1. CIVE2602 - Engineering Mathematics 2.2 (20 credits) Statistics and Probability Lecture 12- Linear regression review Transforming data -Coursework questions? ©Claudio Nunez 2010, sourced from http://commons.wikimedia.org/wiki/File:2010_Chile_earthquake_- _Building_destroyed_in_Concepci%C3%B3n.jpg?uselang=en-gb Available under creative commons license

2. Regression analysis Aim: To predict y from x (‘regressing y on x’) y x ~N(0,σ 2 )

3. Dependant variable (y) Independent or Control variable (x) What’s the “best fit” line? “minimises squared residuals” Find values for a and b (cover this in examples class)

4. How good is the regression? 3 techniques to be aware of – 1) Coefficient of Determination (R 2 ) 2) Examine the residuals (plot and evaluate) 3) Significance tests (find p-values for α and β ) (value between 0 and 1 (i.e. 0-100%) Coursework Question 2- When considering indicators of goodness of fit. These approaches would need to evaluated to get a full picture of how well the model fits the data. Discuss your model assumptions Indicators of goodness of fit.

5. Some data… Amplitude of vibrations measured on a bridge support vs number of cars driving across at any one time Graphs>interactive>scatter> Number of cars Amplitude of vibration (mm) ©Terraplanner 2007, sourced from http://www.flickr.com/photos/halonfury/1977813099/ Available under creative commons license

6. Assumptions of regression Try a scatterplot (again!) or plot residuals… Residuals - 0 + OKBAD The independent (x) variable is measured without error (!) ‘Errors’ in dependent (y) variable are normally distributed Variance in dependent variable is constant Relationship between variables is linear

7. If we suspect that relation between the x and y is NOT linear we can try to apply transforms to x and/or y to see if we can find a relationship

8. Testing the assumptions Variance: not OK Linearity: OK So let’s log transform y variable… Number of cars Amplitude of vibration (mm) R 2 =0.74

9. What happens… R 2 =0.54 Residuals: Not OK Variance: OK Linearity: not OK Number of cars Vibrations (mm) ln(y) log transform y variable applied

10. Transformation affects linearity…. Log (x) Log (y) Log(x), Log(y) BeforeAfter There are lots of other transforms you can try e.g. squaring or cubing x or y or both etc

11. May need to transform x variable as well… R 2 =0.79 Residuals: OK Variance: OK Linearity: OK Number of cars (ln(x)) Vibrations (mm) (ln(y)) ln(y) = ln(1.88)+1.47ln(x) ln(y) = ln(1.88)+ln(x 1.47 ) ln(y) = ln(1.88x 1.47 ) y= 1.88x 1.47

12. R 2 =0.82 Residuals: OK Variance: OK Linearity: OK Sample diameter(ln(x)) Failure strength (KN) (ln(y)) ln(y) = ln(2)+1.5ln(x) ln(y) = ln(2)+ln(x 1.5 ) ln(y) = ln(2x 1.5 ) y= 2x 1.5 Another log log graph What is the equation for y in terms of x?

13. R 2 =0.75 Residuals: OK Variance: OK Linearity: OK Curing time (weeks) Crushing strength (KN) (ln(y)) A different log(y), x graph What is the equation for y in terms of x? ln(y) = 3 + 5x

14. We perform another regression on our transformed data and see if we get a better regression

16. More than 1 independent (‘predictor’) variable: Multiple Regression e.g. bridge vibrations (z) as a function of number of cars (x) and wind speed (y) Z (vibrations) No. of cars Wind speed

17. Multiple Regression Fit best Plane (x,y) to explain z - minimise (residual)² just as in linear Conceptually identical to linear model Can similarly use any number of predictor variables - fitting hyperplanes in multidimensional space… z (amplitude of vibrations) x (no. of cars) Model : z = a + b 1 x + b 2 y Note: effects of x & y are both linear, and are additive y (wind speed)

18. In this week examples class... You will undertake a regression using Excel Download the data from the VLE and open in Excel Work through the work sheet and it will take you through a regression and evaluating it. Make the best use of the time and ask in the class if you don’t understand. Course work hand in extend to 28 th March

19. CIVE1620 - Engineering Mathematics 2.2 feedback TURN ON 1)Enter Character in <> brackets seen on yellow bar (if it asks for a user ID put a “0” – this make you anonymous) 2)When asked a question you can enter a letter or number and press enter (green button) Use the scale below for A-E to answer the TEST question ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree

20. CIVE1620 - Engineering Mathematics 2.2 Feedback on resources for the whole level 2 Engineering Maths Module (Stats and other maths) ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 1 In general I have found the range of online resources useful for this module useful. (e.g. VLE material, Mathlab, e-mail support, links to online resources etc)

21. CIVE1620 - Engineering Mathematics 2.2 Feedback on resources ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 2 I have found the Mathlab tasks have helped with my understanding of the module (Eng Math and Stats).

