(1) securities and portfolio pricing theory
Develop the pricing theory of securities (especially derivatives such as futures and options). The mathematical method used is mainly to put forward a suitable stochastic differential equation or stochastic difference equation model to form the corresponding backward equation. The corresponding nonlinear Feynman-Kac formula is established, and a very general extended Black-Scholes pricing formula is derived from it. The backward equation will be a high-dimensional nonlinear singular equation with constraints.
This paper studies the pricing of portfolio with different maturities and yields. It is necessary to establish a mathematical model combining pricing and optimization. In the study of mathematical tools, it may be necessary to study stochastic programming, fuzzy programming and optimization algorithms.
Under the condition of incomplete market, the pricing theory related to preference is introduced.
(2) Incomplete market economy equilibrium theory (GEI)
It is planned to conduct research in the following aspects:
1. Infinite dimensional space, infinite horizontal space and infinite state.
2. Stochastic economy, no arbitrage equilibrium, change of economic structure parameters, nonlinear asset structure.
3. Innovation and design of asset securitization.
4. Friction economy
5. Corporate Behavior and Production, Bankruptcy and Bad Debt
6. Securities market game.
(3) The application of GEI's plate equilibrium algorithm and Monte Carlo method in the calculation of economic equilibrium point, the application of GEI theory in macro-control of finance, finance and economy, and the study of natural resource asset pricing and sustainable utilization under the framework of the theory of sustainable development under incomplete market conditions.
1. What is an association rule?
Before describing some details about association rules, let's look at an interesting story: "Diapers and Beer".
In a supermarket, there is an interesting phenomenon: diapers and beer are sold together. But this strange move has increased the sales of diapers and beer. This is not a joke, but a real case of Wal-Mart supermarket chain in the United States, which has always been talked about by merchants. Wal-Mart has the largest data warehouse system in the world. In order to accurately understand customers' buying habits in their stores, Wal-Mart analyzes customers' shopping behavior and wants to know what products customers often buy together. Wal-Mart's data warehouse centralizes the detailed raw transaction data of its stores. On the basis of these original transaction data, Wal-Mart uses data mining methods to analyze and mine these data. An unexpected discovery is: "The product most bought with diapers is beer!" After a lot of practical investigation and analysis, it reveals an American behavior pattern hidden behind "diapers and beer": in the United States, some young fathers often go to the supermarket to buy baby diapers after work, and 30% ~ 40% of them will buy some beer for themselves. The reason for this phenomenon is that American wives often tell their husbands to buy diapers for their children after work, and husbands will bring back their favorite beer after buying diapers. According to conventional thinking, diapers have nothing to do with beer. If we don't use data mining technology to mine and analyze a large number of transaction data, it is impossible for Wal-Mart to discover this valuable law inside the data.
Data association is an important discovery knowledge in database. If there is some regularity between the values of two or more variables, it is called correlation. Correlation can be divided into simple correlation, time series correlation and causal correlation. The purpose of association analysis is to find out the hidden association network in the database. Sometimes we don't know the correlation function of the data in the database, and even if we do, it is uncertain, so the rules generated by correlation analysis are credible. Association rule mining finds interesting associations or related relationships between itemsets in a large number of data. Agrawal is equal to 1993. Firstly, the problem of mining association rules between itemsets in customer transaction database is proposed. Later, many researchers did a lot of research on mining association rules. Their work includes optimizing the original algorithm, such as introducing random sampling and parallel thinking to improve the efficiency of algorithm mining rules; Popularize the application of association rules. Mining association rules is an important topic in data mining, which has been widely studied by the industry in recent years.
2. Mining process, classification and related algorithms of association rules.
2. 1 association rule mining process
The process of mining association rules mainly includes two stages: the first stage, all high-frequency itemsets must be found from the data set, and the second stage, association rules are generated from these high-frequency itemsets.
In the first stage of association rule mining, all high-frequency itemsets must be found from the original data set. High frequency means that the frequency of a project group relative to all records must reach a certain level. The frequency with which teams appear is called support. Taking a 2- itemset containing two items A and B as an example, the support degree of the item group containing {A, B} can be obtained by the formula (1). If the support degree is greater than or equal to the set MinimumSupport threshold, {A, B} is called high-frequency project group. K- itemsets that meet the minimum support are called Frequentk- itemsets, which are generally expressed as Largek or frequentke. The algorithm also generates Largek+1from largek's itemsets until no high-frequency itemsets can be found.
