TP7 Econometrie
Jean Charvet
7 mars 2019
Regression lineaire multiple
Modélisation de l’Ozone à l’aide de toutes les variables explicatives
ozo = read.csv("~/data/ozone.txt", header = TRUE, sep =";")
summary(ozo)## O3 T12 T15 Ne12
## Min. : 41.8 Min. : 7.90 Min. :10.10 Min. :0.00
## 1st Qu.: 66.6 1st Qu.:17.40 1st Qu.:18.15 1st Qu.:3.00
## Median : 83.9 Median :19.75 Median :21.05 Median :6.00
## Mean : 86.3 Mean :20.32 Mean :21.39 Mean :5.02
## 3rd Qu.:102.2 3rd Qu.:24.32 3rd Qu.:25.70 3rd Qu.:7.00
## Max. :139.0 Max. :29.50 Max. :30.60 Max. :8.00
## N12 S12 E12 W12 VENT12
## Min. :0.00 Min. :0.00 Min. :0.00 Min. :0.00 E:16
## 1st Qu.:0.00 1st Qu.:0.00 1st Qu.:0.00 1st Qu.:0.00 N: 9
## Median :0.00 Median :0.00 Median :0.00 Median :0.00 S: 7
## Mean :0.94 Mean :0.58 Mean :1.36 Mean :1.54 W:18
## 3rd Qu.:0.00 3rd Qu.:0.00 3rd Qu.:3.00 3rd Qu.:2.75
## Max. :7.00 Max. :6.00 Max. :8.00 Max. :8.00
## Vx maxO3v
## Min. :-27.0600 Min. : 38.00
## 1st Qu.:-10.8000 1st Qu.: 63.40
## Median : -3.2550 Median : 83.40
## Mean : -0.8312 Mean : 84.28
## 3rd Qu.: 9.3500 3rd Qu.:102.75
## Max. : 28.3600 Max. :142.80Objectif : établir un modèle de régression multiple expliquant la pollution. Soit un seuil d’erreur de niveau \(\alpha = 0.05\).
Ici, notre variable cible est O3 et nos features sont les T12, T15, Ne12, N12, S12, E12, W12, Vx.
feat = data.frame(ozo$T12, ozo$T15, ozo$Ne12, ozo$N12, ozo$S12, ozo$E12, ozo$W12, ozo$Vx, ozo$maxO3v)
reg = lm(ozo$O3 ~., feat)
summary(reg)##
## Call:
## lm(formula = ozo$O3 ~ ., data = feat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -24.0532 -7.7178 0.8154 7.9974 27.9616
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 54.72777 17.27890 3.167 0.002943 **
## ozo.T12 -0.35184 1.57308 -0.224 0.824159
## ozo.T15 1.49719 1.53774 0.974 0.336092
## ozo.Ne12 -4.19219 1.06376 -3.941 0.000318 ***
## ozo.N12 1.27549 1.36319 0.936 0.355061
## ozo.S12 3.17112 1.91078 1.660 0.104817
## ozo.E12 0.52773 1.94274 0.272 0.787294
## ozo.W12 2.47488 2.07201 1.194 0.239342
## ozo.Vx 0.60770 0.48575 1.251 0.218187
## ozo.maxO3v 0.24536 0.09649 2.543 0.014964 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.21 on 40 degrees of freedom
## Multiple R-squared: 0.7505, Adjusted R-squared: 0.6944
## F-statistic: 13.37 on 9 and 40 DF, p-value: 1.507e-09var(ozo$O3)## [1] 571.2494Modele de regression multiple :
\(\forall i = 1,.., n\) : \(Y_i = b_1X^1_i+b_2X^2 +..+ b_pX^p_i+u_i\)
Où bien sûr \(u\) est l’erreur aléatoire tel que :
\[\mathbb E[u] = 0_n\\ V(u)= \sigma^2Id_n\]
Ainsi nous avons les paramètres du modèle : \(b \in \mathbb R^p\) et \(\sigma^2 >0\) Nous testerons alors chaque variable une par une :
Probleme de test (Student) : la variable est-elle significative ?
\(H_0\) : la variable n’est pas significative, ou \(b_p = 0\).
\(H_1\) : la variable est significative, ou \(b_p \not = 0\)
Règle de décision : ici, on regarde l’indice statistique \(\mathcal T\). On établit la règle suivante :
t-value > qt(\(\frac{1+ \alpha}{2}\) )
\[ A REMPLIR QUANTILE STUDENT ETFCCC \]
On rappelle :
\[ \mathcal T = \frac{\hat b}{\sqrt {\hat {Var(\hat b)}}} \]
Problème de test : la regression est-elle signficative ?
\(H_0\) : la regression n’est pas significative.
\(H_1\) : la regression est significative.
Règle de décision :
Consolider une sortie R
a/ On remarque le sortie R nous informe qu’il y a 19 degrés de liberté, 5 variables sont impliqués dans la modélisation : nous avions alors 24 observations sur ces données.
b/ 1- Intercept ????
2- \(-1518.18 / 739.93 = -2.05\) Calcul de la T valeur
3- \(34.67 / 0.178 = 194,7753\) Calcul de la T valeur
4- Rien du tout
5- \(14.31 * 2.697 = 38.59407\) Calcul de la T valeur
6- 4 (Nb de variables explicatives)
7- 19 (Degrés de liberté)
c/ SCT = SCE + SCR \(R^2 = \frac{SCE}{SCT}\)
Ici \(R^2 = 0.86\)
Taux de remplissage des trains
Variable d’intérêt est le taux de remplissage d’un train
Audit
train = read.csv("~/data/RemplissageTrain.csv", header = TRUE, sep =";", dec = ",")
