Individual And Pairwise Representativeness Of Sampling Points In Interpolation Tasks Of Heavy Metals Distribution In The Topsoil

INTRODUCTION

The environmental soil monitoring methods often require preliminary data to be sufficient to represent soil–environment relationships throughout the study area (Zhu 2015). A limited quantity of soil sample data to represent the study area is still an issue to predict soil properties and estimate prediction uncertainty. A large number of publications are devoted to the issues of representative sampling of the components of the environment (Malof 2018; Liu 2022). The task of assessing representativeness and constructing a representative set arises when organizing sampling to assess the quality of environmental components, when statistically processing environmental monitoring data, and when choosing a training subset for artificial neural networks (ANNs) that model the spatial distribution of a feature (Nath 2018; Demyanov 2020; Mello 2022). The existing rules for choosing a training subset do not reflect the picture of pollution (Baglaeva 2020; Malof 2018). The formal structure of the training subset must be determined by the rules governing the origin and maintenance of ecological topologies in order to correctly interpret ecological patterns (Prager 2009).

Insufficient attention is paid to the interpretation of the results of a modeling. Often behind the scenes is the connection between the features of environmental data and landscapes (Boussange 2022). The key challenge is understanding how the connectivity and heterogeneity of the model results relate to environmental characteristics. To characterize environmental connectivity, these tasks are proposed to be solved by spatial graph theory methods (O’Brien 2006, Urban 2009). We are interested in how graph topology is combined with the spatial distribution of element contents in topsoil. The graph topology properties demonstrate landscape complexity and allow us to determine a finite size of local basic landscape diversity. Using graph topology, we evaluate a representativeness training subset to build the element content distribution in the topsoil. Our study suggests a formalization of assessment of individual and collective representativeness for sampling points to explain the connected landscape pattern. If there are individual sampling points that are important at some scale but not at another one, then there can be doublets, triplets, or n-lets of the sample that are important for modeling at some scale but not at another one (Dale 2010; Shu 2015).

Monitoring of environmental parameters in the conditions of urban development is not able to provide complete spatial and temporal characteristics of pollution (O’Hare 2020; Pesch 2008; Zhong 2021). For a comprehensive assessment of the levels of environmental pollution in cities (Wang 2020; Xu 2023), monitoring is often combined with other methods of obtaining data, including models based on ANN. In the scientific literature, works have been published when the selection of points in the training subset occurs using information about the distribution of the feature under study, but the gain in the accuracy of the model turns out to be small (Kramm 2020; Fernandez Jaramillo 2018; Gutierrez-Velez 2020). The prediction accuracy of ANN models is greatly influenced by the choice of points used to train the ANN. Random sampling points should only be used on a homogeneous experimental site (Legendre 2004; Prager 2009). As demonstrated by (Wang 2021; Ziggah 2019; Baglaeva 2021), various sampling points make contribute differently to the ANN forecast error, i.e., have different - representativeness for the purposes of the forecast.

Previously, some authors presented a definition of representativeness. Zhu (2018) uses the representativeness of a single sampling point and a sampling point set to other points as the similarity of these points to the sampling point set. The representativeness is a similarity in geographic configuration between sample point k and prediction point i, which is then used as the weight in the prediction of the value of the target variable at prediction point i, together with the other involved sample points whose weights are determined similarly. And this similarity is also used to measure the uncertainty associated with the prediction (Levin 2002; Zhu 2018). By representativeness, we understand the characteristics of points of the studied statistical population to adequately reflect the characteristics of the trait under study. Representative sampling or a representative selection of points in the training subset provides, within a given accuracy, reliable data on the content of a pollutant in an environmental component (air, water, soil etc.) in a selected area at a given point in time.

