CN110826022A - Method for maximum likelihood ranking based on traffic data between entities - Google Patents
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Abstract
The invention discloses a method for maximum likelihood ranking based on traffic data between entities. For the traffic data matrix between the entities, firstly, a Bootstrap technology is adopted to obtain a more stable traffic matrix, and then the sequence between the entities is adjusted based on the idea of maximum likelihood estimation, so that the probability of generating the actual traffic sequence by the system is the maximum. The method can be applied to university ranking, city ranking, country ranking, and the like. Because the hierarchical relationship between the entities in reality is complex, the problems of overhigh calculation cost, difficult index collection and the like caused by excessive factors considered by the conventional ranking method are solved, and meanwhile, the selection of the indexes has subjective factors.
Description
Technical Field
The invention belongs to the field of ranking and evaluation, mainly aims to research entity ranking relations, can be applied to various entity traffic relations such as university ranking, immigration national relations, international trade positions, social positions and the like, and relates to maximum likelihood, simulation iteration and the like on a specific method.
Background
The evaluation and ranking of entities is widespread in social life, covering almost every aspect of modern life, such as sports events, attractions, wine, teacher and school evaluations, and so on. This ranking has a significant impact, which can lead to uneven resource allocation, with more resources tending to be allocated to higher ranked entities.
Take university rank as an example. University ranking is an entity evaluation problem of greater interest to the social and scientific communities, where university ranking can indicate the university's social reputation, is a relative relationship assessment between schools that has a significant impact on educational resource allocation and may ultimately affect school strength, because higher ranking helps universities attract more money, better research opportunities, more teachers, and more outstanding students, both at the national level and at the global level. Taking immigration as an example, the immigration is an important population geographic phenomenon and social phenomenon, which can cause important changes in social, economic, political, cultural, resource and environmental conditions related to the basic elements of the population in the immigration place and the immigration place, thereby causing changes in various service requirements, and the ranking of research countries in the immigration aspect can provide certain guidance and help for people with immigration requirements, and can also affect national immigration policies. In the aspect of trade status, international trade has been rapidly developed in recent years, and each country needs to make clear the status and the role of the country in the international trade system so as to guide the establishment and the modification of trade policies and enhance the prestige and the influence of the country in the international trade. In the aspect of social relations, along with the development of network technologies, social platforms are diversified, the importance of the position of each individual in the social platforms is known, so that schemes can be planned and popularized better, and the information transmission efficiency is improved. Therefore, by analyzing the relationship between the entities, the entity is facilitated to plan future development schemes, and the status or influence of the entity on the whole system is improved.
However, the relationship between the entities is complex, the evaluation index of the entity relates to the aspect of the entity field, high-dimensional data is often adopted, and the calculation is complex and the cost is high; meanwhile, for the setting of indexes, firstly, some indexes are difficult to be defined uniformly, and secondly, the subjectivity of the selection of the indexes is a great defect. In the case of university ranking, each year's university ranking is promulgated by different institutions, and the ranking methods they use in different institutions or research are diverse. Some primary ranking agencies consider funding, research level or influence, level of specialization, heat of enrollment, student selection, level of internationalization, graduate employment status, historical reputation, and other criteria, among others. For example, the world university ranking for higher education in thames ranks universities according to five categories of teaching, research, citation, industry income, and international vision; there are six main categories of indicators ranked by the british QS university: evaluating quality, academic reputation, citation, student-teacher proportion, international student proportion and international teacher proportion, and establishing a ranking method system through complex weighting integration.
Traditional methods of ranking entities have two insurmountable deficiencies: one is the data problem, although the wider the data index relates to the surface, the more information can be contained, so that the final result is more convincing, but the difficulty of collection and the complexity of calculation exist, and the cost can be over-high; another is that the selection of the index may contain many subjective factors, and different people have different selections and importance settings, resulting in lower reliability.
Currently, researchers have been looking to develop an objective, simple and effective ranking method. In 2015, Clauset et al creatively proposed that ranking was calculated based on the Minimum Violation Ranking (MVR) used by the doctor graduate to the university ranking of traffic data to any of various colleges and universities, with method results highly correlated to the us news ranking (USN ranking) and the us national research council ranking (NRC ranking), meaning that there was strong agreement between them. It is important to emphasize that the MVR model saves a lot of strict data query and computation costs, and the data used is also very objective.
Disclosure of Invention
The invention provides an idea based on maximum likelihood ranking, which can be reduced to the calculation of minimum weighted reverse edges under certain assumption. The minimum reverse edge assumption of the original MVR method can be regarded as a special case of the general new method we propose.
