Data mining in practice: DataPreprocessing -The Use of Normalization

In this article, we will explore one of the basic steps in the knowledge discovery process, "Data Preprocessing", an important step that can be considered as a fundamental building block of data mining. The process of preprocessing has many steps, but can be summarized as the extraction, transformation and loading of the data. To be more precise modifying the source data into a different format which:

(a) enables data mining algorithms to be applied easily

(b) improves the effectiveness and the performance of the mining algorithms

(c) represents the data in easily and understandable format for both humans and machines

(d) supports faster data retrieval from databases

(e) makes the data suitable for a specific analysis to be performed.

The real world data can be considered extremely complicated to interpret without Data Preprocessing. I am going to explain this through a simple example based on Normalization. For people who come from database background this Normalization is completely different from 1st, 2nd and 3rd form of normalization used in the relational database design. We are talking about another type of normalization, and it's related to data preprocessing technique. To see how a simple data preprocessing technique could improve the effectiveness of analysis in orders of magnitude, let's move on further and talk about euclidian distance and how it's can be used to evaluate similarities between samples of data.

Euclidian Distance

Consider two points in a two-dimensional space (p1,p2) and (q1,q2) , the distance between these two points is given by the formula shown at the Figure 01.

Figure 01. Euclidean Distance Measure for 2-D Points

This is called the Euclidian Distance between two points. The same concept can be extended easily to multidimensional space. If the points are (p1,p2,p3,p4,...) and (q1,q2,q3,q4,...), the distance between the points is given by the formula (Figure 02).

Figure 02. Euclidean Distance for Multi-Dimensional Points

Now, I am going to introduce a sample data set of mobile profile users details. The task is to find the mobile users who have similar profiles, that is, that has similar use of the phone based on the call and SMS logs.

Table 01. Example Data Set (Mobile users profiles during one month)

In this data set, the first column is an unique identifier and the rest of the columns contain information. Ignoring the ID attribute, if we consider each column (attribute) as a dimension we can assume that each mobile user is represented by a point in 3-D space. The good thing is that we can calculate the distance between each of these points. The distance between points 1 and 2, (user 1 and 2) can be calculated as below:

Figure 03. Euclidean Distance between users 01 and 02.

Look at the data set above, through examination we figure out that the user id 1 and 4 are users with almost similar profiles, that is, use their phones very similar. So in the three dimensional space the distance between them should be less, that is, these two points should be very close.

The following table (Figure 04) gives the euclidean distance calculation for each user with the other users in our given data set.

Figure 04. Euclidean Distance Matrix for all users

We can see from the above list that the distance between the user 1 and 4 is 2000.00, which is less when compared between userId 1 and other users. So our interpretation seems to be correct. However, an alert reader would have noticed a significant observation here. If you see the list again, you can see that the euclidian distance values are very close to the differences in duration calls. Which this means ?

See this, the difference between the duration calls of users 1 and 4 is abs(25000 - 27000) = 2000, and the euclidean distance between one and four is 2000.30303! So it looks like this approach seems to be flawed. The euclidean distances are dominated by the duration calls amount. So guess you can't rely on euclidean distance to find mobile users with similar profile.

How do we get ourselves out of this problem ? We do not want the duration calls attribute to dominate the euclidean distance calculation. We can achieve this by applying one of the Data Preprocessing techniques called Normalization over the data set.

What is Normalization ?

The attribute data is scaled to fit into a specific range. There are many types of normalization available, we will see one technique called Min-Max Normalization. Min-Max Normalization transforms a value A to B which fits in the range [C,D]. It is given by the formula below (Figure 05):

Figure 05. Min-Max Normalization Formula

Consider the example below, the duration calls value is 50000, we want to transform this in to the range [0.0 , 1.0], so first we find the maximum value of duration calls which is 55000 and the minimum value of duration calls, 25000, them the new scaled value for 50000 will be:

Figure 06. Min-Max for Duration Calls Attribute from the User 01

Now let's apply the normalization technique to all the three attributes in our data set. Consider the following maximum and minimum values:

Max duration calls = 55001

Min duration calls = 24999

Min SMS = 23

Max SMS = 33

Max consume data = 8

Min consume data = 3

The attributes need to be scaled to fit in the range [0.0 , 1.0]. Applying the min-max normalization formula above, we get the normalized data set as given below (Figure 07):

Figure 07. Data set after Normalization

Now given the new data set all normalized, We will calculate the euclidean distances for each employee with the other employees. It's given in the table below (Figure 08):

Figure 08. Euclidean Distance Matrix for the data set

Now compare the euclidean distance calculation before and after normalization. We can see that the distances are no more dominated by the duration calls attribute and they make more sense now. The number of messages and data consumed now also contribute to the distance calculation. You can see how the normalization technique and data preprocessing can really help get useful and right information about your data before applying some machine learning or data mining algorithm.

