Parameterized Family and Model Parameters  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  
================================================================================= A parameterized family, in the context of mathematics or statistics, refers to a collection or set of mathematical objects (such as functions, curves, or distributions) that can be uniquely identified and distinguished by one or more parameters. These parameters act as variables that allow you to generate different members of the family by varying their values. For example:
Model parameters, also known as regression coefficients, are the parameters specific to the predictor variables in ML. They determine how changes in the predictor variables affect the expected value of the response variable. In a linear regression model, for example, the model parameters are the coefficients for each predictor variable. For a GLM, these parameters are specific to the chosen link function and probability distribution. Parameterized families are useful in mathematics and statistics because they allow for a flexible and systematic way to describe and analyze a wide range of mathematical objects. By varying the parameters, you can study different instances or variations of the same fundamental concept, making it easier to understand and work with mathematical structures and relationships. 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: [3974a] where,
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: [3974b] 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: [3974c] where,
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 process, where the goal is to minimize the difference between the predicted output (hypothesis(x)) and the actual output (y), is known as "training" or "model training" in machine learning. The difference can be given by (h(x)y)^{2}. In supervised learning, this is a fundamental step where the machine learning algorithm adjusts its parameters to make the predictions as close as possible to the true target values in the training dataset. The process of minimizing the difference between the hypothesis and the actual target is typically achieved through various optimization techniques, such as gradient descent, which iteratively updates the model's parameters to reduce the prediction error. The objective is to find the set of parameters that results in the best possible fit of the model to the training data, allowing it to generalize well to new, unseen data. Parameterized families are a fundamental concept in machine learning, particularly in the context of optimization. In machine learning, models are often defined as parameterized families of functions, and the process of training these models involves optimizing the parameters within the family to fit a given dataset or perform a specific task. Here's how parameterized families are used in machine learning optimization:
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