The main technical points are as follows:
the Bootstrap method is a simulated sampling statistical inference method based on original data, can be used for researching the distribution characteristics of certain statistic of a group of data, and is particularly suitable for the problems that interval estimation, hypothesis test and the like of parameters are difficult to derive by a conventional method. The boottrap method originally stems from the idea of: the population is difficult to obtain, since the sample can be sampled from the population, how can there be no resampling from the sample? The general sampling mode of Bootstrap is "full sampling with put back" (its sample size is also determined according to the situation, and not necessarily equal to the original sample size), and its basic idea is: the re-sampling is performed back within the range of the original data, and the probability that each observation unit in the original data is extracted every time is equal.
Maximum likelihood estimation is interpreted in the Baidu encyclopedia as a statistical method based on the principle of maximum likelihood. The maximum likelihood principle is colloquially said to be that when an experiment is performed, when the probability of a certain condition occurring is the highest, the condition is considered to be the most likely condition to occur, and the result is the most consistent with the experiment. The general steps of maximum likelihood estimation are: 1) setting a likelihood function according to the distribution; 2) taking logarithm of the likelihood function; 3) the likelihood function is made to take an extreme value by adjusting the parameter to be estimated or by deriving the parameter.
The invention forms a maximum likelihood ranking model based on the idea of maximum likelihood estimation. Firstly, taking the distribution of the collected flow data as the empirical distribution of real data, and performing 'sampling with putting back', wherein the probability of each observation unit being extracted is equal, so as to obtain more stable flow data; then according to the concrete situation, setting the probability of occurrence of the flow relation among the entities, wherein the probability is related to the ranking relation among the entities, and according to the size of the ranking difference among the entities, the probability of occurrence also has corresponding difference, and finally obtaining the likelihood function of the entity system through the given probability and the weight thereof; by selecting an appropriate optimization method, the likelihood function is made to be extremal, and since this is a sorting problem, an optimal ranking can be finally obtained by adjusting the sorting relationship between some entities.
The invention discloses a method for maximum likelihood ranking based on traffic data between entities, which comprises the following steps:
step 1, acquiring a robust flow data matrix by using a Bootstrap technology;
step 2, setting reasonable weight and giving a likelihood function;
and 3, selecting a proper optimization method to enable the likelihood function to take an extreme value to obtain the optimal ranking.
Step 1, using Bootstrap technology to obtain robust flow data matrix
1-1) simulating the empirical distribution of real flow data by using a flow adjacency matrix, and giving the occurrence probability of each flow;
1-2) sampling the data with a place back, wherein the sampling times are the total flow quantity, and the probability of each flow being sampled every time is the occurrence probability of each flow;
1-3) repeating the operation for multiple times to obtain a plurality of Bootstrap samples as more robust flow data;
step 2, setting reasonable weight and giving out likelihood function
2-1) setting the relative probability of occurrence of the relationship of traffic between entities (generally, the probability of flowing from a better ranked entity to a worse ranked entity is considered to be higher);
2-2) carrying out normalization processing on the relative probability of each flow to obtain the final jump probability;
2-3) giving proper weight to the probability generated by different traffic grade crossing relations (the probability should be related to grade crossing difficulty); 2-4) the likelihood function of the given system after integrating the data distribution and the weight;
step 3, selecting a proper optimization method to enable the likelihood function to take an extreme value to obtain the optimal ranking
3-1) selecting any one optimization algorithm such as a greedy algorithm, a genetic algorithm, simulated annealing and the like, maximizing the likelihood function of the system as much as possible, and recording the entity ranking relation at the moment;
3-2) if the maximum value of the likelihood function corresponds to a plurality of entity ranking relations, averaging all possible rankings to be used as a final entity ranking relation;
3-3) because the number of entities is often large, it is difficult to achieve the optimization result, so the optimization algorithm is repeated n times, and the final ranking is averaged to be the most likely ranking.
Advantageous effects
1. Compared with the prior entity relationship problem using complex and subjective indexes, the method actually provides a new visual angle for researching the entity relationship: complex systems have a number of entities with which the relationships between them are intricate, have a large number, and are not easily grasped directly, and traffic is often a manifestation of the relationships of the entities. By analyzing a flow, the relationship between entities on a certain feature or side can be reflected. The method mainly researches the ordering problem among the entities by analyzing the simple, objective and very representative index of the flow data reflecting the entity relationship, and has lower cost and calculation complexity;
2. compared with the initial minimum reverse method, the method integrates the idea of maximum likelihood into the method, increases the basis of statistics, applies weights corresponding to override levels of different reverse flows at the same time, is more practical, can be actually developed into the initial minimum reverse method under the condition of not setting the weights, and is more general.