To conclude this tutorial, it's important to notice that there are many measures similar to euclidean distance which can be used to calculate the similarity between two records such as Pearson coefficient, Tanimoto Coefficient, etc. You can try replacing euclidean distance with any of these measures and experiment.

You can read more on these measures in the following link. Take a special look at other normalization techniques such as Xˆ2 (X Square) Normalization and decimal scaling which are also worth trying.

Special thanks to the Blog IntelligenceMining that inspired me to write about this important topic!

Any doubts or suggestions,

Please make yourself welcome to give!

Marcel Pinheiro Caraciolo

Data mining in practice: Learn about Bayesian Classifier Algorithm with Python

Hi all,

In this article we will continue our studies about Data Mining algorithms. Now, i will present a supervised learning algorithm called Bayesian Classification. As same as the previous articles presented in this blog, a simple example of the algorithm will be presented which can be executed with Python Interpreter.

The Algorithm

The Bayesian classification algorithm is called with this name because is based on the Bayes' probability theorem. It's known also by Naïve Bayes classification rule or only by Bayesian Classifier.

The algorithm aims to predict the class membership probabilities, such as the probability that a given tuple or pattern belongs to a particular class,that is, predict the most probable class that the pattern belongs to. This type of prediction is called statistical classification, which is totally based on probabilities.

This classification also is called a simple or Naïve, because it assumes that the effect of an atribute value on a given class is independent of the values of the other attributes. This assumption is called class conditional independence and it's made to simplify the computations involved.

Furthermore about attributes, it's important to notice that the Bayesian classifier gets better results when the attribute values are categorical instead of continuous-valued . Maybe this will be more clear at the example that will be shown soon.

Other characteristic of the algorithm is that it requires data set already classified, that is, a set of patterns and their associated class labels. Based on this data set, called also 'training data set ' , the algorithm receives as input a new pattern (unknown data), that is, data patterns for which the class label is not known, and returns as output the class which the probability calculated for this data is maximum in accordance to probabilistic calculations. Different from the K-means algorithm seen in the previous article, the Bayesian classifier doesn't need a metric to compare the 'distance' between the instances and neither classifies the unknown pattern automatically, since it's necessary a data set already classified (training set). Because of this requirement, the Bayesian Classification algorithm is considered a supervised data mining algorithm.

To see how the algorithm works, let's resume it with the following four steps:

Step 01: Probabilities classes evaluation (Class Prior Probability).

In this step, each class of the training set has its probability calculated. In most of times, we only work with two classes, for instance, one class shows if a certain consumer buys or not a product based on his demographic characteristics. The calculation is done by dividing the number of patterns of a specific class by total number of patterns of the training set.

Step 02: Probabilities evaluation of the training set

Now, each value of each attribute of the data of the training set has your probability calculated for each possible class. This step is where occurs the most consuming time computational processing of the algorithm, since depending on the number of attributes, classes and patterns of the training set, it's possible that many calculations must be done before get some results (probabilities).

It's important to notice that this calculation depends totally on the attribute values of the unknown sample data, that is, the sample that you desire to predict the class label. Supposing that there are k classes in the test set and m attributes in the test set, it must be necessary to calculate k x m probabilities.

Step 03: Evaluate the probabilities of the unknown data.

In this step, the probabilities calculated for the patterns of the unknown data of the same class are multiplied. Thus, the result obtained is multiplied by the probability class calculated at the Step 01.

With the probabilities of each class calculated, then check which class has the maximum value for the probability of the unknown data. The algorithm ends returning the class with the probability that has the maximum value (the predicted class) for the unknown data.

Further information about Bayesian classification can be found at the links below:

http://en.wikipedia.org/wiki/Na%C3%AFve_Bayes
http://www.devmedia.com.br/articles/viewcomp.asp?comp=2637

Now, let's see a practical example of the use of Simple Bayesian Classification algorithm with the probabilities evaluation.

Example of the Algorithm

In this example, let's consider that a bank loans officer wants to predict if the client will be a bank defaulter or not. For this, the bank must consider his historical client profiles and some attributes. To make easy the comprehension of the scenario and the data model, let's use a training set with only 15 rows and 4 columns (attributes). The Figure 01 shows the training set that will be used in this example.

Figure 01. The historic client profiles (Training Set).

The attributes shown at the Figure 01 are described as below:

CLIENT_ID: This column has an unique integer sequential identifier. For the algorithm this attribute is optional, but it may help to organize the rows of the data set.

GENDER : This attribute identifies the gender of the client. The values allowed are only MALE or FEMALE.

MARITAL_STATUS: This attribute brings information about the marital status of the client. It can be only the values MARRIED or SINGLE.

