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DIVISION S-1—SOIL PHYSICS

The Neuro-m Method for Fitting Neural Network Parametric Pedotransfer Functions

Australian Cotton Cooperative Research Centre, Department of Agricultural Chemistry and Soil Science, Ross St. Building A03, The University of Sydney, NSW 2006, Australia

* Corresponding author (budiman{at}acss.usyd.edu.au)

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Parametric pedotransfer functions (PTFs), which predict parameters of a model from basic soil properties are useful in deriving continuous functions of soil properties, such as water retention curves. The common method for deriving parametric water retention PTFs involves estimating the parameters of a soil hydraulic model by fitting the model to the data, and then forming empirical relationships between basic soil properties and parameters. The latter step usually utilizes multiple linear regression or artificial neural networks. Neural network analysis is a powerful tool and has been shown to perform better than multiple linear regression. However neural-network PTFs are usually trained with an objective function that fits the estimated parameters of a soil hydraulic model. We called this the neuro-p method. The estimated parameters may carry errors and since the aim is to be able to estimate water retention, it is sensible to train the network to fit the measured water content. We propose a new objective function for neural network training, which predicts the parameters of the soil hydraulic model and optimizes the PTF to match the measured and observed water content, we called this neuro-m method. This method was used to predict the parameters of the van Genuchten model. Using Australian soil hydraulic data as a training set, neuro-m predicted the water retention from bulk density and particle-size distribution with a mean accuracy of 0.04 m³ m^-3. The relative improvement of neuro-m over neural networks that was optimized to fit the parameters (neuro-p) is 13%. Compared with a published neural network PTF, the new method is 30% more accurate and less biased.

Abbreviations: AIC, Akaike's information criterion • ANN, artificial neural network • MD, mean deviation • MR, mean residuals • neuro-m, neural network PTF with objective function that matches the measured values • neuro-p, neural network PTF with objective function that matches the parameters • P_<2, mass particles <2 mm • P_2–20, mass particles 2 to 20 mm • P_20–2000, mass of particles 20 to 2000 mm • PTF, pedotransfer function • RMSD, root mean squared deviation • RMSR, root mean squared of residuals • SSR, sum of square residuals

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
PEDOTRANSFER FUNCTIONS (Bouma, 1989), predictive functions of certain soil properties from other easily, routinely, or cheaply measured properties, have recently become a popular topic in soil science research. Different types of function have been developed to predict either physical or chemical properties of the soil. Research areas include formulating a better physical model (Arya et al., 1999), finding the most influential soil properties as predictive variables (Timlin et al., 1999), grouping soil into classes to minimize the variance of prediction (Pachepsky and Rawls, 1999), and developing alternative methods to derive or fit the PTFs (Scheinost et al., 1997). Most pedotransfer functions have been developed to predict soil hydraulic properties, especially water retention curves. This is mainly as the response to the urgent need for soil hydraulic properties as inputs to soil-water models.

In probably the first research of its kind, Bloemen (1977)(1980) derived the relationships between parameters of the Brooks-Corey model and particle-size distribution. This approach is now called parametric PTFs. For the water-retention curve, we can assume that the water content, , and the potential, h, relationship can be described adequately by a soil hydraulic model that is a closed-form equation with a certain number of parameters, e.g., Brooks and Corey or van Genuchten equation. A parametric approach is usually preferred to single-point regression (predicting water retention at a specific potential), as it yields a continuous function of the (h) relationship. Water retention at any potential can be estimated. Many soil-water transport models only require the parameters of the soil hydraulic models as inputs, thus the predicted parameters can be used directly to run them.

The usual steps in deriving parametric PTFs are fitting a soil hydraulic model to individual water-retention data, estimating the parameters of the model, and forming empirical relationships between basic soil properties and parameters. The latter step can be achieved by various mathematical methods, e.g., multiple linear regression (Wösten et al., 1995), or artificial neural networks (ANN) (Schaap et al., 1998).

The most widely used soil hydraulic model is the van Genuchten function (van Genuchten, 1980):

[1]

where _r and _s are the residual and saturated water content, is the scaling parameter, n is the curve shape factor and m is an empirical constant, which can be related to n by m = 1 - .