We assume that not only points distinguish in different representativeness for the evaluation of a feature, but also sets of points (doublet, triplets, ..., n-lets) have different representativeness. In the present work, it is proposed to consider the comparison of individual and collective representativeness when points are included in the training subset. Under the individual representativeness of the sampling point, we mean the frequency of its hits in the training subset, training on which provides the smallest model error. Collective representativeness is the frequency of hits of a collection of points (pairs, triples, quadruples, etc.) in the training subset, training on which provides the smallest model error. To build a representative training subset, it is necessary to 1) determine which n-lets are the most representative (n-lets size and representativeness level); 2) reveal the relationship between individual and collective representativeness. Determination of the volume of all representative n-lets requires large computing power of the computer, so the collective representativeness in this work was evaluated by pair.

MATERIALS AND METHODS

Sampling location

200 soil samples were collected in the residential part of Noyabrsk city (N 63.2°, E 75.5°), Russia. The industry of the city is hydrocarbon energy. The climatic zone is subarctic, or Dfc, by Köppen climate classification. The predominant soil type is gley taiga (Gd 23-1ab) on the FAO-UNESCO soil map¹.

The sampling point map was designed so that a given number of samples (200 in the residential area of Noyabrsk) on average evenly filled the study area. The average distance between sampling points was about 300 m. This distance varied depending on the density of buildings, the location of roads, etc. in order to ensure sampling in undisturbed areas of the open soil surface. The sampling depth was 0.05 m, since we were interested in the top layer of soil. The sampling was carried out with a cylindrical sampler with a diameter of 0.05 m. The soil sampling procedure schema was shown in Fig. 1.

The raw data preparation

The soil sampling procedure was previously described in detail for Noyabrsk (Baglaeva 2020). Fig. 1 shows the schema of this procedure. The raw data preparation consisted of sampling 200 specimens from undisturbed soil sites, sample preparation, and chemical analysis using inductively coupled plasma-mass spectrometry.

Preparation of soil samples and chemical analysis were conducted in compliance with actual standard requirements. For quality control, standard reference samples were used, certified for the content of determined elements, similar in composition to the samples under study.

The same total element contents were determined in the topsoil. Further modeling involved the total content of Cuprum and Niccolum in topsoil.

Multilayer perceptron

The input data are the geographic coordinates for the simulation. The output data are the element’s contents.

The multilayer perceptron (MLP) with Levenberg-Marquardt learning algorithm was used to demonstrate the possibilities of the method as the easy-to-understand model ANN for modeling the spatial distribution of the element contents in the topsoil. The construction of the MLP model based on the number of neurons inside the hidden layer was chosen after several training cycles and error estimation for the test subset. We used the tangential activation function, which is best suited for predicting the features of the spatial distribution of element content in the topsoil (Baglaeva 2021). The MLP structure had one input layer consisting of two neurons (spatial coordinates x and y), one hidden layer with 9 neurons, and one output layer with one neuron (element content).

Representativeness assessment

Let the representativeness of the sampling point be an ability of this point to provide: 1) a small root-mean-square error RMSE (Eq. 1) for estimating accuracy; 2) a high correlation coefficient Corr (Eq. 2) to check the synchronism of changes between predicted and observed values with the participation of this point in training.

(1)

where p(i) is predicted data; o(i) is observed data; n is the number of subset points. RMSE (1) tests the accuracy between predicted and observed data.

(2)

where p is predicted average; o is observed average. The correlation coefficient Corr shows the linear statistical relationship between the predicted values and the observed ones, how much the changes in the predicted values repeat the systematic changes in the observed ones.

The individual representativeness for each point considers the set of the best (small RMSE (Eq. 1) and high Corr (Eq. 2)) networks in which the point participated in training. Collective representativeness is the representativeness of the sampling points, which support connections with the neighbors, which makes it possible to provide a small RMSE and a high Corr when participating in the training of a group of sampling points. Collective representativeness considers the various combinations of training sampling points, such as doublets, triplets, and so on.

Individual representativeness

A four-step (4-step) algorithm was used for individual representativeness assessment of the sampling points involved in the training subset (Fig. 2). The raw data were repeatedly divided randomly into training and test subsets in the ratio of 75%/25%, respectively. 200 points were randomly divided into 150 training and 50 test points. The number of divisions was 100,000.