EDUCATION: This attribute brings the information about the education level of the client. It can assume only four different values: HIGHSCHOOL_INCOMPLETE , HIGHSCHOOL_COMPLETE, GRADUATION_INCOMPLETE and GRADUATION_COMPLETE.

INCOMES: This attributes refers to the earnings of the client. It can only has the values: ONE_MIMINUM_SALARY, TWO_MINIMUM_SALARIES and UPPER_THREE_MINIMUM_SALARIES.

DEFAULTER: This column represents the classification label attribute of the patterns. In this example the classification shows if the client is bank defaulter, that is DEFAULTER=YES, or the client is not bank defaulter, that is, DEFAULTER=NO. To clarify the visualisation, the clients of the training set that are defaulters were marked in red and clients that aren't defaulters are marked in blue.

Let's execute the Bayesian Classification to a given unknown pattern. Based on the data shown at the Figure 01, the target is to predict the class label (DEFAULTER) of this new client shown at the Figure 02 by using the Bayesian Classifier.

Figure 02. The new Client to be classified as DEFAULTER or NOT DEFAULTER

Step 01: The Evaluation of the classes probabilities.

There are only two classes, one that shows the client is bank defaulter (DEFAULTER= YES) and another that points the client is not bank defaulter (DEFAULTER= NO). Calculating the probabilities of the classes, we have:

Probability DEFAULTER= YES : 4/15 = 0,2667

Probability DEFAULTER= NO: 11/15 = 0,7334


Step 02: Calculate the probabilities of the training set.

For the first attribute of the unknown data GENDER=MALE, let's calculate the probability of DEFAULTER=YES:

Probability of GENDER=MALE and INADIPLENT=YES : 2/4 = 0,5

And for the case where the client is male and is not defaulter, we have:

Probability GENDER=MALE and DEFAULTER=NO: 4/11 = 0,3636

For the rest of the attribute values of the data set, we have:

Probability of MARITAL_STATUS =SINGLE and DEFAULTER=YES: 1/4 = 0,25
Probability of MARITAL_STATUS=SINGLE and DEFAULTER=NO: 6/11 = 0,5455

Probability of EDUCATION= HIGHSCHOOL_INCOMPLETE an DEFAULTER=YES: 1/4 = 0,25
Probability of EDUCATION= HIGHSCHOOL_INCOMPLETE and DEFAULTER=NO: 4/11 = 0,3636

Probability of INCOMES= ONE_MIMINUM_SALARY and DEFAULTER=YES: 1/4 = 0,25
Probability of INCOMES= ONE_MIMINUM_SALARY and DEFAULTER=NO: 4/11 = 0,3636

Step 03: Calculate the probability of the unknown data.

Multiplying the probabilities of the unknown data for the case of DEFAULTER=YES by the priori probability of DEFAULTER calculated at the Step 01, we have:

0,5 x 0,25 x 0,25 x 0,25 x 0,2667 = 0,0021

Multiplying the probabilities of the unknown data for the case of DEFAULTER= NO by the probability of NOT DEFAULTER calculated at the Step 01, we have:

0,3636 x 0,5455 x 0,3636 x 0,3636 x 0,7334 = 0,0192

As 0,0192 > 0,0021, the algorithm classifies the unkown pattern as INADIPLENT=NO, that is, this new client has higher probability of not becoming a bank defaulter than becoming one, based on the previous data (training set) and the Bayesian classification.

To help classifying those clients, let's use a implementation of the Bayesian Classification algorithm that will work with only many attributes that has nominal (categorical) values. This implementation was written with Python Script 'bayesian_classify.py' .

>>> python bayesian_classify.py 'C:\dataset.txt' 'DEFAULTER' 'MALE;SINGLE;HIGHSCHOOL_INCOMPLETE;ONE_MINIMUM_SALARY' 1

Figure 03. The bayesian_classify.py call

The first parameter that must be passed as argument of the script is the data set file path. The second parameter must indicate the list of columns used at the classification, all then splitted by comma and at one string. The third parameter shows the column that has the classifications. The fourth parameter must receive the unknown data pattern values list split by comma and in the same order of the attributes passed in the second parameter. The Figure 03 shows the call of the script at the console based on the example shown above.

The Script has one more parameter. If this parameter is passed as 0, the script returns all probabilities of each class. If the parameter is passed as 1, the script returns only the classification of the unknown data. The Figure 04 shows the result of the call of the script presented at the Figure 03.

>>> DEFAULTER=NO

Figure 03. Execution of the bayesian_classifier.py returning the classification.

Therefore, it must have to be considered some observations before using the Bayesian Classifier. It's necessaty that the training set must be correct and consistent, since one line that presents some wrong value can compromise the final result. Other drawback of the algorithm is when there is missing value in the attribute, so the probability is assigned to 0, which makes difficult to give the correct classification of certain samples. Anyway, some techniques have been presented to go around these problems, but it's not the scope of this article now.

