Introduction to Regression

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1 Intrductin t Regressin

2 Administrivia Hmewrk 6 psted later tnight. Due Friday after Break. 2

3 Statistical Mdeling Thus far we ve talked abut Descriptive Statistics: This is the way my sample is Inferential Statistics: This is what I can likely cnclude frm my sample Tday we mve twards what we might call Predictive Statistics 3

4 Linear Regressin fr Predictin Examples: Given a persn s age and gender, predict their height Given the area f a huse, predict its sale price Given unemplyment, inflatin, number f wars, and ecnmic grwth, predict the president s apprval rating. Given a persn s brwser histry, predict hw lng they ll stay n a prduct page Given the advertising budget expenditures in varius media markets, predict the number f prducts they ll sell 4

5 Linear Regressin fr Inference Examples: Is a persn s age and gender related t their height Is the area f a huse, related t its sale price Is unemplyment, inflatin, number f wars, and ecnmic grwth related t the president s apprval rating. Is a persn s brwser histry related t hw lng they ll stay n a prduct page Is the advertising budget expenditures in varius media markets related the number f prducts they ll sell 5

6 Area as Predictr fr Huse Price 6

7 Area as Predictr fr Huse Price 7

8 Explratin Open up yur cmputer and lad the Lecture 20 in-class ntebk Y = x + PX + E E ~N( 0,82 ) 8

9 Simple Linear Regressin Mdel Defs and Assumptins f SLR mdel: ( U DATA pints ) 1. Y ; = Xtpxi + Ei 2. EACH OF Ei 's ARE INDEPENDENT 3. ti - Nl, 52 ) 9

10 Simple Linear Regressin Mdel SLR mdel vcabulary: Ye XTPXTE X: the independent variable, the predictr, the explanatry variable, the feature = Y: the dependent variable r the respnse variable = * FIXED : the randm deviatin r randm errr * RANDOM, NOT RANDOM. * RANDOM Variable Questin: What exactly is ding? 10

11 Simple Linear Regressin Mdel (x 1,y 1 ), (x 2,y 2 ),...,(x n,y n ) The pints be scattered abut the true regressin line resulting frm n independent bservatins will YTRUELME Y=x+px El / { ( x. Yi ) 11

12 Simple Linear Regressin Thery Questin: Hw d we knw that the simple linear regressin is apprpriate? * 2 MEASURE...I Experience * EYEBAH Spilers * R ' - VALVE 12

13 Interpreting SLR Parameters Y is a randm variable. What is it s expectatin? EIY ]=E[t+px+E Y=xtpX+E =EIx3tpE[X]tE[ EI = Xtpxt E~N1,I 13

14 Interpreting SLR Parameters Y = a PX is the intercept f the true regressin line (the s-called baseline average) del 00 - ) + 14

15 Interpreting SLR Parameters is the slpe f the true regressin line Y xtpcxti,= ) Tz=xtp. 3 15

16 a Interpreting the Errr Term 2. The variance parameter determines the extent t which each nrmal curve spreads abut the true regressin line Y~NHtP '. Y = xtpxte 16

17 Directinal Cnsideratins S far we ve cme up with a framewrk where we can chse the mdel parameters and then generate randm data. This is called a generative mdel. But really, we want t run this prcess in reverse. We have data, and we want t find/learn/estimate the parameters that explain the data. General Mdel + Parameters sample Data General Mdel + Parameters =FERENCE Data 17

18 Hw Can We Estimate Params frm Data? Plan f Attack: The variance f ur mdel will be smallest if the differences between between the estimate f the true regressin line and each pint is the smallest. This is ur gal: minimize 2 2 (x 1,y 1 ), (x 2,y 2 ),...,(x n,y n ) We ll use ur sample data,, t estimate the parameters f the regressin line What are we assuming abut each f the bservatins?. K ( X. y, ) ( Xu, yz ) OBTAINED independently 18

19 Estimating Mdel Parameters,, E,=(y ))2 - ( xtpx, ' " Y ' l sse=.ie/yi-ldtpxid 2 19

20 Estimating Mdel Parameters The sum f the squared-errrs fr the pints t the regressin line is given by SSE =,, lyi (x 1,y 1 ), (x 2,y 2 ),...,(x n,y n ) - lxtpx ;) ) ' The pint-estimates (estimates frm data) f the slpe and the intercept parameters are called the least-square estimates, and are defined t be the values that minimize the SSE 3sa =, ask } ssimwitan... 20

21 Estimating Mdel Parameters The fitted regressin line r the least-squares line is then the line given by if = xatpx Questin: Hw d we actually find the parameter estimates? 21

22 Estimating Mdel Parameters Sse = age = If, -!yi!2 n latpxi ) E, -21g i. i=, 12 ", yi y - latpxil ) = - - pxi = a - pi= 2 SSE Jp, f2 ily - Httsx ;D = 22

23 Hw Can We D This in Practice? Get yur laptps back ut and let s figure it ut! 2=5 - in = If lxi - II ( yi - g) #ix2 23

24 Residuals y = +154 The fitted r predicted values are btained by plugging in the independent data variables int the fitted mdel The residuals are the differences between the bserved and predicted respnses: r r ; = ye - y : 24

25 Residuals Claim: The residuals are estimates f the true errr TRUE t MODEL riyefei 25

26 Maximum Likelihd Estimatin An alternate methd fr estimating mdel parameters is t create a likelihd functin invlving the mdel parameters and the data, and chse the value f the parameter that maximizes it We ve dne this befre, just haven t called it Maximum Likelihd Estimatin Example: Suppse yu have a biased cin, yu flip it 6 times and get 5 Heads and 1 Tails. Estimate the parameter p fr the cin. 26

27 Maximum Likelihd Estimatin 27

28 Maximum Likelihd Estimatin 28

29 Maximum Likelihd Estimatin 29

30 Maximum Likelihd Estimatin 30

31 OK! Let s G t Wrk! Get in grups, get ut laptp, and pen the Lecture 20 In-Class Ntebk Let s: D sme stuff! 31

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