1. The total raw data set was randomly divided 100,000 times into two non-overlapping sets, training and test subsets, in the ratio of 75%/25%, respectively. Thus, we got 100,000 training and 100,000 test subsets.
2. For each random division, 5 MLP networks were built additionally (500,000 MLP networks in total). For each trained network, the root-mean-square errors (RMSE) of the forecast of the training and test subsets were determined. The network with the minimum RMSE was chosen.
3. RMSE and Corr for the training, test, and general subsets were calculated for 100,000 better networks.
4. Each sample point was assigned a set of the best networks in which it participated in training. For each sample point, we calculated the basic statistics of RMSE and Corr for the training, test, and general subsets for the networks in which the point participated in training.

Individual representativeness was assessed by comparing mean RMSE and Corr values. The best representative point is the one whose inclusion in the training subset provides a lower mean RMSE and a higher mean correlation coefficient with the observed values.

Collective representativeness

To assess the collective representativeness of the sampling points of the training subset, a four-step (4-step) algorithm was also used (Fig. 2). We divided the training and test subsets into 75% and 25%, respectively, used the training results as a set of the best networks for each point, and calculated the corresponding distributions of RMSE and correlation coefficients for the training, test, and general subsets.

Collective (paired) representativeness was assessed using samples of two points out of two hundred. For each pair of sampling points, the basic statistics of RMSE for the training, test, and total subsets were calculated for networks in which both points participated in training. 1000 pairs from these pairs were selected with the lowest mean RMSE for the pair that fell into the training set, which corresponded to the 0.051 quantile. The collective representativeness of the sampling points was assessed by the number of its connections with other points within the 0.051 quantile (1000 point pairs) according to the average RMSE or within the 1 - 0.051 = 0.949 quantile (1000 point pairs) according to the average correlation coefficient. We built graphs for a cutoff threshold of 10 connections with other sampling points.

In this work, due to computational difficulties, we limited ourselves to pairs of sampling points. The hypothesis that the synergy effect exists was tested by comparing individual and paired representativeness to predict element content. For verification, we used the conditional distribution of the correlation coefficients and RMSE means (provided that a pair of sampling points fell into the best training subset).

RESULTS

Table 1 presents the characteristics of Cuprum and Niccolum distributions in the study area. The total Cuprum content is in the range from 5.89 to 69.59 mg/kg, Niccolum from 3.58 to 41.94 mg/kg, which does not exceed Clarke in the urban soil (Shichkin 2018).

For each data split, MLP models were built, and representativeness characteristics were calculated: RMSE and correlation coefficients. Table 2 presents the statistical characteristics of the representativeness assessment. Fig. 3 shows the RMSE means and correlation coefficients obtained for 19,900 models.

As can be seen from Fig. 3, the RMSE distribution is split into two clusters: Cuprum and Niccolum. The lower RMSE cluster is associated with the inclusion in the training subset of a single point 129 for Niccolum (the point with the highest Niccolum content) and 134 for Cuprum (the point with the highest Cuprum content).

Pair representativeness was visualized as RMSE and correlation coefficient graphs for Niccolum and Cuprum contents (Fig. 4). The sampling points are graph vertices. The graph edges were the best links between pairs of sampling the points named by doublet.

Pair representativeness graphs were built by the least RMSE doublets and the largest Corr doublets for Ni and Cu (Fig. 4). When constructing the graphs, we were limited to about 20 doublets. That is, the correlation graphs included the doublets with correlation coefficients greater than 0.2667 for Cu and 0.2983 for Ni. The RMSE graphs consisted of the doublets with RMSEs less than 2.83 for Cu and 3.868 for Ni. Both Niccolum and Cuprum RMSE graphs seem to have a scale-free network structure. The edges of the correlation coefficient graphs are stitching through the area for each element. RMSE as the representativeness indicator define the most “polluted” points, and the correlation coefficient adds “important” points for description of the element distributions in the topsoil. Individual and pairwise representativeness comparisons were shown for Niccolum and Cuprum contents in Table 3.

