Feature-free regression kriging
Peng Luoa, Yilong Wub, Yongze Songb
aSenseable City Lab, Massachusetts Institute of Technology, Cambridge, USA
bSchool of Design and the Built Environment, Curtin University, Perth, Australia
Abstract
Spatial interpolation is a crucial task in geography. As perhaps the most widely used
interpolation methods, geostatistical models—such as Ordinary Kriging (OK)—assume
spatial stationarity, which makes it difficult to capture the nonstationary characteristics
of geographic variables. A common solution is trend surface modeling (e.g., Regression
Kriging, RK), which relies on external explanatory variables to model the trend and
then applies geostatistical interpolation to the residuals. However, this approach re-
quires high-quality and readily available explanatory variables, which are often lacking
in many spatial interpolation scenarios—such as estimating heavy metal concentra-
tions underground. This study proposes a Feature-Free Regression Kriging (FFRK)
method, which automatically extracts geospatial features—including local dependence,
local heterogeneity, and geosimilarity—to construct a regression-based trend surface
without requiring external explanatory variables. We conducted experiments on the
spatial distribution prediction of three heavy metals in a mining area in Australia. In
comparison with 17 classical interpolation methods, the results indicate that FFRK,
which does not incorporate any explanatory variables and relies solely on extracted
geospatial features, consistently outperforms both conventional Kriging techniques and
machine learning models that depend on explanatory variables. This approach effec-
tively addresses spatial nonstationarity while reducing the cost of acquiring explana-
tory variables, improving both prediction accuracy and generalization ability. This
finding suggests that an accurate characterization of geospatial features based on do-
main knowledge can significantly enhance spatial prediction performance—potentially
yielding greater improvements than merely adopting more advanced statistical models.
∗Corresponding author: Yongze Song, yongze.song@curtin.edu.au
Email addresses: pengluo@mit.edu (Peng Luo), 22074689@student.curtin.edu.au (Yilong
Wu)
Preprint submitted to Elsevier
July 11, 2025
arXiv:2507.07382v1  [physics.soc-ph]  10 Jul 2025

Keywords:
Spatial interpolation, regression kriging, spatial statistics
1. Introduction
Spatial interpolation is one of the essential tasks in geography (Lam, 1983; Good-
child, 2004). Based on samples collected from the Earth’s surface (e.g., land and ocean),
a continuous surface is estimated by appropriately modeling the relationships between
the samples (Webster and Oliver, 2007). With the new Earth observation technolo-
gies, such as remote sensing, have emerged in recent years, the availability of many
geographical variables has increased dramatically (Campbell and Wynne, 2011). How-
ever, there are still scenarios where the distribution of variables can only be obtained
through sampling, such as the soil organic matter (Cheng et al., 2024) or the elemental
content under the ocean (Luo et al., 2023). Recently, the potential of spatial interpo-
lation method in deep space exploration, such as Mars mineral prediction, has been
recognized (Jiao et al., 2025). In such cases, spatial interpolation remains a necessary
step.
From the perspective of modern spatial statistics, the earliest interpolation methods
are considered to belong to the category of deterministic interpolation, such as Inverse
Distance Weighting (IDW) (Panigrahi, 2021). They based on an assumption that the
value at a certain location can be obtained by inversely weighting the distances from
surrounding observed points (Tomczak, 1998). However, this approach neglects spatial
dependencies and fails to adequately consider the overall spatial distribution character-
istics and patterns of observed points, limiting its interpolation accuracy. Geostatistical
models were developed with a thorough consideration of spatial dependence (Matheron,
1963). For example, the original geostatistical method was Ordinary Kriging (Cressie,
1988). They interpolate based on the spatial variation patterns in the data, resulting
in more accurate and unbiased estimation results. This approach has found extensive
applications across various fields — from geology and mining, where it is used to pre-
dict unknown geological features or the distribution of mineral deposits, to agriculture,
environmental science, and ecology. The basic assumption of geostatistical models is
second-order stationarity, which assumes that the variance between two samples de-
pends only on their distance and direction, regardless of their absolute locations (Clark
2

et al., 1979).
However, the distribution of geographic variables can exhibit non-stationary char-
acteristics (Cantrell and Cosner, 1991).
For example, when a study area contains
multiple distinct terrain structures, the distribution patterns of geographic variables
often vary (Anselin, 2013).
This phenomenon contradicts the assumptions of ordi-
nary kriging, highlighting the need for interpolation methods specifically designed for
non-stationary features.
Many methods have been developed to enhance the Kriging model to account for
second-order spatial non-stationarity and improve interpolation accuracy. The first
and most straightforward solution for handling spatial second-order non-stationarity is
to divide the non-stationary surface into several homogeneous subregions (Luo et al.,
2023). Within each subregion, spatial stationarity holds, allowing for separate applica-
tion of Ordinary Kriging. A representative method following this approach is Stratified
Kriging (StK). However, modeling at partition boundaries can introduce discontinuities
in the prediction results. To do the stratified modelling while maintain the smooth of
the prediction result over space, some methods have been proposed to comprehensively
model spatial relationships, enabling the construction of semivariogram functions for
different strata. Notable examples include P-MSN (Gao et al., 2020) and the Gen-
eralized Heterogeneous Model (GHM) (Luo et al., 2023). P-MSN and GHM provide
effective stratified modeling solutions with improved spatial continuity and accuracy
compared to ordinary StK. However, they still suffer from the inherent drawbacks of
stratified modeling, namely, the effectiveness of this approach heavily depends on the
reliability of spatial partitioning algorithms. Furthermore, partition-based modeling
reduces the number of available samples within each subregion, which makes it chal-
lenging to fit reliable semivariogram functions (Liu et al., 2021).
The second solution is to model spatial non-stationary patterns by decomposing
the predicted values of geographic variables into two components: a trend surface and
spatial variability (Hengl et al., 2007). The trend surface is assumed to be fitted by a
deterministic function, while the spatial variability represents a random process that
geostatistical methods can predict. Different methods are developed to model the trend
(Hengl et al., 2007). When non-stationarity varies by region, location information can
3

