Interpolasi Spasial Data meuse

Closest Observation, NN, IDW, Ensemble, dan Kriging dengan Evaluasi RMSE (Cross Validation)

Pendahuluan

Interpolasi spasial adalah metode untuk memperkirakan nilai peubah pada lokasi yang belum diamati berdasarkan nilai di lokasi tetangga yang sudah diketahui (Cressie, 1993). Asumsi dasarnya adalah adanya autokorelasi spasial: lokasi yang berdekatan cenderung memiliki nilai yang lebih mirip.

Pada tugas ini digunakan data meuse dari paket sp, yaitu 155 titik contoh tanah di dataran banjir Sungai Meuse (Belanda) yang memuat kadar logam berat. Peubah yang diinterpolasi adalah zinc (kadar seng, ppm). Lima metode dibandingkan:

No	Metode	Ide singkat
1	Closest Observation	Nilai prediksi = nilai titik terdekat (poligon Thiessen/Voronoi)
2	NN (Nearest Neighbour)	Rata-rata dari k tetangga terdekat
3	IDW	Rata-rata terboboti, bobot = 1/jarak^p
4	Kriging (Ordinary)	Bobot optimal dari model semivariogram
5	Ensemble	Gabungan prediksi NN, IDW, dan Kriging

Evaluasi menggunakan RMSE dari k-fold cross validation.

Persiapan Data

library(sp)        # sumber data meuse
library(sf)        # data spasial
library(gstat)     # IDW, NN, variogram, kriging
library(ggplot2)   # visualisasi
library(dplyr)
library(tidyr)
library(knitr)

data(meuse)        # 155 titik pengamatan
data(meuse.grid)   # grid 40 m untuk lokasi prediksi

# Ubah ke objek sf (CRS: Amersfoort / RD New = EPSG:28992)
meuse_sf <- st_as_sf(meuse,      coords = c("x", "y"), crs = 28992, remove = FALSE)
grid_sf  <- st_as_sf(meuse.grid, coords = c("x", "y"), crs = 28992, remove = FALSE)

# Peubah yang diinterpolasi
meuse_sf$vble <- meuse_sf$zinc

dim(meuse_sf)

## [1] 155  16

summary(meuse_sf$vble)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   113.0   198.0   326.0   469.7   674.5  1839.0

Eksplorasi Singkat

ggplot() +
  geom_raster(data = meuse.grid, aes(x, y), fill = "grey90") +
  geom_sf(data = meuse_sf, aes(color = vble, size = vble), alpha = 0.8) +
  scale_color_viridis_c(option = "magma", direction = -1, name = "Zinc (ppm)") +
  scale_size(range = c(1, 5), guide = "none") +
  labs(title = "Sebaran Titik Pengamatan Kadar Zinc (meuse)",
       x = "Easting", y = "Northing") +
  theme_minimal()

ggplot(meuse_sf, aes(vble)) +
  geom_histogram(bins = 25, fill = "#3b6ea5", color = "white") +
  labs(title = "Distribusi Kadar Zinc", x = "Zinc (ppm)", y = "Frekuensi") +
  theme_minimal()

Distribusi zinc menceng ke kanan (sebagian besar titik bernilai rendah, sedikit titik bernilai sangat tinggi di dekat sungai).

Fungsi Bantu

# RMSE
rmse <- function(obs, pred) sqrt(mean((obs - pred)^2))

# Closest Observation (k = 1) dan NN (k tetangga): idp = 0 -> bobot sama rata
pred_knn <- function(train, test, k) {
  g <- gstat(formula = vble ~ 1, data = train, nmax = k, set = list(idp = 0))
  predict(g, newdata = test, debug.level = 0)$var1.pred
}

# IDW dengan pangkat p
pred_idw <- function(train, test, p = 2) {
  g <- gstat(formula = vble ~ 1, data = train, set = list(idp = p))
  predict(g, newdata = test, debug.level = 0)$var1.pred
}

