In this problem, we write a program to find the coefficients for a linear regression model for the dataset provided (data2.txt). Assume a linear model: y = w0 + w1*x. You need to 1) Plot the data (i.e., x-axis for 1 st column, y-axis for 2 nd column), and use Python to implement the following methods to find the coefficients: 2) Normal equation, and 3) Gradient Descent using batch AND stochastic modes respectively: a) Determine an appropriate termination condition (e.g., when cost function is less than a threshold, and/or after a given number of iterations). b) Print the cost function vs. iterations for each mode; compare and discuss batch and stochastic modes in terms of the accuracy and the speed of convergence. c) Choose a best learning rate. For example, you can plot cost function vs. learning rate to determine the best learning rate. Please implement the algorithms by yoursef and do NOT use the fit() function of the library

Linear regression is a statistical modeling technique used to analyze the relationship between a dependent variable and one or more independent variables. In this problem, we are tasked with finding the coefficients for a linear regression model using the dataset provided in “data2.txt”. Our model assumes a linear relationship between the dependent variable “y” and the independent variable “x”, defined as y = w0 + w1*x.

To begin, we should plot the data to visually examine the relationship between x and y. The x-axis will represent the values in the first column of the dataset, while the y-axis will represent the values in the second column. This step allows us to gain insights into the nature of the data and determine if a linear relationship is reasonable.

Next, we will implement two methods to find the coefficients for our linear regression model: the Normal equation and Gradient Descent.

The Normal equation provides us with a closed-form solution for finding the optimal values of the coefficients w0 and w1. By solving a system of linear equations, we can obtain the exact values of these coefficients that minimize the sum of squared errors.

On the other hand, Gradient Descent is an iterative optimization algorithm that finds the optimal coefficients by iteratively updating them based on the gradient of the cost function. We will implement both batch and stochastic modes of Gradient Descent. In the batch mode, we update the coefficients after processing the entire dataset, while in the stochastic mode, we update the coefficients after processing each individual data point.

To determine a termination condition for both methods, we can set a threshold for the cost function. Once the cost function falls below this threshold, or after a given number of iterations, we can terminate the optimization process.

For each mode, we should print the value of the cost function at each iteration. This will allow us to monitor the convergence of the algorithm and compare the accuracy and speed of convergence between the batch and stochastic modes. It is important to note that the batch mode typically converges more slowly but provides more accurate estimates, while the stochastic mode converges faster but may have higher variance in the estimates.

Additionally, we need to choose an appropriate learning rate for the Gradient Descent algorithm. The learning rate determines the size of the step taken in each iteration. If the learning rate is too large, the algorithm may fail to converge. Conversely, if the learning rate is too small, the algorithm may converge slowly. We can plot the cost function against different learning rates to determine the optimal learning rate that balances convergence and speed.

In implementing the algorithms, we should refrain from using pre-built functions or libraries, and instead develop the algorithms ourselves. This will enable us to gain a deeper understanding of the underlying concepts and ensure our ability to customize and adapt the algorithms as needed.

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