be used to model the spatial trend, leading to the development of Universal Kriging
(Stein and Corsten, 1991). In this approach, a deterministic polynomial function is
used to represent the trend surface, while the residual spatial variability is captured
through a stochastic component. Regression Kriging (RK) generalizes this idea by
explicitly modeling the trend using a regression model, which can include multiple
explanatory variables (Hengl et al., 2007). RK first fits a regression model to predict
the trend and then applies Kriging to the residuals (Hengl et al., 2004). Because it
integrates diverse information sources and allows for a flexible choice of trend models,
regression kriging often achieves the highest predictive accuracy among Kriging-based
interpolation methods.
However, incorporating explanatory variables significantly increases the cost of spa-
tial interpolation and introduces more uncertainty. Additionally, for many geostatisti-
cal tasks, obtaining high quality explanatory variables remains a challenge (Yao et al.,
2023). For example, a classic kriging interpolation task involves predicting mineral
distribution. Despite the availability of numerous satellite sensors for obtaining remote
sensing data, collecting explanatory variables for subsurface prediction is difficult. This
presents a challenge for current spatial interpolation methods, including geostatistical
models.
In this study, we are committed to developing a regression kriging method that do
not require explanatory variables and can address spatial non-stationarity. In cases
where explanatory variables cannot be obtained, we can utilize the spatial patterns of
geographic variables to predict the trend surface. Specifically, we propose a feature
free regression kriging model. For each observation point, a series of features can be
extracted from the values and distance relationships of the remaining points in its
neighboring region. These features are then used to construct a regression model for
fitting the trend surface. Finally, the residuals are employed to construct Ordinary
Kriging for spatial interpolation. To validate the performance of FFRK, we conducted
a case study in a selected region of Australia, focusing on the prediction of three heavy
metal concentrations. Furthermore, FFRK was compared against 17 classical spatial
interpolation models.
4

2. Feature-free regression kriging
2.1. Basic of Regression Kriging
Regression Kriging is a hybrid spatial prediction method that combines a determin-
istic regression model with a geostatistical interpolation model. The fundamental idea
of RK is to separately model the trend component and spatially correlated residuals,
allowing for a more flexible and accurate spatial prediction.
2.1.1. General Formulation of Regression Kriging
Given an observed spatial variable Z(x) at location x, RK decomposes the spatial
variation into two components:
Z(x) = m(x) + ε(x)
(1)
where m(x) represents the deterministic trend component, which captures large-scale
variations and can be modeled using regression techniques, ε(x) represents the spatially
correlated residuals, which account for small-scale variations and are modeled using
geostatistical interpolation.
2.1.2. Trend Modeling via Regression
The trend component m(x) is typically modeled using a regression function:
m(x) = g(X(x))
(2)
where g(·) is a regression model, which can be a linear regression, random forest, or
any machine learning algorithm. X(x) is a vector of explanatory variables at location
x, which can include environmental, demographic, or geospatial factors.
The regression model is trained using known data points {xi, Z(xi)}n
i=1.
Once
trained, this regression model can predict the trend component at any location xp:
m(xp) = g(X(xp))
(3)
where m(xp) is the predicted trend at the location xp.
5

2.1.3. Residual Modeling via Kriging
After estimating the trend component, the residuals at known sample locations xi
are computed as:
ε(xi) = Z(xi) −m(xi)
(4)
where the ε(xi) is the residuals at the location xi.
These residuals are spatially correlated and modeled using geostatistical techniques.
Using the fitted semivariogram γ(h), residuals at unknown locations xp are interpolated
using ordinary kriging:
ε∗(xp) =
n
X
i=1
wiε(xi)
(5)
where wi are kriging weights derived from the semivariogram model.
2.1.4. RK Prediction
The predicted value at an unknown location xp is obtained by combining the
regression-predicted trend and the kriging-interpolated residual:
Z∗(xp) = m(xp) + ε∗(xp)
(6)
where Z∗(xp) is the predicted value at location xp, and
2.2. Concept of FFRK
The most important feature of RK is its ability to fully utilize explanatory variables
to fit the trend, thereby improving the accuracy of ordinary kriging.
However, its
limitation is also evident: if explanatory variables are difficult to obtain, the method
becomes unusable.
As is shown in Figure 1, the idea behind our proposed FFRK is straightforward:
in the absence of explanatory variables, we replace them with the spatial distribution
characteristics of the predicted variable itself to perform regression kriging. Therefore,
FFRK follows the same algorithmic foundation as RK, with the only difference being in
trend fitting—rather than constructing a regression model using explanatory variables,
it is built based on geospatial features.
6

