Homework3

2025-10-25

Introduction and Importing

For this homework, I used a dataset on Air Quality from the UCI Archives.
Specifically, the dataset contains the levels of different contaminants in the air in an Italian city such as:
- Carbon Monoxide, Non-metallic Hydrocarbons, Benzene, etc.

library(plotly)

## Loading required package: ggplot2

## 
## Attaching package: 'plotly'

## The following object is masked from 'package:ggplot2':
## 
##     last_plot

## The following object is masked from 'package:stats':
## 
##     filter

## The following object is masked from 'package:graphics':
## 
##     layout

library(ggplot2)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

airq <- read.csv("AirQualityUCI.csv", sep=";", dec=",")
airq$Date_Objects <- as.Date(airq$Date, format="%m/%d/%Y")
head(airq)

##         Date     Time CO.GT. PT08.S1.CO. NMHC.GT. C6H6.GT. PT08.S2.NMHC.
## 1 10/03/2004 18.00.00    2.6        1360      150     11.9          1046
## 2 10/03/2004 19.00.00    2.0        1292      112      9.4           955
## 3 10/03/2004 20.00.00    2.2        1402       88      9.0           939
## 4 10/03/2004 21.00.00    2.2        1376       80      9.2           948
## 5 10/03/2004 22.00.00    1.6        1272       51      6.5           836
## 6 10/03/2004 23.00.00    1.2        1197       38      4.7           750
##   NOx.GT. PT08.S3.NOx. NO2.GT. PT08.S4.NO2. PT08.S5.O3.    T   RH     AH  X X.1
## 1     166         1056     113         1692        1268 13.6 48.9 0.7578 NA  NA
## 2     103         1174      92         1559         972 13.3 47.7 0.7255 NA  NA
## 3     131         1140     114         1555        1074 11.9 54.0 0.7502 NA  NA
## 4     172         1092     122         1584        1203 11.0 60.0 0.7867 NA  NA
## 5     131         1205     116         1490        1110 11.2 59.6 0.7888 NA  NA
## 6      89         1337      96         1393         949 11.2 59.2 0.7848 NA  NA
##   Date_Objects
## 1   2004-10-03
## 2   2004-10-03
## 3   2004-10-03
## 4   2004-10-03
## 5   2004-10-03
## 6   2004-10-03

Nitrogenous Oxides

figure <- plot_ly(x=airq$Date_Objects, y=airq$NOx.GT., type="scatter", mode="markers", name="data",
        width=600, height=350) %>%
        layout(title="Plot_ly plot of Nitrogenous Oxides for Each Date",
               xaxis=list(title="Date"), yaxis=list(title="Nitrogenous Oxides Emissions (in mg/m^3)"))
figure

ggplot Number 1

figure2 <- ggplot(aes(x=NOx.GT., y=PT08.S1.CO.), data=airq) +
  geom_point() +
  geom_smooth(method="lm", se=TRUE) +
  labs(title="Total Hourly Averaged Sensor Response over Averaged NOx Response")
figure2

## `geom_smooth()` using formula = 'y ~ x'

ggplot Number 2

figure3 <- ggplot(aes(x=CO.GT., y=PT08.S1.CO.), data=airq) +
  geom_point() +
  geom_smooth(method="lm", se=TRUE) +
  labs(title="Average Benzene Concentration vs Average Carbon Monoxide Concentration") +
  coord_cartesian(xlim=c(0, 15))
figure3

## `geom_smooth()` using formula = 'y ~ x'

Linear Regression Model For GGplot 1

model <- lm(PT08.S1.CO. ~ NOx.GT., data=airq)
summary(model)

## 
## Call:
## lm(formula = PT08.S1.CO. ~ NOx.GT., data = airq)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1655.52   -92.70    18.74   146.33   922.30 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 988.93322    3.91569  252.56   <2e-16 ***
## NOx.GT.       0.35617    0.01272   27.99   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 316.8 on 9355 degrees of freedom
##   (114 observations deleted due to missingness)
## Multiple R-squared:  0.07728,    Adjusted R-squared:  0.07718 
## F-statistic: 783.5 on 1 and 9355 DF,  p-value: < 2.2e-16

Explanation

Model: \(\text{PT08.S1.CO.} = \beta_0 + \beta_1 \cdot \text{NOx.GT.} + \varepsilon; \hspace{1 cm} \varepsilon \sim N(0;\sigma^2)\)

PT08.S1.CO. is the “Hourly Averaged Sensor Response (when Nominally targetting CO)”
NOx.GT. is the “True hourly averaged NOx (Nitrogenous Oxides) concentration”
The equation above is the linear regression relationship demonstrated on the previous slide
As seen in the previous slide, The R squared value is 0.07728, so the linear relationship is weak

Carbon Monoxide Plot_ly

figure4 <- plot_ly(x=airq$Date_Objects, y=airq$CO.GT., type="scatter", mode="markers", name="data",
        width=600, height=200) %>%
        layout(title="Plot_ly plot of Carbon Monoxide for Each Date",
               xaxis=list(title="Date"), yaxis=list(title="Carbon Monoxide Emissions (in mg/m^3)"))
figure4

Second Example of Linear Regression

model_2 <- lm(NOx.GT. ~ CO.GT.,data=airq)
summary(model_2)

## 
## Call:
## lm(formula = NOx.GT. ~ CO.GT., data = airq)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -439.83 -123.28  -73.58   97.54 1233.41 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 228.31550    2.47262   92.34   <2e-16 ***
## CO.GT.        1.74519    0.02914   59.89   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 218.9 on 9355 degrees of freedom
##   (114 observations deleted due to missingness)
## Multiple R-squared:  0.2772, Adjusted R-squared:  0.2771 
## F-statistic:  3587 on 1 and 9355 DF,  p-value: < 2.2e-16

Explanation

NOx.GT. is the True hourly averaged NOx (Nitrogenous Oxides) concentration
CO.GT. is the true hourly averaged Carbon Monoxide concentration
The R squared value for this regression model is 0.2771 – This is a lot higher than the value for the previous model, which indicates a stronger relationship
Model: \(\text{NOx.GT.} = \beta_0 + \beta_1 \cdot \text{CO.GT.} + \varepsilon; \hspace{1 cm} \varepsilon \sim N(0;\sigma^2)\)