# Fit semivariogram: pilih model terbaik (SSErr terkecil) dari Sph/Exp/Gau
fit_vgm <- function(data) {
  v <- variogram(vble ~ 1, data)
  fits <- lapply(c("Sph", "Exp", "Gau"), function(m) {
    tryCatch(
      suppressWarnings(fit.variogram(
        v, vgm(psill = max(v$gamma) - min(v$gamma), model = m,
               range = max(v$dist) / 2, nugget = min(v$gamma)))),
      error = function(e) NULL)
  })
  fits <- Filter(Negate(is.null), fits)
  ok   <- Filter(function(f) !isTRUE(attr(f, "singular")), fits)
  if (length(ok) == 0) ok <- fits
  ok[[which.min(sapply(ok, function(f) attr(f, "SSErr")))]]
}

# Ordinary Kriging
pred_ok <- function(train, test) {
  m <- fit_vgm(train)
  krige(vble ~ 1, train, test, model = m, debug.level = 0)$var1.pred
}

Penentuan Parameter

NN: Banyaknya Tetangga (k)

k = 1 adalah Closest Observation. Nilai k untuk NN dipilih dengan leave-one-out CV.

ks <- 2:10
rmse_k <- sapply(ks, function(k) {
  g <- gstat(formula = vble ~ 1, data = meuse_sf, nmax = k, set = list(idp = 0))
  rmse(meuse_sf$vble, meuse_sf$vble - gstat.cv(g, nfold = nrow(meuse_sf),
                                                verbose = FALSE)$residual)
})

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tab_k <- data.frame(k = ks, RMSE = round(rmse_k, 2))
k_opt <- tab_k$k[which.min(tab_k$RMSE)]
kable(tab_k, caption = "RMSE LOOCV untuk beberapa nilai k")

RMSE LOOCV untuk beberapa nilai k
k	RMSE
2	283.55
3	245.21
4	259.28
5	263.99
6	268.78
7	267.61
8	269.06
9	276.40
10	278.17

cat("k optimum untuk NN =", k_opt)

## k optimum untuk NN = 3

IDW: Pangkat Jarak (p)

ps <- seq(0.5, 5, by = 0.5)
rmse_p <- sapply(ps, function(p) {
  g <- gstat(formula = vble ~ 1, data = meuse_sf, set = list(idp = p))
  rmse(meuse_sf$vble, meuse_sf$vble - gstat.cv(g, nfold = nrow(meuse_sf),
                                                verbose = FALSE)$residual)
})

## [inverse distance weighted interpolation]
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tab_p <- data.frame(p = ps, RMSE = round(rmse_p, 2))
p_opt <- tab_p$p[which.min(tab_p$RMSE)]
ggplot(tab_p, aes(p, RMSE)) +
  geom_line(color = "#3b6ea5") + geom_point(size = 2) +
  geom_point(data = tab_p[tab_p$p == p_opt, ], color = "red", size = 4) +
  labs(title = "RMSE LOOCV IDW terhadap Pangkat (p)", x = "p", y = "RMSE") +
  theme_minimal()

cat("p optimum untuk IDW =", p_opt)

## p optimum untuk IDW = 3

Kriging: Semivariogram

v_emp <- variogram(vble ~ 1, meuse_sf)
v_fit <- fit_vgm(meuse_sf)
v_fit

##   model     psill    range
## 1   Nug  24800.96   0.0000
## 2   Sph 134745.26 830.9657

plot(v_emp, v_fit, pch = 16, col = "black",
     main = "Semivariogram Empiris dan Model Terpilih (zinc)")

Interpolasi pada Seluruh Area

Seluruh data dipakai untuk memprediksi nilai pada grid 40 m (meuse.grid).