Z(x) = m(x) + ε(x)
(7)
m(x) = g(F(x))
(8)
where: g(·) is a predictive model, F(x) is the extracted geo-feature vector.
Subsequently, the process of FFRK is entirely consistent with that of RK: the trend
surface m(x) is fitted based on geospatial features, followed by residual computation
and kriging interpolation of the residuals.
Figure 1: Schematic overview of the developed feature-free regression kriging (FFRK) for spatial
interpolation, and its different wit the Regression Kriging
2.3. Geofeature extraction
In spatial analysis, the interpolation accuracy highly depends on the feature rep-
resentations of the data. the framework of FFRK can be combined with any kinds of
geofeature, according to the specific application.
In this study, as a showcase of the FFRK framework, we consider three distinct
types of geospatial features to capture both local pattern and global pattern in spatial
distributions (Figure 2). These features are designed to provide a more comprehensive
7

representation of spatial processes and to enhance interpolation performance. The first
is the local trend feature, extracted using IDW, which describes the general directional
pattern of spatial structure in the neighborhood of each sample. This feature reflects
the core principle of Tobler’s First Law of Geography—spatial proximity—and helps
characterize the local continuity of spatial variables. The second is the local hetero-
geneity feature, which captures intra-regional variability and unevenness by computing
statistical descriptors (e.g., quantiles) of neighboring observations. This feature high-
lights the spatial heterogeneity inherent in many geospatial phenomena. The third is
the geosimilarity feature, which measures statistical similarity between sample points
based on the local distributions used in the second feature. Unlike traditional distance-
based methods, this similarity reflects environmental resemblance between distant loca-
tions, allowing the model to account for non-local spatial dependencies and enhancing
its ability to identify globally consistent patterns.
Figure 2: Three types of geofeatures for FFRK-based spatial interpolation used in this study, including
local trend, local heterogeneity, and geosimilarity
8

Each prediction location xp is associated with a feature vector:
F(xp) = [FIDW(xp), FSVD(xp), FGOS(xp)]
(9)
where the three feature sets are computed as follows.
2.3.1. Local Dependence: IDW Feature Extraction
The IDW method estimates the value of a target point xp based on the weighted
average of surrounding observations xi, where the weight is inversely proportional to
the distance:
Z∗(xp) =
Pk
i=1 wiZ(xi)
Pk
i=1 wi
(10)
where the weight wi is defined as:
wi =
1
d(xp, xi)p
(11)
where d(xp, xi) is the Euclidean distance between the target point xp and an observation
xi.
IDW assumes a spatially homogeneous distribution without explicit trend modeling.
However, it may introduce bias when a strong spatial trend exists. The extracted IDW-
based feature for each point is:
FIDW(xp) = [fIDW(xp)]
(12)
2.3.2. Local Heterogeneity: Spatially varying distribution
We refer to the quantile-based characterization of local neighborhoods as the Spa-
tially Varying Distribution (SVD) feature, which captures the local statistical structure
of spatial data. The extracted feature vector FSVD(xp) provides a structured represen-
tation of local heterogeneity. By incorporating SVD into spatial modeling, we can
effectively account for local non-stationarity and spatial variability, making interpola-
tion more adaptive to complex spatial structures.
First, given a target location xp, we define its neighborhood Nk(xp) as the set of its
k nearest observed points:
Nk(xp) = {xi | d(xi, xp) is among the k smallest distances}
(13)
9

where the distance d(xi, xp) is computed using the Euclidean metric.
Second, for each point xp, we compute the empirical quantiles of the variable Z(x)
within its neighborhood Nk(xp):
Qq(xp) = Quantile ({Z(xi) | xi ∈Nk(xp)}, q)
(14)
where Qq(xp) represents the q-th quantile of the observed values in the neighborhood.
Third, the final SVD feature vector for location xp consists of multiple quantile
values across different probability levels:
FSVD(xp) = [Qq1(xp), Qq2(xp), ..., Qqd(xp)]
(15)
where {q1, q2, ..., qd} represents the predefined quantile levels (e.g., q ∈{0, 0.05, 0.10, ..., 1.0}).
2.3.3. Geosimilarity: GOS Feature Extraction
The Geographically Optimal Similarity (GOS) method is selected to being as the
third geofeature to estimate the target variable at unknown locations (Song, 2023).
GOS identifies spatial configurations with similar structures to make predictions.
First, we define the spatial configuration of each observed location based on ex-
tracted features.
For each observed location xi, we define its spatial configuration
based on a set of extracted features FSVD(xp).
Second, the similarity between an unknown location xp and an observed location
xi is defined as:
S(xi, xp) = P {Ej(fSVD,j(xi), fSVD,j(xp))}
(16)
where Ej is the similarity function between the j-th feature of the observed and un-
known locations, P is an aggregation function to determine overall similarity.
For continuous spatial features, we define Ej as:
Ej(xi, xp) = exp