summary(train)## TX HEURE JOUR SEMAINE
## Min. :0.1464 Min. : 7.00 Min. : 1.00 Min. : 1
## 1st Qu.:0.4680 1st Qu.:10.00 1st Qu.: 38.00 1st Qu.: 6
## Median :0.6246 Median :11.00 Median : 74.00 Median :11
## Mean :0.6058 Mean :12.35 Mean : 74.03 Mean :11
## 3rd Qu.:0.7388 3rd Qu.:14.00 3rd Qu.:110.00 3rd Qu.:16
## Max. :1.0000 Max. :21.00 Max. :147.00 Max. :21
## TEMPERATURE PUISSANCE DUREESOLEIL PLUIE
## Min. :18.80 Min. :142.0 Min. : 18 Min. : 0.000
## 1st Qu.:25.60 1st Qu.:753.0 1st Qu.:560 1st Qu.: 0.000
## Median :27.80 Median :845.0 Median :702 Median : 0.000
## Mean :27.58 Mean :807.6 Mean :635 Mean : 1.183
## 3rd Qu.:29.80 3rd Qu.:922.0 3rd Qu.:804 3rd Qu.: 0.000
## Max. :35.20 Max. :997.0 Max. :872 Max. :29.800str(train)## 'data.frame': 777 obs. of 8 variables:
## $ TX : num 0.146 0.325 0.342 0.273 0.24 ...
## $ HEURE : int 10 11 12 14 7 8 10 8 9 10 ...
## $ JOUR : int 1 1 1 1 2 2 2 3 3 3 ...
## $ SEMAINE : int 1 1 1 1 1 1 1 1 1 1 ...
## $ TEMPERATURE: num 23.1 23.1 23.1 23.1 18.8 18.8 18.8 20 20 20 ...
## $ PUISSANCE : int 839 839 839 839 492 492 492 925 925 925 ...
## $ DUREESOLEIL: int 642 642 642 642 319 319 319 745 745 745 ...
## $ PLUIE : num 0 0 0 0 0 0 0 0 0 0 ...Modelisation
trainregresseur = data.frame(train$HEURE, train$JOUR, train$SEMAINE , train$TEMPERATURE, train$PUISSANCE, train$DUREESOLEIL, train$PLUIE)
reg2 = lm(train$TX ~., trainregresseur)
summary(reg2)##
## Call:
## lm(formula = train$TX ~ ., data = trainregresseur)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.233287 -0.041098 0.001505 0.039796 0.191818
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.794e-02 2.438e-02 0.736 0.4621
## train.HEURE 6.798e-03 6.452e-04 10.536 < 2e-16 ***
## train.JOUR 1.997e-03 1.365e-03 1.463 0.1440
## train.SEMAINE 1.170e-02 9.591e-03 1.220 0.2229
## train.TEMPERATURE 5.946e-03 8.010e-04 7.423 3.05e-13 ***
## train.PUISSANCE 8.124e-06 2.386e-05 0.341 0.7335
## train.DUREESOLEIL 9.134e-05 1.604e-05 5.696 1.75e-08 ***
## train.PLUIE -9.883e-04 5.201e-04 -1.900 0.0578 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.06543 on 769 degrees of freedom
## Multiple R-squared: 0.8659, Adjusted R-squared: 0.8647
## F-statistic: 709.2 on 7 and 769 DF, p-value: < 2.2e-16Probleme de test (Student) : les variables sont-elles significatives ?
\(H_0\) : la variable \(X_p\) n’est pas significative, ou \(b_p = 0\).
\(H_1\) : la variable \(X_p\) est significative, ou \(b_p \not = 0\).
Règle de décision : ici, on regarde l’indice statistique : \(\mathcal T = \frac{\hat b}{\sqrt{\hat{Var(\hat b)}}}\ \)~\(\ \mathcal t_{1587}\). On établit la règle suivante :
t-value > \(qt_{1587}\)(\(\frac{1+ \alpha}{2}\))
qt(1.05/2,769)## [1] 0.06272725Probleme de test (Fisher) : la régression est-elle significative ?
\(H_0\) : la regression n’est pas significative.
\(H_1\) : la regression est significative.
Règle de décision : nous regardons l’indice statistique : \(\mathcal F = \frac {\mathbb R^2/1}{(1-\mathbb R^2)/1587}\)
On a \(F = 709.2\) >
qf(0.05,1,769)## [1] 0.003934708Probleme de test (Shapiro) : les résidus sont-ils gaussiens ?
\(H_0\) = les résidus sont gaussiens
\(H_1\) : les résidus ne sont pas gaussiens
shapiro.test(reg2$residuals)##
## Shapiro-Wilk normality test
##
## data: reg2$residuals
## W = 0.99737, p-value = 0.2491Ainsi, la p-value est supérieur au seuil \(\alpha = 0.05\) et la statistique \(W\) est très proche de 1, on accepte alors l’hypothèse nulle et les résidus sont donc problamement issus d’une population normalement distribuée.
Prédiction
index = sample(1:777, 539, replace = FALSE)
TRAIN = train[index,]
TEST = train[-index,]
train## TX HEURE JOUR SEMAINE TEMPERATURE PUISSANCE DUREESOLEIL PLUIE
## 1 0.1464205 10 1 1 23.1 839 642 0.0
## 2 0.3250252 11 1 1 23.1 839 642 0.0
## 3 0.3416184 12 1 1 23.1 839 642 0.0
## 4 0.2734668 14 1 1 23.1 839 642 0.0
## 5 0.2402137 7 2 1 18.8 492 319 0.0
## 6 0.2594631 8 2 1 18.8 492 319 0.0
## 7 0.2351525 10 2 1 18.8 492 319 0.0
## 8 0.2954669 8 3 1 20.0 925 745 0.0
## 9 0.2274551 9 3 1 20.0 925 745 0.0
## 10 0.1474659 10 3 1 20.0 925 745 0.0
## 11 0.4337837 14 3 1 20.0 925 745 0.0
## 12 0.3275769 15 3 1 20.0 925 745 0.0
## 13 0.2035415 18 3 1 20.0 925 745 0.0
## 14 0.3103142 20 3 1 20.0 925 745 0.0
## 15 0.1984047 7 4 1 21.3 806 810 0.0
## 16 0.2939757 9 4 1 21.3 806 810 0.0
## 17 0.3614220 10 4 1 21.3 806 810 0.0
## 18 0.2752958 11 4 1 21.3 806 810 0.0
## 19 0.3544230 12 4 1 21.3 806 810 0.0
## 20 0.3397150 14 4 1 21.3 806 810 0.0
## 21 0.4523794 16 4 1 21.3 806 810 0.0
## 22 0.3155709 18 4 1 21.3 806 810 0.0
## 23 0.3046718 21 4 1 21.3 806 810 0.0
## 24 0.3035910 10 5 1 23.4 945 811 0.0
## 25 0.3158297 11 5 1 23.4 945 811 0.0
## 26 0.4347140 12 5 1 23.4 945 811 0.0
## 27 0.2518922 14 5 1 23.4 945 811 0.0
## 28 0.3233734 16 5 1 23.4 945 811 0.0
## 29 0.3721862 20 5 1 23.4 945 811 0.0
## 30 0.4334932 10 6 1 21.9 833 504 0.2
## 31 0.2119416 11 6 1 21.9 833 504 0.2
## 32 0.3251621 12 6 1 21.9 833 504 0.2
## 33 0.2520132 14 6 1 21.9 833 504 0.2
## 34 0.2288692 8 7 1 22.9 856 582 0.0
## 35 0.4743415 10 7 1 22.9 856 582 0.0
## 36 0.3204868 11 7 1 22.9 856 582 0.0
## 37 0.3867505 14 7 1 22.9 856 582 0.0
## 38 0.4426777 10 8 2 25.1 850 806 0.0
## 39 0.3150748 11 8 2 25.1 850 806 0.0
## 40 0.3006761 12 8 2 25.1 850 806 0.0
## 41 0.3986137 14 8 2 25.1 850 806 0.0
## 42 0.3156704 7 9 2 27.3 917 804 0.0
## 43 0.3933000 8 9 2 27.3 917 804 0.0
## 44 0.3140663 10 9 2 27.3 917 804 0.0
## 45 0.3952174 8 10 2 25.6 845 697 0.0
## 46 0.4370813 9 10 2 25.6 845 697 0.0
## 47 0.4414927 10 10 2 25.6 845 697 0.0
## 48 0.4339489 