Data Mining and Fuzzy Data Mining Using MapReduce Algorithms

1. Introduction

Data mining is an emerging area for knowledge discovery to extract hidden and useful information from large amounts of data. Data mining methods like association rules, clustering, and classification use advanced algorithms such as decision tree and k-means for different purposes and goals. The research fields of data mining include machine learning, deep learning, and sentiment analysis. Information has to be retrieved within a reasonable time period for big data analysis. This may be achieved through the consecutively retrieval (C-R) of datasets for queries. The C-R property was first introduced by Ghosh [1]. After that, the C-R property was extended to statistical databases. The C-R cluster property is a presorting to store the datasets for clusters. In this chapter, C-R property is extended to cluster analysis. MapReduce algorithms are studied for cluster analysis. The time and space complexity shall be reduced through the consecutive retrieval (C-R) cluster property. Security of the data is one of the major issues for data analytics and data science when the original data is not to be disclosed.

The web programming has to handle incomplete information. Web intelligence is an emerging area and performs data mining to handle incomplete information. The incomplete information is fuzzy rather than probability. In this chapter, fuzzy web programming is discussed to deal with data mining using fuzzy logic. The fuzzy algorithmic language, called FUZZYALGOL, is discussed to design queries in data mining. Some examples are discussed for web programming with fuzzy data mining.

2. Data mining

Data mining [2, 3, 4, 5] is basically performed for knowledge discovery process. Some of the well-known data mining methods are frequent itemset mining, association rule mining, and clustering. Data warehousing is the representation of a relational dataset in two or more dimensions. It is possible to reduce the space complexity of data mining with consecutive storage of data warehouses.

The relational dataset is a representation of data with attributes and tuples.

Definition: A relational dataset R or cluster dataset is defined as a collection of attributes A₁, A₂ ,…, Am and tuples t₁, t₂,…, tn and is represented as

R = A₁ x A₂ x …x Am

ti = a_i1 x a_i2 x … x aim are tuples, where i =1,2,.., n

or

R(A₁. A₂. … Am). R is a relation.

R(ti)= (a_i1. a_i2 …. a_im) are tuples, where i =1,2,.., n

or instance, two sample datasets “price” and “sales” are given in Tables 1 and 2, respectively.

The lossless join of the datasets “price” and “sales” is given in Table 3.

In the following, some of the methods (frequency, association rule, and clustering) are discussed.

Consider the “purchase” relational dataset given in Table 4.

3. Data mining using C-R cluster property

The C-R (consecutive retrieval) property [1, 3] is the retrieval of records of database consecutively. Suppose R = {r₁, r₂, …, rn} is the dataset of records and C = {C₁, C₂, …, Cm} is the set of clusters.

The best type of file organization on a linear storage is one in which records pertaining to clusters are stored in consecutive locations without redundancy storing any data of R.

If there exists on such organization of R for C said to have the Consecutive Retrieval Property or C-R cluster property with respect to dataset R. Then C-R cluster property is applicable to linear storage.

The C-R cluster property is a binary relation between a cluster set and dataset.

Suppose if a cluster in a cluster set C is relevant to the data in a dataset R, then the relevancy is denoted by 1 and the irrelevancy is denoted by 0. Thus, the relevancy between cluster set C and dataset R can be represented as (n x m) matrix, as shown in Table 8. The matrix is called dataset-cluster incidence matrix (CIM).

Consider the dataset for customer account given in Table 9.

The dataset given in Table 9 is reorganized in ascending order based on sorting, as shown in Table 10.

Consider the following clusters of queries:

C1 = Find the customers whose sales is greater than or equal to 100.

C2 = Find the customers whose sales is less than 100.

C3 = Find the customers whose sales is greater than or equal average sales.

C4 = Find the customers whose sales is less than average sales.

The CIM is given in Table 11.

The dataset given in Table 11 is reorganized with sort on C₁ in descending order, as shown in Table 12. Thus, C₁ has C-R cluster property.

The dataset given in Table 11 is reorganized with sort on C₂ in descending order, as shown in Table 13. Thus, C₂ has C-R cluster property.

The dataset given in Table 11 is reorganized with sort on C₃ in descending order, as shown in Table 14. Thus, C₃ has C-R cluster property.

The dataset given in Table 11 is reorganized with sort on C₄ in descending order, as shown in Table 15. Thus, C₄ has a C-R cluster property.