−(FSVD,j(xi) −FSVD,j(xp))2
2σ2
j

(17)
where σj represents the standard deviation of feature j.
10

Third, instead of using all observations for prediction, GOS selects only the most
similar locations. The optimal similarity threshold Sλ is determined by minimizing
prediction errors:
λ = arg min
κ RMSE(κ)
(18)
where κ is the proportion of the most similar samples used for prediction, and RMSE
is the root mean square error from cross-validation.
Fourth, the prediction at xp is computed as:
FGOS(xp) =
P
i∈Nλ Sλ(xi, xp)Z(xi)
P
i∈Nλ Sλ(xi, xp)
(19)
where Nλ is the set of selected observations with similarity above Sλ.
2.4. Regression kriging with geofeatures
Combining all three components, we define the final spatial feature vector:
F(xp) = [FIDW(xp), FSVD(xp), FGOS(xp)]
(20)
where: FIDW(xp) captures local dependence. FSVD(xp) captures local heterogeneity
using statistical quantiles.
FGOS(xp) captures global similarity based on regression
over spatially similar regions.
The final feature vector has a total of:
dim(F) = 1 + dSVD + 1
(21)
where: 1 represents the IDW feature. dSVD represents the number of quantile-based
SVD features (which can be adjusted based on resolution).
1 represents the GOS
similarity feature.
A machine learning regression model g(F) is trained on known observations, which
captures the large-scale trend:
Ztrend(x) = g(F(x))
(22)
where Ztrend(x) is the trend value at the location x.
11

The final step is to interpolate the residuals and combine them with the predicted
trend using ordinary kriging.
3. Case study: mapping trace elements with FFRK
3.1. Study area and data
In this work, we use trace element data of Cu, Pb, and Zn, from one region of Aus-
tralia to test the performance of FFRK. We concentrated on the geographic variability
of the three elements for the trace element distribution because these elements are well
known to be important markers of environmental contamination and are essential for
evaluating ecological health. The spatial distributions of trace elements are shown in
Figure 3.
Study_Area
75.01 - 353.50
41.01 - 75.00
26.01 - 41.00
16.01- 26.00
2.00 - 16.00
Zn(ppm)
Study_Area
24.01 - 75.00
17.01 - 24.00
12.01 - 17.00
6.13 - 12.00
0.70 - 6.12
Pb(ppm)
Study_Area
54.01 - 8600.00
24.01 - 54.00
13.01 - 24.00
8.01 - 13.00
0.60 - 8.00
Cu(ppm)
0
25
12.5
km
Cu
Zn
Pb
Figure 3: Spatial distributions of trace elements: Cu, Zn, and Pb
We included nine environmental explanatory variables in alphabetical order to ex-
amine the factors influencing their spatial patterns: slope, road network density (Road),
normalized difference vegetation index (NDVI), distance to major roads (MainRd), dis-
tance to lithology (Dlith), distance to fault lines (Dfault), soil organic carbon (SOC),
soil pH levels, and water distribution. These variables were selected for their relevance
in characterizing and predicting the distribution of trace elements. The calculation of
the variables, such as Dlith, Dfault, is referenced in (Song, 2023). Table 1 summa-
rizes the descriptive statistics of all explanatory variables, including the mean (Mean),
minimum (Min), median, maximum (Max), standard deviation (SD), and coefficient
of variation (CV).
12

Table 1: Descriptive statistics of environmental variables
Variable
Code
Mean
Min
Median
Max
SD
CV
Distance to water(km)
Water
1.114
0.000
0.648
7.696
1.188
1.067
Distance to mainroads(km)
MainRoads
19.991
0.010
17.485
58.371
14.219
0.711
Distance to roads(km)
Roads
9.817
0.002
7.953
50.141
8.363
0.852
Distance to mine(km)
MineKm
14.785
0.025
11.425
56.638
12.117
0.820
Slope(degree)
Slope
0.275
0.008
0.246
1.712
0.161
0.585
Normalized difference vegetation index
NDVI
0.178
0.062
0.180
0.251
0.024
0.135
Soil organic carbon
SOC
0.868
0.686
0.870
1.066
0.053
0.061
Soil pH
pH
5.741
5.113
5.753
6.178
0.178
0.031
Distance to lithology (km) for Cu
DlithCu
9.132
0.000
7.147
39.839
7.769
0.851
Distance to lithology (km) for Zn
DlithZn
8.150
0.000
6.229
39.839
7.637
0.937
Distance to lithology (km) for Pb
DlithPb
3.363
0.000
2.627
15.946
3.020
0.898
Distance to fault(km) for Cu
DfaultCu
16.070
0.001
13.285
54.765
12.150
0.756
Distance to fault(km) for Pb
DfaultPb
12.017
0.003
11.111
43.471
8.090
0.673
Distance to fault(km) for Zn
DfaultZn
14.174
0.001
11.844
52.697
10.851
0.766
Elevation(m)
Elevation
482.699
398.050
485.128
588.649
36.700
0.076
Aspect(degree)
Aspect
171.645
0.751
174.126
358.712
90.576
0.528
3.2. Experiment design
Figure 4 illustrates the experimental design of this study, which aims to validate
the reliability of the proposed FFRK method. For the target variable, we computed
its three-dimensional geofeatures, including dependence, heterogeneity, and similarity.
Subsequently, we selected four types of machine learning models—linear model (LM),
decision tree (DT), random forest (RF), and support vector machine (SVM)—to pre-
dict the trend of the target variable. After obtaining the residuals by comparing the
predicted trend with observations, we applied ordinary kriging for spatial interpolation
of the residuals. Finally, the predicted trend and interpolated residuals were combined
to produce the prediction.
We compare our proposed method with several representative baselines, including
classical geostatistical approaches, machine learning models, and their stratified and
hybrid variants. First, Ordinary Kriging (OK) serves as the foundational geostatistical
method and is used as a baseline for performance comparison. Second, we consider
machine learning models, which directly establish predictive relationships between ex-
planatory variables and the target variable. In this study, we employ four commonly
13