14 10 2 25.6 845 697 0.0
## 49 0.4521556 15 10 2 25.6 845 697 0.0
## 50 0.4346397 18 10 2 25.6 845 697 0.0
## 51 0.4380963 20 10 2 25.6 845 697 0.0
## 52 0.1515455 7 11 2 22.6 333 206 0.0
## 53 0.3975938 9 11 2 22.6 333 206 0.0
## 54 0.2886941 10 11 2 22.6 333 206 0.0
## 55 0.4044745 11 11 2 22.6 333 206 0.0
## 56 0.2743014 12 11 2 22.6 333 206 0.0
## 57 0.2085989 14 11 2 22.6 333 206 0.0
## 58 0.4680677 16 11 2 22.6 333 206 0.0
## 59 0.2457763 18 11 2 22.6 333 206 0.0
## 60 0.4416870 21 11 2 22.6 333 206 0.0
## 61 0.4242405 10 12 2 22.1 972 726 0.0
## 62 0.2920096 11 12 2 22.1 972 726 0.0
## 63 0.2908911 12 12 2 22.1 972 726 0.0
## 64 0.3544566 14 12 2 22.1 972 726 0.0
## 65 0.3519459 16 12 2 22.1 972 726 0.0
## 66 0.4059332 20 12 2 22.1 972 726 0.0
## 67 0.3140495 10 13 2 20.6 925 804 0.0
## 68 0.2956014 11 13 2 20.6 925 804 0.0
## 69 0.3854294 12 13 2 20.6 925 804 0.0
## 70 0.4012454 14 13 2 20.6 925 804 0.0
## 71 0.3150787 8 14 2 20.9 970 846 0.0
## 72 0.3011926 10 14 2 20.9 970 846 0.0
## 73 0.1856966 11 14 2 20.9 970 846 0.0
## 74 0.3592074 14 14 2 20.9 970 846 0.0
## 75 0.3365368 10 15 3 21.9 942 846 0.0
## 76 0.4397412 11 15 3 21.9 942 846 0.0
## 77 0.4010554 12 15 3 21.9 942 846 0.0
## 78 0.4083678 14 15 3 21.9 942 846 0.0
## 79 0.3313498 7 16 3 22.3 933 805 0.0
## 80 0.3666302 8 16 3 22.3 933 805 0.0
## 81 0.3491196 10 16 3 22.3 933 805 0.0
## 82 0.3443388 8 17 3 22.9 945 822 0.0
## 83 0.3728057 9 17 3 22.9 945 822 0.0
## 84 0.4566577 10 17 3 22.9 945 822 0.0
## 85 0.4176864 14 17 3 22.9 945 822 0.0
## 86 0.3935909 15 17 3 22.9 945 822 0.0
## 87 0.3797728 18 17 3 22.9 945 822 0.0
## 88 0.4770466 20 17 3 22.9 945 822 0.0
## 89 0.3531441 7 18 3 22.0 942 816 4.0
## 90 0.3318565 9 18 3 22.0 942 816 4.0
## 91 0.3331939 10 18 3 22.0 942 816 4.0
## 92 0.3841353 11 18 3 22.0 942 816 4.0
## 93 0.4584801 12 18 3 22.0 942 816 4.0
## 94 0.4043989 14 18 3 22.0 942 816 4.0
## 95 0.4472872 16 18 3 22.0 942 816 4.0
## 96 0.4474550 18 18 3 22.0 942 816 4.0
## 97 0.3937723 21 18 3 22.0 942 816 4.0
## 98 0.3073776 10 19 3 20.2 494 42 1.4
## 99 0.3904624 11 19 3 20.2 494 42 1.4
## 100 0.3646948 12 19 3 20.2 494 42 1.4
## 101 0.3581369 14 19 3 20.2 494 42 1.4
## 102 0.2430949 16 19 3 20.2 494 42 1.4
## 103 0.3805241 20 19 3 20.2 494 42 1.4
## 104 0.3241596 10 20 3 25.5 964 690 0.0
## 105 0.4154593 11 20 3 25.5 964 690 0.0
## 106 0.3912074 12 20 3 25.5 964 690 0.0
## 107 0.3629190 14 20 3 25.5 964 690 0.0
## 108 0.4355900 8 21 3 29.9 878 570 0.0
## 109 0.2689895 10 21 3 29.9 878 570 0.0
## 110 0.4393762 11 21 3 29.9 878 570 0.0
## 111 0.4562504 14 21 3 29.9 878 570 0.0
## 112 0.3056330 10 22 4 24.5 142 158 0.4
## 113 0.3330892 11 22 4 24.5 142 158 0.4
## 114 0.3690513 12 22 4 24.5 142 158 0.4
## 115 0.4285945 14 22 4 24.5 142 158 0.4
## 116 0.3577822 7 23 4 23.2 936 774 0.4
## 117 0.2854114 8 23 4 23.2 936 774 0.4
## 118 0.5055170 10 23 4 23.2 936 774 0.4
## 119 0.4500105 8 24 4 24.1 964 822 0.0
## 120 0.4278496 9 24 4 24.1 964 822 0.0
## 121 0.4118585 10 24 4 24.1 964 822 0.0
## 122 0.4429489 14 24 4 24.1 964 822 0.0
## 123 0.3515002 15 24 4 24.1 964 822 0.0
## 124 0.5619255 18 24 4 24.1 964 822 0.0
## 125 0.4482596 20 24 4 24.1 964 822 0.0
## 126 0.2738907 7 25 4 21.5 636 134 0.0
## 127 0.2943538 9 25 4 21.5 636 134 0.0
## 128 0.3074329 10 25 4 21.5 636 134 0.0
## 129 0.3568721 11 25 4 21.5 636 134 0.0
## 130 0.3439922 12 25 4 21.5 636 134 0.0
## 131 0.3136313 14 25 4 21.5 636 134 0.0
## 132 0.3340849 16 25 4 21.5 636 134 0.0
## 133 0.3841724 18 25 4 21.5 636 134 0.0
## 134 0.3427149 21 25 4 21.5 636 134 0.0
## 135 0.3107103 10 26 4 22.4 920 499 0.0
## 136 0.3368635 11 26 4 22.4 920 499 0.0
## 137 0.3216873 12 26 4 22.4 920 499 0.0
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## 139 0.4333149 16 26 4 22.4 920 499 0.0
## 140 0.4605965 20 26 4 22.4 920 499 0.0
## 141 0.2778195 10 27 4 23.2 708 684 0.0
## 142 0.4785666 11 27 4 23.2 708 684 0.0
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## 147 0.4021451 11 28 4 23.5 933 842 0.0
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## 149 0.3236435 10 29 5 25.8 792 728 0.0
## 150 0.3203250 11 29 5 25.8 792 728 0.0
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## 153 0.4550426 7 30 5 27.4 939 720 0.0
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## 165 0.3450034 10 32 5 25.2 939 854 0.0
## 166 0.4540797 11 32 5 25.2 939 854 0.0
## 167 0.4292635 12 32 5 25.2 939 854 0.0
## 168 0.4877978 14 32 5 25.2 939 854 0.0
## 169 0.6138414 16 32 5 25.2 939 854 0.0
## 170 0.6209866 18 32 5 25.2 939 854 0.0
## 171 0.4716904 21 32 5 25.2 939 854 0.0
## 172 0.4064540 10 33 5 26.1 914 852 0.0
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## 182 0.4621242 8 35 5 25.1 836 312 0.2
## 183 0.5018333 10 35 5 25.1 836 312 0.2
## 184 0.3412601 11 35 5 25.1 836 312 0.2
## 185 0.3974669 14 35 5 25.1 836 312 0.2
## 186 0.4942226 10 36 6 25.5 945 846 0.0
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## 190 0.3618345 7 37 6 28.7 836 588 0.0
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## 193 0.4379708 8 38 6 30.2 928 822 0.0
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## 214 0.6614890 20 40 6 33.0 925 804 0.0
## 215 0.3805643 10 41 6 33.8 911 684 0.0
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## 285 0.6036099 12 54 8 30.7 906 758 0.0
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## 287 0.5718594 16 54 8 30.7 906 758 0.0
