Relation entre pression atmosphérique et température

bdd <- read.table("./DATA/bdd.csv", header = TRUE, sep = ",", dec = ".")
bdd$date <- as.POSIXct(
  paste0(
    bdd$year, "-", 
    bdd$month, "-", 
    bdd$day, " ", 
    bdd$hour, ":", 
    bdd$minute, ":", 
    bdd$second), 
  format = "%Y-%m-%d %H:%M:%S"
)
bdd$date <- as.numeric(bdd$date)
set.seed(12345)
sampleP <- sample(1:nrow(bdd), size = 1000)
bdd <- bdd[sampleP, c("temperature", "pressure", "date")]
str(bdd)

## 'data.frame':    1000 obs. of  3 variables:
##  $ temperature: num  26.78 9.79 3.85 6.91 26.58 ...
##  $ pressure   : num  995 1039 1066 1032 997 ...
##  $ date       : num  1.57e+09 1.55e+09 1.55e+09 1.55e+09 1.56e+09 ...

summary(modPoly)

## 
## Call:
## lm(formula = myY ~ poly(myX, polyN))
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -13.0500  -2.1855   0.2745   2.4241  10.2494 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         10.7864     0.1162   92.87   <2e-16 ***
## poly(myX, polyN)1 -231.5078     3.6730  -63.03   <2e-16 ***
## poly(myX, polyN)2   65.3138     3.6730   17.78   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.673 on 997 degrees of freedom
## Multiple R-squared:  0.8114, Adjusted R-squared:  0.811 
## F-statistic:  2144 on 2 and 997 DF,  p-value: < 2.2e-16

On peut voir que quand la pression atmosphérique est basse, parfois la température reste basse, ce qui explique la distribution non Normale des résidus. Nous allons ignorer ce problème ici qui pourrait être résolu en conditionnant les valeurs de pression atmosphérique à certaines heures de la journée.

# normalité résidus : problème !
shapiro.test(resid(modPoly)) 

## 
##  Shapiro-Wilk normality test
## 
## data:  resid(modPoly)
## W = 0.98265, p-value = 1.562e-09

autre exemple

modPoly2 <- getNicePolyLmPlot(
  myX = bdd$date, 
  myY = bdd$temperature, 
  polyN = 2)

summary(modPoly2)

## 
## Call:
## lm(formula = myY ~ poly(myX, polyN))
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -17.2992  -3.9771  -0.4574   3.5839  25.8801 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        10.7864     0.1883  57.293   <2e-16 ***
## poly(myX, polyN)1 189.0865     5.9536  31.760   <2e-16 ***
## poly(myX, polyN)2  14.8321     5.9536   2.491   0.0129 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.954 on 997 degrees of freedom
## Multiple R-squared:  0.5045, Adjusted R-squared:  0.5035 
## F-statistic: 507.5 on 2 and 997 DF,  p-value: < 2.2e-16

summary(modPoly3)

## 
## Call:
## lm(formula = myY ~ poly(myX, polyN))
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -12.5665  -3.5522  -0.1015   2.9821  22.7406 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        10.7864     0.1624  66.402  < 2e-16 ***
## poly(myX, polyN)1 189.0865     5.1369  36.810  < 2e-16 ***
## poly(myX, polyN)2  14.8321     5.1369   2.887  0.00397 ** 
## poly(myX, polyN)3 -95.1672     5.1369 -18.526  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.137 on 996 degrees of freedom
## Multiple R-squared:  0.6315, Adjusted R-squared:  0.6303 
## F-statistic: 568.8 on 3 and 996 DF,  p-value: < 2.2e-16

Model

\[Y=\beta_1 + \beta_2X + \beta_3x^2+...+\beta_px^p+\epsilon\]