Pendahuluan

Hubungan antara variabel lingkungan sering kali bersifat kompleks dan tidak selalu mengikuti pola linear. Pada dataset airquality, misalnya, konsentrasi Ozone diduga dipengaruhi oleh tingkat Solar.R. Secara teoritis, radiasi matahari dapat memicu pembentukan ozon, tetapi pola hubungan yang muncul di data bisa melengkung atau bervariasi pada rentang tertentu. Oleh karena itu, diperlukan analisis regresi dengan berbagai pendekatan—mulai dari model linear sederhana hingga spline—untuk memperoleh gambaran yang lebih akurat mengenai pola hubungan tersebut.

library(tidyverse)

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library(splines)

df_airquality <- datasets::airquality
head(df_airquality)

##   Ozone Solar.R Wind Temp Month Day
## 1    41     190  7.4   67     5   1
## 2    36     118  8.0   72     5   2
## 3    12     149 12.6   74     5   3
## 4    18     313 11.5   62     5   4
## 5    NA      NA 14.3   56     5   5
## 6    28      NA 14.9   66     5   6

datasets::airquality adalah dataset bawaan R yang berisi kualitas udara di New York (Ozone, Solar.R, Solar.R, Solar.R, Month, Day).

# Cek apakah ada missing value
colSums(is.na(airquality))

##   Ozone Solar.R    Wind    Temp   Month     Day 
##      37       7       0       0       0       0

Dari hasil pemeriksaan, dataset airquality memiliki missing value pada variabel Ozone dan Solar.R. Informasi ini penting sebelum melakukan analisis, karena kita perlu menentukan strategi untuk menangani nilai yang hilang, misalnya dengan menghapus baris, mengisi dengan rata-rata/median, atau menggunakan metode imputasi lainnya.

# Ganti NA pada kolom Ozone dengan median
airquality$Ozone[is.na(airquality$Ozone)] <- median(airquality$Ozone, na.rm = TRUE)

# Ganti NA pada kolom Solar.R dengan median
airquality$Solar.R[is.na(airquality$Solar.R)] <- median(airquality$Solar.R, na.rm = TRUE)

# Cek lagi apakah masih ada missing value
colSums(is.na(airquality))

##   Ozone Solar.R    Wind    Temp   Month     Day 
##       0       0       0       0       0       0

median(…, na.rm = TRUE) menghitung median dengan mengabaikan NA. Baris yang NA kemudian diisi dengan nilai median dari kolom tersebut. Setelah diganti, hasil colSums(is.na()) akan menunjukkan 0 → artinya tidak ada lagi missing value.

Data Air Quality

ggplot(df_airquality,aes(x=Solar.R, y=Ozone)) +
                 geom_point(alpha=0.55, color="black") + 
                 theme_bw()

## Warning: Removed 42 rows containing missing values or values outside the scale range
## (`geom_point()`).

Dari grafik terlihat hubungan tidak sepenuhnya linear → ada indikasi kurva.
Hal ini menjadi dasar kita perlu coba model non-linear (tangga, spline).

Regresi Linear

mod_linear = lm(Ozone~Solar.R,data=df_airquality)
summary(mod_linear)

## 
## Call:
## lm(formula = Ozone ~ Solar.R, data = df_airquality)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -48.292 -21.361  -8.864  16.373 119.136 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 18.59873    6.74790   2.756 0.006856 ** 
## Solar.R      0.12717    0.03278   3.880 0.000179 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 31.33 on 109 degrees of freedom
##   (42 observations deleted due to missingness)
## Multiple R-squared:  0.1213, Adjusted R-squared:  0.1133 
## F-statistic: 15.05 on 1 and 109 DF,  p-value: 0.0001793

Model regresi linear sederhana: Ozone = β0 + β1*Solar.R.
summary() menampilkan koefisien, nilai p, R², dll.
Interpretasi: jika β1 signifikan, maka ada hubungan linear antara Solar.R dan Ozone.
Kelemahan: hubungan bisa saja non-linear → model ini bisa terlalu kaku.

ggplot(df_airquality,aes(x=Solar.R, y=Ozone)) +
                 geom_point(alpha=0.55, color="black") +
   stat_smooth(method = "lm", 
               formula = y~x,lty = 1, col = "blue",se = F)+
  theme_bw()

## Warning: Removed 42 rows containing non-finite outside the scale range
## (`stat_smooth()`).

## Warning: Removed 42 rows containing missing values or values outside the scale range
## (`geom_point()`).

Dari grafik terlihat bahwa linear fit cukup oke di bagian tengah, tapi di ekor (nilai kecil & besar) tidak mengikuti pola dengan baik → indikasi perlu model lebih fleksibel.

