STATISTIC. Simple regression

Transcription

1 STATISTIC Simple regression 1

2 Problem : Can we explain the price of the ski pass by the number of ski runs? 2

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6 Data STATION PIS PFJ Max Eaux Bonnes Courette Font Romeu Luz La Mongie Piau Engaly Saint Lary La Bonhomme La Bresse Gérardmer St Maurice Moselle Ventron Métabief Monts Jura Les Rousses Super Besse Mont Dore Le Lioran Auron Beuil Isola Montgenèvre Orcières Merlette Les Orres Pra Loup Puy Saint Vincent Risoul Sauze Super Sauze Serre Chevalier Super Dévoluy La Joue Valberg La Foux Val d'allos Le Seignus Val d'allos Vars Avoriaz Les Carroz Chamonix Chatel La Clusaz Combloux Les Contamines Flaine Les Gets Le Grand Bornand Les Houches Megève Morillon Morzine Praz sur Arly Praz de Lys St Gervais Samoens Thollon Les Aillons Les Arcs Areches Beaufort Aussois Bessans 4 70 Bonneval sur arc Le Corbier Courchevel Crest Voland Corennoz Flumet Les Karellis Les Menuires Meribel La Norma Notre Dame de Belleco La Plagne Pralognan La Rosière Les Saisies St François Longchamp St Martin de Belleville St Sorlin d'arves La Tania Tignes La Toussuire Val Cenis Valfréjus Val d'isère Valloire Val Meinier Valmorel la Belle Val Thorens Alpe d'huez Alpe du Grand Serre Auris Autrans Chamrousse Le Collet d'allevard Les 2 Alpes Lans en Vercors Meaudre St Pierre de Chartreuse Les 7 Laux Vaujany Villard de Lans

7 The fit line 7

8 Least squares method ^ y=b 0 + b 1 x We re looking for Real value Predicted value y i error e i ŷ i * b 0 et b 1 minimizing n i= 1 2 e i 200 Prix Surface x i 8

9 Output SPSS PSP=88, ,873 RUN + residu 9

10 Model s quality Decomposition (y y) = (ŷ y) + i 2 i 2 e 2 i Total Sum of Squares Regression Sum of Squares Residual Sum of Squares So 1 = RegSS/TSS + ResSS/TSS R 2 10

11 R 2 Cor(X,Y) = sign (b1) R 2 11

12 What can we see with the R 2? 1) 0 R 2 1 2) R 2 = 1 3) R 2 = 0 Y * * * * * X Y y * * * * * * * * * * * * X 12

13 What can we see with the Cor(X,Y)? Y * * * * * * * ** * * * * * Y aˆ > 0 X Cor( X, Y) > 0 aˆ < 0 Cor( X, Y ) < 0 X 13

14 Direct calculation for Cor(X,Y) * y = ( y y)/ y y Cor( X, Y ) = ( x x)( y y) i ( x x) ( y y) i i 2 2 i SPSS Results : * x = ( x x)/ x x 14

15 Is Cor(X,Y) significant with a risk α = 0.05? Decision s rule Test : H 0 : Cor(X,Y) = 0 H 1 : Cor (X,Y) 0 We reject H 0 with a risk α = 0.05 of mistakes if Cor( X, Y ) 2 n (we need n > 20) 15

16 Calcul direct de R Nous faisons 0% d erreur en disant que prix forfait jour et nb de pistes sont liées. Nous faisons 41% d erreur en disant que prix forfait jour et kilométrage ski de fond sont liées. 16

17 The statistic s model for the simple regression Model : Y = b 1 X+ b 0 + ε, with ε N(0, σ) Y y = b 1 x + b 0 95% of Y s values Loi de Y * * μ x σ μ x = ax+b μ x σ x X The standard deviation σ is about the scatter s quarter 17

18 Estimation for σ Estimation for σ : ˆ σ 2 ˆ σ = 1 = n 2 ˆ σ 2 n i= 1 2 e i = estimation deσ = estimation deσ 2 18

19 Forecast for Y? Modele : Y = b1x + b0 + ε, with ε N(0, σ) Problem : Calculate a forecast and an intervall with 95 % of Y s values when X =x? 19

