Intermediate Lab PHYS 3870

1. Intermediate Lab PHYS 3870 Lecture 5 Comparing Data and Models—Quantitatively Linear Regression

2. Intermediate Lab PHYS 3870 Comparing Data and Models—Quantitatively Linear Regression References: Taylor Ch. 6, 7, 8 Also refer to “Glossary of Important Terms in Error Analysis”

3. Intermediate Lab PHYS 3870 Errors in Measurements and Models A Review of What We Know

4. Precision and Random (Statistical) Error

5. Standard Deviation

6. Standard Deviation of the Mean

7. Comparison with Other Data Comparison of precision or accuracy?

8. Direct Comparison with Standard Comparison of precision or accuracy?

9. Errors in Models—Error Propagation

10. Specific Rules for Independent Error Propagation

11. General Formula for Error Propagation

12. General Formula for Multiple Variables

13. Mean of Gaussian Distribution as “Best Estimate” Principle of Maximum Likelihood To find the most likely value of the mean (the best estimate of ẋ), find X that yields the highest probability for the data set. Consider a data set {x1, x2, x3 …xN } Each randomly distributed with The combined probability for the full data set is the product Best Estimate of X is from maximum probibility or minimum summation Solve for derivative set to 0 Minimize Sum Best estimate of X Best Estimate of σ

14. “Best Estimates” of Gaussian Distribution Principle of Maximum Likelihood To find the most likely value of the mean (the best estimate of ẋ), find X that yields the highest probability for the data set. Consider a data set {x1, x2, x3 …xN } The combined probability for the full data set is the product Best Estimate of X is from maximum probibility or minimum summation Solve for derivative wrst X set to 0 Minimize Sum Best estimate of X Best Estimate of σ is from maximum probibility or minimum summation Solve for derivative wrstσset to 0 Best estimate of σ Minimize Sum See Prob. 5.26

15. Weighted Averages Best Estimate of χ is from maximum probibility or minimum summation Solve for derivative wrstχset to 0 Minimize Sum Best estimate of χ

16. Intermediate Lab PHYS 3870 Comparing Measurements to Models Linear Regression

17. Question 1: What is the Best Linear Fit (A and B)? Best Estimate of intercept, A , and slope, B, for Linear Regression or Least Squares-Fit for Line

18. “Best Estimates” of Linear Fit Best Estimates of A and B are from maximum probibility or minimum summation Solve for derivative wrstA and B set to 0 Minimize Sum Best estimate of A and B

19. “Best Estimates” of Linear Fit Best Estimates of A and B are from maximum probibility or minimum summation Solve for derivative wrstA and B set to 0 Minimize Sum Best estimate of A and B Minimize Sum

20. Least Squares Fits to Other Curves

21. Least Squares Fits to Other Curves

22. Least Squares Fits to Other Curves

23. Least Squares Fits to Other Curves

24. Problem 8.24

25. Problem 8.24

26. Problem 8.24

27. Problem 8.24

28. Problem 8.24

29. Question 2: Is it Linear?

30. Tabulated Correlation Coefficient

31. Uncertainties in Slope and Intercept Taylor: Relation to R2 value:

32. Schwartz Inequality

33. Intermediate Lab PHYS 3870 Non-Linear Curve Fitting

34. Chi Squared Analysis for Least Squares Fit

35. Weighted Averages

36. Question 1: What is the Best Linear Fit (A and B)?

37. Chi Squared Analysis for Least Squares Fit From the probability table ~99.5% (highly significant) confidence