Attempts have been made to correlate the parameters of the van Genuchten equation (or other soil hydraulic models) to basic soil properties (Vereckeen et al., 1989), however many researchers have encountered difficulties (Tietje and Tapkenhinrichs, 1993; Scheinost et al., 1997). Van den Berg et al. (1997) suggested that this could be caused by interdependency amongst the parameters. The problems are also likely because the model does not always fit the data, and overparametization of the model (too many parameters to fit over a limited water-retention data). Hence the fitted parameters may carry errors and bear no significant physical meaning. To overcome this problem, Van den Berg et al. (1997) suggested the following approach: fit the model to observed data, apply multiple regression analysis to one of the parameters, fit the model again by fixing the parameter calculated from the regression, continue to fit a regression to another parameter and repeat the process until all the parameters are fitted. Alternatively, Scheinost et al. (1997) proposed: set-up the expected relationship between the parameters of the model and soil properties, and then insert the relationship into the model and estimate the parameters of the relationship by fitting the extended model using nonlinear regression.

The extended nonlinear regression method of Scheinost et al. (1997) has been found to provide good estimates of water retention data (Minasny et al., 1999). However there are limitations, relationships between the parameters and basic soil properties have to be formulated first. Furthermore, since we are fitting the whole data set to a single extended version of the van Genuchten function, the predictions regress to a mean curve. This is demonstrated using the data from the study by Minasny et al. (1999). Figure 1a shows the plot of scaled water content S_e = and scaled potential (h) of the measured data. Where the symbol represents a parameter estimated from the raw water retention data and represents a parameter predicted using extended nonlinear regression. If we plot the values predicted from extended nonlinear regression _e = vs. h. (Fig. 1b), the large variations in the original data are scaled to a mean curve. This method would be inadequate if we wish to use the PTFs to characterize the spatial variability within a field.

View larger version (26K): [in this window] [in a new window]   Fig. 1. The scaled water content Se and scaled potential h of the (a) measured data, and (b) predicted using extended nonlinear regression.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Artificial Neural Networks
A neural network is an attempt to build a mathematical model that supposedly works in an analogous way to the human brain. A network consists of many elements or neurons that are connected by communication channels or connectors. These connectors carry numeric data arranged by a variety of means and organized into layers. The neural networks can perform a particular function when certain values are assigned to the connections or weights between elements. To describe a system, there is no assumed structure of the model, instead the networks are adjusted or trained so that a particular input leads to a specific target output (Gershenfeld, 1999).

The mathematical model of a neural network comprises of a set of simple functions linked together by weights. The network consists of a set of input units x, output units y, and hidden units z, which link the inputs to outputs (Fig. 2) . The hidden units extract useful information from inputs and use them to predict the outputs. The type of ANN considered here is called the multilayer perceptron (Nørgaard, 2000). A network with an input vector of elements x_l (l = 1, ... , N_i) is transmitted through a connection that is multiplied by weight, w_jl, to give the hidden unit z_j(j = 1, ... , N_h):

[2]

where N_h is the number of hidden units and N_i is the number of input units. The hidden units consist of the weighted input and a bias (w_j0). A bias is simply a weight with constant input of 1 that serves as a constant added to the weight. These inputs are passed through a layer of activation function f which produces:

[3]

View larger version (37K): [in this window] [in a new window]   Fig. 2. The structure of a neural network.

[4]

The outputs from hidden units pass another layer of filters:

[5]

and fed into another activation function F to produce output y (k = 1, ... , N_o):

[6]

The weights are adjustable parameters of the network and are determined from a set of data through the process of training (Nørgaard, 2000). The training of a network is accomplished using an optimization procedure (such as nonlinear least squares). The objective is to minimize the sum of squares of the residuals (SSR) between the measured and predicted output.

Schaap et al. (1998) estimated van Genuchten parameters for 1209 soil samples from the USA using a neural network. They distinguished their PTFs based on the level of available information: texture (clay, silt, and sand); texture and bulk density; texture, bulk density, and measured at -33 (_-33) and -1500 kPa (_-1500). They found that their PTFs performed better than four previously published multiple-regression PTFs.