Regresi Fungsi Tangga

mod_tangga = lm(Ozone ~ cut(Solar.R,5),data=df_airquality)
summary(mod_tangga)

## 
## Call:
## lm(formula = Ozone ~ cut(Solar.R, 5), data = df_airquality)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -45.743 -20.647  -6.437  14.853 112.257 
## 
## Coefficients:
##                           Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                 15.167      7.071   2.145 0.034254 *  
## cut(Solar.R, 5)(72.4,138]   10.271     10.308   0.996 0.321338    
## cut(Solar.R, 5)(138,203]    37.214      9.637   3.862 0.000194 ***
## cut(Solar.R, 5)(203,269]    40.576      8.702   4.663 9.13e-06 ***
## cut(Solar.R, 5)(269,334]    29.690      9.637   3.081 0.002629 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 30 on 106 degrees of freedom
##   (42 observations deleted due to missingness)
## Multiple R-squared:  0.2167, Adjusted R-squared:  0.1871 
## F-statistic: 7.331 on 4 and 106 DF,  p-value: 2.984e-05

cut(Solar.R, 3) membagi Solar.R ke dalam 3 interval sama lebar.
Artinya model mengestimasi rata-rata Ozone berbeda-beda untuk tiap interval Solar.R.
Cocok kalau kita hanya ingin melihat perbedaan rata-rata antar kelompok.
Kelemahan: tidak halus (stepwise), prediksi bisa “loncat-loncat”.

ggplot(df_airquality,aes(x=Solar.R, y=Ozone)) +
                 geom_point(alpha=0.55, color="black") +
  stat_smooth(method = "lm", 
               formula = y~cut(x,5), 
               lty = 1, col = "blue",se = F)+
  theme_bw()

## Warning: Removed 42 rows containing non-finite outside the scale range
## (`stat_smooth()`).

## Warning: Removed 42 rows containing missing values or values outside the scale range
## (`geom_point()`).

Membagi Solar.R jadi 5 kategori (lebih detail).
Garis biru menunjukkan rata-rata Ozone per interval.
Hasilnya “berundak” (fungsi tangga).
Mudah dipahami, tapi kehilangan informasi variasi halus.

Regresi Spline

mod_spline3 = lm(Ozone ~ bs(Solar.R, knots = c(5, 10, 20, 30, 40)),data=df_airquality)
summary(mod_spline3)

## 
## Call:
## lm(formula = Ozone ~ bs(Solar.R, knots = c(5, 10, 20, 30, 40)), 
##     data = df_airquality)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -45.255 -18.910  -3.842  15.718 109.745 
## 
## Coefficients: (1 not defined because of singularities)
##                                            Estimate Std. Error t value Pr(>|t|)
## (Intercept)                                  12.665     13.828   0.916  0.36189
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))1    3.279     32.834   0.100  0.92065
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))2  -22.451     51.623  -0.435  0.66454
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))3   28.804     44.766   0.643  0.52136
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))4  -15.621     31.049  -0.503  0.61596
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))5    4.717     22.568   0.209  0.83485
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))6    4.984     23.393   0.213  0.83171
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))7   98.325     35.535   2.767  0.00671
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))8       NA         NA      NA       NA
##                                              
## (Intercept)                                  
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))1   
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))2   
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))3   
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))4   
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))5   
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))6   
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))7 **
## bs(Solar.R, knots = c(5, 10, 20, 30, 40))8   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 29.78 on 103 degrees of freedom
##   (42 observations deleted due to missingness)
## Multiple R-squared:  0.2502, Adjusted R-squared:  0.1993 
## F-statistic:  4.91 on 7 and 103 DF,  p-value: 8.159e-05

mod_spline3ns = lm(Ozone ~ ns(Solar.R, knots = c(5, 10, 20, 30, 40)),data=df_airquality)
summary(mod_spline3ns)