## 288 0.5427208 20 54 8 30.7 906 758 0.0
## 289 0.5369691 10 55 8 29.6 906 774 6.1
## 290 0.6045436 11 55 8 29.6 906 774 6.1
## 291 0.5021482 12 55 8 29.6 906 774 6.1
## 292 0.4617631 14 55 8 29.6 906 774 6.1
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## 308 0.4535867 15 59 9 28.3 775 540 0.2
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## 324 0.5000971 16 61 9 27.5 970 786 0.0
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## 332 0.5501488 11 63 9 29.0 933 674 0.0
## 333 0.4525263 14 63 9 29.0 933 674 0.0
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## 342 0.5253393 9 66 10 29.3 928 828 0.0
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## 386 0.5736414 9 74 11 27.9 856 702 0.0
## 387 0.6273971 10 74 11 27.9 856 702 0.0
## 388 0.6899031 11 74 11 27.9 856 702 0.0
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## 390 0.6086143 14 74 11 27.9 856 702 0.0
## 391 0.7256696 16 74 11 27.9 856 702 0.0
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## 396 0.6771520 12 75 11 30.7 900 854 0.0
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## 398 0.7255050 16 75 11 30.7 900 854 0.0
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## 410 0.6307686 12 78 12 31.7 931 846 0.0
## 411 0.5741613 14 78 12 31.7 931 846 0.0
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## 413 0.7640830 8 79 12 31.7 903 798 0.0
## 414 0.7078188 10 79 12 31.7 903 798 0.0
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## 417 0.5706883 10 80 12 35.2 786 444 0.8
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## 419 0.7385609 15 80 12 35.2 786 444 0.8
## 420 0.7520111 18 80 12 35.2 786 444 0.8
## 421 0.6194029 20 80 12 35.2 786 444 0.8
## 422 0.5918728 7 81 12 31.6 681 126 5.1
## 423 0.6537163 9 81 12 31.6 681 126 5.1
## 424 0.5683006 10 81 12 31.6 681 126 5.1
## 425 0.6224316 11 81 12 31.6 681 126 5.1
## 426 0.6468763 12 81 12 31.6 681 126 5.1
## 427 0.7077232 14 81 12 31.6 681 126 5.1
## 428 0.6254035 16 81 12 31.6 681 126 5.1
## 429 0.6433080 18 81 12 31.6 681 126 5.1
## 430 0.7381576 21 81 12 31.6 681 126 5.1
## 431 0.7345759 10 82 12 28.2 775 636 0.0
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## 447 0.7179530 12 85 13 31.2 895 744 0.0
## 448 0.7335032 14 85 13 31.2 895 744 0.0
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## 450 0.5529707 8 86 13 29.3 686 252 2.2
## 451 0.5657877 10 86 13 29.3 686 252 2.2
## 452 0.6640624 8 87 13 31.4 875 703 0.0
## 453 0.6434162 9 87 13 31.4 875 703 0.0
## 454 0.6830478 10 87 13 31.4 875 703 0.0
## 455 0.6921618 14 87 13 31.4 875 703 0.0
## 456 0.7560488 15 87 13 31.4 875 703 0.0
## 457 0.7987324 18 87 13 31.4 875 703 0.0
## 458 0.6430104 20 87 13 31.4 875 703 0.0
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## 461 0.6928630 10 88 13 31.7 922 842 0.0
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## 476 0.6323564 12 90 13 25.1 886 474 0.0
## 477 0.7143081 14 90 13 25.1 886 474 0.0
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## 479 0.6730412 10 91 13 28.7 895 812 0.0
## 480 0.8704198 11 91 13 28.7 895 812 0.0
## 481 0.6256517 14 91 13 28.7 895 812 0.0
## 482 0.6936931 10 92 14 30.9 911 834 0.0
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## 484 0.6996029 12 92 14 30.9 911 834 0.0
## 485 0.7228226 14 92 14 30.9 911 834 0.0
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## 487 0.6947116 8 93 14 30.2 920 674 0.6
## 488 0.6628000 10 93 14 30.2 920 674 0.6
## 489 0.6491526 8 94 14 28.8 886 648 25.9
## 490 0.6027973 9 94 14 28.8 886 648 25.9
## 491 0.6373730 10 94 14 28.8 886 648 25.9
## 492 0.5691886 14 94 14 28.8 886 648 25.9
## 493 0.6880269 15 94 14 28.8 886 648 25.9
## 494 0.6315388 18 94 14 28.8 886 648 25.9
## 495 0.7721474 20 94 14 28.8 886 648 25.9
## 496 0.6976548 7 95 14 28.8 753 774 0.0
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## 500 0.7067332 12 95 14 28.8 753 774 0.0
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## 504 0.8116484 21 95 14 28.8 753 774 0.0
## 505 0.6784142 10 96 14 30.6 619 738 0.0
## 506 0.7878936 11 96 14 30.6 619 738 0.0
## 507 0.6679720 12 96 14 30.6 619 738 0.0
## 508 0.7953406 14 96 14 30.6 619 738 0.0
## 509 0.7376694 16 96 14 30.6 619 738 0.0
## 510 0.8644023 20 96 14 30.6 619 738 0.0
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## 514 0.7786377 14 97 14 29.5 942 804 0.0
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## 518 0.6997189 14 98 14 31.9 903 812 0.0
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## 521 0.5753125 12 99 15 31.1 856 648 0.0
## 522 0.6350748 14 99 15 31.1 856 648 0.0
## 523 0.6840372 7 100 15 30.1 789 804 10.0
## 524 0.6799068 8 100 15 30.1 789 804 10.0
## 525 0.7713728 10 100 15 30.1 789 804 10.0
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## 527 0.5875407 9 101 15 30.7 803 534 0.0
## 528 0.7237893 10 101 15 30.7 803 534 0.0
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## 531 0.8753513 18 101 15 30.7 803 534 0.0
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## 538 0.7924202 14 102 15 32.3 795 714 0.0
## 539 0.7172010 16 102 15 32.3 795 714 0.0
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## 541 0.6336578 21 102 15 32.3 795 714 0.0
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## 544 0.5856856 12 103 15 32.3 836 762 0.0
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## 546 0.6590749 16 103 15 32.3 836 762 0.0
