Regression Analysis

Airquality: New York Air Quality Measurements

https://www.rdocumentation.org/packages/datasets/versions/3.6.2/topics/airquality

head(airquality)

summary(airquality)

     Ozone           Solar.R           Wind             Temp           Month            Day      
 Min.   :  1.00   Min.   :  7.0   Min.   : 1.700   Min.   :56.00   Min.   :5.000   Min.   : 1.0  
 1st Qu.: 18.00   1st Qu.:115.8   1st Qu.: 7.400   1st Qu.:72.00   1st Qu.:6.000   1st Qu.: 8.0  
 Median : 31.50   Median :205.0   Median : 9.700   Median :79.00   Median :7.000   Median :16.0  
 Mean   : 42.13   Mean   :185.9   Mean   : 9.958   Mean   :77.88   Mean   :6.993   Mean   :15.8  
 3rd Qu.: 63.25   3rd Qu.:258.8   3rd Qu.:11.500   3rd Qu.:85.00   3rd Qu.:8.000   3rd Qu.:23.0  
 Max.   :168.00   Max.   :334.0   Max.   :20.700   Max.   :97.00   Max.   :9.000   Max.   :31.0  
 NA's   :37       NA's   :7                                                                      

df <- na.omit(airquality)
summary(df)

     Ozone          Solar.R           Wind            Temp           Month            Day       
 Min.   :  1.0   Min.   :  7.0   Min.   : 2.30   Min.   :57.00   Min.   :5.000   Min.   : 1.00  
 1st Qu.: 18.0   1st Qu.:113.5   1st Qu.: 7.40   1st Qu.:71.00   1st Qu.:6.000   1st Qu.: 9.00  
 Median : 31.0   Median :207.0   Median : 9.70   Median :79.00   Median :7.000   Median :16.00  
 Mean   : 42.1   Mean   :184.8   Mean   : 9.94   Mean   :77.79   Mean   :7.216   Mean   :15.95  
 3rd Qu.: 62.0   3rd Qu.:255.5   3rd Qu.:11.50   3rd Qu.:84.50   3rd Qu.:9.000   3rd Qu.:22.50  
 Max.   :168.0   Max.   :334.0   Max.   :20.70   Max.   :97.00   Max.   :9.000   Max.   :31.00  

plot(df$Temp, df$Ozone, xlab = "Temperature", ylab = "Ozone")

Assumption:

y = exp(a + b * x)

x’, y’ transformation:

log(y) = log(exp(a + b*x))
log(y) = a + b*x

Fitting linear model

df$LogOzone <- log(df$Ozone)

linModel <- lm(formula = df$LogOzone ~ df$Temp)
linModel

Call:
lm(formula = df$LogOzone ~ df$Temp)

Coefficients:
(Intercept)      df$Temp  
   -1.84852      0.06767  

summary(linModel)

Call:
lm(formula = df$LogOzone ~ df$Temp)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.14417 -0.32555  0.02066  0.34234  1.49100 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -1.848518   0.455080  -4.062  9.2e-05 ***
df$Temp      0.067673   0.005807  11.654  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.5804 on 109 degrees of freedom
Multiple R-squared:  0.5548,    Adjusted R-squared:  0.5507 
F-statistic: 135.8 on 1 and 109 DF,  p-value: < 2.2e-16

linModel$coefficients

(Intercept)     df$Temp 
-1.84851790  0.06767266 

Nonlinear Least Squares

ozone <- df$Ozone
temp <- df$Temp
a <- linModel$coefficients[1]
b <- linModel$coefficients[2]
model <- nls(ozone ~ exp(a + b*temp), start = list(a=a, b=b))
summary(model)

Formula: ozone ~ exp(a + b * temp)

Parameters:
   Estimate Std. Error t value Pr(>|t|)    
a -1.175445   0.539823  -2.177   0.0316 *  
b  0.061241   0.006215   9.854   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 22.9 on 109 degrees of freedom

Number of iterations to convergence: 5 
Achieved convergence tolerance: 1.541e-06

plot(ozone ~ temp)
curve(exp(coef(model)["a"] + coef(model)["b"] * x), add = T, col = "red")

plot(ozone, predict(model), xlab="True Ozone", "ylab" = "Ozone Forecast")
abline(c(0, 1), col="red")

resid <- residuals(model)
pred <- predict(model)
plot(pred, resid)
abline(0, 0, col="red")

qqnorm(resid)
qqline(resid, col="red")