The dataset is given for C₁ ⋈ C₂ has C-R cluster property (Table 16).

The dataset is given for C₃ ⋈ C₄ has C-R cluster property (Table 17).

The dataset is given for C₁ ⋈ C₃ has C-R cluster property (Table 18).

The dataset is given for C₂ ⋈ C₄ has C-R cluster property (Table 19).

The dataset is given for C₂ ⋈ C₃ has C-R cluster property (Table 20).

The cluster sets {C₁ ⋈ C₂, C₃ ⋈ C₄, C₁ ⋈ C₃, C₂ U⋈ C₄, C₂ U⋈ C₃} has C-R cluster property. Thus, the cluster sets have C-R cluster properties with respect to dataset R.

3.2 Visual design for parallel cluster

The C-R cluster property is studied with graphical approach. This graphical approach can be studied for designing parallel cluster processing (PCP).

Suppose Vi is the vertex of RICM of C. The G(C) is defined by vertices Vi, i=1,2,…, and n, and two vertices have an edge Eij associated with interval Ii={Vi, V_i+1} i=1,…,n-1.

If G(C) has C-R cluster property, the vertices of G(C) have consecutive 1’s or 0’s.

Consider the cluster set {C₁, C₂}. The G(C1) has the vertices (1,1,1,0,0,0,0), and the G(C₂) has the vertices (0,0,0,1,1,1,1), G(C₃) has the vertices (1,1,1,1, 0,0,0), and G(C₄) has vertices (0,0,0,0,1,1,1).

The parallel cluster property exists if G(Ci) ∩G(Cj)=Ф.

For instance, consider the G(C₁) and G(C₂). G(C₁) ∩G(C₂)=Ф, so that the cluster set {C₁, C₂} has parallel cluster property. The graphical representation is shown in Figure 1.

Similarly the cluster set {C₃, C₄} has the parallel cluster property (PCP). The cluster set {C₃, C₄} has no PCP because it is G(C₂) ∩ G(C₃) # Ф

The graph G(C₁) ∩ G(C₂) = Ф have consecutive cluster property.

The graph G(C₃) ∩ G(C₄) = Ф have consecutive cluster property. The graphical representation is shown in Figure 2.

The graph G(C₂) ∩ G(C₃) # Ф does not have consecutive cluster property. The graphical representation is shown in Figure 3.

4. Design of retrieval of cluster using blackboard system

Retrieval of clusters from blackboard system [8] is the direct retrieval of data sources. When the query is being processed, the entire database has to bring to main memory but in blackboard architecture, the data item source is direct from the blackboard structure. For the retrieval of information for a query, data item is directly retrieved from the blackboard which contains data item sources. Hash function may be used to store the data item set in the blackboard.

The blackboard systems may be constructed with data structure for data item sources.

Consider the account (AC-No, AC-Name, AC-Balance)

Here AC-No is key of datasets.

Each data item is data sourced which is mapped by h(x).

These data items are stored in blackboard structure.

When the transaction is being processed, there is no need to take the entire database into the main memory. It is sufficient to retrieval of particular data item of particular transaction from the blackboard system (Figure 4).

The advantage of blackboard architecture is highly secured for blockchain transaction. The blockchain technology has no third-party interference.

5. Fuzzy data mining

Sometimes, data mining is unable to deal with incomplete database and unable to combine the data and reasoning. Fuzzy data mining [6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] will combine the data and reasoning by defining with fuzziness. The fuzzy MapReducing algorithms have two functions: mapping reads fuzzy datasets and reducing writes the after operations.

Definition: Given some universe of discourse X, a fuzzy set is defined as a pair {t, μ_d(t)}, where t is tuples and d is domains and membership function μ_d(x) is taking values on the unit interval [0,1], i.e., μ_d(t)➔[0,1], where tiЄX is tuples (Table 28).

The sale is defined intermittently with fuzziness (Tables 29–32).

μ _Demand(x)=0.9/90+0.85/80+0.8/75+0.65/70

or

Fuzziness may be defined with function

μ _Demand(x)= (1+(Demand-100)/100) ⁻¹ Demand <=100

=1 Demand>100

1. Negation

1. Union

Union of 1105 = max{0.8,0.7}=0,8

Fuzzy semijoin is given by sales ⋈ items-sale as shown in Table 33.