## 547 0.9187561 20 103 15 32.3 836 762 0.0
## 548 0.6228977 10 104 15 30.4 842 782 3.4
## 549 0.7388381 11 104 15 30.4 842 782 3.4
## 550 0.7331244 12 104 15 30.4 842 782 3.4
## 551 0.7128196 14 104 15 30.4 842 782 3.4
## 552 0.6039336 8 105 15 26.5 711 457 28.3
## 553 0.6315993 10 105 15 26.5 711 457 28.3
## 554 0.6667751 11 105 15 26.5 711 457 28.3
## 555 0.6857075 14 105 15 26.5 711 457 28.3
## 556 0.7803991 10 106 16 26.5 903 816 0.0
## 557 0.6899182 11 106 16 26.5 903 816 0.0
## 558 0.8099785 12 106 16 26.5 903 816 0.0
## 559 0.7551993 14 106 16 26.5 903 816 0.0
## 560 0.7213884 7 107 16 26.5 900 794 0.0
## 561 0.8177505 8 107 16 26.5 900 794 0.0
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## 563 0.7147520 8 108 16 25.7 878 805 0.0
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## 566 0.7459819 14 108 16 25.7 878 805 0.0
## 567 0.7128137 15 108 16 25.7 878 805 0.0
## 568 0.8838398 18 108 16 25.7 878 805 0.0
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## 571 0.7332834 9 109 16 26.8 867 805 0.0
## 572 0.7909574 10 109 16 26.8 867 805 0.0
## 573 0.7797779 11 109 16 26.8 867 805 0.0
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## 575 0.6522294 14 109 16 26.8 867 805 0.0
## 576 0.8942852 16 109 16 26.8 867 805 0.0
## 577 0.6912246 18 109 16 26.8 867 805 0.0
## 578 0.8441400 21 109 16 26.8 867 805 0.0
## 579 0.7754382 10 110 16 27.0 870 560 16.5
## 580 0.7896342 11 110 16 27.0 870 560 16.5
## 581 0.7928367 12 110 16 27.0 870 560 16.5
## 582 0.6969074 14 110 16 27.0 870 560 16.5
## 583 0.7109562 16 110 16 27.0 870 560 16.5
## 584 0.7377907 20 110 16 27.0 870 560 16.5
## 585 0.7811088 10 111 16 27.4 806 606 0.0
## 586 0.6951484 11 111 16 27.4 806 606 0.0
## 587 0.8303590 12 111 16 27.4 806 606 0.0
## 588 0.6737145 14 111 16 27.4 806 606 0.0
## 589 0.6893804 8 112 16 25.7 642 618 0.0
## 590 0.7138775 10 112 16 25.7 642 618 0.0
## 591 0.7278216 11 112 16 25.7 642 618 0.0
## 592 0.6867447 14 112 16 25.7 642 618 0.0
## 593 0.7354036 10 113 17 26.7 864 780 0.0
## 594 0.7753078 11 113 17 26.7 864 780 0.0
## 595 0.7558726 12 113 17 26.7 864 780 0.0
## 596 0.7289392 14 113 17 26.7 864 780 0.0
## 597 0.6716213 7 114 17 26.1 556 360 0.0
## 598 0.6939379 8 114 17 26.1 556 360 0.0
## 599 0.6162415 10 114 17 26.1 556 360 0.0
## 600 0.7932847 8 115 17 26.9 850 678 0.0
## 601 0.7157439 9 115 17 26.9 850 678 0.0
## 602 0.6680988 10 115 17 26.9 850 678 0.0
## 603 0.7473105 14 115 17 26.9 850 678 0.0
## 604 0.7345105 15 115 17 26.9 850 678 0.0
## 605 0.7386898 18 115 17 26.9 850 678 0.0
## 606 0.8136783 20 115 17 26.9 850 678 0.0
## 607 0.7099837 7 116 17 25.9 845 570 0.0
## 608 0.6397167 9 116 17 25.9 845 570 0.0
## 609 0.8666843 10 116 17 25.9 845 570 0.0
## 610 0.7006496 11 116 17 25.9 845 570 0.0
## 611 0.6650261 12 116 17 25.9 845 570 0.0
## 612 0.7765726 14 116 17 25.9 845 570 0.0
## 613 0.9344392 16 116 17 25.9 845 570 0.0
## 614 0.6777440 18 116 17 25.9 845 570 0.0
## 615 0.8869084 21 116 17 25.9 845 570 0.0
## 616 0.6866325 10 117 17 27.0 756 690 0.0
## 617 0.6787618 11 117 17 27.0 756 690 0.0
## 618 0.7727583 12 117 17 27.0 756 690 0.0
## 619 0.7886367 14 117 17 27.0 756 690 0.0
## 620 0.7844984 16 117 17 27.0 756 690 0.0
## 621 0.8255917 20 117 17 27.0 756 690 0.0
## 622 0.6938159 10 118 17 27.2 447 110 1.0
## 623 0.7544482 11 118 17 27.2 447 110 1.0
## 624 0.5943475 12 118 17 27.2 447 110 1.0
## 625 0.7239158 14 118 17 27.2 447 110 1.0
## 626 0.7002824 8 119 17 29.8 828 744 0.0
## 627 0.7868521 10 119 17 29.8 828 744 0.0
## 628 0.7741394 11 119 17 29.8 828 744 0.0
## 629 0.7654719 14 119 17 29.8 828 744 0.0
## 630 0.6399019 10 120 18 31.3 817 738 0.0
## 631 0.8887344 11 120 18 31.3 817 738 0.0
## 632 0.8348524 12 120 18 31.3 817 738 0.0
## 633 0.9559551 14 120 18 31.3 817 738 0.0
## 634 0.8868143 7 121 18 28.3 764 516 3.4
## 635 0.6599052 8 121 18 28.3 764 516 3.4
## 636 0.7570421 10 121 18 28.3 764 516 3.4
## 637 0.6615473 8 122 18 28.6 753 571 0.0
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## 639 0.8005566 10 122 18 28.6 753 571 0.0
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## 642 0.7413952 18 122 18 28.6 753 571 0.0
## 643 0.8298010 20 122 18 28.6 753 571 0.0
## 644 0.6886930 7 123 18 29.5 822 768 0.0
## 645 0.7637654 9 123 18 29.5 822 768 0.0
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## 652 0.9690202 21 123 18 29.5 822 768 0.0
## 653 0.9385650 10 124 18 25.5 842 768 0.0
## 654 0.7637261 11 124 18 25.5 842 768 0.0
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## 656 0.8450733 14 124 18 25.5 842 768 0.0
## 657 0.7827613 16 124 18 25.5 842 768 0.0
## 658 0.8571505 20 124 18 25.5 842 768 0.0
## 659 0.7378133 10 125 18 27.6 822 763 0.4
## 660 0.8079559 11 125 18 27.6 822 763 0.4
## 661 0.7358932 12 125 18 27.6 822 763 0.4
## 662 0.8299745 14 125 18 27.6 822 763 0.4
## 663 0.6704877 8 126 18 28.1 828 756 0.0
## 664 0.8182851 10 126 18 28.1 828 756 0.0
## 665 0.6731402 11 126 18 28.1 828 756 0.0
## 666 0.8447561 14 126 18 28.1 828 756 0.0
## 667 0.7396069 10 127 19 28.3 803 726 0.2
## 668 0.8721409 11 127 19 28.3 803 726 0.2
## 669 0.7408937 12 127 19 28.3 803 726 0.2
## 670 0.8361233 14 127 19 28.3 803 726 0.2
## 671 0.7057038 7 128 19 28.6 800 606 0.0
## 672 0.7991792 8 128 19 28.6 800 606 0.0
## 673 0.7955757 10 128 19 28.6 800 606 0.0
## 674 0.6718131 8 129 19 29.3 767 698 0.0
## 675 0.8034531 9 129 19 29.3 767 698 0.0
## 676 0.8517653 10 129 19 29.3 767 698 0.0
## 677 0.8384532 14 129 19 29.3 767 698 0.0
## 678 0.8099156 15 129 19 29.3 767 698 0.0
## 679 0.8734136 18 129 19 29.3 767 698 0.0
## 680 0.8253290 20 129 19 29.3 767 698 0.0
## 681 0.8527863 7 130 19 28.5 747 654 0.0
## 682 0.7494986 9 130 19 28.5 747 654 0.0
## 683 0.8621444 10 130 19 28.5 747 654 0.0
## 684 0.8261986 11 130 19 28.5 747 654 0.0
## 685 0.7864409 12 130 19 28.5 747 654 0.0
## 686 0.8631558 14 130 19 28.5 747 654 0.0
## 687 0.8941797 16 130 19 