Math 4030 – 11b Method of Least Squares. Model: Dependent (response) Variable Independent (control) Variable Random Error Objectives: Find (estimated)

Slides:

Advertisements

Similar presentations

Modeling of Data. Basic Bayes theorem Bayes theorem relates the conditional probabilities of two events A, and B: A might be a hypothesis and B might.

Advertisements

Topic 12: Multiple Linear Regression

Kin 304 Regression Linear Regression Least Sum of Squares

Chapter 12 Simple Linear Regression

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 ~ Curve Fitting ~ Least Squares Regression Chapter.

1 Simple Linear Regression and Correlation The Model Estimating the Coefficients EXAMPLE 1: USED CAR SALES Assessing the model –T-tests –R-square.

Simple Regression. Major Questions Given an economic model involving a relationship between two economic variables, how do we go about specifying the.

Ch11 Curve Fitting Dr. Deshi Ye

Definition  Regression Model  Regression Equation Y i =  0 +  1 X i ^ Given a collection of paired data, the regression equation algebraically describes.

Simple Linear Regression

Chapter 12 Simple Linear Regression

Read Chapter 17 of the textbook

LINEAR REGRESSION: What it Is and How it Works Overview What is Bivariate Linear Regression? The Regression Equation How It’s Based on r.

LINEAR REGRESSION: What it Is and How it Works. Overview What is Bivariate Linear Regression? The Regression Equation How It’s Based on r.

Lesson #32 Simple Linear Regression. Regression is used to model and/or predict a variable; called the dependent variable, Y; based on one or more independent.

Lecture 12 Projection and Least Square Approximation Shang-Hua Teng.

Probability & Statistics for Engineers & Scientists, by Walpole, Myers, Myers & Ye ~ Chapter 11 Notes Class notes for ISE 201 San Jose State University.

Chapter 6 (cont.) Regression Estimation. Simple Linear Regression: review of least squares procedure 2.

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 ~ Curve Fitting ~ Least Squares Regression Chapter.

Regression Analysis What is regression ?What is regression ? Best-fit lineBest-fit line Least squareLeast square What is regression ?What is regression.

Stats for Engineers Lecture 9. Summary From Last Time Confidence Intervals for the mean t-tables Q Student t-distribution.

Linear Regression James H. Steiger. Regression – The General Setup You have a set of data on two variables, X and Y, represented in a scatter plot. You.

CISE301_Topic41 CISE301: Numerical Methods Topic 4: Least Squares Curve Fitting Lectures 18-19: KFUPM Read Chapter 17 of the textbook.

Section 5.2: Linear Regression: Fitting a Line to Bivariate Data.

MTH 161: Introduction To Statistics

Thomas Knotts. Engineers often: Regress data  Analysis  Fit to theory  Data reduction Use the regression of others  Antoine Equation  DIPPR.

Regression Regression relationship = trend + scatter

1 Multiple Regression A single numerical response variable, Y. Multiple numerical explanatory variables, X 1, X 2,…, X k.

Chapter 11 Correlation and Simple Linear Regression Statistics for Business (Econ) 1.

Chapter 6 (cont.) Difference Estimation. Recall the Regression Estimation Procedure 2.

Math 4030 – 13a Correlation & Regression. Correlation (Sec. 11.6):  Two random variables, X and Y, both continuous numerical;  Correlation exists when.

Economics 173 Business Statistics Lecture 10 Fall, 2001 Professor J. Petry

Linear Prediction Correlation can be used to make predictions – Values on X can be used to predict values on Y – Stronger relationships between X and Y.

Curve Fitting Pertemuan 10 Matakuliah: S0262-Analisis Numerik Tahun: 2010.

© 2001 Prentice-Hall, Inc.Chap 13-1 BA 201 Lecture 18 Introduction to Simple Linear Regression (Data)Data.

5.4 Third Order Determinants and Cramer’s Rule. Third Order Determinants To solve a linear system in three variables, we can use third order determinants.

Chapter 11: Linear Regression and Correlation Regression analysis is a statistical tool that utilizes the relation between two or more quantitative variables.

