Weighted Least Squares

Table of Contents

Weighted Least Square is an estimate used in regression situations where the error terms are heteroscedastic or has non constant variance.

To get a better understanding about Weighted Least Squares, lets first see what Ordinary Least Square is and how it differs from Weighted Least Square.

What is Ordinary Least Square(OLS)?

In a simple linear regression model of the form,

where

$y\nocache$ is the independent variable

$x_{i} \nocache$ is the independent variable

$_{\beta_{0}} \nocache$ and $_{\beta_{1}} \nocache$ are the regression coefficients

$\epsilon\nocache$ is the random error or the residual.

The goal is to find a line that best fits the relationship between the outcome variable $y\nocache$ and the input variable $x_{i} \nocache$ . With OLS, the linear regression model finds the line through these points such that the sum of the squares of the difference between the actual and predicted values is minimum.

i.e., to find $_{\beta_{0}} \nocache$ and $_{\beta_{1}} \nocache$ such that

$\sum_{i=1}^{n}[y_{i} - (\beta_{0} + \beta_{1}x_{i})]^2\nocache$

is minimum.

In such linear regression models, the OLS assumes that the error terms or the residuals (the difference between actual and predicted values) are normally distributed with mean zero and constant variance. This constant variance condition is called homoscedasticity.

If this assumption of homoscedasticity does not hold, the various inferences made with this model might not be true.

To check for constant variance across all $y\nocache$ values along the regression line, a simple plot of the residuals and the fitted outcome values and the histogram of residuals such as below can be used.

In an ideal case with normally distributed error terms with mean zero and constant variance , the plots should look like this.

Residuals vs Fitted Values Plot

histogram of residuals

From the above plots its clearly seen that the error terms are evenly distributed on both sides of the reference zero line proving that they are normally distributed with mean=0 and has constant variance.

The histogram of the residuals also seems to have datapoints symmetric on both sides proving the normality assumption.

In some cases, the variance of the error terms might be heteroscedastic, i.e., there might be changes in the variance of the error terms with increase/decrease in predictor variable.

In those cases of non-constant variance Weighted Least Squares (WLS) can be used as a measure to estimate the outcomes of a linear regression model.

Now let’s see in detail about WLS and how it differs from OLS.

#python

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Buddha Community

Camilla Nicolas

1604056260

In-depth analysis of the regularized least-squares algorithm

This article will introduce key concepts about Regularized Loss Minimization (RLM) and Empirical Risk Minimization (ERM), and it’ll walk you through the implementation of the least-squares algorithm using MATLAB. The models obtained using RLM and ERM will then be compared and discussed against each other.

We’ll use a polynomial curve-fitting problem to predict the best polynomial for this data. The least-squares algorithm will be implemented step-by-step using MATLAB.

By the end of this post, you’ll understand the least-squares algorithm and be aware of the advantages and downsides of RLM and ERM. Additionally, we’ll discuss some important concepts about overfitting and underfitting.

Dataset

We’ll use a simple one input dataset with N = 100 data points. This dataset was originally proposed by Dr. Ruth Urner on one of her assignments for a machine learning course. In the repository below, you’ll find two TXT files: dataset1_inputs.txt and dataset1_outputs.txt.

These files contain the input and output vectors. Using MATLAB, we’ll plot these data points in a chart. On MATLAB, I imported them in Home > Import Data. Then, I created the flowing script for plotting the data points.

#data-science #programming #polynomial-regression #least-squares #machine-learning

Phoebe Shanahan

1598299980

How to find the least squares plane from a cloud of point using Excel, Numbers etc…

When using a spreadsheet or any kind of scientific software, you often find the possibility to do a linear regression, this allows you to find the line that fits the best a cloud of point. To do so softwares use what is known as the method of the least squares. Unfortunately spreadsheets are only capable of doing this method for a 2D space.

In this article I will attempt to explain how we can use a spreadsheet to to find the least squares plane from a cloud of point in a 3D space. And hopefully you will have a better understanding of this method.

