1. Bayesian Classification Using P-tree
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Proprietary material.
Patents pending on bSQ, Ptree technology
Classification
Classification is a process of predicting an
unknown attribute-value in a relation
Given a relation, R(k1..kn, A1, …, An, C),
where ki’s are the structural attribute
A1, …, An, C are attributes and
C is the class label attribute.
Given an unclassified data sample (no C-value present), classification predicts the C-value for the given sample and thus determine its class.
There are two types of classification techniques
Eager classifier: Build a classifier from training sample ahead of classifying a new sample.
Lazy classifier: No classifier is built ahead of time, training data is used directly to classify new sample.
Stream Data: comes continuously or in fixed time intervals.
E.g., weather data for a particular area or images taken from a satellite within fixed intervals.

2. Preparing the data for Classification
Data Cleaning
Involves the handling of noisy data and missing values.
Noise could be removed or reduce by applying "smoothing" and missing values could be replaced with most common or some statistically determined value.
Relevance Analysis
In the given data not all its attributes are relevant to the classification task.
To reduce the task of classification these attribute should be identified and remove from classification task.
Data transformation
Data can be generalized using a concept hierarchy from low level to high level.
For spatial data, values of different bands are continuous numerical values.
We may intervalize them as high, medium, low etc. using the concept hierarchy.

3. Bayesian Classification
A Bayesian classifier is a statistical classifier, which is based on following theorem known as Bayes theorem: 
Bayes theorem:
Let X be a data sample whose class label is unknown. Let H be a hypothesis (i.e., X belongs to class, C). P(H|X) is the posterior probability of H given X. P(H) is the probability of H, then
P(H|X) = P(X|H)P(H)/P(X)
Where P(X|H) is the posterior probability of X given H and P(X) is the probability of X.

4. Naïve Bayesian Classification
Given a relation R(K, A1..An, C) where K is the structure attribute and Ai and C are feature attributes. Also C is the class label attribute.     
Each data sample is represented by feature vector, X=(x1..,xn) depicting the measurements made on the sample from A1,..An, respectively.
Given classes, C1,...Cm, the naive Bayesian Classifier will predict the class of unknown data sample, X, to be class, Cj having the highest posterior probability, conditioned on X 
P(Cj|X) > P(Ci|X), where i  j (called the maximum posteriori hypothesis),
From the Bayes theorem:
P(Cj|X) = P(X|Cj)P(Cj)/P(X)
P(X) is constant for all classes so we maximize P(X|Cj)P(Cj). If we assume equal likelihood of classes, maximize P(X|Cj) otherwise we maximize the whole product.
To reduce the computational complexity of calculating all P(X|Cj)'s the naive assumption of class conditional independence of values is used.

5. Naïve Bayesian Classification
Class Conditional Independence:
This assumption says that the values of the attributes are conditionally independent of one another. So, P(X|Cj)=P(X1|Cj)*..*P(Xn|Cj).
Now the P(Xi|Cj)’s can be calculated directly from data sample.
Calculating P(Xi|Cj) from P-trees: P(Xi|Cj) = sjxi/sj
where sj = # of samples in class Cj and
sjxi = # of training samples of class Cj, having Ai-value xi.
These values can be calculated by:
sjxi = RootCount [ (Pi(xi) ^ (PC(Cj) ],
sj= RootCount [ PC(Cj)]

6. Non-Naive Bayesian Classifier (Cont.)
One problem with Non-Naïve-Bayesian P-tree classifiers:
If the rc(Ptree(X))=0 then we will not get a class label for that tuple.
It could happen if the whole tuple is not present in the training data
X1 X2 … Xk-1 Xk Xk+1 … Xn
Whole tuple
Solution (Partial Naïve): So in that case we can divide the whole tuple into two parts separating one attribute from the whole tuple. e.g.
Whole tuple
X1 X2 … Xk-1 Xk Xk+1 … Xn
X1 X2 … Xk-1 Xk+1 … Xn Xk
Separated tuple
X’k
P(X|Ci)=rc[Ptree(X’k) ^ PC(Ci)] * rc[Pk(Xk) ^ PC(Ci)]
Now the problem is how to select the attribute Xk
One way to use the information gain theory.
Calculate the info gain of all the attributes Xi
then Xk is the one having lowest information gain

7. Information Gain Let C have m different classes, C1 to Cm
The information needed to classify a given sample is: I(s1..sm) = -(i =1..m)[pi*log2(pi)]
where pi=si/s is the probability that a sample belongs to Ci.
A, an attrib, having values, {a1...av}. The entropy of A is E(A) = (j=1..v i=1..m sij/ s ) * I(s1j..smj)
I(s1j..smj) = -i=1..m pij*log2(pij)
where pij=sij/Sj is the probability that a sample in Ci belongs to Ai
Information gain of A: Gain(A) = I(s1..sm) - E(A)
si = rc(PC(ci)
Sj = rc(PA(aj)
sij = rc( PC(ci) ^ PA(aj) )

8. Performance of Ptree AND operation
Performance of Classification: Comparison Performance for 4 classification classes
Bits
NBC
BCIG
Succ
IG Use 2
.14
.48
.40 3
.19
.50
.34 4
.27
.51
.32 5
.26
.52
.31 6
.24 7
.23
.20
Performance of Classification: Classification success rate comparisons
IG Use - Proportion of the number of times the information gain was used for successful classification.

9. Performance in Data Stream Application
Data Stream mining should have the following criteria
It must require a small constant time per record.
It must use only a fixed amount of main memory
It must be able to build a model at most one scan of the data
It must make a usable model available at any point of time.
It should produce a model that is equivalent to the one that would be obtained by the corresponding database-mining algorithm.
When the data-generating phenomenon is changing over time, the model at any time should be up-to-date but also include the past information.
Data Stream mining Using P-tree
It must require a small constant time per record.
P-tree require small and constant time.
It must use only a fixed amount of main memory
Ok for P-tree
It must be able to build a model at most one scan of the data
To build the P-tree only one scan is required
It must make a usable model available at any point of time.
It should produce a model that is equivalent to the one that would be obtained by the corresponding database-mining algorithm.
Any conventional algorithm is also implementable with P-tree
When the data-generating phenomenon is changing over time, the model at any time should be up-to-date but also include the past information.