Mapping — SQalytics

1 Introduksi

1.1 Latar Belakang

Spatial mapping atau geostatistical interpolation mengkonversi pengukuran titik diskrit (mis. kandungan logam berat di 30 lokasi tambak) menjadi continuous map atas area target — esensial untuk environmental monitoring, agricultural precision farming, food safety risk mapping (Krige, 1951; Cressie, 1990). Metode klasik kriging (Krige, 1951; Matheron, 1963) memberikan prediksi unbiased + estimated uncertainty berbasis spatial autocorrelation (semivariogram).

Aplikasi sains pangan: mapping kontaminasi mikotoksin di petak gudang, mapping pesticide residue di kebun, mapping kualitas tanah untuk crop precision, atau distribution heating zone dalam oven besar (thermal mapping for sterilization validation).

1.2 Tujuan Modul

Modul Mapping di SQalytics ditujukan untuk:

Menerima point measurement data: $(x, y)$ koordinat + value.
Fit semivariogram untuk spatial autocorrelation.
Menjalankan Ordinary Kriging untuk interpolasi.
Memvisualisasikan continuous map + uncertainty (kriging variance) map.
Audiens: peneliti environmental sains, precision agriculture, food safety risk mapping.

1.3 Posisi di Antara Alternatif

Pilih Mapping untuk continuous spatial interpolation. Untuk scatter plot saja, pakai Graph Studio. Untuk regression multivariate non-spatial, pakai Regression Studio. Untuk PCA spatial pattern, pakai PCA Explorer (akan datang).

2 Metode

2.1 Dasar Teoretis

Semivariogram $\gamma(h)$ — measure spatial dissimilarity:

$$\gamma(h) = \frac{1}{2 |N(h)|} \sum_{(i,j) \in N(h)} (z_i - z_j)^2$$

dengan $h$ jarak (lag), $N(h)$ pasangan poin dengan jarak ≈ $h$. Plot $\gamma(h)$ vs $h$ memberikan experimental variogram yang di-fit ke theoretical model: spherical, exponential, Gaussian, dst.

Spherical model (paling umum):

$$\gamma(h) = c_0 + c \left[\frac{3h}{2a} - \frac{1}{2}\left(\frac{h}{a}\right)^3\right] \quad \text{for } 0 \leq h \leq a$$

dengan nugget $c_0$ (measurement error), sill $c_0 + c$ (total variance), range $a$ (jarak di mana autocorrelation $\to 0$).

Ordinary Kriging predictor untuk lokasi $x_0$:

$$\boxed{\, \hat{z}(x_0) = \sum_{i=1}^{n} \lambda_i z(x_i) \,}$$

dengan weights $\lambda_i$ dari solving kriging system:

$$\begin{pmatrix} \Gamma & \mathbf{1} \\ \mathbf{1}^T & 0 \end{pmatrix} \begin{pmatrix} \lambda \\ \mu \end{pmatrix} = \begin{pmatrix} \gamma_0 \\ 1 \end{pmatrix}$$

dengan $\Gamma$ matrix semivariogram antar sample points, $\gamma_0$ vector semivariogram dari $x_0$ ke sample points, $\mu$ Lagrange multiplier.

Kriging variance $\sigma_K^2$ — uncertainty prediksi:

$$\sigma_K^2(x_0) = \sum_i \lambda_i \gamma(x_0, x_i) + \mu$$

Area dengan kriging variance tinggi = prediction unreliable → tambah sampling di sana.

2.2 Persamaan Inti

Experimental variogram: $\gamma(h) = \frac{1}{2 |N(h)|} \sum (z_i - z_j)^2$
Spherical model: spherical parameters $c_0$ (nugget), $c$, $a$ (range).
Kriging predictor: $\hat{z}(x_0) = \sum \lambda_i z_i$
Kriging variance: $\sigma_K^2$ provides uncertainty per point.

2.3 Asumsi & Batas Validitas

Asumsi	Konsekuensi jika dilanggar	Cara cek di SQalytics
Stationarity (mean konstan spatial)	Bias prediksi	Cek trend dengan moving window
Isotropy (autocorrelation sama segala arah)	Anisotropic — pakai directional variogram	Cek 4-direction variogram
$\geq 30$ sample points untuk variogram fit	Variogram noisy	Modul flag
Spatial autocorrelation ada (nugget < sill)	Kriging = average	Cek variogram shape
Tidak ada outlier ekstrem	Variogram distorted	Pre-screen

3 Cara Kerja

3.1 Step-by-Step di SQalytics

Buka Mapping dari domain Mutu dan Analisis Lanjutan.
Muat tabel: Sample, X (longitude or coord), Y (latitude or coord), Value.
(Opsional) Set boundary polygon atau pakai bounding box auto.
Klik Compute Variogram.
Pilih variogram model: spherical (default), exponential, Gaussian.
Atur grid resolution untuk interpolated map (default 50 × 50).
Klik Run Kriging Interpolation.
Tinjau hasil:
- Tab Variogram — experimental + fitted curve.
- Tab Interpolated Map — colormap value over area.
- Tab Uncertainty Map — kriging variance.
- Tab Cross-validation — leave-one-out RMSE.

