Abstract
The spatial analysis of soil properties by means of quantitative methods is useful to make predictions at sampled and unsampled locations. Two most important characteristics are tackled, namely the option of using complex and nonlinear models in contrast with (also very simple) linear approaches, and the opportunity to build spatial inference tools using horizons as basic soil components. The objective is to perform the spatial analysis of clay content for validation purposes in order to understand whether nonlinear methods can manage soil horizons, and to quantitatively measure how much they outperform simpler methods. This is addressed in a case study in which relatively few records are available to calibrate (train) such complex models. We built three models which are based on artificial neural networks, namely single artificial neural networks, median neural networks and bootstrap aggregating neural networks with genetic algorithms and principal component regression (BAGAP). We perform a validation procedure at three different levels of soil horizon aggregations (i.e. topsoil, profile and horizon pedological supports). The results show that neurocomputing performs best at any level of pedological support even when we use an ensemble of neural nets (i.e. BAGAP), which is very data intensive. BAGAP has the lowest RMSE at any level of pedological support with \(\hbox {RMSE}_\mathrm{BAGAP}^{Topsoil} = 7.2\,\%\), \(\hbox {RMSE}_\mathrm{BAGAP}^{Profile} = 7.8\,\%\) and \(\hbox {RMSE}_\mathrm{BAGAP}^{Horizon} = 8.8\,\%\). We analysed indepth artificial neural parameters, and included them in the “Appendix”, to provide the best tuned neuralbased model to enable us to make suitable spatial predictions.
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Appendices
Appendix: Definition of specific terms
Activation function It is a function that converts the neuron’s weighted input to the its output.
Artificial neural network (ANN) It is a network of connected artificial neurons resembling how biological neurons are arranged in the animal brain. Neurons are distributed in (input, hidden and output) layers defining the structure of the neural network.
Bagging Bootstrap aggregating is a technique used to improve the accuracy of a regression (or a classification) task. It combines models trained with randomly generated sets (sampling with replacement).
Early stopping It is a method used during the training phase to improve the generalization ability of the neural network. Two data subsets are required during the training phase: the training set which is used to update the network weights (model calibration), and the validation set which is used to stop the training phase if the validation error increases for a specified number of times.
Initialization (of ANN) Each weight and bias of all network neurons are set to a random initial value used to start the training phase.
Overfitting A statistical model overfits when it learns more noise and random error rather than the underlying function to be approximated. In case of overfitting, a model cannot reproduce signals other than the training ones without considerably losing accuracy.
Principal component regression (PCR) It is a statistical technique in which a subset of loadings of the principal component analysis is used as regressor of the dependent variable.
Training phase In this work, a supervised learning paradigm is used for training which is based on the widely used backpropagation algorithm. Weights and biases are updated to fit the input/output couples.
Artificial neural networks (ANN)
Neural networks are highly complex and nonlinear systems which are capable of learning (adaptivity) and representing (generalization) realworld problems (Haykin 1999). Paralleldistributed and fully connected artificial neurons are elementary structural constituents. An artificial neuron (or perceptron) is an informationprocessing unit which consists of a few basic elements: the input signals \(\left( x_{j}\right)\) scaled by synaptic weights \(\left( w_{kj}\right)\), the neuronal bias \(\left( b_k\right)\), the induced local field, \(\left( I_k\right)\) which is calculated as the weighted sum of the inputs plus the bias, and the activation function \(\left( \phi \left( I_k\right) \right)\) with which the i−th neuron generates an equalized output (\(y_k\), Fig. 9).
A feedforward neural network (FFNN) is a multilayer perceptron made by a set of neurons organized in the input layer in zero or more hidden layers and in the output layer. Signals propagate on a layer basis along one direction at a time. The supervised learning process is based on the popular twopass error backpropagation algorithm: in the forward pass, the input vector propagates through the network, generating outputs while keeping fixed synaptic weights. During the backward pass, the error signal (given by the difference between the desired target and th network output) backpropagates to adjust the synaptic weights. We used the slower convergent LevenbergMarquardt algorithm to carry out the training phase, with or without the early stopping criterion, which consists of the use/nonuse of an independent testing subset during the training phase. Every neuron is equipped with the following hyperbolic tangent sigmoid transfer function:
In this work, we used fully connected multilayer FFNNs to address the input/output mapping. During the training phase, we used input/target couples from the training subset to adjust the weights and biases (free parameters) in order to best fit target signals with network outputs (learning ability by means of neural plasticity). Then, we carried out the simulation phase to measure (through validation procedure) the generalization capability of the trained network on the validation input/target couples which were left out during the onefold splitting procedure described in “Study region and soil data” section.
In the “Appendix” (mANN), we discuss different aspects of the training phase during a sensitivity analysis on bootstrap artificial neural nets. We selected the trained neural network (the median net) from the bestperforming setting to represent the single neural network model in order to explain the clay spatial variability. The procedure employed to develop the BAGAP model is proposed in the “Appendix”, where groups of neural nets are used to validate whether an aggregated response performs better than the response by a single neural network or by linear models.
Sensitivity analysis and the median ANN (mANN)
We carried out a sensitivity analysis by varying three main constitutive elements in order to measure the supervised learning ability under distinctive ANN conditions. The complete procedure is explained in more detail in the following steps:

