formula.
3. Fuzzy comprehensive evaluation method
●Basic idea:
●Based on fuzzy mathematics, applying the principle of fuzzy relationship synthesis, quantifying some unclear and difficult-to-quantify factors, and comprehensively evaluating the status of the evaluation object's membership level (or comment set) from multiple factors a method of According to the given conditions, the comprehensive evaluation assigns a non-negative real number evaluation index to each object, and then sorts and selects the best according to the given conditions.
●Basic steps:
● Determine the factor set and comment set; construct the fuzzy relationship matrix; determine the weight of the index for fuzzy synthesis and evaluation.
●Advantages:
●The mathematical model is simple and easy to grasp, and it has a good evaluation effect on complex problems with multiple factors and multiple levels. The fuzzy evaluation model can not only evaluate and sort the evaluation objects according to the size of the comprehensive score, but also evaluate the grade of the object according to the principle of the maximum membership degree according to the value on the fuzzy evaluation set, and the result contains rich information. The evaluation is carried out pair by pair, and has a unique evaluation value for the evaluated object, and is not affected by the object set where the evaluated object is located. It is close to the thinking habits and description methods of Orientals, 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
synthetic algorithm needs to be further explored. The evaluation process uses a lot of human subjective judgment, because the determination of the weight of each factor has
a certain degree of subjectivity, so in general, fuzzy comprehensive evaluation is a comprehensive evaluation method based on subjective information.
●Application range:
●Widely used in economic management and other fields. The reliability and accuracy of the comprehensive evaluation results depend on the selection factors of indicators, the weight
distribution of factors, and the synthetic operators of comprehensive evaluation.
●Improvement method:
●Adopt combination weighting method: According to the combination of objective weighting method and subjective weighting method, the weighting coefficient is obtained.
4. BP neural network comprehensive evaluation method
●Basic idea:
●It is an interactive evaluation method, which 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 in line with the actual situation.
●Advantages:
●Neural network has self-adaptive ability, and can give an objective evaluation to multi-indicator comprehensive evaluation problems, which is very beneficial for weakening the human factors in weight determination. In the previous evaluation methods, the traditional weight design has a lot of ambiguity, and the influence of human factors in the weight determination is also great. With the passage of time and space, the degree of influence of each index on its corresponding question may also change, and the determined initial weight may not 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 exactly the advantages of artificial neural networks. In view of the limitations of the variable selection method in the comprehensive evaluation modeling process, the principle of neural network can be used to analyze the contribution of variables, and then eliminate the insignificant and unimportant factors to establish a simplified model, which can avoid the interference of subjective factors on variable selection .
●Disadvantages:
The biggest problem encountered in the application of ANN is that it cannot provide an analytical expression, and the weight value cannot be interpreted as a regression coefficient, nor can it be used to analyze the causal relationship. At present, it is not possible to explain the ANN theoretically or practically. meaning of value. A large number of training samples are required, the accuracy is not high, and the application range is limited. The biggest application obstacle is the complexity of the evaluation algorithm, which can only be processed by computers, and commercial software in this area is not yet mature enough.
●Scope of application:
●The neural network evaluation model has self-adaptability and fault tolerance, and can deal with nonlinear and non-local large-scale complex systems. In the training of learning samples, there is no need to consider the weight coefficients between the input factors. ANN automatically adjusts and adapts along the original connection weights through the error comparison between the input value and the expected value. Therefore, this method reflects the interaction between factors. effect.
●Improvement method:
●Adopt combination evaluation method: For the results obtained by other evaluation methods, select a part as training samples and a part as samples to be tested for inspection, so that the neural network can be trained until the requirements are met, and better results can be obtained.
2. Forecasting algorithms
1. Gray forecasting model
●Basic idea:
●Grey prediction is a method to predict the system 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 constructs a gray prediction model with a series of quantitative values reflecting the characteristics of the predicted object observed at equal time intervals, 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, judging whether the level ratio of the data column falls within the coverage, so as to judge whether the known data column can be gray predicted; 2) Establish a gray model according to the prediction algorithm to obtain the predicted value; 3) Test the predicted value----residual error test, scale deviation test; 4) Give the forecast or conclusion.
●Advantages:
●When dealing with less eigenvalue data, the sample space of the data does not need to be large enough to solve the
problems of less historical data, sequence integrity and low reliability, and can generate irregular original data to obtain Generate sequences with strong regularity.