DISCUSSION

Individual representativeness is not enough to determine the best training subset. A point with low individual representativeness in pairs may be “good” for learning. By analogy with individual characteristics, there can be “good” pairs for training (these are all 199 pairs with 129 points for Niccolum, and 134 points for Cuprum), there can be “bad” ones, which increase the model error when included in pairs in the training subset. The results are shown in Table 3. 9 points were selected: 8, 12, 14, 63, 67, 116, 165, 168, 199 for Cuprum and 49, 52, 84, 102, 103, 104, 118, 146, 192 for Niccolum. The elemental content at these points is below or close to the average.

Pair representativeness is not always limited to individual. There is a synergy effect, i.e., taking into account the collective (n-let) representativeness makes it possible to reduce the model error. Paired (collective) representativeness characterizes the interaction of pairs of points, i.e., the ability of pairs of points, when included in the training subset, to provide RMSE and correlation coefficient characteristics that exceed the best individual characteristics. The value added by the pair {94; 129} for Niccolum and a pair of {134; 135} for Cuprum is higher than the value contributed by individual points {94} and {129} for Niccolum and {134} and {135} for Cuprum. This synergy effect is created through the mutual influence between the points. Table 3 shows the best pairs with the lowest RMSE and the highest correlation coefficient and the worst pairs with the highest RMSE and the lowest correlation coefficient for Cu and Ni. Along with the synergy effect, there are relationships that reduce the value of the model. In this case, pairs of points provide less information to describe the element content distribution than individual points included in a pair as the effect of antisynergy. This may be due to the redundant use of points to describe the distribution of the feature.

As can be seen from Table 3, for example, this is {19; 56} for Ni and {139; 165} for Cu. RMSE pair {19; 56} for Niccolum and {139; 165} for Cuprum is greater than the RMSE of individual points {19} and {56} for Niccolum and {139} and {165} for Cuprum. Conversely, the correlation coefficients of a pair {88; 145} for Ni and {126; 127} for Cu are smaller than the correlation coefficients of individual points {88} and {145} for Ni and {126} and {127} for Cu.

The effects found here (synergy and anti-synergy) seem to be useful for predicting spatial variability and predicting the content of elements in the topsoil in areas with complex geographical conditions. This benefit may be expressed as a reduction in the uncertainty of the results of future field studies when they are planned.

The scale-free network structures RMSE graphs of the pair’s representativeness are the same for both Niccolum and Cuprum (Fig. 4), and the central points of these graphs are territorial characteristics. For each pair of the points, the best graph topology characteristic of the territory is identified. This topology can be explained by man-made activity.

The obtained results do not contradict the hypothesis that different locations (geolocations) carry different information and an unequal value for the interpolation of the feature distribution. Evaluation of the representativeness of the points will allow you to choose the most representative points for the areas.

CONCLUSIONS

Comparison of individual and pair (collective) representativeness when points were included in the training subset showed their unequal value for interpolating the distribution of heavy metals in the topsoil. The most representative in terms of individual representativeness were the points with the maximum element content in the selected area. Including these points in the ANN training subset reduces the error and increases the correlation between the results of model calculations and field measurements on the test subset. The graph topology of the best collective representativeness (it looks like a constellation) can be used as territory characteristics associated with man-made activity. The volume of the n-let can be analogous to the dimension of the phase space. Although it is impossible to predict every detail of the evolution of such a system, it is possible to develop statistical mechanics with heterogeneous ensembles of interacting agents (Levin 2002), similar to the description of statistical ensembles in gas dynamics.

In this work, we have limited ourselves to pair representativeness; determining the volume of all representative n-s requires huge computational costs and remains a task for future research. Complex adaptive systems are limited in their predictability because multiscale interactions and evolutionary processes are linked through non-linear interactions.