Saturday, May 29, 2010

Constructing Decision Tree using WEKA


Introduction

Decision tree learning is a method for assessing the most likely outcome value by taking into account the known values of the stored data instances. This learning method is among the most popular of inductive inference algorithms and has been successfully applied in broad range of tasks such as assessing the credit risk of applicants and improving loyality of regular customers.

Information, Entropy and Information Gain


There are several software tools already available which implement various algorithms for constructing decision trees which optimally fit the input dataset. Nevertheless it is useful to clarify the underlying principle.

Information

First let's briefly discuss the concept of information. In simplified form we could say that the information is a measure of uncertainty or surprise; the more we are surprised about the news the bigger is the information. If there was a newspaper which pubblished a news about the morning sunrise certainly none of the readers would be interested in such story because it doesn't hold any information - the sunrise is a fact and we know it will happen tomorrow again. On the other hand if a story in a newspaper is about something noone expected it's a big news. An example is Michael Jackson's death. It happened during the time of one of the worst recessions which affected the whole world but according to the statistics the news about Michael Jackson were far more interesting than the economic facts of the time.

Entropy

In General entropy is the measure of the uncertainty associated with random variable. In case of stored instances entropy is the measure of the uncertainty associated with the selected attribute.
where p(x_i)\, is the probability mass function of outcome x_i\,.

Information Gain

Information gain is the expected reduction in entropy caused by partitioning the examples according to the attribute. In machine learning this concept can be used to define a preferred sequence of attributes to investigate to most rapidly narrow down the state of the selected attribute. Such a sequence (which depends on the outcome of the investigation of previous attributes at each stage) forms a decision tree. Usually an attribute with high information gain should be preferred to other attributes.



An Example

The process of decision tree construction is described by the following example: There are 14 instances stored in the database described with 6 attributes: day, outlook, temperature, humidity, wind and playTennis. Each instance describes the facts of the day and the action of the observed person (played or not played tennis). Based on the given record we can assess which factors affected the person's decision about playing tennis.

For the root element an attribute with the highest entropy is selected. In the case of this example it is the outlook because it is the most uncertain parameter.
After the root parameter is selected the information gain for every possible combination of pair is calculated and then the branches are selected upon the information gain value; the higher is the information gain the more reliable the branch is.

Entropy(S) is the entropy of classificator PlayTennis - there are 14 instances stored in the dataset from which a person decides do play tennis 9 times (9+) and not to play 5 times (5-).

The above example shows the calculation of the information gain value of classificator Wind: There are 8 instances of value Weak. In case of weak wind a person decides to play tennis 6 times and not to play 2 times [6+, 2-]. In case of strong wind (6 instances) the user decides to play 3 times and not to play 3 times as well. Considering the given instances the calculated value of information gain of attribute wind is 0.48.

Information gain values of all attributes are calculated in the same way and the results are the following:


Gain(S, Outlook) = 0.246
Gain(S, Humidity) = 0.151
Gain(S, Wind) = 0.048
Gain(S, Temperature) = 0.029


Using WEKA

The example is taken from the Book Data Mining: Practical Machine Learning Tools and Techniques, Second Edition (Morgan Kaufmann Series in Data Management Systems)

The .arff file

One of possible options for importing the record in WEKA tool is writing records in .arff file format. In this example the file looks like the following:


@relation PlayTennis

@attribute day numeric

@attribute outlook {Sunny, Overcast, Rain}
@attribute temperature {Hot, Mild, Cool}

@attribute humidity {High, Normal}

@attribute wind {Weak, Strong}
@attribute playTennis {Yes, No}

@data
1,Sunny,Hot,High,Weak,No,?
2,Sunny,Hot,High,Strong,No,?

3,Overcast,Hot,High,Weak,Yes,?
4,Rain,Mild,High,Weak,Yes,?

5,Rain,Cool,Normal,Weak,Yes,?

6,Rain,Cool,Normal,Strong,No,?

7,Overcast,Cool,Normal,Strong,Yes,?
8,Sunny,Mild,High,Weak,No,?
9,Sunny,Cool,Normal,Weak,Yes,?
10,Rain,Mild,Normal,Weak,Yes,?
11,Sunny,Mild,Normal,Strong,Yes,?
12,Overcast,Mild,High,Strong,Yes,?

13,Overcast,Hot,Normal,Weak,Yes,?

14,Rain,Mild,High,Strong,No,?



The file can then be imported using WEKA explorer. When the above file is imported the interface looks like the following:


Figure: Available attributes (left) and graphical representation (right) from the file data

after the data is imported there is a set of classification methods available under the classification tab
. In this case the c4.5 algorithm has been chosen which is entitled as j48 in Java and can be selected by clicking the button choose and select trees->j48.

Figure: after running Start The decision tree is created in this case using c4.5 algorithm

The tree is then created by selecting start and can be displayed by selecting the output from the result list with the right-mouse button choosing the option "Visualize Tree".

Figure: The decision tree constructed by using the implemented C4.5 algorithm

The algorithm implemented in WEKA constructs the tree which is consistent with the information gain values calculated above: the attribute Outlook has the highest information gain thus it is the root attribute. Attributes Humidity and Wind have lower information gain than Outlook and higher than Temperature and thus are placed below Outlook. The information gain of a specific branch can be calculated the same way as the example above. For example if we were interested what is the information gain of branch (Outlook, Temperature) the attributes Outlook and Temperature should be placed in the equation for calculating the information gain.

References

Data Mining and KnowledgeDiscovery; Prof. Dr. Nada Lavra─Ź

Data Mining: Practical Machine Learning Tools and Techniques, Second Edition;
Ian H. Witten, Eibe Frank


Sunday, May 23, 2010

LDPC Block Code Simulator: GUI Preview

In the past days I've created a LDPC Block Code Simulator in Matlab. It is possible to enter the parity matrix H, the information sequence D which is encoded into codeword C and then add Gaussian Noise of a chosen variance which simulates the transmission channel charcteristics.

The simulator then calculates the original codeword and enables user to switch between the steps of the iterative decoding process. Each step displays the current message values that are passed between symbol and parity nodes and the next step operation.

Below are the screenshots of three steps. In this example H is 4 x 8 parity-check matrix ([1 0 1 0 1 0 0 0; 0 0 0 1 0 1 0 0; 1 0 0 1 0 0 1 0; 0 1 1 0 0 0 0 1]). The information sequence D ([1 0 1 1]) is encoded into codeword C (1 0 1 1 0 1 0 1). The original unipolar codeword is transformed to bipolar code (1 is transformed to -1 and 0 is transformed to 1) and the noise of variance 0.5 is added to the transformed code which results Y (-1.0371 0.12971 -1.1976 -1.808 1.3784 -1.3998 1.0451 -1.2452), the simulated received sequence. After 5 steps the algorithm determines the codeword C' (1 0 1 1 0 1 0 1) which is equal to the original codeword.