## 
## Call:
## lm(formula = Ozone ~ ns(Solar.R, knots = c(5, 10, 20, 30, 40)), 
##     data = df_airquality)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -40.020 -23.768  -3.773  14.152 115.599 
## 
## Coefficients: (1 not defined because of singularities)
##                                            Estimate Std. Error t value Pr(>|t|)
## (Intercept)                                   80.99      39.20   2.066   0.0413
## ns(Solar.R, knots = c(5, 10, 20, 30, 40))1   -50.30      49.37  -1.019   0.3106
## ns(Solar.R, knots = c(5, 10, 20, 30, 40))2   -66.02      49.84  -1.325   0.1882
## ns(Solar.R, knots = c(5, 10, 20, 30, 40))3   -83.57      38.60  -2.165   0.0327
## ns(Solar.R, knots = c(5, 10, 20, 30, 40))4    35.06      19.99   1.753   0.0824
## ns(Solar.R, knots = c(5, 10, 20, 30, 40))5  -110.05     107.17  -1.027   0.3069
## ns(Solar.R, knots = c(5, 10, 20, 30, 40))6       NA         NA      NA       NA
##                                             
## (Intercept)                                *
## ns(Solar.R, knots = c(5, 10, 20, 30, 40))1  
## ns(Solar.R, knots = c(5, 10, 20, 30, 40))2  
## ns(Solar.R, knots = c(5, 10, 20, 30, 40))3 *
## ns(Solar.R, knots = c(5, 10, 20, 30, 40))4 .
## ns(Solar.R, knots = c(5, 10, 20, 30, 40))5  
## ns(Solar.R, knots = c(5, 10, 20, 30, 40))6  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 30.29 on 105 degrees of freedom
##   (42 observations deleted due to missingness)
## Multiple R-squared:  0.2089, Adjusted R-squared:  0.1712 
## F-statistic: 5.544 on 5 and 105 DF,  p-value: 0.000143

Spline memecah rentang Solar.R menjadi beberapa segmen dengan knot (titik belok), lalu pasang polinomial di tiap segmen.
bs() → B-spline: fleksibel, bisa melengkung mengikuti data.
ns() → Natural spline: mirip, tapi di luar knot paling luar dipaksa linear (lebih stabil untuk ekstrapolasi).
quantile() dipakai supaya knot sesuai distribusi data (tidak sembarang angka).
Hasil summary() menunjukkan koefisien basis spline (tidak mudah diinterpretasi langsung, tapi penting untuk prediksi).

ggplot(df_airquality,aes(x=Solar.R, y=Ozone)) +
                 geom_point(alpha=0.55, color="black") +
  stat_smooth(method = "lm", 
               formula = y~bs(x, knots = c(5, 10, 20, 30, 40)), 
               lty = 1, aes(col = "Cubic Spline"),se = F)+
    stat_smooth(method = "lm", 
               formula = y~ns(x, knots = c(5, 10, 20, 30, 40)), 
               lty = 1, aes(col = "Natural Cubic Spline"),se = F)+labs(color="Tipe Spline")+
  scale_color_manual(values = c("Natural Cubic Spline"="red","Cubic Spline"="blue"))+theme_bw()

## Warning: Removed 42 rows containing non-finite outside the scale range
## (`stat_smooth()`).
## Removed 42 rows containing non-finite outside the scale range
## (`stat_smooth()`).

## Warning: Removed 42 rows containing missing values or values outside the scale range
## (`geom_point()`).

Garis biru = cubic spline, merah = natural spline.
Keduanya mengikuti pola data lebih baik daripada regresi linear.
Natural spline lebih “jinak” di ekor → cocok untuk data lingkungan yang rawan outlier.
Dari sini terlihat: spline memberi fleksibilitas untuk menangkap hubungan non-linear.

