CIVE Engineering Mathematics 2.2 (20 credits) Statistics and Probability Lecture 12- Linear regression review Transforming data -Coursework questions? ©Claudio Nunez 2010, sourced from _Building_destroyed_in_Concepci%C3%B3n.jpg?uselang=en-gb Available under creative commons license

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

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)

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 %) 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.

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 Available under creative commons license

Assumptions of regression Try a scatterplot (again!) or plot residuals… Residuals 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

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

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

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

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

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

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?

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

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

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

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)

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

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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)

More than one answer is allowed!

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".

Where S x is the larger

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

z or t