28.5 747 654 0.0
## 688 0.8610041 18 130 19 28.5 747 654 0.0
## 689 0.9579926 21 130 19 28.5 747 654 0.0
## 690 0.7974133 10 131 19 29.2 706 432 0.0
## 691 0.8295692 11 131 19 29.2 706 432 0.0
## 692 0.8306497 12 131 19 29.2 706 432 0.0
## 693 0.8812060 14 131 19 29.2 706 432 0.0
## 694 0.8139049 16 131 19 29.2 706 432 0.0
## 695 0.9937466 20 131 19 29.2 706 432 0.0
## 696 0.8610258 10 132 19 28.9 692 618 0.2
## 697 0.8626351 11 132 19 28.9 692 618 0.2
## 698 0.9333624 12 132 19 28.9 692 618 0.2
## 699 0.7164334 14 132 19 28.9 692 618 0.2
## 700 0.9245519 8 133 19 30.3 733 558 0.0
## 701 0.8214036 10 133 19 30.3 733 558 0.0
## 702 0.8680415 11 133 19 30.3 733 558 0.0
## 703 0.8599776 14 133 19 30.3 733 558 0.0
## 704 0.8631920 10 134 20 29.2 789 726 0.0
## 705 0.8128517 11 134 20 29.2 789 726 0.0
## 706 0.7741631 12 134 20 29.2 789 726 0.0
## 707 0.8486847 14 134 20 29.2 789 726 0.0
## 708 0.7873592 7 135 20 27.1 775 734 0.0
## 709 0.8731402 8 135 20 27.1 775 734 0.0
## 710 0.8336146 10 135 20 27.1 775 734 0.0
## 711 0.9355312 8 136 20 26.7 745 714 0.0
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## 713 0.9189098 10 136 20 26.7 745 714 0.0
## 714 0.8701497 14 136 20 26.7 745 714 0.0
## 715 0.8651185 15 136 20 26.7 745 714 0.0
## 716 0.8661117 18 136 20 26.7 745 714 0.0
## 717 0.9702422 20 136 20 26.7 745 714 0.0
## 718 0.9034049 7 137 20 27.2 750 667 1.6
## 719 0.9040255 9 137 20 27.2 750 667 1.6
## 720 0.7489925 10 137 20 27.2 750 667 1.6
## 721 0.7804357 11 137 20 27.2 750 667 1.6
## 722 0.8862587 12 137 20 27.2 750 667 1.6
## 723 0.8028192 14 137 20 27.2 750 667 1.6
## 724 0.7617844 16 137 20 27.2 750 667 1.6
## 725 0.8750366 18 137 20 27.2 750 667 1.6
## 726 0.9215890 21 137 20 27.2 750 667 1.6
## 727 0.7856311 10 138 20 27.5 572 324 0.8
## 728 0.8865290 11 138 20 27.5 572 324 0.8
## 729 0.7520506 12 138 20 27.5 572 324 0.8
## 730 0.6605511 14 138 20 27.5 572 324 0.8
## 731 0.7769170 16 138 20 27.5 572 324 0.8
## 732 0.8809513 20 138 20 27.5 572 324 0.8
## 733 0.8386404 10 139 20 27.5 628 619 0.0
## 734 0.8950005 11 139 20 27.5 628 619 0.0
## 735 0.8023136 12 139 20 27.5 628 619 0.0
## 736 0.8672133 14 139 20 27.5 628 619 0.0
## 737 0.8035335 8 140 20 29.2 636 270 5.8
## 738 0.8182326 10 140 20 29.2 636 270 5.8
## 739 0.7230936 11 140 20 29.2 636 270 5.8
## 740 0.7336964 14 140 20 29.2 636 270 5.8
## 741 0.7967502 10 141 21 29.6 617 384 0.0
## 742 0.6707501 11 141 21 29.6 617 384 0.0
## 743 0.8409421 12 141 21 29.6 617 384 0.0
## 744 0.7766300 14 141 21 29.6 617 384 0.0
## 745 0.8326606 7 142 21 29.1 722 372 6.0
## 746 0.9174286 8 142 21 29.1 722 372 6.0
## 747 0.8427307 10 142 21 29.1 722 372 6.0
## 748 0.8944423 8 143 21 28.2 494 570 0.0
## 749 0.8072688 9 143 21 28.2 494 570 0.0
## 750 0.8083631 10 143 21 28.2 494 570 0.0
## 751 0.9171793 14 143 21 28.2 494 570 0.0
## 752 1.0000000 15 143 21 28.2 494 570 0.0
## 753 0.8417462 18 143 21 28.2 494 570 0.0
## 754 0.8457075 20 143 21 28.2 494 570 0.0
## 755 0.7669963 7 144 21 27.9 403 288 0.0
## 756 0.8541606 9 144 21 27.9 403 288 0.0
## 757 0.8411742 10 144 21 27.9 403 288 0.0
## 758 0.7176004 11 144 21 27.9 403 288 0.0
## 759 0.7763630 12 144 21 27.9 403 288 0.0
## 760 0.8407902 14 144 21 27.9 403 288 0.0
## 761 0.8178732 16 144 21 27.9 403 288 0.0
## 762 0.7614821 18 144 21 27.9 403 288 0.0
## 763 0.6561115 21 144 21 27.9 403 288 0.0
## 764 0.6850695 10 145 21 23.9 731 558 0.0
## 765 0.8139428 11 145 21 23.9 731 558 0.0
## 766 0.9531650 12 145 21 23.9 731 558 0.0
## 767 0.8387946 14 145 21 23.9 731 558 0.0
## 768 0.9078340 16 145 21 23.9 731 558 0.0
## 769 0.9987230 20 145 21 23.9 731 558 0.0
## 770 0.7126008 10 146 21 26.1 392 96 0.4
## 771 0.8166395 11 146 21 26.1 392 96 0.4
## 772 0.8193923 12 146 21 26.1 392 96 0.4
## 773 0.7941865 14 146 21 26.1 392 96 0.4
## 774 0.7408087 8 147 21 20.3 342 18 8.1
## 775 0.7625812 10 147 21 20.3 342 18 8.1
## 776 0.7499970 11 147 21 20.3 342 18 8.1
## 777 0.8473657 14 147 21 20.3 342 18 8.1Modélisons sur notre variable TRAIN:
TRAINREG = data.frame(TRAIN$HEURE, TRAIN$JOUR, TRAIN$SEMAINE , TRAIN$TEMPERATURE, TRAIN$PUISSANCE, TRAIN$DUREESOLEIL, TRAIN$PLUIE)
regTrain = lm(TRAIN$TX~.,TRAINREG)
summary(regTrain)##
## Call:
## lm(formula = TRAIN$TX ~ ., data = TRAINREG)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.172392 -0.040987 -0.000245 0.039838 0.190201
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.164e-02 2.867e-02 1.104 0.270
## TRAIN.HEURE 6.975e-03 7.658e-04 9.109 < 2e-16 ***
## TRAIN.JOUR 1.985e-03 1.594e-03 1.245 0.214
## TRAIN.SEMAINE 1.274e-02 1.120e-02 1.138 0.256
## TRAIN.TEMPERATURE 5.090e-03 9.681e-04 5.258 2.11e-07 ***
## TRAIN.PUISSANCE 3.724e-06 2.644e-05 0.141 0.888
## TRAIN.DUREESOLEIL 9.380e-05 1.786e-05 5.252 2.18e-07 ***
## TRAIN.PLUIE -5.122e-04 5.706e-04 -0.898 0.370
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.06335 on 531 degrees of freedom
## Multiple R-squared: 0.8799, Adjusted R-squared: 0.8783
## F-statistic: 555.7 on 7 and 531 DF, p-value: < 2.2e-16Probleme de test (Student) : les variables sont-elles significatives ?
\(H_0\) : la variable \(X_p\) n’est pas significative, ou \(b_p = 0\).
\(H_1\) : la variable \(X_p\) est significative, ou \(b_p \not = 0\).
Règle de décision : ici, on regarde l’indice statistique : \(\mathcal T = \frac{\hat b}{\sqrt{\hat{Var(\hat b)}}}\ \)~\(\ \mathcal t_{1587}\). On établit la règle suivante :
t-value > \(qt_{1587}\)(\(\frac{1+ \alpha}{2}\)) pour rejeter \(H_0\)
qt(1.05/2,531)## [1] 0.06273642Les variables sont représentatives mais moins que sur la modélisation globale.