The second stage of association rule mining is to generate association rules. Generating association rules from high-frequency itemsets is to generate rules by using the high-frequency K itemsets in the previous step. Under the conditional threshold of MinimumConfidence, if the credibility of a rule meets the minimum confidence, the rule is called association rule. For example, the reliability of rule AB generated by high-frequency k-item group {A, B} can be obtained by formula (2). If the reliability is greater than or equal to the minimum reliability, AB is called an association rule.
As far as the case of Vuormaa is concerned, using association rule mining technology to mine records in the transaction database, two thresholds of minimum support and minimum trust should be set at first, assuming that the minimum support min_support=5% and the minimum trust min_confidence=70%. Therefore, the association rules that meet the needs of this supermarket must meet the above two conditions at the same time. If the association rule "diaper, beer" found through the mining process meets the following conditions, the association rule of "diaper, beer" will be accepted. Support (diapers, beer) can be described by the formula >: =5%, confidence (diapers, beer) > =70%. Among them, support (diapers, beer) >: the significance of =5% in this application example is that at least 5% of all transaction records show that diapers and beer were purchased at the same time. In this application example, confidence (diapers, beer) > =70% means that at least 70% of all transaction records including diapers will buy beer at the same time. Therefore, if the consumer buys diapers in the future, the supermarket will be able to recommend the consumer to buy beer at the same time. This product recommendation behavior is based on the association rule of "diapers, beer", because the past transaction records of supermarkets support the consumption behavior that "most transactions that buy diapers will buy beer at the same time".
As can be seen from the above introduction, association rule mining is usually more suitable for the case that the indicators in the record take discrete values. If the index values in the original database are continuous data, the data should be discretized properly before mining association rules (in fact, the values in a certain interval correspond to a certain value). Data discretization is an important step before data mining, and whether the discretization process is reasonable will directly affect the mining results of association rules.
2.2 Classification of Association Rules
According to different situations, association rules can be classified as follows:
1. Association rules can be divided into Boolean and numerical types according to the types of variables handled in the rules.
The values processed by Boolean association rules are all discrete and classified, which shows the relationship between these variables. Numerical association rules can be combined with multi-dimensional association rules or multi-layer association rules to deal with numerical fields and dynamically divide them, or they can directly deal with original data. Of course, numeric association rules can also contain category variables. For example: gender = "female" => occupation = "secretary", which is a Boolean association rule; Gender = "female" => avg (income) = 2300, and the income involved is numerical, so it is a numerical association rule.
2. According to the abstract level of data in rules, it can be divided into single-layer association rules and multi-layer association rules.
In the single-level association rule, all variables do not take into account that there are many different levels of actual data; In multi-level association rules, the multi-level nature of data has been fully considered. For example, IBM desktop => Sony printer is a single-layer association rule for detailed data; Desktop => Sony printer is a multi-layer association rule between higher level and detail level.
3. According to the dimensions of data involved in rules, association rules can be divided into one-dimensional and multi-dimensional.
In one-dimensional association rules, we only involve one dimension of data, such as items purchased by users; In multidimensional association rules, the data to be processed will involve multiple dimensions. In other words, one-dimensional association rules deal with some relationships in a single attribute; Multidimensional association rules deal with some relationships among various attributes. For example: beer => diapers, this rule only involves the items purchased by users; Gender = "female" => occupation = "secretary". This rule involves information in two fields and is a two-dimensional association rule. Apriori algorithm
2.3 Association rule mining algorithm
1.Apriori algorithm: frequent itemsets are discovered by using candidate itemsets.
Apriori algorithm is the most influential algorithm for mining frequent itemsets of Boolean association rules. Its core is a recursive algorithm based on the idea of two-stage frequency set. This association rule belongs to single-dimensional, single-layer and Boolean association rules in classification. Here, all itemsets with support greater than the minimum support are called frequent itemsets, which is called frequency sets for short.
