Latihan dan Tugas
Angga Pratama
Machine Learningt
Tugas dan Latihan
1 Materi 2
1.1 Latihan 2.11
- Ubah fungsi sebenarnya menjadi 𝑓(𝑥)=sin(2𝜋𝑥) dan ulangi analisis. Bagaimana berubah bias–variance?
- Ubah ukuran sampel n dari 40 menjadi 200. Apa yang terjadi pada variance model fleksibel?
- Coba beberapa nilai lambda pada ridge. Kapan model mulai underfit?
- (Opsional) Coba deg=5 vs deg=15. Jelaskan perubahan kompleksitas.
1.1.1 Soal 1
1. Simulasi Data Mengubah fungsi menjadi f(x)=(2x)
set.seed(123)
f_true <- function(x) sin(2*pi*x)
sigma <- 0.8
n <- 40
x_grid <- seq(-2,2,length.out=120)
f_grid <- f_true(x_grid)
x <- runif(n,-2,2)
y <- f_true(x) + rnorm(n,0,sigma)
plot(x,y,pch=16,main="Training sample: Y = sin(2πx) + noise")
lines(x_grid,f_grid,lwd=2)
2. Fit 2 model
# model sederhana
m_lm <- lm(y ~ x)
# model fleksibel
deg <- 15
m_poly <- lm(y ~ poly(x, deg, raw = TRUE))
plot(x, y, pch=16, main="Fit pada satu dataset")
lines(x_grid, f_grid, lwd=2)
lines(x_grid,
predict(m_lm, newdata=data.frame(x=x_grid)),
lwd=2, lty=2)
lines(x_grid,
predict(m_poly, newdata=data.frame(x=x_grid)),
lwd=2, lty=3)
legend("topleft",
legend=c("f(x) true",
"Linear (OLS)",
paste0("Polynomial deg=",deg)),
lty=c(1,2,3),
bty="n")
Interpretasi:
• Model linear tidak mampu mengikuti pola sinus → bias tinggi • Model polynomial lebih fleksibel → bias lebih kecil tetapi berpotensi variance tinggi
3.Simulasi bias–variance
B <- 300
pred_lm <- matrix(NA_real_,B,length(x_grid))
pred_poly <- matrix(NA_real_,B,length(x_grid))
set.seed(2026)
for(b in 1:B){
x_b <- runif(n,-2,2)
y_b <- f_true(x_b) + rnorm(n,0,sigma)
fit_lm <- lm(y_b ~ x_b)
fit_poly <- lm(y_b ~ poly(x_b,deg,raw=TRUE))
pred_lm[b,] <- predict(fit_lm,
newdata=data.frame(x_b=x_grid))
pred_poly[b,] <- predict(fit_poly,
newdata=data.frame(x_b=x_grid))
}
bv_summary <- function(pred_mat,f_grid,sigma2){
mu_hat <- colMeans(pred_mat)
bias2 <- (mu_hat - f_grid)^2
varhat <- apply(pred_mat,2,var)
msehat <- bias2 + varhat + sigma2
list(
mu_hat = mu_hat,
bias2 = bias2,
variance = varhat,
msehat = msehat
)
}
s_lm <- bv_summary(pred_lm,f_grid,sigma^2)
s_poly <- bv_summary(pred_poly,f_grid,sigma^2)
par(mfrow=c(2,2),mar=c(4,4,2,1))
plot(x_grid,s_lm$bias2,type="l",lwd=2,
main="Bias^2 vs x (Linear)",
ylab="Bias^2",xlab="x")
plot(x_grid,s_lm$variance,type="l",lwd=2,
main="Variance vs x (Linear)",
ylab="Variance",xlab="x")
plot(x_grid,s_poly$bias2,type="l",lwd=2,
main=paste0("Bias^2 vs x (Poly ",deg,")"),
ylab="Bias^2",xlab="x")
plot(x_grid,s_poly$variance,type="l",lwd=2,
main=paste0("Variance vs x (Poly ",deg,")"),
ylab="Variance",xlab="x")
4.Ringkasan
avg_metrics <- function(s) {
c(
Bias2 = mean(s$bias2),
Variance = mean(s$variance),
Irreducible = sigma^2,
MSE = mean(s$msehat)
)
}
rbind(
Linear_OLS = avg_metrics(s_lm),
Poly_high_degree = avg_metrics(s_poly)
)## Bias2 Variance Irreducible MSE
## Linear_OLS 0.4781825 5.881990e-02 0.64 1.177002e+00
## Poly_high_degree 578.6735751 1.912306e+05 0.64 1.918099e+05Interpretasi
Ketika fungsi sebenarnya menjadi sinus:
• model linear memiliki bias besar
• model polynomial memiliki bias lebih kecil tetapi variance tinggi
Ini memperjelas bias–variance tradeoff.