Probleme de test (Fisher) : la régression est-elle significative ?
\(H_0\) : la regression n’est pas significative.
\(H_1\) : la regression est significative.
Règle de décision : nous regardons l’indice statistique : \(\mathcal F = \frac {\mathbb R^2/1}{(1-\mathbb R^2)/1587}\)
On a \(F = 471.3\) >
qf(0.05,1,531)## [1] 0.003935859La regression est significative.
Probleme de test (Shapiro) : les résidus sont-ils gaussiens ?
\(H_0\) = les résidus sont gaussiens
\(H_1\) : les résidus ne sont pas gaussiens
shapiro.test(regTrain$residuals)##
## Shapiro-Wilk normality test
##
## data: regTrain$residuals
## W = 0.99664, p-value = 0.3223Ainsi, la p-value est supérieur au seuil \(\alpha = 0.05\) et la statistique \(W\) est très proche de 1, on accepte alors l’hypothèse nulle et les résidus sont donc problamement issus d’une population normalement distribuée.
x=1:100
pred.train = predict(regTrain,data.frame(TRAIN.HEURE = TEST$HEURE,TRAIN.JOUR = TEST$JOUR, TRAIN.SEMAINE =TEST$SEMAINE ,TRAIN.TEMPERATURE = TEST$TEMPERATURE,TRAIN.PUISSANCE = TEST$PUISSANCE,TRAIN.DUREESOLEIL = TEST$DUREESOLEIL, TRAIN.PLUIE = TEST$PLUIE), level = 0.95, interval = 'prediction')
mean(pred.train[,1]) - mean(TEST$TX)## [1] 0.004715725plot(pred.train[,1], TEST$TX, lwd =3, main = "Prediction en fonction des valeurs empiriques")
abline(c(0,1), col = "red", lwd = 2)
legend("topleft", c("Prev.~ Val. Emp.", "x=y"), col = c("Black","Red"), lwd = c(3,2), lty = c(3,1))
(mean(pred.train[,1]) - mean(TEST$TX))^2 + var(pred.train[,1])## [1] 0.02871678pred.train## fit lwr upr
## 1 0.3249492 0.1996059 0.4502924
## 2 0.2316041 0.1053260 0.3578823
## 3 0.2455549 0.1193750 0.3717348
## 4 0.2952188 0.1698027 0.4206350
## 5 0.3510219 0.2253222 0.4767217
## 6 0.3649727 0.2390223 0.4909231
## 7 0.3234258 0.1980426 0.4488090
## 8 0.3227612 0.1976643 0.4478582
## 9 0.3367120 0.2116552 0.4617688
## 10 0.3925151 0.2668963 0.5181339
## 11 0.2877950 0.1625914 0.4129986
## 12 0.3093501 0.1840728 0.4346274
## 13 0.3492878 0.2240144 0.4745611
## 14 0.3632385 0.2379907 0.4884864
## 15 0.3771893 0.2518947 0.5024839
## 16 0.3385849 0.2136384 0.4635313
## 17 0.4013633 0.2761545 0.5265721
## 18 0.4153141 0.2898484 0.5407797
## 19 0.2843125 0.1578335 0.4107915
## 20 0.3261648 0.1995501 0.4527794
## 21 0.3610417 0.2338240 0.4882594
## 22 0.3349069 0.2097511 0.4600627
## 23 0.3628085 0.2376688 0.4879481
## 24 0.3503483 0.2249042 0.4757925
## 25 0.3440167 0.2182979 0.4697355
## 26 0.3509921 0.2253066 0.4766776
## 27 0.3719183 0.2462246 0.4976120
## 28 0.3916301 0.2661175 0.5171428
## 29 0.3638704 0.2386469 0.4890940
## 30 0.3705489 0.2454811 0.4956167
## 31 0.3444037 0.2189900 0.4698173
## 32 0.3723052 0.2471222 0.4974882
## 33 0.4071821 0.2818810 0.5324833
## 34 0.4211329 0.2956582 0.5466076
## 35 0.3549694 0.2285753 0.4813636
## 36 0.4181363 0.2926587 0.5436138
## 37 0.3637834 0.2355724 0.4919944
## 38 0.3963882 0.2713853 0.5213911
## 39 0.4382405 0.3132413 0.5632398
## 40 0.4731174 0.3476243 0.5986106
## 41 0.3263589 0.2008605 0.4518573
## 42 0.3682112 0.2426188 0.4938037
## 43 0.4379113 0.3125639 0.5632587
## 44 0.4044336 0.2794336 0.5294336
## 45 0.4532613 0.3283352 0.5781874
## 46 0.4823442 0.3573893 0.6072991
## 47 0.4962950 0.3711619 0.6214281
## 48 0.4886298 0.3636500 0.6136097
## 49 0.5165314 0.3911257 0.6419370
## 50 0.4425313 0.3175639 0.5674987
## 51 0.4015885 0.2761716 0.5270054
## 52 0.4347138 0.3095735 0.5598541
## 53 0.5154534 0.3905403 0.6403665
## 54 0.5224288 0.3974690 0.6473885
## 55 0.4968286 0.3713364 0.6223208
## 56 0.5038040 0.3783559 0.6292520
## 57 0.5107794 0.3853574 0.6362013
## 58 0.5177547 0.3923408 0.6431687
## 59 0.5456563 0.4200941 0.6712184
## 60 0.5040753 0.3787564 0.6293942
## 61 0.5319768 0.4066600 0.6572937
## 62 0.5459276 0.4205034 0.6713518
## 63 0.5738291 0.4479746 0.6996836
## 64 0.5058000 0.3801847 0.6314153
## 65 0.5267262 0.4010905 0.6523618
## 66 0.5027212 0.3767560 0.6286864
## 67 0.5166720 0.3908203 0.6425236
## 68 0.5196076 0.3937363 0.6454790
## 69 0.5335584 0.4077141 0.6594027
## 70 0.5475092 0.4216199 0.6733984
## 71 0.5076485 0.3785166 0.6367805
## 72 0.4070965 0.2813393 0.5328537
## 73 0.4210472 0.2953708 0.5467237
## 74 0.4889299 0.3641190 0.6137408
## 75 0.4970295 0.3721257 0.6219334
## 76 0.5040049 0.3791165 0.6288933
## 77 0.5179557 0.3930437 0.6428676
## 78 0.4424806 0.3159868 0.5689743
## 79 0.5531110 0.4280455 0.6781764
## 80 0.5261323 0.4012164 0.6510482
## 81 0.5400831 0.4152758 0.6648904
## 82 0.5576989 0.4329761 0.6824217
## 83 0.5716497 0.4468217 0.6964777
## 84 0.5482283 0.4233699 0.6730867
## 85 0.6110068 0.4855992 0.7364143
## 86 0.5689475 0.4437562 0.6941389
## 87 0.5759229 0.4507385 0.7011074
## 88 0.5461419 0.4210658 0.6712180
## 89 0.5531173 0.4281210 0.6781136
## 90 0.5167125 0.3918706 0.6415545
## 91 0.5236879 0.3989097 0.6484661
## 92 0.5655402 0.4407638 0.6903167
## 93 0.6004171 0.4751444 0.7256899
## 94 0.5040101 0.3790602 0.6289599
## 95 0.5179609 0.3931622 0.6427595
## 96 0.5249362 0.4001861 0.6496864
## 97 0.5388870 0.4141794 0.6635946
## 98 0.5667885 0.4419484 0.6916287
## 99 0.5807393 0.4557243 0.7057544
## 100 0.6016655 0.4762531 0.7270778
## 101 