Drawings
FIG. 1 is a flow diagram of a maximum likelihood ranking model;
FIG. 2 illustrates a Bootstrap operation flow diagram;
FIG. 3 is a diagram illustrating the relative occurrence of traffic between different entitiesA probability schematic diagram; (A, B, C, D, E, F, G and H in the figure denote different institution numbers, p1,p2,p3Represents the relative probability of occurrence of traffic from an institution to institutions ranked 1,2, and 3 lower, respectively, q1,q2,q3Representing the relative probability of occurrence of traffic from an institution to an institution ranked 1,2, 3 higher, respectively)
Figure 4 is a flow chart of an optimization likelihood function.
Detailed Description
The technical scheme of the invention is explained in detail by taking university ranking as an example in combination with the accompanying drawings as follows:
the idea of the invention is to perform Bootstrap processing on the collected flow data so as to obtain more stable flow data; then according to the concrete situation, setting the probability of occurrence of the flow relation among the entities, wherein the probability is related to the ranking relation among the entities, and according to the size of the ranking difference among the entities, the probability of occurrence also has corresponding difference, and finally obtaining the likelihood function of the entity system through the given probability and the weight thereof; and selecting a proper optimization method to enable the likelihood function to take an extreme value, wherein the entity ranking under the condition is the final ranking.
The basic flow of the method of the invention is shown in fig. 1, and specifically comprises the following steps:
step 1, using Bootstrap technology to obtain robust flow data matrix
When the acquired flow data is huge in quantity, the flow data can be often represented in the form of an adjacent matrix, when matlab is used for conducting Bootstrap on the data, firstly, a flow matrix is used as the overall distribution of the research data, the probability that the flow appears in each sampling is used as the proportion of the volume of each flow in the overall flow, then, the data with the same volume as the original data sample is extracted to be used as a Bootstrap sample to analyze the data, the operation is repeated for multiple times to acquire a plurality of Bootstrap samples, and the specific operation process is shown in FIG. 2.
Step 2, setting reasonable weight and giving out likelihood function
There is some relationship between the ranking relationships between entities and the traffic relationships between entities, and typically traffic is directed from higher ranked entities to lower ranked entities, e.g., university doctor graduates are more likely to attend lower ranked institutions than their graduates. Thus, when the probability is set, a doctor graduate is considered to flow from a higher ranked entity to a same or lower ranked entity with a probability of p, whereas the doctor graduate flows from a lower ranked entity to a higher ranked entity with a probability of q. For example, fig. 3 shows a probability diagram based on the flow direction of doctor graduate employment, 8 schools are listed in the diagram, the schools are ranked from left to right and are sequentially decreased (the left school is better), wherein the thickness of the flow arc represents the size of the flow, the flow above the coordinate axis represents the forward flow conforming to the basic assumption, the flow from the better school flows to the poorer school, the flow below the coordinate axis represents the reverse flow violating the basic assumption, the flow from the poorer school flows to the better school, and the flow marked with emphasis in the diagram represents the flow direction from the graduation of the university of the E institution to the employment of other schools.
2-1) as the difference between the rankings of the two entities increases, both p and q gradually decrease, and how to decrease the selection is related to the selection corresponding to the weight, considering that the difficulty of flowing from the school with better ranking to the school with less ranking than the school with less ranking is different, and conversely, the difficulty of flowing from the school with less ranking to the school with more ranking than the school with less ranking is different, so p is decomposed into p0,p1,p2,...,pN-1Decomposing q into q1,q2,...,qN-1Where the relative probability of flow in an institution of the same rank is set to p0And the other means the relative probability of the flow to the institutions with the lower or higher ranking of 1,2, … … and N-1 respectively, and finally the relative probability of the occurrence of each flow is shown as the formula (1).
Wherein, Pj(u → v) represents the probability that graduate j graduates from college u and goes to college v for employment, and u represents the graduateThe school number, v represents the number of the employment institution, piiRepresenting the rank of institution i, smaller values represent higher ranks, and q is assumed to be1>q2>...>qN-1，p0>p1>p2>...>pN-1。
2-2) is provided with ZuTo normalize constant, have
Normalizing the relative probability to obtain the final jump probability Pj_z(u → v) is as shown in formula (2).
2-3) setting given an appropriate, rational weight, setting p1,p2,...,pN-1,q1,q2,...,qN-1And p0The likelihood function of the whole system is obtained by combining the probability of each person in the working flow direction and the number of people in the flow direction, as shown in formula (3).
Where A is the set of all graduates. Omega is the set of universities and colleges, nu→vRepresents the number of persons from graduation u of an institution and leaving v of the institution, nu→allRepresents the number of persons who go from the graduation of an institution u to the incumbent of all institutions (including u), u-(N-1),...,u-1,u0,u+1,...,u+(N-1)Respectively represent institutions which are 1, …, N-1 lower than u, institutions which are the same rank as u, and collections of institutions which are 1, …, N-1 higher than u.
Let τ bei＝qi/p0,i＝1,2,...N-1，υi＝pi/p0N-1, the normalization constants are simplified, some
Then
The likelihood function is subjected to simplified operation to obtain equation (4).
