CS4378V CS5369L Homework 2 Assigned

Cs4378v Cs5369l Homework 2cs4378v Cs5369l Homework 2assigned

Cs4378v Cs5369l Homework 2cs4378v Cs5369l Homework 2assigned

CS4378V_CS5369L: Homework #2 CS4378V_CS5369L: Homework #2 Assigned: Tuesday, October 9, 2012 Due: Tuesday, October 23, points)CS4378V_CS5369L: Homework #2 1. (40 pts) Implement the linear regression algorithm (gradient descent) and run the linear regression algorithm on the training samples (linear_regression_data.txt). You can find the dataset on Tracs under “Resources: Dataâ€. Get the learned linear function and the predicted results of the testing samples (linear_regression_data.txt). 2. (40 pts.) Learn the probabilities for Naive Bayes for the following training examples. Use Laplace’s smoothing to estimate conditional probabilities.

How well does your Naive Bayes hypothesis perform on the following test examples? Show your work. Attributes Class x1 x2 x3 x4 y . (20 pts) Design an n-input perceptron that implements the function: if k or more of the inputs are true, the output is true. (Suppose the input value can only be “1†(true) or “0†(false))

Paper For Above instruction

The assignment comprises three key tasks: implementing linear regression using gradient descent, applying Naive Bayes classifier with Laplace smoothing, and designing an n-input perceptron for a threshold function. This paper details the methodologies, implementation strategies, and expected outcomes for each part, providing comprehensive insights into fundamental machine learning algorithms.

Implementing Linear Regression with Gradient Descent

Linear regression is a supervised learning algorithm used for predicting continuous variables by modeling the relationship between dependent and independent variables. The core idea is to minimize the residual sum of squares between observed and predicted values through optimization. Gradient descent is an iterative method that adjusts model parameters to find the minimum of the cost function, typically the mean squared error.

Implementation begins with initializing the weights, often to zero or small random values. The dataset, 'linear_regression_data.txt', must be loaded, which contains features and output values. The algorithm then computes predicted outputs for each sample, calculates the error, and updates the weights in the direction that decreases the error—scaled by a learning rate.

Mathematically, the weight update rule in gradient descent for linear regression is:

= - α * ∇J(w)

where α is the learning rate and ∇J(w) is the gradient of the cost function with respect to weights. The process repeats until convergence criteria are met, such as minimal change in errors or reaching maximum iterations.

Post-training, the learned linear function is used to predict outputs on test data. The quality of the model can be evaluated using metrics like Mean Squared Error (MSE) or R-squared value.

Applying Naive Bayes with Laplace Smoothing

Naive Bayes classifiers are probabilistic models based on Bayes’ theorem, assuming attribute independence. For training data, conditional probabilities P(X_i | Y) are estimated, along with the prior probability P(Y). Laplace’s smoothing (add-one smoothing) is used to handle zero-frequency problems, ensuring that probabilities are non-zero, which prevents the model from being overly confident or undefined.

The estimation process involves counting the occurrences of each attribute value for each class, adding one to each count, and normalizing by the total counts plus the number of possible attribute values. This ensures a smooth probability distribution.

For classifying test examples, the model computes the posterior probability for each class given the attributes using:

P(Y | X) ∝ P(Y) * Π P(X_i | Y)

where the product runs over all attributes. The class with the highest posterior probability is selected as the predicted class.

Designing an n-Input Perceptron for Threshold Function

A perceptron is a simple linear classifier that computes a weighted sum of its inputs and outputs true if this sum exceeds a threshold. To implement a function where the output is true if k or more of the inputs are true, the perceptron needs to have appropriately assigned weights and a bias term.

The design involves setting each input weight to 1. The bias (or threshold) is set to (k - 1) to fulfill the condition: if at least k inputs are true, the sum exceeds (k - 1), triggering a true output.

Mathematically:

Output = 1 if (∑ w_i * x_i) ≥ (k - 1)

with w_i = 1 for all i, and the bias term incorporated into the threshold calculation.

This perceptron effectively acts as a threshold gate, activating when the count of true inputs reaches the specified k.

Conclusion

This assignment integrates fundamental machine learning algorithms—linear regression, Naive Bayes classification, and perceptron design—demonstrating their implementation and applicability. Linear regression models continuous outcomes through gradient descent optimization, Naive Bayes provides probabilistic classification with smoothing to handle sparse data, and the n-input perceptron implements a threshold function based on summation of binary inputs. Mastery of these techniques facilitates a deeper understanding of predictive modeling and classification tasks in machine learning.

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