Figure 4: Experimental design for evaluating the performance of FFRK in comparison with other
interpolation models
used models: Linear Regression (LM), Decision Tree Regression (DT), Random Forest
Regression (RF), and Support Vector Machine Regression (SVM).
Third, stratified models are introduced to better account for spatial heterogeneity.
These models partition the study area into multiple subregions based on homogeneity
analysis, with separate machine learning models trained within each region. The same
four algorithms (LM, DT, RF, and SVM) are applied in each subregion. The spatial
partitioning is conducted using a decision tree, which segments the domain based on
geographic coordinates (longitude and latitude) as well as trace element values. In ad-
dition to stratified machine learning models, we also include stratified kriging, resulting
in a total of five stratified models.
Fourth, we evaluate Regression Kriging (RK), a hybrid method that combines trend
modeling and spatial interpolation. Specifically, a machine learning model (LM, DT,
RF, or SVM) is used to estimate the trend, and the residuals are then interpolated
using kriging. Finally, we present our proposed method, Feature-Filtered Regression
Kriging (FFRK). Similar to RK, FFRK follows a two-step structure but incorporates a
feature selection process prior to regression. The four variants of FFRK correspond to
the four regression models used: FFRK (LM), FFRK (DT), FFRK (RF), and FFRK
(SVM).
14

In total, we compared 18 models. Among them, only Ordinary Kriging (OK), Strat-
ified Kriging (StK) and four FFRK models do not require any explanatory variables,
whereas the other three model categories require nine explanatory variables as inputs.
In the FFRK model, two hyperparameters are involved.
The first is K, which
defines the number of neighboring sample points used to compute geo-features and fit
the semivariogram. The second is the quantile interval, used in the second geo-feature
to characterize the statistical information within the neighborhood of each sample
point. In this study, we set K = 15 and the quantile interval = 5%, following the
settings adopted in a previous study (Song, 2023). A sensitivity analysis of these two
hyperparameters and their impact on the performance of FFRK is presented in Section
3.4.2.
3.3. Cross-validation and Model Evaluation
In this study, we adopted 10-fold cross-validation to evaluate the prediction per-
formance and generalization capability of the proposed FFRK method and baseline
models. Specifically, the dataset was randomly divided into ten subsets of equal size.
In each iteration, nine subsets served as the training set, while the remaining subset
was reserved as the validation set. This training-validation procedure was repeated ten
times, with the averaged results across the ten iterations representing the final model
performance. This approach effectively mitigates the instability arising from a single
random partition and enhances the robustness of model evaluation.
In particular, for Stratified Models, we employed a regression tree to partition
the study area spatially, and subsequently performed stratified sampling based on the
distribution of the target variable. This ensured consistent internal data distribution
across each of the ten folds constructed for cross-validation.
Moreover, the cross-validation process was utilized for model parameter optimiza-
tion. Optimal parameters for each machine learning model were determined by min-
imizing the mean RMSE obtained from cross-validation. Finally, each model was re-
trained once using the complete dataset to derive globally optimal parameters, and the
resulting optimized models were preserved for subsequent global spatial predictions.
15