The fuzzy k-means clustering algorithm (FKCA) is optimization algorithm for fuzzy datasets (Table 34).

Fuzzy k-means cluster algorithm (FKAC) is given by, using FAD

best=R

K=means=best

for i range(1,n)

for j range(1,n)

ti=fuzzy union(r_i.RU r_i.R_j), if r_i.R=r_j.R

C reduce best

k-means < best

return

The fuzzy multivalued association property of data mining may be defined with multivalued fuzzy functional dependency.

The fuzzy multivalued association (FMVD) is the multivalve dependency (MVD). The association multivalve dependency (FAMVD) may be defined by using Mamdani fuzzy conditional inference [3].

If EQ(t₁(X),t₂(X),t₃(X)) then EQ(t₁(Y) ,t₂(Y)) or EQ(t₂(Y) ,t₃(Y)) or EQ(t₁(Y) ,t₃(Y))

= min{EQ(t₁(Y) ,t₂(Y)) EQ(t₂(Y) ,t₃(Y)) EQ(t₁(Y) ,t₃(Y))}

= min{min(t₁(Y) ,t₂(Y)) , min(t₂(Y) ,t₃(Y)) , min(t₁(Y) ,t₃(Y))}

= min(t₁(Y) ,t₂(Y). t₃(Y))

The fuzzy k-means clustering algorithm (FKCA) is the optimization algorithm for fuzzy datasets (Table 35).

Fuzzy k-means cluster algorithm (FKAC) is given by, using FAMVD

best=R

K=means=best

for i range(1,n)

for j range(1,n)

for k range(1,n)

ti=fuzzy union(ri.R U rj.R U r_k.R), if ri.R=r_j.R=rk.R

C reduce best

k-means<best

return

The fuzzy k-means clustering algorithm (FKCA) is the optimization algorithm for fuzzy datasets.

K=means=n

for i range(1,n)

for j range(1,n)

ti=fuzzy union(ri.R U si.Sj), if ri.R=sj.S

C =best

k-means < best

return

For example, consider the sorted fuzzy sets of Table 5 is given in Table 36.

6. Fuzzy security for data mining

Security methods like encryption and decryption are used cryptographically. These security methods are not secured. Fuzzy security method is based on the mind and others do not descript. Zadeh [16] discussed about web intelligence, world knowledge, and fuzzy logic. The current programming is unable to deal question answering containing approximate information. For instance “which is the best car?” The fuzzy data mining with security is knowledge discovery process with data associated.

The fuzzy relational databases may be with fuzzy set theory. Fuzzy set theory is another approach to approximate information. The security may be provided by approximate information.

Definition: Given some universe of discourse X, a relational database R1 is defined as pair {t, d}, where t is tuple and d is domain (Table 37).

Price = 0.4/50+0.5/60+07/80+0.8/100

The fuzzy security database of price is given in Table 38.

Demand = 0.4/50+0.5/60+0.7/80+0.8/100

The fuzzy security database of demand is given in Table 39.

The lossless natural join of demand and price is union and is given in Table 40.

Table 40.

Lossless join.

The actual data has to be disclosed for analysis on the web. There is no need to disclose the data if the data is inherently define with fuzziness.

“car with fuzziness >07” may defined as follows:

For instance,

XML data may be defined as

<CAR>

<COMPANY>

<NAME> Benz <NAME>

<FUZZ> 0.8 <FUZZ>

</COMPANY>

<COMPANY>

<NAME> Suzuki <NAME>

<FUZZ> 0.9<FUZZ>

</COMPANY>

<COMPANY>

<NAME> Toyoto<NAME>

<FUZZ> 0.6<FUZZ>

</COMPANY>

<COMPANY>

I<NAME> Skoda<NAME>

<FUZZ> 0.7<FUZZ>

</COMPANY>

Xquery may define using projection operator for demand car is given as

Name space default =http://www.automoble.com/company

Validate <CAR> {

For $name in COMPANY/CAR

where $company/ Max($demand>0.7)}

return <COMPANY> {$company/name, $company/fuzzy}</COMPANY>

</CAR>

The fuzzy reasoning may be applied for fuzzy data mining.

Consider the more demand fuzzy database by decomposition (Tables 41 and 42).