1.
Target data subsetting (for any level of pedological support)
Data were firstly stratified using the onefold random splitting procedure described in “Study region and soil data” section. Hence, 20 % of clay data composed the validation (Va) subset, which was made by the same signals within each level of pedological support. We used it to measure the generalization capability by trained ANNs and to compare neurocomputing with other prediction models.
The residual 80 % of data were randomly split into training (Tr) and testing (Te) subsets during bootstrapping. The training phase for any ANN takes place on Tr. If necessary, we remove a random 12.5 % Te subset from the training data to test the generalization capability using the ES method (step 3). In Table 2, a summary of target data dimensionality is reported according to the use (ES = 1) or nonuse (ES = 0) of the ES criterion.

2.
Input terms gathering
Digital terrain analysis of the topographic surface and remote sensing are sources for spatially continuous input covariates. The best tradeoff between the ANN size (i.e. number of inputs and hence the number of free parameters) and sample size is investigated for all levels of pedological support. The \(p_{io}^{b}\) inputs used in the sensitivity analysis are defined in the ith subscript in Table 3, where i denotes one of three possible sets of inputs; o indicates one of three target subsets {Tr, Va and Te}, as defined in step 1 b is the ANN bootstrap resample presented afterwards (see also Eq. 9).

3.
Bootstrapping (for any setting)
Preliminary trials suggested that the size of the input layer (=n) should equal the number of input terms and that both inputs and hidden layers should have the same size as a good tradeoff ability between fitting and generalization. We carried out a sensitivity analysis by bootstrapping ANNs for each tobeinvestigated neural network setting. The list of feasible settings is given in Table 4 for three varying network conditions: (1) the number and type of input variable (\(p_{io}^{b}\)), (2) the topology (the number of hidden layers ranges from zero to one) and (3) the use (ES = 1) or nonuse (ES = 0) of ES. The need to study the effect of each neural net’s setting on learning and generalization capabilities drove the sensitivity analyses to one thousand bootstrapped ANNs for each setting type. The sensitivity sheet mapped out in Table 4 was repeated for each level of pedological support.

4.
Single neural network architectures
As an example, setting #8 in Table 4 uses the \(p_{3o}^{b}\) input of Table 3 at Horizon, which is made of DEM, SPI, NDVIS5, NDVID16, ASP, \(N_\mathrm{LAY}\). The corresponding 6:6:1 network topology is depicted in Fig. 10, and its mathematical expression is:
$$\begin{aligned} y= \phi \left( ob + \sum _{r = 1}^{6}ow_r\cdot \phi \left( hb_r + \sum _{s = 1}^{6}hw_{rs}\cdot \phi \left( ib_s + \sum _{v = 1}^{6}iw_{sv}\cdot x_v \right) \right) \right) \end{aligned}$$(8)iw, hw, ow input, hidden and output weights, ib, hb, ob input, hidden and output biases, \(\phi\) activation function (Eq. 7), x, y input and output signals, v, s, r subscripts typifying network layers.