Curve Fitting Introduction Least-Squares Regression Linear Regression Polynomial Regression Multiple Linear Regression Today’s class Numerical Methods.

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 ~ Curve Fitting ~ Least Squares Regression.

1 Simple Linear Regression and Correlation Least Squares Method The Model Estimating the Coefficients EXAMPLE 1: USED CAR SALES.

Adam Q. Colley EMIS8381 March 31,  One Dependent Variable  One or more (n) Independent Variables  (Independent to the Dependent Variable)  More.

Differential Equations Linear Equations with Variable Coefficients.

Dependent (response) Variable Independent (control) Variable Random Error XY x1x1 y1y1 x2x2 y2y2 …… xnxn ynyn Raw data: Assumption:  i ‘s are independent.

1 1 Slide The Simple Linear Regression Model n Simple Linear Regression Model y =  0 +  1 x +  n Simple Linear Regression Equation E( y ) =  0 + 

Correlation and Regression Chapter 9. § 9.2 Linear Regression.

Chapter 14 Introduction to Regression Analysis. Objectives Regression Analysis Uses of Regression Analysis Method of Least Squares Difference between.

Statistics 350 Lecture 2. Today Last Day: Section Today: Section 1.6 Homework #1: Chapter 1 Problems (page 33-38): 2, 5, 6, 7, 22, 26, 33, 34,

Chapter 11 Linear Regression and Correlation. Explanatory and Response Variables are Numeric Relationship between the mean of the response variable and.

Chapter 13 Nonlinear and Multiple Regression. Polynomial Regression For k th-degree model: Y=  0 +  1 x +  2 x 2 +…+  k x k +  Where  is N(0, 

Regression and Correlation of Data Summary

Part 5 - Chapter

Probability Theory and Parameter Estimation I

Kin 304 Regression Linear Regression Least Sum of Squares

Ch12.1 Simple Linear Regression

BPK 304W Regression Linear Regression Least Sum of Squares

Warmup: Find the product, if possible. −6 4 − 

Solving Equations by Factoring and Problem Solving

LESSON 21: REGRESSION ANALYSIS

Linear Regression.

Lesson 13-3: Determinants & Cramer’s Rule

Linear regression Fitting a straight line to observations.

Nonlinear Fitting.

Discrete Least Squares Approximation

Least Square Regression

CALCULATING EQUATION OF LEAST SQUARES REGRESSION LINE

Multivariate Analysis Regression

Ch 4.1 & 4.2 Two dimensions concept

Adequacy of Linear Regression Models

Regression and Correlation of Data

Regression and Correlation of Data

Presentation transcript:

Math 4030 – 11b Method of Least Squares

Model: Dependent (response) Variable Independent (control) Variable Random Error Objectives: Find (estimated) values of the coefficients  and  (based on the sample data) Evaluate the model efficiency; Predict or estimate the Y values for “un-tested” x values. XY x1x1 y1y1 x2x2 y2y2 …… xnxn ynyn Raw data:

Regression Line (Best Fitting Line) Estimated Y value for given x Estimated coefficients for  and  Error Finding the equation of the best fitting line is to find the coefficients a and b, the estimated values of  and . How?

The Method of Least Squares Method of Least Squares finds the line (or a and b) such that sum of squared errors is minimized (among all choices of a and b.

Calculation: Solve a system of 2 linear equations.

Solution by Cramer’s Rule:

Square-Sum Notations:

Solutions: Estimate  : Estimate  : Equation for the Regression Line: Residual sum of squares (or error sum of squares): Note: exchange x and y will end in different regression line.

Curvilinear Regression (Sec. 11.3): Exponential: Reciprocal: Power:

Objectives: Find (estimated) values of the coefficients  i ‘s (based on the sample data) Predict or estimate the Y values for “un-tested” x values. XY x1x1 y1y1 x2x2 y2y2 …… xnxn ynyn Raw data: Polynomial Regression:

Calculation: Solve a system of p + 1 linear equations to solve p + 1 unknowns. If all x i values are distinct, the system has the unique solution.