grid_pred <- st_drop_geometry(grid_sf)[, c("x", "y")]

grid_pred$`Closest Observation` <- pred_knn(meuse_sf, grid_sf, 1)
grid_pred$NN                    <- pred_knn(meuse_sf, grid_sf, k_opt)
grid_pred$IDW                   <- pred_idw(meuse_sf, grid_sf, p_opt)
grid_pred$Kriging               <- pred_ok(meuse_sf, grid_sf)

Cross Validation

Skema yang dipakai: 5-fold CV diulang 10 kali (seed berbeda) agar hasil perbandingan stabil. Pada setiap lipatan, seluruh metode (termasuk pemilihan model semivariogram) dibangun ulang hanya dari data latih, lalu diuji pada data uji. Model Null (rata-rata data latih) dipakai sebagai pembanding dasar.

K <- 5; REP <- 10

cv_once <- function(seed) {
  set.seed(seed)
  fold <- sample(rep(1:K, length.out = nrow(meuse_sf)))
  out <- data.frame(rep = seed, obs = meuse_sf$vble, fold = fold)
  for (f in 1:K) {
    tr <- meuse_sf[fold != f, ]; te <- meuse_sf[fold == f, ]; i <- fold == f
    out$Null[i]                    <- mean(tr$vble)
    out$`Closest Observation`[i]   <- pred_knn(tr, te, 1)
    out$NN[i]                      <- pred_knn(tr, te, k_opt)
    out$IDW[i]                     <- pred_idw(tr, te, p_opt)
    out$Kriging[i]                 <- pred_ok(tr, te)
  }
  out
}

cv_res <- bind_rows(lapply(1:REP, cv_once))

Ensemble

Ensemble dibentuk dari tiga metode: NN, IDW, dan Kriging. Dua versi dicoba:

Ensemble Rata-rata: rata-rata sederhana ketiga prediksi.
Ensemble Terboboti: bobot proporsional terhadap 1/RMSE tiap metode (metode yang lebih akurat mendapat bobot lebih besar).

komp <- c("NN", "IDW", "Kriging")

rmse_komp <- sapply(komp, function(m) rmse(cv_res$obs, cv_res[[m]]))
bobot     <- (1 / rmse_komp) / sum(1 / rmse_komp)
kable(data.frame(Metode = komp, RMSE = round(rmse_komp, 2), Bobot = round(bobot, 3)),
      caption = "Bobot Ensemble Terboboti")

Bobot Ensemble Terboboti
	Metode	RMSE	Bobot
NN	NN	259.57	0.324
IDW	IDW	264.12	0.318
Kriging	Kriging	234.80	0.358

# Ensemble pada hasil CV
cv_res$`Ensemble Rata-rata` <- rowMeans(cv_res[, komp])
cv_res$`Ensemble Terboboti` <- as.matrix(cv_res[, komp]) %*% bobot

# Ensemble pada grid
grid_pred$`Ensemble Rata-rata` <- rowMeans(grid_pred[, komp])
grid_pred$`Ensemble Terboboti` <- as.matrix(grid_pred[, komp]) %*% bobot

Hasil RMSE

metode <- c("Null", "Closest Observation", "NN", "IDW", "Kriging",
            "Ensemble Rata-rata", "Ensemble Terboboti")

rmse_rep <- cv_res %>%
  pivot_longer(all_of(metode), names_to = "Metode", values_to = "pred") %>%
  group_by(Metode, rep) %>%
  summarise(RMSE = rmse(obs, pred), .groups = "drop")

tab_rmse <- rmse_rep %>%
  group_by(Metode) %>%
  summarise(`RMSE Rata-rata` = mean(RMSE), `Simpangan Baku` = sd(RMSE),
            `RMSE Minimum` = min(RMSE), `RMSE Maksimum` = max(RMSE)) %>%
  arrange(`RMSE Rata-rata`) %>%
  mutate(Peringkat = row_number(),
         `Perbaikan thd Null (%)` = 100 * (1 - `RMSE Rata-rata` /
                                    `RMSE Rata-rata`[Metode == "Null"]))

kable(tab_rmse, digits = 2, caption = "Perbandingan RMSE (5-fold CV, 10 ulangan)")