A New Objective Function for Neural Network Pedotranfer Functions
Neural network analysis is quite powerful and according to Gershenfeld (1999) with one hidden layer that has enough hidden units, it can describe any continuous function. Conventionally, parametric PTFs train the network to fit the estimated van Genuchten parameters. Problems using this method are:

1. The van Genuchten equation does not necessarily fit the measured data, hence optimizing the neural networks to fit the parameters cannot guarantee to provide good estimates.
   
2. When fitting the van Genuchten equation to water-retention data, we minimize the difference between the predicted and measured water content, . However, we train the neural networks to minimize the difference between of the predicted and estimated parameter values p.
   
3. Van Genuchten (or other hydraulic) equation and its parameters are nonlinear. A PTF that has improved predictions for one of the van Genuchten parameters does not necessarily perform better in predicting (Schaap et al., 1998).
   

Since our aim is to predict water retention it will be sensible to train the network to fit the measured water content. We propose a new objective function for neural network training, which predicts the van Genuchten parameters and minimizes the difference between the measured water content and the one calculated from the predicted parameters. This is illustrated in Fig. 3 .

View larger version (25K): [in this window] [in a new window]   Fig. 3. Structure of a neural network predicting the van Genuchten parameters and water retention.

1. Fit the individual water-retention curve to the van Genuchten function and estimate the parameters _r, _s, , and n.
   
2. Train the neural network to predict the parameter vector p = from basic soil properties by minimizing objective function:
   

   [7]

   
   where Ns is the number of soil samples, N_o is the number of outputs (parameters to predict), W and U are the weight of the hidden and output layer respectively, and (x) is the predicted parameter from inputs x.
   

3. The above steps are usually used for parametric PTFs, we term this neuro-p, a neural network with an objective function that matches the parameters. The proposed method continues with fine-tuning steps described below:
   
4. Use the trained weights as an initial guess for the second training, which fine tunes the estimates.
   
5. For each soil sample, predict the hydraulic parameters with the trained weights, and calculate the water content using the van Genuchten equation at each of the measured potentials.
   
6. Adjust the weights, W and U, to minimize the difference between the predicted and measured water content with the optimization routine. The objective function is:
   

   [8]

   where N_s is the number of soil samples, N_d(i) is the number of water retention data in soil sample i, and (x, h, p) is the predicted water content at potential h using the parameters p, which is calculated from inputs x. We term this neuro-m, a neural network with an objective function that matches the measured or observed values.
   

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Data Set
The Australian data set was compiled from previously published water retention and basic soil properties data across Australia (Minasny et al., 1999). The data were screened and dubious data were discarded, which left us with 862 soil samples. The data were split randomly into a prediction (484 samples) and a validation set (378 samples). The data available are water-retention curves, which range between 5 to 10 -h pairs, and basic soil properties: field bulk density (_b) and particle-size distribution, comprising the mass of particles <2 (P_<2), 2 to 20 (P_2–20), and 20 to 2000 µm (P_20–2000), which were normalized to sum to 100%. Assuming a log-normal distribution, the geometric mean particle-size diameter (d_g) of the particle-size distribution can be calculated according to Shirazi and Boersma (1984). Usually the data are only reported in terms of clay (P_<2), silt (P_2–20), and sand (P_20–2000) which sums to 100%, d_g can be simply calculated as:

[9]

The water retention data was fitted to the van Genuchten equation using nonlinear regression with the Levenberg-Marquardt algorithm (Marquardt, 1963). The _r, and n parameters of van Genuchten were estimated with the following constraints imposed: _r 0 m³ m^-3, 0.0001 1 hPa^-1, and 1.01 n 10. If _r < 0.0001 then its value is fixed at 0. The _s is fixed at water content of 0 kPa.

The prediction set alone was used to train the neural network. Since the prediction and validation data came from a similar source, it will be inequitable to compare the performance of other PTFs that have training sets containing exotic (non-Australian) soil sample. As an independent test of the PTFs, we used a published database GRIZZLY (Haverkamp et al., 1997) from Laboratoire d'Étude des Transfers en Hydrologie et Environnement (LTHE), Grenoble. The data consist of 660 soil samples collected from laboratories and field experiments in different countries. The mass of particles P_2–20, P_20–2000, P_2–50, and P_50–2000 were calculated from the cumulative particle-size distribution function, which has similar form as the van Genuchten equation (Haverkamp and Parlange, 1996), provided in the database. The statistics of the data are given in Table 1.