20 prix forfait jour Observations atypiques nb pistes 20

21 Outlier A value y is an outlier if it doesn t belong to his own forecast s intervall : y [ ŷ ± t (n 2) σ ˆ 1+ 1 n + n i = 1 (x x) (x i 2 x) 2 ] So : y [ŷ ± 2σˆ ] soit e = y ŷ > 2σˆ Conclusion : A value is an outlier if its standardized e i residual is above 2. σˆ 21

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23 Multiple regression : Maybe are there other variables to explain PFJ 23

24 explain PFJ by variables Multiple regression Objective : 1. AST Altitude station 2. REM Nb of Ski tows (remonte pente) 3. API Altitude Ski run 4. PIS Nb of ski run (pistes) 5. KMF Cross country run (Km) (ski de fond) 6. LIT Nb bed 7. HOT Nb hotel What is the best model available? First, we try to make a linear model as : Y = b 0 + b 1 X 1 + b 2 X b k X k + ε 24

25 O n définitles vecteurs suivants Y = Y 1 Y 2. Y n β = β 0. β k ɛ= ɛ 1 ɛ 2. ɛ n et lamatrice X = 1 x 11 x 1k x i1 x ik x n1 x nk Y =Xβ+ɛ. 25

26 Ilfaut rechercher b 0,b 1,,b k telque soitminimum. n i= 1 (y i b 0 b 1 x i1 b k x ik ) 2 oùw estle sous espace vectorielengendréparles vecteurs 1,x 1,x 2,,x k. 26

27 (y Xb) x j = 0pourj= 0,,n. Donc(y Xb) X = 0 soitx (y Xb)= 0 etdoncx y =X Xb. Ilsuffi t donc de calculer b =(X X) 1 X y. 27

28 Ilfaut rechercher b 0,b 1,,b k telque soitminimum. n i= 1 (y i b 0 b 1 x i1 b k x ik ) 2 oùw estle sous espace vectorielengendréparles vecteurs 1,x 1,x 2,,x k. 28

29 Résultats logiciel PSP= AST TOW ARUN RUN KCC BED HOT + residu 29

30 The global quality of a model between Y and X 1,,X k? Model : Y = β 0 + β 1 X β k X k + ε Test : H 0 : β 1 = = β k = 0 H 1 : At least one β j 0 Statistic : (ŷi y) / k F = 2 e /(n k 1) i 2 Reject of H 0 with a risk α : Reject of H 0 if F F 1-α (k, n-k-1) Fractil of Fisher-Snedecor distribution 30

31 What is the signification level (SIG)? The smallest value for α with a reject of H 0 Distribution F α SIG F 1-α (k,n-k-1) F calculated 31

32 Outputs SPSS The global quality of the model is good. 32

33 Estimation of σ (standard deviation of residual) Estimation of σ 2 : σˆ 2 = n n i= 1 e 2 i k 1 Estimation of σ : σ ˆ = σˆ 2 33

34 Forecast intervall for y i Model : Y i =b 0 + b 1 x 1i + + b k x ki + ε i Formule simplifiée : ŷ i ± 2 σˆ 34

35 But there is a problem! 35

36 Is the contribution of X j significant? Model : Y = β 0 + β 1 X β j X j + + β k X k + ε Test : H 0 : β j = 0 H 1 : β j 0 Statistic used : t j = βˆ s j j Reject of H 0 with risk α : ˆ 1 où s = écart-type( β ) = j j R (X j;autres X) (xji x j) i Reject of H 0 if t j t 1-α/2 (n-k-1) σˆ 2 Fractil of a Student distribution 36

37 What is the signification level (SIG)? The smallest value for α with a reject of H 0 Student distribution Sig/2 α/2 Sig/2 - t j 0 t j t 1-α/2 (n-k-1) We can reject «H 0 : β j = 0» with a risk α if Sig α 37

38 Which are the significant variables in this model (α = 0.05)? 38

39 Selection of variables Method Backward Step 1 : model complete Each step : I will take out the variable X j with the smallest contribution : t j minimum or Sig(t j ) maximum while there are some variables non significant (while Sig(t j ) 0.1 = default value in SPSS). I compute a new model 39

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43 Bon ski!!! 43