1 The Following Attributes Are Measured For Members Of A Herd Of Asia ✓ Solved
1. The following attributes are measured for members of a herd of Asian elephants: weight, height, tusk length, trunk length, and ear area . 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 . 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 P ( A| +), P ( B| +), P ( C| +), P ( A|- ), P ( B|- ), and P ( C|- ). (b) Use the estimate of conditional probabilities given in the previous question to predict the class label for a test sample ( A = 0 , B = 1 , C = 0) using the naıve Bayes approach. (c) Estimate the conditional probabilities using the m-estimate approach, with p = 1 / 2 and m = 4.
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Similarity Measures for Asian Elephants
In studying a herd of Asian elephants, it is essential to measure various attributes such as weight, height, tusk length, trunk length, and ear area. When it comes to comparing and grouping these elephants based on these attributes, choosing the right similarity measure is crucial. One suitable similarity measure for this task would be the Euclidean distance.
Justification for Using Euclidean Distance
Euclidean distance is a well-established similarity measure that is particularly useful when dealing with quantitative, continuous attributes like those mentioned (weight, height, tusk length, trunk length, and ear area). The formula for calculating Euclidean distance in an n-dimensional space is given by:
\[
d(x, y) = \sqrt{\sum_{i=1}^{n}(x_i - y_i)^2}
\]
Where \(x\) and \(y\) are the attribute vectors for two elephants. The advantages of using the Euclidean distance include:
1. Sensitivity to Scale: Since all the measured attributes (weight, height, etc.) can be represented as continuous numeric values, Euclidean distance effectively captures the relative differences among them.
2. Conceptual Intuition: Visualizing the similarity between two elephants can intuitively be considered as how far apart they are in a multi-dimensional space. The closer the points are, the more similar the elephants are.
3. Bidirectional Measurement: Unlike some similarity measures that might be asymmetric, Euclidean distance is symmetric in that the distance from \(x\) to \(y\) is the same as from \(y\) to \(x\).
Special Circumstances
However, it is essential to note some considerations when employing this measure:
1. Normalization: If the attributes are on different scales (for instance, weight in kilograms and ear area in square centimeters), normalization should occur. This ensures that no attribute disproportionately influences the similarity measure.
2. Handling Missing Data: If any of the attributes have missing values, they can affect the Euclidean distance significantly. Imputation or other techniques should be employed to handle such cases.
3. Dimensionality: In high-dimensional space, the phenomenon known as “curse of dimensionality” can arise. Here, the distance between points becomes less meaningful as the number of dimensions increases. Thus, it may be necessary to reduce dimensions incorporating methods like principal component analysis (PCA) before using Euclidean distance.
Gini Index Calculations
In the context of the binary classification problem presented in a training set, the Gini index serves as a measure of impurity or purity used to decide the attributes' effectiveness in classification.
For an overall collection of training examples, the Gini index can be computed using the formula:
\[
Gini(D) = 1 - \sum_{k=1}^{K}(p_k)^2
\]
Where \(p_k\) is the probability of class \(k\) in the dataset \(D\).
(A) Overall Gini Index
Assuming a dataset with classes 0 and 1, the computation involves counting the number of instances of each class. If, for example, there are 40 instances of class 0 and 60 of class 1, the probabilities \(p_0 = 0.4\) and \(p_1 = 0.6\):
\[
Gini(D) = 1 - (0.4^2 + 0.6^2) = 1 - (0.16 + 0.36) = 1 - 0.52 = 0.48
\]
(B) Gini Index for the Customer ID Attribute
For the Customer ID attribute, the Gini index measures how the dataset is split based on unique customer IDs. If customer IDs are not directly related to the class distinction, the Gini index may frail in meaning.
(C) Gini Index for the Gender Attribute
To compute the Gini index for the Gender attribute, one would categorize the instances by Gender (male and female). If there are 30 males and 70 females, the respective probabilities would form the basis for calculating the Gini index.
(D) Gini Index for the Car Type Attribute
Finally, when calculating the Gini index for the Car Type attribute with a multi-way split (e.g., SUV, Sedan, Truck), one must account for each unique car type contributing to the Gini index summarily based on their proportions.
Naive Bayes Probability Estimation
Moving to the dataset's estimation of conditional probabilities based on provided attributes, we can frame it akin to a Naive Bayes model.
(A) Conditional Probabilities
To estimate probabilities such as \(P(A|+)\), \(P(B|+)\), etc., one computes the ratio of the occurrences of the feature value given the class label.
(B) Predicting Class Labels Using Naive Bayes
For a test sample \(A = 0, B = 1, C = 0\), we would multiply the respective conditional probabilities:
\[
P(Class | Data) = P(A|Class) \times P(B|Class) \times P(C|Class)
\]
Where the chosen class with the highest probability constitutes the predicted label.
(C) M-Estimate Conditional Probabilities
Using the m-estimate approach with \(p = 1/2\) and \(m = 4\) gives a smoothed conditional probability by adjusting counts based on total counts and prior probabilities.
\[
P(A|+) = \frac{n_A + p}{N + m \cdot p}
\]
Where \(n_A\) is the count of attribute occurrences while the denominator adjusts for the number of instances.
Conclusion
Utilizing measures such as the Euclidean distance in classifying and comparing elephants, alongside Gini indexes in binary classification and Naive Bayes probabilistic models, renders a comprehensive view. Enhancements via normalization and addressing high dimensionality remain crucial for meaningful analysis.
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