1.1.2 Soal 2
set.seed(123)
f_true <- function(x) sin(2*pi*x)
sigma <- 0.8
n <- 200
x_grid <- seq(-2,2,length.out=120)
f_grid <- f_true(x_grid)
x <- runif(n,-2,2)
y <- f_true(x) + rnorm(n,0,sigma)
plot(x,y,pch=16,main="Training sample: Y = sin(2πx) + noise")
lines(x_grid,f_grid,lwd=2)
# model sederhana
m_lm <- lm(y ~ x)
# model fleksibel
deg <- 15
m_poly <- lm(y ~ poly(x, deg, raw = TRUE))
plot(x, y, pch=16, main="Fit pada satu dataset")
lines(x_grid, f_grid, lwd=2)
lines(x_grid,
predict(m_lm, newdata=data.frame(x=x_grid)),
lwd=2, lty=2)
lines(x_grid,
predict(m_poly, newdata=data.frame(x=x_grid)),
lwd=2, lty=3)
legend("topleft",
legend=c("f(x) true",
"Linear (OLS)",
paste0("Polynomial deg=",deg)),
lty=c(1,2,3),
bty="n")
B <- 300
pred_lm <- matrix(NA_real_,B,length(x_grid))
pred_poly <- matrix(NA_real_,B,length(x_grid))
set.seed(2026)
for(b in 1:B){
x_b <- runif(n,-2,2)
y_b <- f_true(x_b) + rnorm(n,0,sigma)
fit_lm <- lm(y_b ~ x_b)
fit_poly <- lm(y_b ~ poly(x_b,deg,raw=TRUE))
pred_lm[b,] <- predict(fit_lm,
newdata=data.frame(x_b=x_grid))
pred_poly[b,] <- predict(fit_poly,
newdata=data.frame(x_b=x_grid))
}
bv_summary <- function(pred_mat,f_grid,sigma2){
mu_hat <- colMeans(pred_mat)
bias2 <- (mu_hat - f_grid)^2
varhat <- apply(pred_mat,2,var)
msehat <- bias2 + varhat + sigma2
list(
mu_hat = mu_hat,
bias2 = bias2,
variance = varhat,
msehat = msehat
)
}
s_lm <- bv_summary(pred_lm,f_grid,sigma^2)
s_poly <- bv_summary(pred_poly,f_grid,sigma^2)
par(mfrow=c(2,2),mar=c(4,4,2,1))
plot(x_grid,s_lm$bias2,type="l",lwd=2,
main="Bias^2 vs x (Linear)",
ylab="Bias^2",xlab="x")
plot(x_grid,s_lm$variance,type="l",lwd=2,
main="Variance vs x (Linear)",
ylab="Variance",xlab="x")
plot(x_grid,s_poly$bias2,type="l",lwd=2,
main=paste0("Bias^2 vs x (Poly ",deg,")"),
ylab="Bias^2",xlab="x")
plot(x_grid,s_poly$variance,type="l",lwd=2,
main=paste0("Variance vs x (Poly ",deg,")"),
ylab="Variance",xlab="x")
avg_metrics <- function(s) {
c(
Bias2 = mean(s$bias2),
Variance = mean(s$variance),
Irreducible = sigma^2,
MSE = mean(s$msehat)
)
}
rbind(
Linear_OLS = avg_metrics(s_lm),
Poly_high_degree = avg_metrics(s_poly)
)## Bias2 Variance Irreducible MSE
## Linear_OLS 0.477622715 0.01100145 0.64 1.1286242
## Poly_high_degree 0.001518939 0.14745757 0.64 0.7889765Interpretasi
• variance model fleksibel menurun
• model menjadi lebih stabil
• overfitting berkurang
1.1.3 Soal 3
# fungsi ridge
ridge_fit_predict <- function(x_train, y_train, x_grid, deg=15, lambda=1){
Xtr <- sapply(0:deg, function(k) x_train^k)
Xg <- sapply(0:deg, function(k) x_grid^k)
fit <- glmnet(Xtr, y_train, alpha=0, lambda=lambda, standardize=TRUE)
as.numeric(predict(fit, newx=Xg, s=lambda))
}
# simulasi ridge
lambda <- 3
pred_ridge <- matrix(NA_real_, nrow=B, ncol=length(x_grid))
set.seed(2026)
for(b in 1:B){
x_b <- runif(n,-2,2)
y_b <- f_true(x_b) + rnorm(n,0,sigma)
pred_ridge[b,] <- ridge_fit_predict(
x_b,
y_b,
x_grid,
deg=deg,
lambda=lambda
)
}
# bias-variance ridge
s_ridge <- bv_summary(pred_ridge, f_grid, sigma^2)
# ringkasan model
rbind(
Linear_OLS = avg_metrics(s_lm),
Poly_high_degree = avg_metrics(s_poly),
Ridge_poly_features = avg_metrics(s_ridge)
)## Bias2 Variance Irreducible MSE
## Linear_OLS 0.477622715 0.01100145 0.64 1.1286242
## Poly_high_degree 0.001518939 0.14745757 0.64 0.7889765
## Ridge_poly_features 0.455969500 0.01160909 0.64 1.1075786# plot rata-rata prediksi
plot(x_grid, f_grid, type="l", lwd=2,
main="Mean Prediction E[f^(x)] vs f(x)",
ylab="y", xlab="x")
lines(x_grid, s_lm$mu_hat, lwd=2, lty=2)
lines(x_grid, s_poly$mu_hat, lwd=2, lty=3)
lines(x_grid, s_ridge$mu_hat, lwd=2, lty=4)
legend("topleft",
legend=c("f(x) true",
"Linear",
paste0("Poly ",deg),
paste0("Ridge (lambda=",lambda,")")),
lty=c(1,2,3,4),
bty="n")
Interpretasi
• Model polynomial derajat tinggi (tanpa regularization) memiliki variance sangat besar, sehingga prediksi sangat sensitif terhadap perubahan data training.