3.2 Template Tabel Input + Contoh Data Sintetis

Kolom	Tipe	Wajib	Catatan
`Sample`	category	✓	ID titik
`X_m`	numeric	✓	Koordinat X (m atau lon)
`Y_m`	numeric	✓	Koordinat Y
`Value`	numeric	✓	Pengukuran (mis. Pb mg/kg)

Contoh data sintetis (kontaminasi Pb di petak tambak udang, 30 titik):

Sample	X_m	Y_m	Pb_mg_per_kg
P01	50	80	0.45
P02	120	100	0.62
P03	200	150	0.85
P04	280	180	1.20 (FDA limit 0.5)
… (30 titik total)

SYNTHETIC Data sintetis kontaminasi Pb di petak tambak udang — 30 titik pengukuran dengan koordinat X/Y dan nilai Pb (mg/kg). CSV setara: docs/assets/example-data/id/quality-advanced/template_quality_mapping_kriging.csv.

3.3 Contoh Luaran

Fitted spherical variogram:

Parameter	Value
Nugget $c_0$	0.02 (mg/kg)²
Sill $c_0 + c$	0.35
Range $a$	150 m
Model	Spherical

Interpolated map summary:

Region	Mean Pb	Max Pb	% area > FDA limit (0.5 mg/kg)
Northwest	0.42	0.85	28%
Northeast	1.05	1.42	78% ✗
Southwest	0.38	0.72	22%
Southeast	0.85	1.20	65%

Cross-validation (leave-one-out):

Metric	Value
RMSE	0.12 mg/kg
Mean error	0.01
Correlation observed vs predicted	0.92

Mapping — figure 01 — Gambar 1. *Panel* (a) Semivariogram eksperimental + fitted spherical model (nugget 0.02, sill 0.35, range 150 m); *panel* (b) kriging interpolation *heatmap* Pb contamination (mg/kg) — Northeast hotspot terhighlight.

Kesimpulan ringkas: "Kriging map mengungkap hotspot Pb di Northeast quadrant (mean 1.05 mg/kg, 78% area > FDA limit 0.5). RMSE cross-validation 0.12 mg/kg = excellent prediction accuracy. Spatial autocorrelation range 150 m — pencemaran cluster jarak menengah, kemungkinan upstream contamination source. Recommendation: (i) trace upstream Northeast pollution source, (ii) tambah sampling di high-uncertainty area, (iii) untuk udang harvest, avoid Northeast quadrant atau lakukan filtration. Lanjut ke Compare Many Groups untuk uji statistik between quadrants atau ke SPC Xbar / R Charts untuk longitudinal monitoring."

4 Kesimpulan

4.1 Relevansi Real-World

Food safety risk mapping — mikotoksin, residue pestisida.
Precision agriculture — soil nutrient mapping untuk variable rate fertilization.
Thermal validation sterilization vessel — temperature mapping.
Air quality — particulate spatial.
Environmental remediation — contamination map.

4.2 Where to Go from Here

Compare Many Groups — region-by-region ANOVA.
SPC Xbar / R Charts — temporal monitoring at hotspot.
Cressie (1993) untuk advanced spatial statistics.

⚙ Troubleshooting Cepat

Variogram noisy. Tambah sample points; bin lag distance lebih besar.

Nugget = sill (pure random). Spatial autocorrelation tidak ada — kriging ≈ ordinary mean.

Anisotropic pattern. Pakai directional variogram + anisotropic kriging.

i Riwayat Revisi

Tanggal	Revisi	Penulis
2026-05-12	Draft v2 publikasi (KaTeX variogram + Ordinary Kriging + APA Krige/Matheron/Cressie)	Claude
2026-05-12	Konversi MD → HTML (W5 quality-advanced batch)	Claude

4 Referensi

Krige, D. G. (1951). A statistical approach to some basic mine valuation problems on the Witwatersrand. Journal of the Chemical, Metallurgical and Mining Society of South Africa, 52(6), 119–139.
Matheron, G. (1963). Principles of geostatistics. Economic Geology, 58(8), 1246–1266. https://doi.org/10.2113/gsecongeo.58.8.1246
Cressie, N. A. C. (1993). Statistics for spatial data (Rev. ed.). John Wiley & Sons. https://doi.org/10.1002/9781119115151
Goovaerts, P. (1997). Geostatistics for natural resources evaluation. Oxford University Press.
Isaaks, E. H., & Srivastava, R. M. (1989). An introduction to applied geostatistics. Oxford University Press.