Estimation of the weights were initially carried out using program MATLAB ver. 5.3 with freeware toolbox NNSYSID ver. 2.0 (Nørgaard, 2000). This is the neuro-p method, which minimized objective function Eq. [7]. The trained weights were used as an initial guess for neuro-m. The fine-tuning steps of the neuro-m method were programmed in Fortran. Minimization of the new objective function (Eq. [8]) was carried out using the Levenberg-Marquardt nonlinear least-squares (Marquardt, 1963). We use different initial estimates and select the solution which gives lowest SSR, nevertheless most solutions give similar SSR.

The performance of the proposed method in predicting water retention was compared with a published neural-network PTF. We used the neural network PTF developed by Schaap et al. (1998) which was developed from a wide range of soil types from the USA and was optimized to fit the hydraulic parameters. The network has six hidden units and calculation was performed using the program Rosetta available from the world-wide web (Schaap, 2000). Imam et al. (1999) have found that Rosetta performs better than other multiple regression parametric PTFs for a wide range of soil types from a global soil database. They also found that it is less sensitive to soil texture classes, implying that it is applicable to a wide range of soil materials. Two levels of input information were used (i) three inputs, which only used particle-size distributions: P_<2, P_2–50, P_50–2000, (ii) four inputs, where bulk density is included: P_<2, P_2–50, P_50–2000, _b. Particle-size data of the Australian system (P_2–20, P_20–2000), were converted to the USDA system (P_2–50, P_50–2000) using the empirical equations of Minasny and McBratney (2001):

[10]

Perfomance Criteria
The performance of the PTFs predicting the individual water retention curve was assessed using four criteria. Mean residuals (MR) measures the bias, indicating the tendency of under- or overestimation:

[11]

where N_d is the number of (h) pairs in a water-retention curve. Negative values of MR indicate that the PTFs underestimate the water content and positive values indicate overestimation. The root mean square of residuals (RMSR) calculates mean accuracy of prediction, which represents the expected magnitude of error:

[12]

The mean deviation (MD) and root mean squared deviation (RMSD) of Tietje and Tapkenhinrichs (1993) calculate the area difference between the predicted and measured water retention curves:

[13]

[14]

where a and b are the integration boundaries set at a = 0.1 hPa and b is set according the smallest (most negative) potential measured in the sample. Calculation of MD and RMSD was done on the logarithm of the |potential|. The integral was numerically evaluated using Gaussian quadrature.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
There are no general rules for selecting the number of hidden units in the network. The larger number of hidden units, the more parameters are required. We would like to choose a model that has an optimum number of parameters, as too few hidden units cause underfitting (network cannot describe the data) and too many hidden units cause overfitting (network is fitting the noise of the data). As a compromise between goodness-of-fit and parsimony, Akaike's Information Criterion (AIC) (Akaike, 1973), a statistical Ockham's razor was used:

[15]

which can be estimated by (Webster and McBratney, 1989):

[16]

where N is the total number of water-retention data. For neural networks, np is the number of weights used:

[17]

where 1 is due to bias. Theoretically, the best model is the one that has the smallest AIC. Figure 4 shows the AIC calculated from prediction using neuro-m with three and four inputs, as the number of hidden units increases the AIC decreases rapidly until five units. The neural network will decrease the SSR with increasing the number of hidden units, but the improvement may not be significant and may cause overfitting. From the plot, we pick four hidden units as it produces a low AIC with moderate number of parameters (36 for three inputs and 40 for four inputs) and it is also the point where the AIC reduction starts to slow down.

View larger version (13K): [in this window] [in a new window]   Fig. 4. Akaike's Information Criterion (AIC) as a function of the number of hidden units.

1. Neuropath, a neural-network PTF that was trained using Australian prediction data (N_s = 484), four hidden units, and optimized to fit the parameters (neuro-p method).
   
