linear Algebra least squares
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This document provides an overview of least squares approximation, a linear algebra technique for fitting a linear or polynomial model to data. It explains that least squares approximation finds the "best fit" line or curve to describe the relationship between variables by minimizing the error between predicted and actual values. The document also discusses how the least squares method works by estimating coefficients to create a linear or polynomial expression that best describes the data using techniques like minimizing the error vector and using the inverse and transpose of matrices. Real-world applications to fields like econometrics are also briefly mentioned.
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Ex. It has been suggested that there is a relationship between sleep deprivation of employees and the ability to complete simple tasks. To evaluate this hypothesis, 12 people were asked to solve simple tasks after having been without sleep for 15, 18, 21, and 24 hours. The sample data are shown below.
Subject
Hours without sleep, x
Tasks completed, y
1
15
13
2
15
9
3
15
15
4
18
8
5
18
12
6
18
10
7
21
5
8
21
8
9
21
7
10
24
3
11
24
5
12
24
4
Draw a scatterplot and describe the relationship. Lay a straight-edge on top of the plot and move it around until you find what you think might be a “line of best fit.” Then try to predict the number of tasks completed for someone having been without sleep 16 hours.
Was your line the same as that of the classmate sitting next to you? Probably not. We need a method that we can use to find the “best” regression line to use for prediction. The method we will use is called least-squares. No line will pass exactly through all the points in the scatterplot. When we use the line to predict a y for a given x value, if there is a data point with that same x value, we can compute the error (residual):
Our goal is going to be to make the vertical distances from the line as small as possible. The most commonly used method for doing this is the least-squares method.
The least-squares regression line of y on x is the line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible.
Equation of the Least-Squares Regression Line
· Least-Squares Regression Line:
· Slope of the Regression Line:
· Intercept of the Regression Line:
Generally, regression is performed using statistical software. Clearly, given the appropriate information, the above formulas are simple to use.
Once we have the regression line, how do we interpret it, and what can we do with it?
The slope of a regression line is the rate of change, that amount of change in when x increases by 1.
The intercept of the regression line is the value of when x = 0. It is statistically meaningful only when x can take on values that are close to zero.
To make a prediction, just substitute an x-value into the equation and find .
To plot the line on a scatterplot, just find a couple of points on the regression line, one near each end of the range of x in the data. Plot the points and connect them with a line. .
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We can define heteroscedasticity as the condition in which the variance of the error term or the residual term in a regression model varies. As you can see in the above diagram, in the case of homoscedasticity, the data points are equally scattered while in the case of heteroscedasticity, the data points are not equally scattered.
Two Conditions:
1] Known Variance
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Linear regression by Kodebay
- Linear regression is a predictive modeling technique used to establish a relationship between two variables, known as the predictor and response variables.
- The residuals are the errors between predicted and actual values, and the optimal regression line is the one that minimizes the sum of squared residuals.
- Linear regression can be used to predict variables like salary based on experience, or housing prices based on features like crime rates or school quality. Co-relation analysis examines the relationships between predictor variables.
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- 1. Least Squares Approximation: A Linear Algebra Technique By Larry Wong and James Sfregola
- 2. Soo……You have a bunch of Data. Attendence Big 4 Derby League Position per capita income adjusted population 39,968 1 0 11 35000 1582564 21,852 1 0 10 36725 137471 24,409 1 0 7 36725 371847 26,770 1 0 19 43600 852013 41,917 1 1 2 43600 1351187 37,162 1 0 6 36725 623160 24,510 1 1 16 43600 724121 43,958 1 1 3 36725 738873 40,699 1 0 14 36725 630043 75,595 1 1 1 36725 1194415 What can we do with this? Use Least Squares Approximation (LSE) to estimate the relationship of the dependent variable and the independent variable(s).
- 3. What is Least Squares? • Approximates solutions for inconsistent over determined systems – What does inconsistent mean? – Over determined, what’s that? • In other words LSE allows you to describe a model as being represented by the line of best fit. • Where the “best fit” line, curve or polynomial is created from the approximations.
- 4. Least Squares it’s Actually Pretty Useful Least Squares Approximations are highly relevant in various fields Really, how so? Still don’t believe us? Fine we’ll give examples…but not now.
- 5. So How Does it Estimate a Solution? Least Squares Approximation estimates a best fit solution for a system by Minimizing the magnitude error vector, e. Estimating the coefficients of the best fit equation
- 6. How Do We Ensure Positive Error? Least Squares – Eliminates negative terms – More susceptible to outlying data Least Absolute Difference – Difficult to work with in linear algebra
- 7. `Petal L.` `Petal W.` 1 2 3 4 5 6 7 0.5 1.0 1.5 2.0 2.5
- 8. The Least Squares Theorem For an over determined inconsistent system Ax=b: This is easily solved using inverse and transpose multiplication the resulting vector x can be determined in the form x=(AT A) -1 AT b where x will correspond to the coefficients in your linear or polynomial expression.
- 9. How to Approximate the Error Term? • e = b-Ax , where e is the error term • Where e1= the distance from our data point to the best fit approximation given by the orthogonal projection
- 10. Application to Economics Econometrics utilizes Ordinary Least Squares Approximations (OLS) Similar to LSE except it follows 7 classical assumptions Estimated Regression Equation
- 11. -30000 -20000 -10000 0 10000 20000 30000 0 20000 40000 60000 80000 50 100 150 200 250 300 350 Residual Actual Fitted
- 13. Bibliography • Poole, David Linear Algebra: A Modern Introduction. Canada: Thomson Brooks/Cole, 2006. • Studenmund, A.H. Using Econometrics: A Practical Guide. New York: Pearson Education, 2006.