Komparasi Model

# Fungsi MSE
MSE <- function(pred, actual) {
  mean((pred - actual)^2, na.rm = TRUE)
}

# Buat prediksi dari masing-masing model
pred_linear  <- predict(mod_linear)
pred_tangga  <- predict(mod_tangga)
pred_spline  <- predict(mod_spline3)
pred_nspline <- predict(mod_spline3ns)

compare_stats <- data.frame(
  Model   = c("Linear","Tangga","Spline","Natural Spline"),
  MSE     = c(MSE(pred_linear, df_airquality$Ozone),
              MSE(pred_tangga, df_airquality$Ozone),
              MSE(pred_spline, df_airquality$Ozone),
              MSE(pred_nspline, df_airquality$Ozone)),
  AIC     = c(AIC(mod_linear),
              AIC(mod_tangga),
              AIC(mod_spline3),
              AIC(mod_spline3ns)),
  Adj_R2  = c(summary(mod_linear)$adj.r.squared,
              summary(mod_tangga)$adj.r.squared,
              summary(mod_spline3)$adj.r.squared,
              summary(mod_spline3ns)$adj.r.squared)
)

## Warning in pred - actual: longer object length is not a multiple of shorter
## object length
## Warning in pred - actual: longer object length is not a multiple of shorter
## object length
## Warning in pred - actual: longer object length is not a multiple of shorter
## object length
## Warning in pred - actual: longer object length is not a multiple of shorter
## object length

compare_stats

##            Model      MSE      AIC    Adj_R2
## 1         Linear 1293.729 1083.714 0.1132809
## 2         Tangga 1405.729 1076.964 0.1871312
## 3         Spline 1429.093 1078.108 0.1992623
## 4 Natural Spline 1404.240 1080.069 0.1711837

Tugas Analisis

1. Dengan menggunakan model dan syntax yang sama, lakukan perbandingan antara variabel Ozone dengan Temp.

library(splines)

df_airquality <- na.omit(airquality)
head(df_airquality)

##   Ozone Solar.R Wind Temp Month Day
## 1  41.0     190  7.4   67     5   1
## 2  36.0     118  8.0   72     5   2
## 3  12.0     149 12.6   74     5   3
## 4  18.0     313 11.5   62     5   4
## 5  31.5     205 14.3   56     5   5
## 6  28.0     205 14.9   66     5   6

MSE <- function(actual, predicted) {
  mean((actual - predicted)^2, na.rm = TRUE)
}

mod_linear_temp <- lm(Ozone ~ Temp, data=df_airquality)
mod_tangga_temp <- lm(Ozone ~ cut(Temp, 5), data=df_airquality)
mod_spline_temp <- lm(Ozone ~ bs(Temp, knots=c(70, 80, 90)), data=df_airquality)
mod_nspline_temp <- lm(Ozone ~ ns(Temp, df=3), data=df_airquality)

pred_linear_temp   <- predict(mod_linear_temp)
pred_tangga_temp   <- predict(mod_tangga_temp)
pred_spline_temp   <- predict(mod_spline_temp)
pred_nspline_temp  <- predict(mod_nspline_temp)

# Bandingkan performa model
compare_temp <- data.frame(
  Model   = c("Linear", "Tangga", "Spline", "Natural Spline"),
  MSE     = c(MSE(df_airquality$Ozone, pred_linear_temp),
              MSE(df_airquality$Ozone, pred_tangga_temp),
              MSE(df_airquality$Ozone, pred_spline_temp),
              MSE(df_airquality$Ozone, pred_nspline_temp)),
  AIC     = c(AIC(mod_linear_temp),
              AIC(mod_tangga_temp),
              AIC(mod_spline_temp),
              AIC(mod_nspline_temp)),
  Adj_R2  = c(summary(mod_linear_temp)$adj.r.squared,
              summary(mod_tangga_temp)$adj.r.squared,
              summary(mod_spline_temp)$adj.r.squared,
              summary(mod_nspline_temp)$adj.r.squared)
)

compare_temp

##            Model      MSE      AIC    Adj_R2
## 1         Linear 535.8932 1401.637 0.3567682
## 2         Tangga 493.3074 1394.968 0.3958816
## 3         Spline 471.2354 1391.965 0.4150063
## 4 Natural Spline 474.4948 1387.020 0.4228200