The basic idea of the algorithm is: first, find out all the frequency sets, and the frequencies of these itemsets are at least the same as the predefined minimum support. Then, strong association rules are generated from the frequency set, and these rules must meet the minimum support and minimum credibility. Then, the expected rules are generated by using the frequency set found in step 1, and all rules containing only set items are generated, in which there is only one item in the right half of each rule, and the definition of intermediate rules is adopted here. Once these rules are generated, only those rules that are greater than the minimum credibility given by the user are left behind. In order to generate all frequency sets, a recursive method is used.
There may be a large number of candidate sets, and it may be necessary to scan the database repeatedly, which are two major shortcomings of Apriori algorithm.
2. Algorithm based on partition: Savasere and others designed an algorithm based on partition. Firstly, the algorithm logically divides the database into several disjoint blocks, considers one block at a time and generates all frequency sets for it, then generates all possible frequency sets, and finally calculates the support of these item sets. Here, the size of each block is selected so that each block can be put into the main memory and only needs to be scanned once in each stage. The fact that every possible frequency set is a frequency set in at least one block ensures the correctness of the algorithm. The algorithm can be highly parallel, and each block can be assigned to a processor to generate a frequency set. After each cycle of generating the frequency set, the processors communicate with each other to generate a global candidate set of k items. Usually the communication process here is the main bottleneck of algorithm execution time; On the other hand, the time for each independent processor to generate frequency sets is also a bottleneck.
3.FP- tree frequency set algorithm: Aiming at the inherent defects of Apriori algorithm, J.Han and others put forward a method of not generating candidate mining frequent itemsets: FP- tree frequency set algorithm. Take the strategy of divide and conquer. After the first scan, the frequency set in the database is compressed into a frequent pattern tree (FP-tree), and the relevant information is still preserved. Then the FP-tree is divided into several conditional bases, each of which is related to a frequency set with a length of 1, and then these conditional bases are mined separately. When the amount of original data is large, FP-tree can be put into main memory in combination with partition method. Experiments show that FP-growth has good adaptability to rules with different lengths, and its efficiency is much higher than that of Apriori algorithm.
3. Application in this field at home and abroad
3. 1 Application of Association Rules Mining Technology at Home and Abroad
At present, association rule mining technology has been widely used in western financial enterprises, and can successfully predict the needs of bank customers. Once this information is obtained, banks can improve their marketing. Now, banks are developing new ways to communicate with customers every day. Banks bind the information of their products that customers may be interested in to their ATM machines for users to understand. If the database shows that a customer with a high credit limit has changed his address, it is likely that this customer has recently bought a bigger house, so he may need a higher credit limit, a new high-end credit card, or a home improvement loan. These products can be mailed to customers through credit card bills. When customers call for advice, the database can effectively help telemarketing representatives. The computer screen of the sales representative can show the characteristics of the customer and what products the customer will be interested in.
At the same time, some well-known e-commerce websites also benefit from powerful association rules mining. These e-shopping websites use the rules in association rules to mine, and then set the bundles that users intend to buy together. There are also some shopping websites that use them to set up corresponding cross-selling, that is, customers who buy one product will see advertisements for another related product.
However, at present, in China, "massive data and lack of information" is a common embarrassment faced by commercial banks after data concentration. At present, most databases implemented in the financial industry can only realize the bottom-level functions such as data entry, query and statistics, but can't find all kinds of useful information in the data, such as analyzing these data, discovering their data patterns and characteristics, and then discovering the financial and commercial interests of a customer, consumer group or organization, and observing the changing trend of the financial market. It can be said that the research and application of association rule mining technology in China is not very extensive and deep.
3.2 Some research on association rule mining technology in recent years
Because many application problems are often more complicated than supermarket purchasing problems, a lot of research has expanded association rules from different angles, and integrated more factors into association rule mining methods, thus enriching the application fields of association rules and broadening the scope of supporting management decisions. For example, consider the hierarchical relationship between attributes, temporal relationship, multi-table mining and so on. In recent years, the research on association rules mainly focuses on two aspects, namely, expanding the scope that classical association rules can solve problems and improving the efficiency and interest of classical association rules mining algorithms.