0.5684147 0.4436709 0.6931585
## 102 0.5475494 0.4226576 0.6724412
## 103 0.5624016 0.4373314 0.6874717
## 104 0.5910200 0.4660264 0.7160136
## 105 0.5131992 0.3874320 0.6389665
## 106 0.6458783 0.5208463 0.7709104
## 107 0.6598291 0.5345456 0.7851127
## 108 0.5544889 0.4293763 0.6796015
## 109 0.6521443 0.5265157 0.7777729
## 110 0.5205008 0.3951868 0.6458148
## 111 0.5274762 0.4021765 0.6527759
## 112 0.5414270 0.4161015 0.6667524
## 113 0.5774344 0.4526406 0.7022281
## 114 0.6007444 0.4755709 0.7259179
## 115 0.6266345 0.5019687 0.7513003
## 116 0.5925071 0.4678342 0.7171800
## 117 0.6413348 0.5166186 0.7660510
## 118 0.6762117 0.5509215 0.8015019
## 119 0.6411810 0.5161913 0.7661707
## 120 0.6532928 0.5281324 0.7784532
## 121 0.6367204 0.5116501 0.7617908
## 122 0.6506712 0.5257080 0.7756343
## 123 0.6224717 0.4966025 0.7483408
## 124 0.6294471 0.5036411 0.7552530
## 125 0.6364224 0.5106618 0.7621831
## 126 0.6712994 0.5454952 0.7971035
## 127 0.6922255 0.5661796 0.8182714
## 128 0.5946362 0.4688012 0.7204711
## 129 0.6643900 0.5378889 0.7908911
## 130 0.6301131 0.5054807 0.7547456
## 131 0.6370885 0.5124689 0.7617082
## 132 0.6510393 0.5263907 0.7756878
## 133 0.6393043 0.5144878 0.7641208
## 134 0.6462797 0.5214791 0.7710802
## 135 0.6899968 0.5647915 0.8152021
## 136 0.6676816 0.5425608 0.7928024
## 137 0.6816324 0.5565454 0.8067194
## 138 0.6966509 0.5718192 0.8214827
## 139 0.7385033 0.6131376 0.8638689
## 140 0.6994252 0.5746089 0.8242416
## 141 0.7273268 0.6023756 0.8522780
## 142 0.7412775 0.6161504 0.8664047
## 143 0.7622037 0.6366778 0.8877296
## 144 0.6562963 0.5309355 0.7816571
## 145 0.6702470 0.5447787 0.7957154
## 146 0.6281714 0.5030257 0.7533172
## 147 0.6351468 0.5100200 0.7602737
## 148 0.6592930 0.5340728 0.7845131
## 149 0.6732437 0.5481472 0.7983403
## 150 0.7011453 0.5760788 0.8262117
## 151 0.7012914 0.5761821 0.8264007
## 152 0.7291930 0.6040797 0.8543063
## 153 0.6567911 0.5290383 0.7845440
## 154 0.6916680 0.5640249 0.8193112
## 155 0.7195696 0.5916953 0.8474439
## 156 0.6976121 0.5722976 0.8229267
## 157 0.7083666 0.5834438 0.8332893
## 158 0.7182691 0.5931172 0.8434210
## 159 0.7391952 0.6140157 0.8643748
## 160 0.7034987 0.5780538 0.8289437
## 161 0.7174495 0.5921189 0.8427801
## 162 0.7023371 0.5774409 0.8272333
## 163 0.7499113 0.6250262 0.8747964
## 164 0.8057144 0.6802804 0.9311484
## 165 0.7563322 0.6314181 0.8812463
## 166 0.7443974 0.6191412 0.8696535
## 167 0.7583481 0.6330589 0.8836373
## 168 0.7164060 0.5912023 0.8416098
## 169 0.7571209 0.6321098 0.8821319
## 170 0.7850224 0.6597390 0.9103058
## 171 0.7158366 0.5907510 0.8409221
## 172 0.7925658 0.6674362 0.9176954
## 173 0.8134920 0.6879712 0.9390128
## 174 0.7083454 0.5823380 0.8343528
## 175 0.7527954 0.6279482 0.8776426
## 176 0.7047893 0.5793650 0.8302136
## 177 0.7545782 0.6292989 0.8798575
## 178 0.7824797 0.6572065 0.9077529
## 179 0.6920405 0.5666598 0.8174213
## 180 0.7778482 0.6529467 0.9027496
## 181 0.8057497 0.6805776 0.9309218
## 182 0.8197005 0.6942850 0.9451160
## 183 0.7157662 0.5905468 0.8409857
## 184 0.7924955 0.6672558 0.9177352
## 185 0.7691519 0.6443443 0.8939595
## 186 0.7831027 0.6582703 0.9079350
## 187 0.7857327 0.6606198 0.9108456
## 188 0.8005149 0.6753333 0.9256964
## 189 0.8074903 0.6823356 0.9326449
## 190 0.7608874 0.6359285 0.8858462
## 191 0.7748381 0.6499954 0.8996809
## 192 0.8027397 0.6779117 0.9275676
## 193 0.8306412 0.7055383 0.9557441
## 194 0.7931643 0.6682825 0.9180461
## 195 0.8071150 0.6823147 0.9319154
## 196 0.8768689 0.7513899 1.0023478
## 197 0.7888130 0.6637114 0.9139145
## 198 0.8097391 0.6846334 0.9348448
## 199 0.7937640 0.6687124 0.9188155
## 200 0.8007394 0.6757231 0.9257556
## 201 0.8077148 0.6827157 0.9327138
## 202 0.8216655 0.6966465 0.9466845
## 203 0.8048402 0.6796523 0.9300280
## 204 0.8105991 0.6853631 0.9358351
## 205 0.7820205 0.6567557 0.9072853
## 206 0.8029466 0.6778817 0.9280116
## 207 0.8170014 0.6921205 0.9418823
## 208 0.8728045 0.7476596 0.9979494
## 209 0.8107125 0.6858699 0.9355551
## 210 0.8386140 0.7137909 0.9634372
## 211 0.7952848 0.6703270 0.9202426
## 212 0.8092356 0.6843278 0.9341433
## 213 0.8237487 0.6985970 0.9489004
## 214 0.8446748 0.7195224 0.9698272
## 215 0.8597223 0.7346397 0.9848049
## 216 0.8102154 0.6850309 0.9353998
## 217 0.8311415 0.7061662 0.9561169
## 218 0.8450923 0.7201659 0.9700187
## 219 0.8022277 0.6771301 0.9273252
## 220 0.8092031 0.6841395 0.9342667
## 221 0.8719815 0.7464101 0.9975530
## 222 0.8144390 0.6890035 0.9398744
## 223 0.8353651 0.7099395 0.9607907
## 224 0.8378184 0.7123159 0.9633208
## 225 0.8447937 0.7193193 0.9702682
## 226 0.8517691 0.7263047 0.9772335
## 227 0.8657199 0.7402214 0.9912184
## 228 0.8125243 0.6868287 0.9382199
## 229 0.8446748 0.7191694 0.9701802
## 230 0.9214040 0.7954118 1.0473962
## 231 0.8253180 0.6998107 0.9508254
## 232 0.8392688 0.7138030 0.9647346
## 233 0.8671704 0.7415713 0.9927694
## 234 0.8811211 0.7553475 1.0068947
## 235 0.9020473 0.7758776 1.0282169
## 236 0.8334885 0.7080504 0.9589265
## 237 0.8404638 0.7150621 0.9658656
## 238 0.7698601 0.6429695 0.8967507On a erreur quadratique moyenne qui est égale à 0.02835339