2. Neuroman, a neural-network PTF that was trained using Australian prediction data (N_s = 484), four hidden units, and optimized to fit water content (neuro-m method). Neuroman used the weights of Neuropath as initial guess in the fine-tuning steps.
   
3. Rosetta, a published neural-network that was trained using U.S. data (N_s = 1209), six hidden units, and optimized to fit the parameters (neuro-p method).
   

The elements of the neural network's weights, W and U, for Neuroman are given in Table 2. With these weights, the van Genuchten parameters can be calculated from basic soil properties (sand, silt, clay, and bulk density) using Eq. [6], or by a simple matrix operation on a spreadsheet. So that researchers can potentially use these PTFs, an example is given in the Appendix. There are different techniques proposed for the interpretation of the neural networks (Abrahart et al., 2001), but the weights are usually difficult to interpret. Neural networks that were trained using different initial values will produce different weights but might yield identical performance measures.

[18]

Table 3 shows the RMSR of the three methods applied to the Australian prediction, Australian validation and independent GRIZZLY data sets. In the prediction set, Neuroman clearly performs better than other methods. The relative improvement over Neuropath is 19%. Obviously Rosetta will not perform better as it is trained outside the Australian data set. However in the validation set, which is not used in training, using three inputs (sand, silt, and clay), Rosetta performs better than Neuroman. But when incorporating bulk density (four inputs) the performance of Neuroman is enhanced. The improvement over Neuropath is 11 and 29% over Rosetta. In the independent GRIZZLY data set, which does not come from the same population as the Australian data set, with three inputs, the improvement of Neuroman over Neuropath is 12% and only 5% over Rosetta. Using four inputs, the improvement is 25% over Rosetta. As the prediction improves significantly when incorporating bulk density, it suggests that bulk density is an important factor in predicting volumetric water retention. Overall the improvement of Neuroman is 13% over Neuropath and 30% over Rosetta.

View larger version (38K): [in this window] [in a new window]   Fig. 5. Measured and predicted water content using Neuroman and Rosetta with four inputs for the Australian prediction, Australian validation, and independent GRIZZLY data sets.

Rosetta, which has more hidden units (six) and was trained using a larger data set, does not perform as well as Neuroman (Fig. 6) . This is confirmed from the analysis on an independent data set GRIZZLY (Tables 3 and 4), which shows that the RMSD is higher (Rosetta = 0.03 m³ m^-3, Neuroman = 0.02 m³ m^-3). Moreover, it has the tendency to underestimate the water content as shown by the negative values of MD and MR. Figure 5 shows that the predictions tend to lie below the 1:1 line. Figure 6 shows the mean of MR and RMSR as evaluated at each soil sample. A desired PTF should have MR near 0 and small RMSR, thus showing the relative better performance of Neuroman.

View larger version (16K): [in this window] [in a new window]   Fig. 6. The mean residuals (MR) plotted against root mean squared residuals (RMSR) using Neuroman (Neu) and Rosetta (Ros) for the Australian prediction (P), Australian validation (V), and independent GRIZZLY (G) data sets.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

Example of Using Neural-network Weights from Neuroman4 to Predict van Genuchten Parameters
If we have a soil with bulk density (_b) of 1.30 Mg m^-3, clay (P_<2) content of 45% and sand (P_20–2000) content of 30%, we can calculate ln(d_g) according to Eq. [9] which gives us ln = -4.234. Using the weights from Neuroman4 (Table 2), we apply Eq. [6] as a matrix operation to predict the van Genuchten parameters. We arrange the input in as a vector: x = ^T = ^T, where 1 serve as bias. We multiply the input vector x by the weight matrix W to give us vector z:

We apply activation function f(z) = tanh(z) to each element of vector z to form vector r = ^T. We add a constant 1 to the last row of r to serve as bias. We multiply weight matrix U with r to give v. Since the output activation is linear F = v, output vector y is equal to v.

Transforming the result gives us an estimate of _r = 0.0002 m³ m^-3, _s = 0.497, = 0.027 hPa^-1 and n = 1.102.

	ACKNOWLEDGMENTS

 
This research was funded by the Australian Cotton Cooperative Research Centre.

Received for publication March 6, 2001.

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TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 

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