To download the script of the Bayesian Classification algorithm and the data set used at this article, click here.

I expect you enjoyed and learned more about data mining algorithms!

See you next time,

Marcel Pinheiro Caraciolo

Data mining in practice: Learn about Linear Regression with Python

Hi all,

Let's continue our studies in data mining algorithms. In this article, we will see how to use the linear regression to predict values of data series with a simple implementation in Python programming language.

Linear regression is more connected to statistics than computing, it can be used to fit a predictive model to an observed data set of y and x values. After developing such a model, if an additional value of X is then given without its accompanying value of y, the fitted model can be used to make a prediction of the value of y. This model can be shown as a line which best represents the data set. Generally, the problems that linear regression may help are related to prediction of quantity of items at certain moment.

To better understand how the linear regression works, let's see an example. Consider the table 01, which shows the year evolution of the unit price of a product and also the quantity of units sold of this product.

Table 01. History of the unit cost and quantity of sells of a product.

Based on the data set shown at the Table 01, the goal is predict the quantity of products sold when the price (unit cost) of the product achieve the value 2,0. This prediction must consider only the data provided at the table, without considering other possible factors. It's important to notice that all the data provided are fictional, used here only to illustrate the use of the linear regression to solve prediction problems.

To answer this question, we can apply the linear regression. However, nothing guarantees that the Linear Regression will make a "good"prediction, that is, the linear model will fit well to the data. To better understand how it works, let's plot the data shown at the Table 01 into a scatter plot. The Figure 01 shows this graph where the values of the column Unit Cost are placed at the axis X (horizontal) and the values of the quantity sold place at axis y (vertical).

Fig. 01. Plot with the unit price x Quantity of products sold (1990-2004)

Analyzing visually the data plot of the Figure 01, we can mentally trace a line that adjust itself to the points. The linear regression just does that: it analyzes the data and mount an line equation so we can predict the next points. One important question to be discussed is: How good it must be this line ? Not only a line that can be generated from analysis of the data. We can imagine a curve (maybe generated by exponential equation) that also adapts itself to the data. In the cases where not ever a curve fit well to the data, we must use a non-linear model. To verify if the data fit well to the linear model, you must use the test called Rˆ2 that checks numerically if it worths or not to use the linear model to the data subset.

There are many ways to use the linear regression. The usual way is use the Microsoft Excel software, which allows to add a line at the plot based on the data points created, as also its correspondent line equation. In this article, we will use a python script implementation of the linear regression algorithm. For further information about the calculation behind of the linear regression, i recommend to the readers a visit to the Kardi Teknomo's website at the link:

http://people.revoledu.com/kardi/tutorial/Regression/RegresionExample.html

First let's suppose that we have the table 01 stored at a simple text file named 'dataset.txt' . Just remembering as we talked earlier, our goal is to obtain the prediction of the quantity of units sold when the price achieve the value of $$ 2,00 per unit.

To evaluate this prediction, we use the python script 'linear_reg.py' and at the terminal type the following commands:

>>> python linear_reg.py 'C:/dataset.txt' 'quant_sold' 2.0

The result of the execution of the program is shown at the Figure 02.

>>>y = ax + b (y = 82.9842x + 23.5645)

>>>x = 2 y = 189.533

>>>RR: 0.912826

>>>Linear Model

Figure 02. Result of the execution of the linear regression

It can be noticed that the result of the script printed at the console three lines. The first one showing the line equation, the second the value predicted for the quantity of items sold when the unit cost is $$ 2,00 and finally, the third line brings the evaluated Rˆ2 metrics. The fourth line shows the interpretation of the Rˆ2 metrics: if this value is below than 0.8, it's recommended to use a non-linear model. Otherwise, the linear regression can be used to this prediction problem.

One detail that must be considered is the numerical precision of the python implementation, that can be different from the equation presented by the Excel. Other important observation is that we cannot forget that this generated equation not necessarily provide all the scatters of the plot, that is, by using the equation we can not obtain exactly the same values of the previously data, thus the equation generated by the linear regression creates an approximation of the values. The Figure 03 shows the plot with the new value predicted, which it's represented by the red dot.

Figure 03. The Quantity of products sold predicted when the price = $2,00

In this example, we considered that the quantity sold depends only of the unit cost of the product. Based on this supposition, we worked with a price of $$ 2,00 for each unit and calculated the quantity approximated by the sells model.

To download the script with all the archives used at this example, just click here.

I expect you enjoyed this article,

Any doubts, please comment !

See you next time,

Marcel P. Caraciolo

References:

How to do a Simple Linear Regression with Python

Wikipedia