Dynamic Online learning Algorithm For Three-way decision

. Three-way decision is an important theory for solving uncertain problems. Online computing is a new dynamic Stream computing form . How to execute three-way decision quickly in online computing is a challenging topic. In this paper, Online computing process is divided into incremental computing portion and decreasing computing portion. And a three-way decision dynamic incremental and decreasing learning algorithm for online computing is proposed. Firstly, the dynamic incremental and decreasing learning models is studied for stream computing based on probabilistic rough set . Then, the logical reasoning for three-way decision regions changing are discussed based on the dynamic incremental and decreasing learning models. And a novel dynamic online learning algorithm for three-way decision online computing is proposed based on the above theory. Finally, the experiment by UCI data set show that the proposed algorithms are superior than classical static three-way decision method in time efficiency.


Instruction
Three-way decision is one of the important uncertainty decision theory [1]. In recent years, Three-way decision theory was applied to many application fields, like spam filtering, text emotion and image recognition [2][3] [4].These application have proved the superiority of three-way decision. The probabilistic rough set dynamic computation is another important research field. Stream data [5] computing is a novel online computing era for dynamic big data. The main characteristic of Online Stream computing is that the dynamic data through memory and CPU by sliding window without external memory cache. From the view of memory, we can find that the essence of Online Stream computing that can be implemented both in incremental computing and decreasing computing in memory [6]. With the development of dynamic Online Stream computing platform, such as Twitter, Kafka, Storm, YahooS4, the importance of dynamic Online Stream computing became more prominent. In a dynamic system, incremental learning method learns new knowledge constantly from new samples and the previous learned knowledge, without relearning all the data. Thus, incremental learning can reduce the demand of time and space, and more able to be meet the actual requirements. Incremental learning of rough set [7] has been extensively studied in recent years, the main research contents involve approximation [8], attribute reduction [9] and decision rule extracting [10]. However, Online Stream computing is indeed a new research topic in the field of rough set research. In this study, Online Stream computing process is divided into incremental and decreasing computing portion. Then, a dynamic incremental and decreasing learning algorithm for Threeway decision Online Stream computing is proposed.

Basic theory
Probabilistic rough set is the basic prototypes for constructing Three-way decision theory. The following definitions are basic theory of Three-way decision theory [11]. An information system is defined as:  When an object x is added to an information system in memory, the new object is denoted as x + . After the increase of the information system, the equivalence classes of each condition attribute equivalence class and each decision attribute equivalence class can be updated by the following formula.
When an object x is deleted from the memory information system, the deleted object is denoted as x − . When the information system is deleted, the equivalence classes of each condition attribute and the equivalence classes of each decision attribute can be updated by the following formula.
The superscript t denotes the initial time, and the superscript +1 t denotes the time after the object is deleted and added.

Single object incremental updating strategy for Three-way decision
For a given decision equivalence class, an object is added, and the positive, negative, and boundary regions of three-way decision are changed as follows.
Theorem 1: Given an information system IS , when Theorem 2: Given an information system IS , when Theorem 3：Given an information system IS , when Theorem 4: Given an information system IS , when The proof of these above theorem is brief.

Single object decreasing updating strategy for Three-way decision
For a given decision equivalence class, an object is deleted, and its positive, negative, and boundary regions change as follows.
. Theorem 6: Given an information system IS , when . The proof of these above theorem is brief.

Online incremental and decreasing learning
algorithm for three-way decision 4

.1 Algorithm
In the online Stream computing model, the changing data can be existed at the same time in memory. According to the idea of time division multiplexing, the dynamic incremental and decreasing learning algorithm based on Three-way decision are proposed to deal with the online Stream computing problem.
Dynamic incremental and decreasing learning algorithms for online Stream computing as follows: Algorithm input: The data + x andx , all decision equivalence classes t j D and their corresponding positive region Step 1: Reduce learning, remove datax , and update Three-way decision areas of t j D .
Step 1.1: remove the datax and choose between theorem 5-6 based on the relation between thex and the conditional equivalence class (contained in the 3 decision regions) and the decision equivalence class t j D .
Step 1.2: According to the conditionsx equivalence class t i R belongs to the decision domain, and select the sub theorem in the theorem obtained in step 1.1.
Step 1.3: The conditional probability According to the relation between the conditional probability value and the threshold value, the specific case of the neutron theorem of step 1.2 is selected, and the transformation of the decision region is carried out.
Step 2: Incremental learning, adding data, + x , and updating Three-way decision areas of 1 t j D + after removing datax .
Step 2.1: Add data + x and choose between theorem 1-4 based on the relation between + x and the conditional equivalence class (contained in 3 decision regions) and the decision equivalence class 1 t j D + .
Step 2.2: According to the decision region of the conditional equivalence class . According to the relation between the conditional probability value and the threshold value, select the specific case of the neutron theorem of step 2.2 and transform the decision domain.

Time complexity analysis of algorithm
The time complexity analysis of dynamic incremental and decreasing learning algorithm based on Three-way decision is deducted as followed: Assume there are M data in memory, =| | M U . In the decreasing learning stage, the time complexity of reducing learning is  This experiment simulates the inflow and outflow of the memory data. The algorithm process is given as followed: Firstly, the amount of data in memory is fixed. Then, a new data is added into the memory, and other data object is deleted at the same time from the memory. The stream simulation process is repeated until all stream data is computed.
In this experiment, the amount of data stored in the set memory is 1000, and the thresholds =0.75 =0.35 α β ， respectively. Dynamic incremental and decreasing learning algorithm as algorithm1. Classical nonincremental learning algorithms of Three-way decision as algorithm2 . The two algorithms was done 30000 times and the time cost result is shown in table 2. Table 2 Time cost for algorithm1 and algorithm2  Data set  Algorithm1  Algorithm2   1  8s  350s  2  14s  600s   3  9s  350s  4  9s  350s  5  10s  400s  6  11s  500s  7 12.5s 450s 8 13s 510s The time cost of the two algorithms both showed a linear growth trend, and the proposed dynamic learning algorithm has great reduced of the time cost. The experimental results and the time complexity analysis are matched with the time complexity of section

Conclusion
With the development of big data, the proportion of stream data in machine learning and big data applications is more evidently. It is of great significance to do research on Online Stream computing based on three-way decisions. In this paper, the incremental and decreasing mode of single object insertion and deletion was analyzed , and then the computing time complexity for the Three-way decision is deducted. The experiments by UCI data sets show that the proposed online computing algorithms are superior than classical rough set three-way decision method in time efficiency.