●Disadvantages:
●It is only suitable for short-term and medium-term forecasts, and is only suitable for forecasts that approximate exponential growth.
●Scope of application:
● The model uses not a sequence of raw data, but a sequence of generated data. The core system is Gray Model. It is
a method of accumulating raw data (or generated by other processing) to obtain an approximate exponential law and then modeling.
2. Regression prediction method
●Regression forecasting method is 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 unary regression prediction and multiple regression prediction. At the same time, according to the correlation between the independent variable and the dependent variable, it can be divided into linear regression prediction method and nonlinear regression method. The learning of regression problems is equivalent to function fitting: choose a function curve to fit the known data well and predict the unknown data well.
●Advantages:
●1) The regression analysis method is more simple and convenient when analyzing multi-factor models; 2) Using the regression model, as long as the model and data used are the same, the only result can be calculated by 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 level of the degree improves the effect of the forecasting equation;
●Disadvantages:
Sometimes in regression analysis, which factor to choose and which expression to use is just a guess, which 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 this 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 denoted as AR|MA(p, d, q), which is the most common model used for time series forecasting in statistical models.
●Modeling process:
●1) Import experimental data. 2) Determine the order of the ARMA model. 3) Residual error test. 4) Give the result
●Scope of application:
●According to the continuous regularity of the development of objective things, 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.
●Advantages:
●Generally use ARMA model to fit time series and predict the future value of the time series. DanielI test for stationarity. Automatic regression AR (Auto regressive) and moving average MA (MovingAverage) forecasting model, the forecasting accuracy is relatively high, suitable for medium and long-term forecasting problems
●Disadvantages:
●When encountering large changes in the outside world, there will often be large deviations. The time series forecasting method is better for short-term forecasting than long-term forecasting.
4. Differential Equations
●The differential equation model is a common and important model in our daily life. We often involve this type of question in our usual courses. For example, we often encounter Newton’s second law to related issues.
●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 relationship 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-term, medium-term and long-term forecasts. Such as the prediction model of infectious diseases, the prediction model of economic growth (or population), the Lanchester war prediction model.
●Disadvantages:
●Reflect the internal laws and internal relations of things, but since the establishment of the equation is based on the assumption of the independence of local laws, when it is used as a long-term forecast, the error is large, and the solution of the differential equation is difficult to obtain
●Scope of application:
●The causal prediction model applicable to the principle of basic correlation is mostly a typical problem in physics or geometry, assuming conditions, expressing laws with mathematical symbols, listing equations, and the result of solving is the answer to the problem.
Three, optimization algorithm
What is an optimization problem?
●The so-called optimization refers to the discipline of choosing the most reasonable one from all possible solutions to achieve the optimal goal. In real life, when people do anything, whether it is analyzing a problem or making a decision, they must use a standard to measure whether it is optimal (such as investment by funders). In various scientific issues, engineering issues, production management, and social and economic issues, people always hope to obtain the greatest gains (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 so that the value of the objective function is minimized (or maximized) under the condition of satisfying the constraints.
● Particle swarm optimization is mostly used in optimization problems where the decision variable is a continuous variable (its convergence speed is fast, but its ability to jump out of the local optimal solution is
relatively weak.
●Genetic Algorithm (GeneticAIgorithm) is a calculation model of the biological evolution process that simulates the natural selection of the biological evolution theory and the genetic mechanism
, and is a method to search for the optimal solution by simulating the natural evolution process.
It is mainly used for optimization problems where the decision variable is a discrete variable , such as integer programming, 0-1 programming, etc. Its convergence speed is relatively slow, but its ability to jump out of local optimal solutions is strong.
●The simulated annealing algorithm is born out of the physical process in nature and comes from the principle of solid annealing. It is a probability-based algorithm. The solid is heated to a sufficient temperature and
then allowed to cool slowly. When heating, the particles inside the solid rise with the temperature When it becomes disordered, the internal energy increases, and
when it cools slowly, the particles gradually become orderly, and reach an equilibrium state at each temperature, and finally reach the ground state at room temperature, and the internal energy decreases to the minimum.
Relatively speaking, the simulated annealing algorithm has no constraints on the type of decision variables, whether it is a continuous variable or a discrete variable, it can be
solved and the ability to jump out of the local optimal solution is very good, and it is easy to find the global optimal solution. It can work in a single thread, and cannot
search in a large range. When the dimension of the decision variable is high, the convergence speed of the algorithm is very slow.