Perbandingan RMSE (5-fold CV, 10 ulangan)
Metode	RMSE Rata-rata	Simpangan Baku	RMSE Minimum	RMSE Maksimum	Peringkat	Perbaikan thd Null (%)
Kriging	234.69	7.53	227.01	253.08	1	36.31
Ensemble Terboboti	245.51	8.84	234.21	265.80	2	33.37
Ensemble Rata-rata	246.15	8.90	234.60	266.49	3	33.20
NN	259.35	11.09	238.95	280.51	4	29.62
IDW	264.01	8.19	254.27	281.98	5	28.35
Closest Observation	318.33	12.77	295.76	335.80	6	13.61
Null	368.49	1.33	366.72	370.87	7	0.00

tab_plot <- tab_rmse %>%
  mutate(Metode = reorder(Metode, `RMSE Rata-rata`),
         terbaik = `RMSE Rata-rata` == min(`RMSE Rata-rata`))

ggplot(tab_plot, aes(Metode, `RMSE Rata-rata`, fill = terbaik)) +
  geom_col(width = 0.7, show.legend = FALSE) +
  geom_errorbar(aes(ymin = `RMSE Rata-rata` - `Simpangan Baku`,
                    ymax = `RMSE Rata-rata` + `Simpangan Baku`), width = 0.2) +
  geom_text(aes(label = round(`RMSE Rata-rata`, 1)), vjust = -2.2, size = 3.5) +
  scale_fill_manual(values = c("grey70", "#d1495b")) +
  coord_flip() +
  labs(title = "Perbandingan RMSE antar Metode Interpolasi",
       subtitle = "Rata-rata 5-fold CV dengan 10 ulangan (garis = ± 1 simpangan baku)",
       x = NULL, y = "RMSE (ppm)") +
  theme_minimal()

cv_res %>%
  filter(rep == 1) %>%
  pivot_longer(all_of(setdiff(metode, "Null")), names_to = "Metode", values_to = "pred") %>%
  mutate(Metode = factor(Metode, levels = setdiff(metode, "Null"))) %>%
  ggplot(aes(obs, pred)) +
  geom_point(alpha = 0.6, color = "#3b6ea5") +
  geom_abline(color = "red", linetype = 2) +
  facet_wrap(~Metode) +
  labs(title = "Nilai Aktual vs Prediksi (CV, ulangan ke-1)",
       x = "Aktual (ppm)", y = "Prediksi (ppm)") +
  theme_minimal()

Peta Hasil Interpolasi

urut <- c("Closest Observation", "NN", "IDW", "Kriging",
          "Ensemble Rata-rata", "Ensemble Terboboti")

grid_pred %>%
  pivot_longer(all_of(urut), names_to = "Metode", values_to = "Zinc") %>%
  mutate(Metode = factor(Metode, levels = urut)) %>%
  ggplot(aes(x, y, fill = Zinc)) +
  geom_raster() +
  geom_point(data = meuse, aes(x, y), inherit.aes = FALSE, size = 0.4, alpha = 0.5) +
  scale_fill_viridis_c(option = "magma", direction = -1, name = "Zinc (ppm)") +
  coord_equal() +
  facet_wrap(~Metode, ncol = 3) +
  labs(title = "Prediksi Kadar Zinc pada Grid 40 m", x = NULL, y = NULL) +
  theme_minimal() +
  theme(axis.text = element_blank(), panel.grid = element_blank())

Peta Ketidakpastian Kriging

Kelebihan Kriging dibanding metode lain adalah menghasilkan ragam prediksi (ukuran ketidakpastian).