First of all let’s consider our set of point. Each point has three coordinates, x, y and z where x, y, z are three reals. You can see the set of point use in this exemple in the following picture.

Now that we have our points we can start digging into the math. Let me remind you what a plan is mathematicaly speaking. A plan is the result of a linear fonction of two variables, these fancy words simply means that for a couple of two numbers _x _and y, the function gives you a third number z, thus the link between x, y and z is linear, meaning the operations done on x and y are only multiplication with a real or addition with a real. Finally we can write this equation for a plan P’ :
Image for post

Any given couple of _x _and y has a corresponding z value, we can use our x and y values in this equation to express the gap between the z value of our point and the z value it should have if it was part of this “ideal” plan. We will call this new function g for gap.

#numbers #mathematics #excel #least-squares #spreadsheets #cloud

Alec Nikolaus

1596337740

Weighted Least Squares

Table of Contents

Weighted Least Square is an estimate used in regression situations where the error terms are heteroscedastic or has non constant variance.

To get a better understanding about Weighted Least Squares, lets first see what Ordinary Least Square is and how it differs from Weighted Least Square.

What is Ordinary Least Square(OLS)?

In a simple linear regression model of the form,

where

$y\nocache$ is the independent variable

$x_{i} \nocache$ is the independent variable

$_{\beta_{0}} \nocache$ and $_{\beta_{1}} \nocache$ are the regression coefficients

$\epsilon\nocache$ is the random error or the residual.

The goal is to find a line that best fits the relationship between the outcome variable $y\nocache$ and the input variable $x_{i} \nocache$ . With OLS, the linear regression model finds the line through these points such that the sum of the squares of the difference between the actual and predicted values is minimum.

i.e., to find $_{\beta_{0}} \nocache$ and $_{\beta_{1}} \nocache$ such that

$\sum_{i=1}^{n}[y_{i} - (\beta_{0} + \beta_{1}x_{i})]^2\nocache$

is minimum.

In such linear regression models, the OLS assumes that the error terms or the residuals (the difference between actual and predicted values) are normally distributed with mean zero and constant variance. This constant variance condition is called homoscedasticity.

If this assumption of homoscedasticity does not hold, the various inferences made with this model might not be true.

To check for constant variance across all $y\nocache$ values along the regression line, a simple plot of the residuals and the fitted outcome values and the histogram of residuals such as below can be used.

In an ideal case with normally distributed error terms with mean zero and constant variance , the plots should look like this.

Residuals vs Fitted Values Plot

histogram of residuals

From the above plots its clearly seen that the error terms are evenly distributed on both sides of the reference zero line proving that they are normally distributed with mean=0 and has constant variance.

The histogram of the residuals also seems to have datapoints symmetric on both sides proving the normality assumption.

In some cases, the variance of the error terms might be heteroscedastic, i.e., there might be changes in the variance of the error terms with increase/decrease in predictor variable.

In those cases of non-constant variance Weighted Least Squares (WLS) can be used as a measure to estimate the outcomes of a linear regression model.

Now let’s see in detail about WLS and how it differs from OLS.

#python

Ahebwe Oscar

1624114920

Self hosted FLOSS fitness/workout, nutrition and weight tracker written with Django

wger

wger (ˈvɛɡɐ) Workout Manager is a free, open source web application that help you manage your personal workouts, weight and diet plans and can also be used as a simple gym management utility. It offers a REST API as well, for easy integration with other projects and tools.

workout

Mobile app

Installation

These are the basic steps to install and run the application locally on a Linux

system. There are more detailed instructions, other deployment options as well

as an administration guide available at https://wger.readthedocs.io or locally

in your code repository in the docs folder.

Please consult the commands’ help for further information and available

parameters.

Production

If you want to host your own instance, take a look at the provided docker

compose file. This config will persist your database and uploaded images:

https://github.com/wger-project/docker

…

#django #self hosted floss fitness/workout #weight tracker written #self hosted floss fitness/workout, nutrition and weight tracker written with django #floss #nutrition

parivartan ayurveda

1605870015

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