3.4. Results
3.4.1. Accuracy evaluation
Figure 5 presents the spatial interpolation results of FFRK alongside several other
regression-based models. We selected Linear Model (LM) as the representative machine
learning regression method, and also included Regression Kriging (RK) based on LM,
as well as Stratified Kriging (StK). Additionally, we compared the results of Ordinary
Kriging (OK) and StK. Our observations show that OK and StK produce the smoothest
interpolation surfaces among the methods evaluated. However, this smoothness comes
at the cost of lacking fine-grained spatial detail. In contrast, FFRK produces spatial
patterns that closely resemble those of LM and RK, capturing more nuanced local
variations.
Figure 6 presents the accuracy results of 13 global modeling approaches for predict-
ing the concentrations of three heavy metals, evaluated using R², RMSE, and MAE.
Among these methods, 12 involve the use of machine learning (ML) models, with the
exception of ordinary kriging (OK). To ensure a fair comparison, we categorized the
methods into four groups based on the ML model used for trend modeling: LM-based,
DT-based, RF-based, and SVM-based.
The results indicate that the FFRK method, despite not incorporating any explana-
tory variables, achieves significantly higher accuracy than both standalone ML models
and regression models based on ML modeling. The latter two approaches utilized nine
explanatory variables and a large dataset but failed to yield higher prediction accu-
racy. For example, in Cu prediction, the R² values of FFRK with LM-, DT-, RF-,
and SVM-based trend modeling exceeded those of regression kriging using the same
ML models by 73.72%, 40.31%, 26.87%, and 21.27%, respectively. The most signifi-
cant improvement was observed in Pb prediction, where FFRK with DT-based trend
modeling achieved an R² of 0.27, surpassing regression kriging and the DT model by
0.07 and 0.03, respectively.
In Table 2, we present the results of five stratified modeling approaches. First,
we partitioned the spatial domain into multiple homogeneous subregions based on the
values of explanatory variables.
Within each subregion, we decomposed and fitted
separate models to perform spatial predictions. Finally, we evaluated the overall pre-
16

Figure 5: Spatial distributions of the predicted results for three trace elements—Cu, Zn, and Pb—are
shown using six prediction models: FFRK, LM, RK, Stratified RK, OK, and STK
17

LM RK
(LM)
FFRK
(LM)
OK
DT RK
(DT)
FFRK
(DT)
RF RK
(RF)
FFRK
(RF)
SVM RK
(SVM)
FFRK
(SVM)
LM RK
(LM)
FFRK
(LM)
OK
DT RK
(DT)
FFRK
(DT)
RF RK
(RF)
FFRK
(RF)
SVM RK
(SVM)
FFRK
(SVM)
LM RK
(LM)
FFRK
(LM)
OK
DT RK
(DT)
FFRK
(DT)
RF RK
(RF)
FFRK
(RF)
SVM RK
(SVM)
FFRK
(SVM)
LM-based 
models
DT-based 
models
RF-based 
models
SVM-based 
models
LM-based 
models
DT-based 
models
RF-based 
models
SVM-based 
models
LM-based 
models
DT-based 
models
RF-based 
models
SVM-based 
models
0.4
0.3
0.2
0.1
R2
0.7
0.5
0.3
0.1
RMSE
0.5
0.3
0.1
MAE
0.7
0.4
0.1
MAE
1.0
0.8
0.6
0.4
0.2
RMSE
0.4
0.3
0.2
0.1
0.5
R2
0.4
0.3
0.2
0.1
R2
0.8
0.6
0.4
0.2
RMSE
0.5
0.6
0.4
0.3
0.2
0.1
MAE
Cu
Zn
Pb
Regression Kriging
(RK)
Feature-free Regression Kriging
(FFRK)
Machine learning regression
Figure 6: Evaluation results (R2, RMSE, and MAE) for the 13 global models
diction accuracy across the entire study area. To assess the impact of stratification,
we experimented with different numbers of partitions, ranging from 2 to 10, and cal-
culated the average accuracy. The results indicate that compared to global modeling
approaches, including OK and various machine learning models, stratified modeling
can effectively improve prediction accuracy, as discussed in the Introduction. How-
ever, its performance still falls short of that achieved by the FFRK method. In some
cases, stratification resulted in over-segmentation, where certain subregions contained
too few observations, leading to model underfitting and significantly reduced predic-
tive performance. Additionally, stratified modeling introduced abrupt and unrealistic
discontinuities in the final spatial interpolation results, which arose due to artificial
boundaries between partitions.
The scatter plots comparing the observed and predicted values for all models are
presented in the appendix (Figures S1, S2, and S3). It can be seen that the scatter
plots of the FFRK models, which exhibit the highest accuracy, are the closest to the
1:1 line.
18

Table 2: Mean R², MAE, and RMSE for Different Models and Elements (K=2 to 10)
Element
Model
Mean R²
Mean MAE
Mean RMSE
Cu
Stratified Kriging
0.3584
0.6720
0.8964
Stratified LM
0.3738
0.6699
0.8978
Stratified DT
0.3009
0.7114
0.9486
Stratified RF
0.4115
0.6406
0.8705
Stratified SVM
0.4205
0.6318
0.8637
Zn
Stratified Kriging
0.3791
0.4830
0.6287
Stratified LM
-6.4581
0.5559
1.5290
Stratified DT
0.3546
0.5054
0.6613
Stratified RF
0.4367
0.4575
0.6179
Stratified SVM
0.4147
0.4725
0.6298
Pb
Stratified Kriging
0.2747
0.5277
0.6958
Stratified LM
0.1043
0.5040
0.7462
Stratified DT
0.2178
0.5296
0.7041
Stratified RF
0.4063
0.4399
0.6137
Stratified SVM
0.3776
0.4474
0.6283
3.4.2. Sensitivity analysis for the hyperparameter in FFRK
In Kriging-based methods, an hyperparameter is the number of neighboring points
used when fitting the semivariogram and performing interpolation. In the previous
experiments, to ensure a fair comparison among different methods, we set the number
of nearest neighbors (k) to 15 for all models. In this section, we assess the sensitivity
of different methods to k. Figure 7 presents the accuracy of FFRK and RK models
under varying numbers of neighboring points.
The results indicate that for the same type of ML model used in trend modeling,
FFRK (dashed line) consistently outperforms RK (solid line) across most k values,
demonstrating higher R² and lower RMSE. However, there are a few exceptions. For
instance, when k is less than 11 in Zn prediction, the RF-based RK model achieves
higher accuracy than the RF-based FFRK model. Despite this, FFRK based on LM
and DT still significantly outperforms RK when k is below 11. This discrepancy may
19