• Ridge regression menambahkan penalti pada koefisien model, sehingga koefisien menjadi lebih kecil dan model menjadi lebih stabil.
• Dengan λ sedang (misalnya λ = 3), variance model menurun cukup besar sementara bias hanya meningkat sedikit. Hal ini biasanya menghasilkan MSE yang lebih kecil dibanding polynomial biasa.
• Ketika λ terlalu kecil, ridge hampir sama dengan model polynomial biasa sehingga variance masih tinggi.
• Ketika λ terlalu besar, koefisien model ditekan terlalu kuat sehingga model kehilangan fleksibilitas dan mulai underfit.
• Hasil ini menunjukkan bahwa regularization membantu mengontrol tradeoff antara bias dan variance dan dapat meningkatkan kemampuan generalisasi model.
1.1.4 Soal 4
Deg 5
# degree 5
deg <- 5
# fit model
m_poly5 <- lm(y ~ poly(x, deg, raw=TRUE))
# plot fit pada satu dataset
plot(x, y, pch=16, main="Polynomial Fit (deg=5)")
lines(x_grid, f_grid, lwd=2)
lines(x_grid,
predict(m_poly5, newdata=data.frame(x=x_grid)),
lwd=2, lty=3)
legend("topleft",
legend=c("f(x) true","Poly deg=5"),
lty=c(1,3), bty="n")
# simulasi bias-variance
pred_poly5 <- matrix(NA_real_, B, length(x_grid))
set.seed(2026)
for(b in 1:B){
x_b <- runif(n,-2,2)
y_b <- f_true(x_b) + rnorm(n,0,sigma)
fit_poly5 <- lm(y_b ~ poly(x_b,5,raw=TRUE))
pred_poly5[b,] <- predict(fit_poly5,
newdata=data.frame(x_b=x_grid))
}
# hitung bias variance
s_poly5 <- bv_summary(pred_poly5, f_grid, sigma^2)Deg 15
# degree 15
deg <- 15
# fit model
m_poly15 <- lm(y ~ poly(x, deg, raw=TRUE))
# plot fit pada satu dataset
plot(x, y, pch=16, main="Polynomial Fit (deg=15)")
lines(x_grid, f_grid, lwd=2)
lines(x_grid,
predict(m_poly15, newdata=data.frame(x=x_grid)),
lwd=2, lty=3)
legend("topleft",
legend=c("f(x) true","Poly deg=15"),
lty=c(1,3), bty="n")
# simulasi bias-variance
pred_poly15 <- matrix(NA_real_, B, length(x_grid))
set.seed(2026)
for(b in 1:B){
x_b <- runif(n,-2,2)
y_b <- f_true(x_b) + rnorm(n,0,sigma)
fit_poly15 <- lm(y_b ~ poly(x_b,15,raw=TRUE))
pred_poly15[b,] <- predict(fit_poly15,
newdata=data.frame(x_b=x_grid))
}
# hitung bias variance
s_poly15 <- bv_summary(pred_poly15, f_grid, sigma^2)Ringkasan perbandingan
rbind(
Poly_deg5 = avg_metrics(s_poly5),
Poly_deg15 = avg_metrics(s_poly15)
)## Bias2 Variance Irreducible MSE
## Poly_deg5 0.442995918 0.03558981 0.64 1.1185857
## Poly_deg15 0.001518939 0.14745757 0.64 0.7889765Interpretasi
Polynomial deg = 5
• model lebih sederhana
• bias lebih besar
• variance lebih kecilPolynomial deg = 15
• model sangat fleksibel
• bias kecil
• variance sangat besarKesimpulan:
meningkatkan kompleksitas model mengurangi bias tetapi meningkatkan variance (bias–variance tradeoff).