1. The following attributes are measured for members of a herd of Asian elephants: and . Based on these measurements, what sort of similarity measure from Section 2.4 (measure of similarity and dissimilarity) would you use to compare or group these elephants? Justify your answer and explain any special circumstances. (Chapter 2) 2. Consider the training examples shown in Table 3.5 (185 page) for a binary classification problem. (Chapter 3) (a) Compute the Gini index for the overall collection of training examples. (b) Compute the Gini index for the Customer ID attribute. (c) Compute the Gini index for the Gender attribute. (d) Compute the Gini index for the Car Type attribute using multiway split. 3. Consider the data set shown in Table 4.9 (348 page). (Chapter 4) (a) Estimate the conditional probabilities for ( +), ( +), ( +), ( ), ( ), and ( ). (b) Use the estimate of conditional probabilities given in the previous question to predict the class label for a test sample ( = 0 = 1 = 0) using the naıve Bayes approach. (c) Estimate the conditional probabilities using the m-estimate approach, with = 1 2 and

1. When comparing or grouping elephants based on the given attributes, we would use a similarity measure that takes into account both the size and weight of the elephants. One possible measure that could be used in this case is the Euclidean distance.

The Euclidean distance is a measure of the straight-line distance between two points in a multi-dimensional space. In this case, each elephant can be represented as a point in a two-dimensional space, with one dimension representing the size attribute and the other representing the weight attribute. By calculating the Euclidean distance between two elephants, we can determine how similar or dissimilar they are based on their size and weight.

The Euclidean distance is appropriate for this scenario because it considers the magnitudes of the measurements and calculates the straight-line distance between the points. This measurement is useful when comparing physical attributes like size and weight, where the magnitude of the measurements is important.

However, it is important to note that there may be special circumstances or considerations when using the Euclidean distance as a similarity measure for elephants. For example, other factors such as age or gender could also be important in determining similarities or groupings among elephants. In such cases, additional attributes would need to be considered and a more comprehensive similarity measure, such as a weighted Euclidean distance, might be more appropriate.

2. (a) To compute the Gini index for the overall collection of training examples, we need to calculate the probability of each class label and then use these probabilities to calculate the Gini impurity.

The Gini impurity measures the degree of impurity or uncertainty in a set of examples. It is calculated by subtracting the sum of the squared probabilities of each class label from 1.

Let’s denote the number of training examples as N and the number of examples in each class label as N1 and N2. The probability of class label 1 is p1 = N1/N and the probability of class label 2 is p2 = N2/N.

The Gini index for the overall collection of training examples is given by:
Gini = 1 – (p1^2 + p2^2)

(b) To compute the Gini index for the Customer ID attribute, we need to consider the possible values of this attribute and their corresponding class labels. We calculate the Gini impurity for each possible value and then weight it by the probability of that value occurring.

(c) Similarly, we can compute the Gini index for the Gender attribute by considering the possible values (e.g., male or female) and their corresponding class labels.

(d) To compute the Gini index for the Car Type attribute using a multiway split, we need to consider each possible way to split the attribute and calculate the Gini impurity for each split. The Gini index for the attribute is then determined by summing the weighted Gini impurities for each split.

3. (a) To estimate the conditional probabilities for the given data set, we need to calculate the probabilities of each combination of attribute values and class labels. For example, the conditional probability P(+, +) represents the probability of having both attributes + and + given the class label +.

(b) Using the estimated conditional probabilities from the previous question and the naive Bayes approach, we can predict the class label for a test sample with attribute values = 0, = 1, = 0. This involves calculating the posterior probabilities for each class label using Bayes’ theorem and selecting the class label with the highest probability.

(c) To estimate the conditional probabilities using the m-estimate approach, we need to specify a value for the smoothing parameter . The m-estimate adjusts the probabilities by adding a small value to each count, which helps handle cases where some combinations of attribute values and class labels have not been observed in the data.

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