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The generalization error bounds found by current error models using the number of effective parameters of a classifier and the number of training samples are usually very loose. These bounds are intended for the entire input space. However, support vector machine (SVM), radial basis function neural network (RBFNN), and multilayer perceptron neural network (MLPNN) are local learning machines for solving problems and treat unseen samples near the training samples to be more important. In this paper, we propose a localized generalization error model which bounds from above the generalization error within a neighborhood of the training samples using stochastic sensitivity measure. It is then used to develop an architecture selection technique for a classifier with maximal coverage of unseen samples by specifying a generalization error threshold. Experiments using 17 University of California at Irvine (UCI) data sets show that, in comparison with cross validation (CV), sequential learning, and two other ad hoc methods, our technique consistently yields the best testing classification accuracy with fewer hidden neurons and less training time.

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Abstract. The number of samples used in the calibration data set affects the quality of the generated predictive models using visible, near and shortwave infrared (VIS–NIR–SWIR) spectroscopy for soil attributes. Recently, the convolutional neural network (CNN) has been regarded as a highly accurate model for predicting soil properties on a large database. However, it has not yet been ascertained how large the sample size should be for CNN model to be effective. This paper investigates the effect of the training sample size on the accuracy of deep learning and machine learning models. It aims at providing an estimate of how many calibration samples are needed to improve the model performance of soil properties predictions with CNN as compared to conventional machine learning models. In addition, this paper also looks at a way to interpret the CNN models, which are commonly labelled as a black box. It is hypothesised that the performance of machine learning models will increase with an increasing number of training samples, but it will plateau when it reaches a certain number, while the performance of CNN will keep improving. The performances of two machine learning models (partial least squares regression – PLSR; Cubist) are compared against the CNN model. A VIS–NIR–SWIR spectra library from Brazil, containing 4251 unique sites with averages of two to three samples per depth (a total of 12 044 samples), was divided into calibration (3188 sites) and validation (1063 sites) sets. A subset of the calibration data set was then created to represent a smaller calibration data set ranging from 125, 300, 500, 1000, 1500, 2000, 2500 and 2700 unique sites, which is equivalent to a sample size of approximately 350, 840, 1400, 2800, 4200, 5600, 7000 and 7650. All three models (PLSR, Cubist and CNN) were generated for each sample size of the unique sites for the prediction of five different soil properties, i.e. cation exchange capacity, organic carbon, sand, silt and clay content. These calibration subset sampling processes and modelling were repeated 10 times to provide a better representation of the model performances. Learning curves showed that the accuracy increased with an increasing number of training samples. At a lower number of samples (< 1000), PLSR and Cubist performed better than CNN. The performance of CNN outweighed the PLSR and Cubist model at a sample size of 1500 and 1800, respectively. It can be recommended that deep learning is most efficient for spectra modelling for sample sizes above 2000. The accuracy of the PLSR and Cubist model seems to reach a plateau above sample sizes of 4200 and 5000, respectively, while the accuracy of CNN has not plateaued. A sensitivity analysis of the CNN model demonstrated its ability to determine important wavelengths region that affected the predictions of various soil attributes.

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BackgroundThe use of visible-near infrared (vis-NIR) spectroscopy for rapid soil characterisation has gained a lot of interest in recent times. Soil spectra absorbance from the visible-infrared range can be calibrated using regression models to predict a set of soil properties. The accuracy of these regression models relies heavily on the calibration set. The optimum sample size and the overall sample representativeness of the dataset could further improve the model performance. However, there is no guideline on which sampling method should be used under different size of datasets.MethodsHere, we show different sampling algorithms performed differently under different data size and different regression models (Cubist regression tree and Partial Least Square Regression (PLSR)). We analysed the effect of three sampling algorithms: Kennard-Stone (KS), conditioned Latin Hypercube Sampling (cLHS) and k-means clustering (KM) against random sampling on the prediction of up to five different soil properties (sand, clay, carbon content, cation exchange capacity and pH) on three datasets. These datasets have different coverages: a European continental dataset (LUCAS, n = 5,639), a regional dataset from Australia (Geeves, n = 379), and a local dataset from New South Wales, Australia (Hillston, n = 384). Calibration sample sizes ranging from 50 to 3,000 were derived and tested for the continental dataset; and from 50 to 200 samples for the regional and local datasets.ResultsOverall, the PLSR gives a better prediction in comparison to the Cubist model for the prediction of various soil properties. It is also less prone to the choice of sampling algorithm. The KM algorithm is more representative in the larger dataset up to a certain calibration sample size. The KS algorithm appears to be more efficient (as compared to random sampling) in small datasets; however, the prediction performance varied a lot between soil properties. The cLHS sampling algorithm is the most robust sampling method for multiple soil properties regardless of the sample size.DiscussionOur results suggested that the optimum calibration sample size relied on how much generalization the model had to create. The use of the sampling algorithm is beneficial for larger datasets than smaller datasets where only small improvements can be made. KM is suitable for large datasets, KS is efficient in small datasets but results can be variable, while cLHS is less affected by sample size.

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