kr <- krige(vble ~ 1, meuse_sf, grid_sf, model = v_fit, debug.level = 0)
grid_pred$Kriging_sd <- sqrt(kr$var1.var)

ggplot(grid_pred, aes(x, y, fill = Kriging_sd)) +
  geom_raster() +
  geom_point(data = meuse, aes(x, y), inherit.aes = FALSE, size = 0.5, color = "white") +
  scale_fill_viridis_c(option = "viridis", name = "Simpangan\nbaku") +
  coord_equal() +
  labs(title = "Simpangan Baku Prediksi Ordinary Kriging",
       subtitle = "Semakin jauh dari titik pengamatan, semakin tidak pasti",
       x = NULL, y = NULL) +
  theme_minimal() +
  theme(axis.text = element_blank(), panel.grid = element_blank())

Interpretasi

Parameter. Dari leave-one-out CV, NN terbaik dengan k = 3 tetangga dan IDW terbaik dengan pangkat p = 3. Nilai p yang cukup besar menandakan bahwa pengaruh titik yang jauh perlu diredam. Semivariogram terbaik berupa model Spherical dengan range ± 831 m: pasangan titik yang berjarak kurang dari ± 831 m masih berkorelasi spasial, dan di atas jarak itu nilainya praktis saling bebas. Nilai nugget (± 24.8 ribu) jauh lebih kecil daripada partial sill (± 134.7 ribu), sehingga sebagian besar ragam zinc bersifat terstruktur secara spasial dan interpolasi memang layak dilakukan.

Perbandingan RMSE. Semua metode lebih baik daripada model Null (RMSE ± 368.5 ppm), artinya posisi spasial memang informatif. Urutan dari terbaik:

Kriging (± 234.7 ppm), metode terbaik. RMSE-nya turun ± 36% dari model Null dan stabil antar ulangan CV (simpangan baku kecil). Kriging unggul karena bobotnya dibentuk dari struktur autokorelasi data (semivariogram), bukan sekadar jarak.
Ensemble (± 245.5 ppm terboboti; ± 246.2 ppm rata-rata). Lebih baik daripada NN dan IDW, tetapi tidak mengalahkan Kriging. Ini wajar: ensemble hanya unggul bila komponennya sama-sama baik dan galatnya saling meniadakan, sedangkan di sini Kriging jauh lebih baik sehingga mencampurnya dengan NN dan IDW justru menarik hasilnya naik. Pembobotan 1/RMSE hanya memperbaiki sedikit dibanding rata-rata biasa karena RMSE ketiga komponen tidak berbeda jauh.
NN (± 259.4 ppm) dan IDW (± 264.0 ppm), setara. Keduanya hanya memakai jarak, tanpa mempertimbangkan pola korelasi spasial.
Closest Observation (± 318.3 ppm), terburuk di antara metode interpolasi. Hanya memakai satu titik terdekat sehingga sangat sensitif terhadap pencilan dan menghasilkan peta berpetak-petak (blok Thiessen) tanpa transisi yang halus.

Peta. Semua metode menunjukkan pola yang sama: kadar zinc tinggi mengelompok di sepanjang aliran sungai (area dekat sungai kemungkinan terdampak luapan) dan menurun ke arah yang lebih jauh. Closest Observation tampak paling kasar, IDW menghasilkan pola “mata sapi” di sekitar titik pengamatan, sedangkan Kriging dan Ensemble paling halus. Peta simpangan baku Kriging memperlihatkan ketidakpastian meningkat di area yang jauh dari titik pengamatan.

Kesimpulan. Ordinary Kriging adalah metode terbaik untuk menginterpolasi kadar zinc pada data meuse karena menghasilkan RMSE terendah. Catatan: parameter k dan p dipilih memakai data penuh dan bobot ensemble dihitung dari hasil CV yang sama, sehingga RMSE NN, IDW, dan Ensemble bisa sedikit terlalu optimis. Keunggulan Kriging tetap bertahan meskipun begitu.