Week 5
Farid
2025-09-24
Week 5 - Analisis Regresi untuk Data Lingkungan
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)## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr 1.1.4 ✔ readr 2.1.5
## ✔ forcats 1.0.0 ✔ stringr 1.5.2
## ✔ ggplot2 3.5.2 ✔ tibble 3.3.0
## ✔ lubridate 1.9.4 ✔ tidyr 1.3.1
## ✔ purrr 1.1.0
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorslibrary(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 6datasets::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 0Dari 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 0median(…, 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-05mod_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.
# 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 lengthcompare_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 0.1711837Tugas Analisis
- Dengan menggunakan model dan syntax yang sama, lakukan perbandingan antara variabel Ozone dengan Temp.
- Ulangi analisis untuk hubungan antara Ozone dengan Wind.
- Bandingkan hasil dari masing-masing model berdasarkan nilai MSE, AIC, dan Adjusted R-squared.
- Berikan insight dan pendapatmu.
1. Model Ozone ~ Temp
mod_linear_temp <- lm(Ozone ~ Temp, data = df_airquality)
mod_tangga_temp <- lm(Ozone ~ cut(Temp, 5), data = df_airquality)knots_temp <- as.numeric(quantile(df_airquality$Temp, probs = c(0.2, 0.4, 0.6, 0.8)))
mod_spline_temp <- lm(Ozone ~ bs(Temp, knots = knots_temp), data = df_airquality)
mod_nspline_temp <- lm(Ozone ~ ns(Temp, knots = knots_temp), data = df_airquality)2. Ozone ~ Wind
mod_linear_wind <- lm(Ozone ~ Wind, data = df_airquality)
mod_tangga_wind <- lm(Ozone ~ cut(Wind, 5), data = df_airquality)knots_wind <- as.numeric(quantile(df_airquality$Wind, probs = c(0.2, 0.4, 0.6, 0.8)))
mod_spline_wind <- lm(Ozone ~ bs(Wind, knots = knots_wind), data = df_airquality)
mod_nspline_wind <- lm(Ozone ~ ns(Wind, knots = knots_wind), data = df_airquality)3. Evaluasi / Bandingkan
MSE <- function(pred, actual) mean((pred - actual)^2, na.rm = TRUE)
# Prediksi
pred_lin_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)
pred_lin_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)# Tabel perbandingan (MSE, AIC, Adj R2)
compare_stats <- data.frame(
Model = c("Temp_Linear","Temp_Tangga","Temp_Spline","Temp_NSpline",
"Wind_Linear","Wind_Tangga","Wind_Spline","Wind_NSpline"),
MSE = c(
MSE(pred_lin_temp, df_airquality$Ozone),
MSE(pred_tangga_temp, df_airquality$Ozone),
MSE(pred_spline_temp, df_airquality$Ozone),
MSE(pred_nspline_temp, df_airquality$Ozone),
MSE(pred_lin_wind, df_airquality$Ozone),
MSE(pred_tangga_wind, df_airquality$Ozone),
MSE(pred_spline_wind, df_airquality$Ozone),
MSE(pred_nspline_wind, df_airquality$Ozone)
),
AIC = c(
AIC(mod_linear_temp), AIC(mod_tangga_temp), AIC(mod_spline_temp), AIC(mod_nspline_temp),
AIC(mod_linear_wind), AIC(mod_tangga_wind), AIC(mod_spline_wind), AIC(mod_nspline_wind)
),
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,
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
)
)## 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
## 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 lengthprint(compare_stats)## Model MSE AIC Adj_R2
## 1 Temp_Linear 1476.416 1067.706 0.4832134
## 2 Temp_Tangga 1365.421 1063.844 0.5125039
## 3 Temp_Spline 1449.200 1056.850 0.5520587
## 4 Temp_NSpline 1427.128 1055.752 0.5490621
## 5 Wind_Linear 1330.958 1093.187 0.3562605
## 6 Wind_Tangga 1324.261 1085.829 0.4107761
## 7 Wind_Spline 1427.255 1072.999 0.4851519
## 8 Wind_NSpline 1387.885 1072.035 0.4811073Insight yang didapat
- Model spline (baik biasa maupun natural) memberikan hasil terbaik untuk kedua variabel: hubungan Ozone dengan Temp/Wind bersifat non-linear.
- Suhu (Temp) lebih berpengaruh terhadap Ozone daripada angin, ditunjukkan oleh Adj R² yang lebih tinggi dan AIC yang lebih rendah.
- Secara ilmiah, ini mendukung pemahaman bahwa kenaikan suhu mempercepat pembentukan Ozone, sementara angin lebih berperan dalam distribusi, bukan pembentukan.