22. CIVE1620 - Engineering Mathematics 2.2 Feedback on resources ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 2b I have found the weekly Examples classes useful for the Statistics part of the module.

23. CIVE1620 - Engineering Mathematics 2.2 Feedback on resources ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 3 I have (or intend to) make use of the online lecture slides or lecture videos that are posted on the VLE (the ones of the lectures).

24. CIVE1620 - Engineering Mathematics 2.2 Feedback on resources ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 4 I have made use of some of the other online links to Maths resources that have been linked to from the VLE page.

25. CIVE1620 - Engineering Mathematics 2.2 Feedback on resources ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 5 I feel the approaches used in the Engineering Maths module (which include working through examples in lecture and using directed out of lecture tasks) has helped to improve my understanding of the maths material.

26. CIVE1620 - Engineering Mathematics 2.2 Module feedback ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 6 It is difficult to read/follow the text added to the slides using the Tablet

27. CIVE1620 - Engineering Mathematics 2.2 ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 7 I find it easier to follow mathematical material when it is written during the lecture on the tablet

28. CIVE1620 - Engineering Mathematics 2.2 ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 8 The use of the A, B, C, D cards during a lecture is helpful for feeding back understanding of lecture material

29. CIVE1620 - Engineering Mathematics 2.2 ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 9 The use of the A, B, C, D cards and can be useful for helping me to engage with a lecture

30. CIVE1620 - Engineering Mathematics 2.2 ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 10 The general interactive elements in the lectures help me to engage with the lecture material.

31. CIVE1620 - Engineering Mathematics 2.2 ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 11 I feel I understand the majority of the material covered in this module.

32. CIVE1620 - Engineering Mathematics 2.2 ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 12 I would have liked this module to have covered more Civil Engineering Maths Examples

33. CIVE1620 - Engineering Mathematics 2.2 ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 13 I feel I am confident with my maths ability

34. CIVE1620 - Engineering Mathematics 2.2 Feedback on resources ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 4 I have found the weekly Examples classes useful for the Statistics part of the module.

35. CIVE1620 - Engineering Mathematics 2.2 Feedback on resources ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 5 For the Limits and series section of the module I found the Problem Sheets useful for developing my understanding of the material.

36. CIVE1620 - Engineering Mathematics 2.2 Module feedback ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 6 Having material hand written using the Tablet computer was difficult to read.

37. CIVE1620 - Engineering Mathematics 2.2 ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 7 I find it easier to follow maths material when it is written during the lecture on the tablet

38. CIVE1620 - Engineering Mathematics 2.2 ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 8 I can see the value of using the A, B, C, D cards during the lecture.

39. CIVE1620 - Engineering Mathematics 2.2 ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 9 Interactive elements in lectures have helped me to engage with the material

40. CIVE1620 - Engineering Mathematics 2.2 ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 10 I can see the relevance of the mathematical content of this module to my degree course.

41. CIVE1620 - Engineering Mathematics 2.2 Feedback on resources ABCDE Strongly agree AgreeNeither agree or disagree DisagreeStrongly disagree Question 11 I feel I understand the majority of the material covered in this module.

42. Coursework Due in on Tuesday 16 th March 12 sides maximum Need to submit online (VLE) and hardcopy by 4pm (late penalties apply until both submitted) Major coursework rules apply. Please take care not to plagiarise! –It will be taken very seriously, group work on this major coursework would constitute plagiarism (plagiarism software is a now used as normal practice on all submissions) Final lecture tomorrow is in Computer cluster 504. If you have any questions regarding coursework etc- I will make time to answer them) (examples class continue into next week)

43. More than one answer is allowed!

47. Definition of a mutually exclusive event If event A happens, then event B cannot, or vice-versa. The two events "it rained on Tuesday" and "it did not rain on Tuesday" are mutually exclusive events. Independent events The outcome of event A, has no effect on the outcome of event B. Such as "It rained on Tuesday" and "My chair broke at work".

48. Where S x is the larger

49. Is F 20,20 < F calculated if no, then NO significant difference

52. z or t