be attributed to the distribution characteristics of Zn: when k is small, the geofeatures
used by FFRK may not provide sufficient information to properly fit the RF model.
In contrast, the RK method, which utilizes nine explanatory variables, has enough
information for RF to achieve better fitting. However, for the relatively simpler LM and
DT models, the geofeatures in FFRK provide sufficient information, enabling FFRK
to surpass RK in predictive performance.
6
8
10
12
14 16
18
20
6
8
10
12
14 16
18
20
6
8
10
12
14 16
18
20
6
8
10
12
14 16
18
20
6
8
10
12
14 16
18
20
6
8
10
12
14 16
18
20
0.50
0.40
0.30
0.25
0.45
0.35
0.80
0.85
0.90
0.95
0.60
0.62
0.64
0.66
0.68
0.30
0.35
0.40
0.45
R2
R2
R2
0.10
0.20
0.30
0.40
0.50
RMSE
RMSE
RMSE
0.55
0.60
0.65
0.70
0.75
LM
DT
RF
SVM
OK
RK
FFRK
Cu
Zn
Pb
Figure 7: The sensitivity analysis for the K value
In the second category of geofeatures, we sample the value distribution of surround-
ing sample points for each predicted location. This sampling is based on fixed quantile
intervals, such as 1% or 5%. In the previous experiments, we selected a 5% quantile
interval, resulting in 20 features in this geofeature category. To assess the sensitivity
of FFRK to this parameter, we varied the quantile interval from 5% to 50% and eval-
uated its impact on the results. Figure 8 shows that the FFRK method is relatively
robust to changes in the quantile sampling interval. The maximum fluctuation in R²
is approximately 0.05, while the highest variation in RMSE is only around 0.02.
20

LM
DT
RF
SVM
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0.5
R2
RMSE
0.80
0.84
0.88
0.92
0.35
0.40
0.45
0.50
0.62
0.64
0.66
RMSE
R2
0.36
0.40
0.44
0.60
0.64
RMSE
0.40
0.45
0.30
0.35
R2
Cu
Zn
Pb
FFRK
Figure 8: The sensitivity analysis for the quantile step
3.4.3. Generalized regression kriging
In previous experiments, we have demonstrated that FFRK, despite not requiring
any explanatory variables, can outperform RF models that utilize multiple explana-
tory variables solely through the extraction of geofeatures. Here, we propose a new
question: if geofeatures and explanatory variables are integrated together for trend
modeling, can the predictive performance be further improved? We define this ap-
proach, which incorporates both geofeatures and explanatory variables, as generalized
regression kriging (GRK).
The results, shown in Figure 9, indicate that across three different datasets and
12 prediction tasks (based on four ML models for trend modeling), GRK outperforms
FFRK in 10 cases and significantly surpasses RK in all cases. The two exceptions
occur in Cu and Zn prediction tasks when DT is used for trend modeling, where
FFRK, relying solely on geofeatures, achieves a slightly higher R² than GRK, which
includes explanatory variables.
These findings suggest that incorporating more information into trend modeling
enhances the performance of regression kriging models. The proposed GRK model is
21

