Decision Tree - Information Gain & Gain Ratio

ID3 uses information gain as its attribute selection measure. The attribute with the highest information gain is chosen as the splitting attribute for node N.

The average amount of information needed to identify the class label,

where p_i - non zero probability

p_i is estimated by |C_i,D| / |D|

Info(D) is also known as the entropy of D.

The required more information is derived from

where

|D_j| / |D| - weight of jth partition

Information gain is defined as the difference between the original information requirement and the new requirement.

Gain(A) = Info(D) - Info_A(D)

The information gain measure is biased toward tests with many outcomes. C4.5, a successor of ID3, uses an extension to information gain known as gain ratio, which attempts to overcome this bias.

Gain ratio differs from information gain, which measures the information with respect to classification that is acquired based on the same partitioning. Gain ration can be calculated by

The attribute with the maximum gain ratio is selected as the splitting attribute.

Gaussian Elimination - Row reduction Algorithm

Gaussian elimination is a method for solving matrix equations of the form, Ax=b. This method is also known as the row reduction algorithm. Back Substitution Solving the last equation for the variable and then work backward into the first equation to solve it. The fundamental idea is to add multiples of one equation to the others in order to eliminate a variable and to continue this process until only one variable is left. Pivot row The row that is used to perform elimination of a variable from other rows is called the pivot row. Example: Solving a linear equation The augmented matrix for the above equation shall be The equation shall be solved using back substitution. The eliminating the first variable (x1) in the first row (Pivot row) by carrying out the row operation. As the second row become zero, the row will be shifted to bottom by carrying out partial pivoting. Now, the second variable (x2) shall be eliminated by carrying out the row operation again. ...

Decision Tree - Gini Index

The Gini index is used in CART. The Gini index measures the impurity of the data set, where p i - probability that data in the data set, D belong to class, C i and pi = |C i,D |/|D| There are 2 v - 2 possible ways to form two partitions of the data set, D based on a binary split on a attribute. Each of the possible binary splits of the attribute is considered. The subset that gives the minimum Gini index is selected as the splitting subset for discrete valued attribute. The degree of Gini index varies between 0 and 1. The value 0 denotes that all elements belong to a certain class or if there exists only one class, and the value 1 denotes that the elements are randomly distributed across various classes. A Gini Index of 0.5 denotes equally distributed elements into some classes. The Gini index is biased toward multivalued attributes and has difficulty when the number of classes is large.

Data Cleaning Process

Data cleaning attempt to fill in missing values, smooth out noise while identifying outliers, and correct inconsistencies in the data. Data cleaning is performed as an iterative two-step process consisting of discrepancy detection and data transformation. The missing values of the attribute can be addressed by Ignoring the value filling the value manually Using global constant to fill the value Using a measure of central tendency (mean or median) of value Using attribute mean or median belonging to same class Using the most probable value Noise is a random error or variance in a measured variable. The noisy data can be smoothened using following techniques. Binning methods smooth a sorted data value by consulting the nearby values around it. smoothing by bin means - each value in a bin is replaced by the mean value of the bin. smoothing by bin medians - each bin value is replaced by the bin median smoothing by bin boundaries - the minimum and maximum values in a ...

The Data Ilm

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