The fuzzy reasoning [14] may be performed using Zadeh fuzzy conditional inference

The Zadeh [14] fuzzy conditional inference is given by

if x is P₁ and x is P₂ …. x is P_n then x is Q =

min 1, {1-min(μ_P1(x), μ_P2(x), …, μ_Pn(x)) +μ_Q(x)}

The Mamdani [7] fuzzy conditional inference s given by

if x is P₁ and x is P₂ …. x is P_n then x is Q =

min {μ_P1(x), μ_P2(x), …, μ_Pn(x) , μ_Q(x)}

The Reddy [12] fuzzy conditional inference s given by

= min(μ_P1(x), μ_P2(x), …, μ_Pn(x))

If x is Demand then x is price

x is more demand

------------------------------------

x is more Demand o (Demand➔Price)

x is more Demand o min{1, 1-Demand+Price}Zadeh

x is more Demand o min{Demand, Price} Mamdani

x is more Demand o {Demand} Reddy

“If x is more demand, then x is more prices” is given in Tables 43 and 44.

The inference for price is given in Table 45.

So the business administrator (DA) can take decision to increase the price or not.

7. Web intelligence and fuzzy data mining

Let C and D be the fuzzy rough sets (Tables 46–51).

Table 48.

Intersect of demand and price.

The operations on fuzzy rough set type 2 are given as

1-C= 1- μ_C(x) Negation

CVD=max{μ_C(x), μ_D(x)} Union

CΛD=min{μ_C(x) , μ_D(x)} Intersection

XML data may be defined as

<SOFTWARE>

<COMPANY>

<NAME> IBM <NAME>

<FUZZ> 0.8 <FUZZ>

</COMPANY>

<COMPANY>

<NAME> Microsoft <NAME>

<FUZZ> 0.9<FUZZ>

</COMPANY>

<COMPANY>

<NAME> Google<NAME>

<FUZZ> 0.75<FUZZ>

</COMPANY>

Xquery may define using projection operator for best software company is given as

Name space default =http://www.software.cm/company

Validate <SOFTWARE> {For $name in COMPANY/SOFTWARE where $company/ Max($fuzz)}

return <COMPANY> {$company/name, $company/fuzzy} </COMPANY>

</SOFTWARE>

Similarly, the following problem may be considered for web programming.

Let P is the fuzzy proposition in question-answering system.

P=Which is tallest buildings City?

The answer is “x is the tallest buildings city.”

For instance, the fuzzy set “most tallest buildings city” may defined as

most tallest buildings city = 0.6/Hoang-Kang + 0.6/Dubai + 0.7/New York +0.8/Taipei+ 0.5/Tokyo

For the above question, output is “tallest buildings city”= 0.8/Taipei by using projection.

The fuzzy algorithm using FUZZYALGOL is given as follows:

BEGIN

Variable most tallest buildings City = 0.6 / Hoang-Kang + 0.6 / Dubai + 0.7 / New York + 0.8 / Taipei + 0.5 / Tokyo

most tallest buildings City =0.8 / Taipei

Return URL, fuzziness=Taipei, 0.8

END

The problem is to find “most pdf of type-2 in fuzzy sets”

The Fuzzy algorithm is

Go to most visited fuzzy set cites

Go to most visited fuzzy sets type-2

Go to most visited fuzzy sets type -2 pdf

The web programming gets “the most visited fuzzy sets” and put in order

The web programming than gets “the most visited type-2 in fuzzy sets”

The web programming gets “the most visited pdf in type-2”

8. Conclusion

Data mining may deal with incomplete information. Bayesian theory needs exponential complexity to combine data. Defining datasets with fuzziness inherently reduce complexity. In this chapter, fuzzy MapReduce algorithms are studied based on functional dependencies. The fuzzy k-means MapReduce algorithm is studied using fuzzy functional dependencies. Data mining and fuzzy data mining are discussed. A brief overview on the work on business intelligence is given as an example.

Most of the current web programming studies are unable to deal with incomplete information. In this chapter, the web intelligence system is discussed for fuzzy data mining. In addition, the fuzzy algorithmic language is discussed for design fuzzy algorithms for data mining. Web intelligence system for data mining is discussed. Some examples are given for web intelligence and fuzzy data mining.

Acknowledgments

The author thanks the reviewer and editor for revision and review suggestions made in this work.