particularly suitable for spatial interpolation tasks where explanatory variables are
available.
Compared to conventional regression kriging, which relies solely on ex-
planatory variables, integrating geofeatures into trend modeling leads to substantial
improvements in predictive accuracy.
LM
DT
RF
SVM
Trend modeling method
RK
FFRK
GRK
Framework
0.260
0.278
0.396
0.383
0.443
0.355
0.510
0.459
0.445
0.345
0.521
0.497
0.30
0.35
0.40
0.45
0.50
RK
FFRK
GRK
0.177
0.070
0.407
0.311
0.375
0.273
0.471
0.419
0.410
0.309
0.517
0.475
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
RK
FFRK
GRK
0.274
0.311
0.433
0.378
0.397
0.380
0.451
0.401
0.417
0.357
0.490
0.437
0.275
0.300
0.325
0.350
0.375
0.400
0.425
0.450
0.475
RK
FFRK
GRK
0.973
0.961
0.879
0.888
0.844
0.908
0.792
0.832
0.843
0.915
0.783
0.802
0.800
0.825
0.850
0.875
0.900
0.925
0.950
RK
FFRK
GRK
0.720
0.766
0.611
0.659
0.628
0.677
0.578
0.605
0.610
0.660
0.552
0.575
0.575
0.600
0.625
0.650
0.675
0.700
0.725
0.750
RK
FFRK
GRK
0.701
0.684
0.620
0.649
0.640
0.649
0.610
0.637
0.629
0.660
0.588
0.618
0.60
0.62
0.64
0.66
0.68
0.70
LM
DT
RF
SVM
LM
DT
RF
SVM
LM
DT
RF
SVM
LM
DT
RF
SVM
LM
DT
RF
SVM
Trend modeling method
Trend modeling method
Trend modeling method
Trend modeling method
Trend modeling method
Framework
Framework
Framework
Framework
Framework
Cu
Zn
Pb
RMSE
R2
R2
R2
RMSE
RMSE
Figure 9: Accuracy of three framework: regression kriging, FFRK, and Generalized Regression Kriging
(GRK)
4. Discussion
Regression Kriging is a highly effective spatial interpolation model for addressing
second-order spatial non-stationarity, which fits a trend surface based on explanatory
variables and then performs kriging on the residuals. However, in many spatial inter-
polation tasks, it is often difficult to obtain explanatory variables, or to find variables
that are sufficiently suitable, which greatly limits the applicability of regression kriging.
This work proposes a novel interpolation method for regression kriging under spatial
non-stationarity without relying on explanatory variables, called FFRK. FFRK substi-
tutes the traditional need for explanatory variables in trend surface fitting by creating
features—termed geofeatures—based on the spatial distribution patterns of geographic
variables. In this study, we introduce three categories of geofeatures, derived respec-
tively from spatial dependence, spatial heterogeneity, and geographic similarity. We
22

apply FFRK to a case study involving the spatial distribution of three heavy metal
concentrations in a region of Australia. We also compare the performance of FFRK
against 17 other interpolation methods of various types, demonstrating that FFRK
achieves the highest predictive accuracy.
FFRK achieves superior performance without using any explanatory variables, out-
performing other methods that do use explanatory variables, including machine learn-
ing models and regression kriging.
This showcases the advantage of using features
derived from the spatial distribution itself in regression kriging.
However, the ad-
vantage does not imply that such spatial distribution features can entirely replace
information from traditional explanatory variables. For the specific case study in this
paper—predicting heavy metal concentrations—it is particularly difficult to identify
suitable, large-scale, readily available explanatory variables. In most spatial prediction
tasks, large-scale remote sensing observations are typically relied upon. However, sub-
surface distributions are often poorly represented by surface-level features detectable
through remote sensing. This is precisely the application scenario that FFRK aims
to address. As shown in Figure 9, when both explanatory variables and geofeatures
are used together (a model we refer to as GRK), the prediction accuracy generally
improves further compared to FFRK alone.
5. Conclusion
This study proposes FFRK, a regression kriging approach that eliminates the need
for explanatory variables by extracting geospatial features from the response variable
itself. Addressing key limitations in existing methods—such as reliance on external
variables and assumptions of spatial stationarity—FFRK captures spatial dependence,
local heterogeneity, and geographic similarity to construct a robust trend surface. We
demonstrate the effectiveness of FFRK through a spatial prediction task involving the
distribution of three heavy metal concentrations in a region of Australia. FFRK outper-
forms traditional kriging, machine learning, and stratified methods, offering a practical
solution for large-scale, non-stationary spatial interpolation tasks where explanatory
data are scarce or unavailable. This work contributes to GIScience by demonstrat-
ing that spatial features alone can effectively support predictive modeling. However,
23

FFRK’s performance depends on the spatial density of observations. Future work will
explore its extension to spatiotemporal settings and automated feature learning for
broader applicability.
Disclosure Statement
No conflict of interest exists in this manuscript, and the manuscript was approved
by all authors for publication.
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Appendix
25

STK
OK
Stratified_LM
Stratified_DT
Stratified_RF
Stratified_SVM
LM
DT
RF
SVM
RK_LM
RK_DT
RK_RF
RK_SVM
FFRK_LM
FFRK_DT
FFRK_RF
FFRK_SVM
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
Observed
Predicted
# of data
0.1
0.2
0.3
0.4
0.5
Figure S1: Scatter plot of the prediction results for Cu
26

STK
OK
Stratified_LM
Stratified_DT
Stratified_RF
Stratified_SVM
LM
DT
RF
SVM
RK_LM
RK_DT
RK_RF
RK_SVM
FFRK_LM
FFRK_DT
FFRK_RF
FFRK_SVM
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
Observed
Predicted
# of data
0.25
0.50
0.75
Figure S2: Scatter plot of the prediction results for Zn
27

STK
OK
Stratified_LM
Stratified_DT
Stratified_RF
Stratified_SVM
LM
DT
RF
SVM
RK_LM
RK_DT
RK_RF
RK_SVM
FFRK_LM
FFRK_DT
FFRK_RF
FFRK_SVM
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
Observed
Predicted
# of data
0.2
0.4
0.6
0.8
Figure S3: Scatter plot of the prediction results for Pb
28