2. Ulangi analisis untuk hubungan antara Ozone dengan Wind.

library(splines)

df_airquality <- na.omit(airquality)
head(df_airquality)

##   Ozone Solar.R Wind Temp Month Day
## 1  41.0     190  7.4   67     5   1
## 2  36.0     118  8.0   72     5   2
## 3  12.0     149 12.6   74     5   3
## 4  18.0     313 11.5   62     5   4
## 5  31.5     205 14.3   56     5   5
## 6  28.0     205 14.9   66     5   6

MSE <- function(actual, predicted) {
  mean((actual - predicted)^2, na.rm = TRUE)
}

mod_linear_wind <- lm(Ozone ~ Wind, data=df_airquality)
mod_tangga_wind <- lm(Ozone ~ cut(Wind, 5), data=df_airquality)
mod_spline_wind <- lm(Ozone ~ bs(Wind, knots=c(5, 10, 15)), data=df_airquality)
mod_nspline_wind <- lm(Ozone ~ ns(Wind, df=3), data=df_airquality)

pred_linear_wind   <- predict(mod_linear_wind)
pred_tangga_wind   <- predict(mod_tangga_wind)
pred_spline_wind   <- predict(mod_spline_wind)
pred_nspline_wind  <- predict(mod_nspline_wind)

compare_wind <- data.frame(
  Model   = c("Linear", "Tangga", "Spline", "Natural Spline"),
  MSE     = c(MSE(df_airquality$Ozone, pred_linear_wind),
              MSE(df_airquality$Ozone, pred_tangga_wind),
              MSE(df_airquality$Ozone, pred_spline_wind),
              MSE(df_airquality$Ozone, pred_nspline_wind)),
  AIC     = c(AIC(mod_linear_wind),
              AIC(mod_tangga_wind),
              AIC(mod_spline_wind),
              AIC(mod_nspline_wind)),
  Adj_R2  = c(summary(mod_linear_wind)$adj.r.squared,
              summary(mod_tangga_wind)$adj.r.squared,
              summary(mod_spline_wind)$adj.r.squared,
              summary(mod_nspline_wind)$adj.r.squared)
)

compare_wind

##            Model      MSE      AIC    Adj_R2
## 1         Linear 601.3750 1419.276 0.2781705
## 2         Tangga 545.7385 1410.423 0.3316730
## 3         Spline 459.0189 1387.946 0.4301719
## 4 Natural Spline 521.1360 1401.365 0.3660851

3. Bandingkan hasil dari masing-masing model berdasarkan nilai MSE, AIC, dan Adjusted R-squared.

compare_temp_vs_wind <- data.frame(
  Variabel = c("Temp", "Wind"),
  MSE      = c(MSE(df_airquality$Ozone, pred_linear_temp),
               MSE(df_airquality$Ozone, pred_linear_wind)),
  AIC      = c(AIC(mod_linear_temp), AIC(mod_linear_wind)),
  Adj_R2   = c(summary(mod_linear_temp)$adj.r.squared,
               summary(mod_linear_wind)$adj.r.squared)
)
compare_temp_vs_wind

##   Variabel      MSE      AIC    Adj_R2
## 1     Temp 535.8932 1401.637 0.3567682
## 2     Wind 601.3750 1419.276 0.2781705

4. Berikan insight dan pendapatmu.

Berdasarkan hasil analisis, model dengan Temp lebih baik dibandingkan Wind karena menghasilkan MSE lebih kecil, AIC lebih rendah, dan Adjusted R² lebih tinggi. Untuk Temp, model terbaik adalah Natural Spline, sedangkan untuk Wind model terbaik adalah Spline. Menurut saya, hal ini wajar karena suhu lebih berpengaruh terhadap pembentukan Ozone, sementara angin lebih berperan dalam penyebaran polutan.

Statistika Lingkungan Week 5

Naura Aiswarya

2025-09-24