Wherein n is+1，n-1Representing the number of all doctor graduates ranked 1 higher and 1 lower in the employment institution than in the graduate institution, respectively, n+2，n-2Representing the number of all doctor graduates ranked 2 higher and 2 lower in the employment institution than in the graduate institution, n+(N-1)，n-(N-1)Representing the number of all doctor graduates ranking N-1 higher and N-1 lower in the employment institution than in the graduate institution, in fact N-iN-1 denotes all "forward flows" that meet the assumption, N+iN-1 represents all "reverse flows" that violate the assumption. By this simplification, the original flow problem is converted into a weighted flow problem, and L and p in the final simplified result0Independently of the weight coefficient τ onlyiAnd upsiloniIn this regard, by setting an appropriate weight, a final likelihood function of the system can be obtained.
Step 3, selecting a proper optimization method to enable the likelihood function to take an extreme value to obtain the optimal ranking
3-1) selecting any one optimization algorithm such as a greedy algorithm, a genetic algorithm, simulated annealing and the like, and enabling the likelihood function of the system to be maximum as much as possible, namely as shown in a formula (5).
Where r represents the rank of the institution, nrRepresents the total number of persons who have a division of the institution with the row name of r and go to any institution (including the colleges).
Given a target likelihood function, running an optimization program by using matlab, wherein the optimization program is shown in fig. 4, and the specific steps are as follows: 1) giving an entity sequence, and calculating a likelihood function value under the sequence by integrating the probability and the weight set in the step 2; 2) randomly exchanging sequences of some two entities, calculating likelihood function values under the new sequences, comparing the likelihood function values with the new sequences, and taking a larger likelihood function value and a sequence corresponding to the larger likelihood function value; 3) repeating the step 2) for a plurality of times until the more optimal likelihood function value can not be generated after enough exchanges are carried out, and taking the sequence corresponding to the current optimal likelihood function value as the optimal sequence.
3-2) if the maximum value of the likelihood function obtained finally corresponds to a plurality of entity sequences, taking the average value of all possible sequences as the final entity ranking relation;
3-3) because the number of entities is often large, it is difficult to find the optimal result by traversing all the sequences, in order to avoid trapping a local optimal mechanism, a method of repeating the optimization algorithm for n times is adopted, and the average value of all the ranking results is taken as the final ranking.
Claims (4)
1. The method is characterized in that firstly, the distribution of the collected flow data is taken as the empirical distribution of real data, sampling with putting back is carried out, and the probability of each observation unit being extracted is uniform, so that more stable flow data is obtained; then according to the concrete situation, setting the probability of occurrence of the flow relation among the entities, wherein the probability is related to the ranking relation among the entities, and according to the size of the ranking difference among the entities, the probability of occurrence also has corresponding difference, and finally obtaining the likelihood function of the entity system through the given probability and the weight thereof; by selecting a proper optimization method, the likelihood function is enabled to take an extreme value, and the optimal ranking is finally obtained by adjusting the ordering relation among the entities, and the method comprises the following specific steps:
step 1, acquiring a robust flow data matrix by using a Bootstrap technology;
step 2, setting reasonable weight and giving a likelihood function;
and 3, selecting a proper optimization method to enable the likelihood function to take an extreme value to obtain the optimal ranking.
2. The method of claim 1, wherein step 1 comprises:
2-1) simulating the empirical distribution of real flow data by using a flow adjacency matrix, and giving the occurrence probability of each flow;
2-2) sampling the data with the place back, wherein the sampling times are the total flow quantity, and the probability of each flow being sampled every time is the occurrence probability;
2-3) repeating the operation for multiple times to obtain a plurality of Bootstrap samples as more robust flow data.
3. The method of claim 1, wherein the step 2 comprises:
3-1) setting the relative probability of occurrence of the relationship of the traffic among the entities, wherein the probability of flowing from the entity with the better rank to the entity with the poorer rank is generally considered to be high;
3-2) carrying out normalization processing on the relative probability of each flow to obtain the final jump probability;
3-3) giving proper weight to the relative probability generated by different override relationships, wherein the relative probability is related to the override difficulty;
3-4) likelihood function of given system after integrating data distribution and weight.
4. The method of claim 1, wherein step 3 comprises:
4-1) selecting a greedy algorithm, a genetic algorithm and a simulated annealing algorithm, so that the likelihood function of the system takes the maximum value, and recording the entity ranking relation at the moment;
4-2) if the maximum value of the likelihood function corresponds to a plurality of entity ranking relations, averaging all possible rankings to be used as a final entity ranking relation;
4-3) when the number of entities is too large to achieve the optimization result, the optimization algorithm needs to be repeated for many times, and the optimization algorithm stops for a long enough time, and then the iteration results of all the rankings are averaged to be used as the final ranking.
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