5.
Median artificial neural net (mANN)
The objective of the sensitivity analysis was to analyse the effects on generalization of the network conditions mapped out in the sensitivity sheet (step 3), i.e. to determine the feasibility of obtaining a welltrained single ANN to be employed in realworld simulations. We assumed that the lowest mANN from within all settings could represent a good tradeoff between learning and generalization. We selected mANN as the ANN whose RMSE is nearest to the median value from within the setting having the lowest median. It was selected as the reference singleANN modelling class for any level of pedological support. To compare the models, we performed simulations with mANNs on Va. Two corollaries of the assumption of building mANNs are: (1) an ANN whose RMSE is below mANN is overtrained and characterized by a lesser generalization capability, and (2) an ANN whose RMSE is above mANN is poorly trained and is unreliable for the simulation.
Bagging neural networks (BAGAP)
The training phase aims to adjust weights and biases in order to supply the generalization capability (a method used to correctly simulate realworld signals) to a single neural network. The generalization capability can be significantly improved using groups of neural networks, where several ANNs are trained and their outputs combined to form a combined response. A collective approach has two major components: a method for training individual ANNs and a method for combining (a selection of) ANNs.
In the particular context of our study domain, single ANNs were not highly suited for simulating clay values for each level of pedological support because a small learning set was available. This was the premise for the development of a sensitivity analysis procedure which afforded for setting up a prototype finetuned ANN (and allowed the mANN) in the previous section. This was the method used to train individual ANNs. We realized the arrangement of the combined ANN (i.e. of more complicated models made by more ANNs) using a bagging procedure (Breiman 1996).
Bagging aims to improve prediction accuracy by combining multiple models. In order to calibrate multiple repeated neural network models, b bootstrap samples of the learning set
with
were drawn at random with replacements to amplify realworld signals. The i subscript denotes three possible sets of predictors (Table 3): o indicates one of three target subsets (step 1); s accounts for three levels of pedological support; \({\mathbf {p}}_{io}^{b}\) are input terms of the bth resample; \({\mathbf {T}}_{so}^{b}\) are the bth resample target data. Each bootstrap replicates \({\mathbf {L}}_{ios}^{b}\) with a unique set of pairs of inputs, and target signals were used to calibrate a different neural network component \({{\mathbf {N}}}{{\mathbf {N}}}_{is}^{b}\) during the sensitivity analysis for a fixed level of pedological support. Next, we used a technique for building robust nonlinear models by aggregating multiple and randomly selected neural network components \(\hbox {NN}_{is}^{\cdot } = \left\{ \left( \hbox {NN}_{is}^{1}\right) , \left( \hbox {NN}_{is}^{2}\right) ,\ldots , \left( \hbox {NN}_{is}^{100}\right) \right\}\) (Langella et al. 2010).
It is proposed that the algorithm structures and calibrates groups of ANNs, called BAGAP, as well as provides a solution to choose which (by virtue of genetic algorithms) and how (using PCR) ANNs should be aggregated. We performed this using the following steps, which are a natural continuation of the steps elucidated in the previous section:

6.
We randomly selected a set of 100 ANNs from within the sensitivity sheet setting (step 3) characterized by the lowest RMSE median (the same setting to which mANN belongs). They represented the mixture of experts used as input for BAGAP training algorithm. For instance, 100 \(NN_{3,hor}^{\cdot }\) were selected at random for Horizon support from setting #9 (Fig. 11, left).

7.
We used genetic algorithms (GAs) to arrange an optimization problem in order to find the best candidate ANN population. At each GA epoch, we obtained a new 1bit population, and the which task was fulfilled. This means that the population of selected neural nets was identified from within 100 \(NN_{3,hor}^{\cdot }\) at each GA epoch (e.g. at the first GA epoch 57 \(NN_{3,hor}^{\cdot }\) were selected as the current best population for Horizon, and so forth at each GA epoch).

8.
The stack of selected \(NN_{is}^{\cdot }\) components at the current GA epoch underwent aggregation through PCR weights calculated using training data (Zhang 1999). The BAGAP training internally requires a crossvalidation procedure to avoid overfitting. As a consequence, we selected the first few eigenvectors which minimized the error function on the testing subset to ensure BAGAP generalization capability, and the how task was solved.

9.
BAGAP yields a combined aggregated response at each epoch. When a stopping criteria was met, the 1bit population and PCR loadings of the current epoch were stored to represent the BAGAP model for the level of pedological support at hand. It was assumed that this particular PCR aggregation of the GAselected ANN population is characterized by the best combined fitness.

10.
At the end of the calibration process, we simulated BAGAP using Va for model comparison.
The GASEN mixture model presented by Zhou et al. (2002) provided the core framework for our BAGAP. According to GASEN, we employed a GA to evolve randomly initialized weights. However, unlike GASEN, which cuts floating weights on an arbitrary threshold after optimizing the search of the minimum error function, BAGAP uses binary (1bit) weights during the GA search in order to select only the population of candidate ANNs. The composition of the combined response is weighted by PCR loadings, and thus, BAGAP has a novel structural component compared to GASEN.
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Langella, G., Basile, A., Bonfante, A. et al. Spatial analysis of clay content in soils using neurocomputing and pedological support: a case study of Valle Telesina (South Italy). Environ Earth Sci 75, 1357 (2016). https://doi.org/10.1007/s1266501661637
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Keywords
 Artificial neural network
 Pedological support
 Soil variability
 Spatial analysis
 Bagging