Linear Regression and its Algorithm  Python for Integrated Circuits   An Online Book  

Python for Integrated Circuits http://www.globalsino.com/ICs/  


Chapter/Index: Introduction  A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z  Appendix  
================================================================================= Linear regression is a fundamental statistical and machine learning technique used for modeling the relationship between a dependent variable (also called the target or response variable) and one or more independent variables (predictors or features). It assumes a linear relationship between the independent variables and the dependent variable. The primary goal of linear regression is to find the bestfitting linear equation that describes the relationship between the variables. This equation is often referred to as the regression model or the linear equation. It has the following general form for a simple linear regression with one independent variable:  [3910a] Where:
In multiple linear regression, when there are multiple independent variables, the equation becomes:  [3910b] Where:
The goal of linear regression is to estimate the values of the coefficients (b_{0}, b_{1}, b_{2}, … , b_{p}) that minimize the sum of squared errors (the squared differences between the predicted values and the actual values of the dependent variable). This process is often performed using mathematical optimization techniques. Linear regression is widely used in various fields, including economics, finance, social sciences, and machine learning, to model and analyze relationships between variables, make predictions, and infer patterns in data. There are also variations of linear regression, such as ridge regression and lasso regression, which introduce regularization to handle multicollinearity and prevent overfitting in more complex models. When you have multiple training samples (also known as a dataset with multiple data points), the equations for the hypothesis and the cost function change to accommodate the entire dataset. This is often referred to as "batch" gradient descent, where you update the model parameters using the average of the gradients computed across all training samples. Hypothesis (for multiple training samples): The hypothesis for linear regression with multiple training samples is represented as a matrix multiplication. Let be the number of training samples, be the number of features, be the feature matrix, and be the target values. The hypothesis can be expressed as: [3910c] where,
Equation :3910c can be rewritten by,  [3910d] Cost Function (for multiple training samples): The cost function in linear regression is typically represented using the mean squared error (MSE) for multiple training samples. The cost function is defined as: [3910e] where,
Gradient Descent (for updating ):To train the linear regression model, you typically use gradient descent to minimize the cost function. The update rule for � in each iteration of gradient descent is as follows: [3910f] where,
Equation 3910f can be simplified by,  [3910g] The negative sign in Equations 3910f and 3910g indicates that we need to minimize the error. Here, Batch Gradient Descent (BGD) is used in the logistic regression because BGD is one of the optimization techniques commonly used to train logistic regression models. In this case, BGD is an optimization algorithm used to find the optimal parameters for the logistic regression model. In each iteration, each parameter is updated simultaneously using the gradients calculated over the entire training dataset. This process is repeated until the cost function converges to a minimum.This batch gradient descent process allows you to find the optimal parameters that minimize the cost function, making your linear regression model fit the training data as closely as possible.The Normal Equation is a mathematical formula used in linear regression to find the coefficients (parameters) of a linear model that best fits a given set of data points. Linear regression is a statistical method used to model the relationship between a dependent variable (the target or output) and one or more independent variables (predictors or features) by fitting a linear equation to the observed data. By solving the Normal Equation, we can obtain the values of the coefficients θ that minimize the sum of squared differences between the predicted values of the dependent variable and the actual observed values. These coefficients define the bestfitting linear model for the given data. While the Normal Equation provides a closedform solution for linear regression, there are also iterative optimization methods like gradient descent that can be used to find the coefficients, especially when dealing with more complex models or large datasets. Nonetheless, the Normal Equation is a valuable tool for understanding the fundamental principles of linear regression and for solving simple linear regression problems analytically. When you use the Normal Equation to solve for the coefficients (θ) in linear regression, you are essentially finding the values of θ that correspond to the global minimum of the cost function in a single step. In linear regression, the goal is to find the values of θ that minimize a cost function, often represented as J(θ). This cost function measures the error or the difference between the predicted values (obtained using the linear model with θ) and the actual observed values in your dataset. To find the values of θ that minimize this cost function, you can use the Normal Equation, which provides an analytical solution. When you solve the Normal Equation, you find the exact values of θ that minimize J(θ) by setting the gradient of J(θ) with respect to θ equal to zero. The key point is that this solution is obtained directly, without the need for iterative optimization algorithms like gradient descent. Gradient descent, for example, iteratively adjusts the parameters θ to minimize the cost function, which may take many steps to converge to the global minimum. In contrast, the Normal Equation provides a closedform solution that directly computes the optimal θ values in a single step by finding the point where the gradient is zero. However, note that the Normal Equation has some limitations:
The key steps and components of the linear regression algorithm are:
Linear regression is a statistical modeling technique that relies on certain assumptions about the relationship between the independent (predictor) variables and the dependent (target) variable. These assumptions include:
Figure 3910 and Equation 3910h shows the linear learning model interaction with input and distribution. During learning process, a model learns parameters like θ through the learning process but the ditribution is not learnt. These parameters capture the relationships between input features and the target variable. the distribution of the data, which represents the underlying statistical properties of the dataset, is typically not learned explicitly in many machine learning models. Instead, the model makes certain assumptions about the distribution (e.g., assuming a normal distribution) but doesn't directly estimate the entire distribution. This separation of learning parameters and modeling the data distribution is a common practice in various machine learning algorithms. Figure 3910. Linear learning model.  [3910h] As discussed in supportvector machines (SVM), we have,  [3910i] where, g is the activation function. n is the number of input features. In linear regression, a goal is to minimize the least squares (OLS) or mean squared error (MSE) term, which measures the error between the predicted values and the actual values ^{(i)}, below, [3910ia] This is the cost function of linear regression without regularization, often referred to as Ordinary Least Squares (OLS) regression. This is the basic form of linear regression that aims to minimize the mean squared error. Term 3910ia with L2 regularization (Ridge Regression) becomes the term below,  [3910ib] Equation 3910i is a basic representation of a singlelayer neural network, also known as a perceptron or logistic regression model, depending on the choice of the activation function g. From Equation 3910i, we can derive different forms or variations by changing the activation function, the number of layers, or the architecture of the neural network as shown in Table 3910a. Table 3910a. Different forms or variations of Equation 3910i.
Table 3910b. Applications of Linear Regression.
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