Sensitivity analysis of multilayer perceptron to input and weightperturbations

An important issue in the design and implementation of a neural network is the sensitivity of its output to input and weight perturbations. In this paper, we discuss the sensitivity of the most popular and general feedforward neural networks-multilayer perceptron (MLP). The sensitivity is defined as the mathematical expectation of the output errors of the MLP due to input and weight perturbations with respect to all input and weight values in a given continuous interval. The sensitivity for a single neuron is discussed first and an analytical expression that is a function of the absolute values of input and weight perturbations is approximately derived. Then an algorithm is given to compute the sensitivity for the entire MLP. As intuitively expected, the sensitivity increases with input and weight perturbations, but the increase has an upper bound that is determined by the structural configuration of the MLP, namely the number of neurons per layer and the number of layers. There exists an optimal value for the number of neurons in a layer, which yields the highest sensitivity value. The effect caused by the number of layers is quite unexpected. The sensitivity of a neural network may decrease at first and then almost keeps constant while the number increases.     

Sensitivity analysis of multilayer perceptron with differentiableactivation functions

In a neural network, many different sets of connection weights can approximately realize an input-output mapping. The sensitivity of the neural network varies depending on the set of weights. For the selection of weights with lower sensitivity or for estimating output perturbations in the implementation, it is important to measure the sensitivity for the weights. A sensitivity depending on the weight set in a single-output multilayer perceptron (MLP) with differentiable activation functions is proposed. Formulas are derived to compute the sensitivity arising from additive/multiplicative weight perturbations or input perturbations for a specific input pattern. The concept of sensitivity is extended so that it can be applied to any input patterns. A few sensitivity measures for the multiple output MLP are suggested. For the verification of the validity of the proposed sensitivities, computer simulations have been performed, resulting in good agreement between theoretical and simulation outcomes for small weight perturbations.     

Achieving a More Robust Neural Network Model for Control of a MR Damper by Signal Sensitivity Analysis

Most neural network models can work accurately on their trained samples, but when encountering noise, there could be significant errors if the trained neural network is not robust enough to resist the noise. Sensitivity to perturbation in the control signal due to noise is very important for the prediction of an output signal. The goal of this paper is to provide a methodology of signal sensitivity analysis in order to enable the selection of an ideal Multi-Layer Perception (MLP) neural network model from a group of MLP models with different parameters, i.e. to get a highly accurate and robust model for control problems. This paper proposes a signal sensitivity which depends upon the variance of the output error due to noise in the input signals of a single output MLP with differentiable activation functions. On the assumption that noise arises from additive/multiplicative perturbations, the signal sensitivity of the MLP model can be easily calculated, and a method of lowering the sensitivity of the MLP model is proposed. A control system of a magnetorheological (MR) fluid damper, which is a relatively new type of device that shows the future promise for the control of vibration, is modelled by MLP. A large number of simulations on the MR damper’s MLP model show that a much better model is selected using the proposed method.     

Application of neural networks in forecasting engine systems reliability

This paper presents a comparative study of the predictive performances of neural network time series models for forecasting failures and reliability in engine systems. Traditionally, failure data analysis requires specifications of parametric failure distributions and justifications of certain assumptions, which are at times difficult to validate. On the other hand, the time series modeling technique using neural networks provides a promising alternative. Neural network modeling via feed-forward multilayer perceptron (MLP) suffers from local minima problems and long computation time. The radial basis function (RBF) neural network architecture is found to be a viable alternative due to its shorter training time. Illustrative examples using reliability testing and field data showed that the proposed model results in comparable or better predictive performance than traditional MLP model and the linear benchmark based on Box–Jenkins autoregressive-integrated-moving average (ARIMA) models. The effects of input window size and hidden layer nodes are further investigated. Appropriate design topologies can be determined via sensitivity analysis.     

A quantified sensitivity measure for multilayer perceptron to input perturbation

The sensitivity of a neural network's output to its input perturbation is an important issue with both theoretical and practical values. In this article, we propose an approach to quantify the sensitivity of the most popular and general feedforward network: multilayer perceptron (MLP). The sensitivity measure is defined as the mathematical expectation of output deviation due to expected input deviation with respect to overall input patterns in a continuous interval. Based on the structural characteristics of the MLP, a bottom-up approach is adopted. A single neuron is considered first, and algorithms with approximately derived analytical expressions that are functions of expected input deviation are given for the computation of its sensitivity. Then another algorithm is given to compute the sensitivity of the entire MLP network. Computer simulations are used to verify the derived theoretical formulas. The agreement between theoretical and experimental results is quite good. The sensitivity measure can be used to evaluate the MLP's performance.     

Centroid based Multilayer Perceptron Networks

In this study we investigate a hybrid neural network architecture for modelling purposes. The proposed network is based on the multilayer perceptron (MLP) network. However, in addition to the usual hidden layers the first hidden layer is selected to be a centroid layer. Each unit in this new layer incorporates a centroid that is located somewhere in the input space. The output of these units is the Euclidean distance between the centroid and the input. The centroid layer clearly resembles the hidden layer of the radial basis function (RBF) networks. Therefore the centroid based multilayer perceptron (CMLP) networks can be regarded as a hybrid of MLP and RBF networks. The presented benchmark experiments show that the proposed hybrid architecture is able to combine the good properties of MLP and RBF networks resulting fast and efficient learning, and compact network structure.     

Sensitivity of feedforward neural networks to weight errors

An analysis is made of the sensitivity of feedforward layered networks of Adaline elements (threshold logic units) to weight errors. An approximation is derived which expresses the probability of error for an output neuron of a large network (a network with many neurons per layer) as a function of the percentage change in the weights. As would be expected, the probability of error increases with the number of layers in the network and with the percentage change in the weights. The probability of error is essentially independent of the number of weights per neuron and of the number of neurons per layer, as long as these numbers are large (on the order of 100 or more).     

A quantitative study of fault tolerance, noise immunity, and generalization ability of MLPs

An analysis of the influence of weight and input perturbations in a multilayer perceptron (MLP) is made in this article. Quantitative measurements of fault tolerance, noise immunity, and generalization ability are provided. From the expressions obtained, it is possible to justify some previously reported conjectures and experimentally obtained results (e.g., the influence of weight magnitudes, the relation between training with noise and the generalization ability, the relation between fault tolerance and the generalization ability). The measurements introduced here are explicitly related to the mean squared error degradation in the presence of perturbations, thus constituting a selection criterion between different alternatives of weight configurations. Moreover, they allow us to predict the degradation of the learning performance of an MLP when its weights or inputs are deviated from their nominal values and thus, the behavior of a physical implementation can be evaluated before the weights are mapped on it according to its accuracy.     

A general insight into the effect of neuron structure on classification

This paper gives a general insight into how the neuron structure in a multilayer perceptron (MLP) can affect the ability of neurons to deal with classification. Most of the common neuron structures are based on monotonic activation functions and linear input mappings. In comparison, the proposed neuron structure utilizes a nonmonotonic activation function and/or a nonlinear input mapping to increase the power of a neuron. An MLP of these high power neurons usually requires a less number of hidden nodes than conventional MLP for solving classification problems. The fewer number of neurons is equivalent to the smaller number of network weights that must be optimally determined by a learning algorithm. The performance of learning algorithm is usually improved by reducing the number of weights, i.e., the dimension of the search space. This usually helps the learning algorithm to escape local optimums, and also, the convergence speed of the algorithm is increased regardless of which algorithm is used for learning. Several 2-dimensional examples are provided manually to visualize how the number of neurons can be reduced by choosing an appropriate neuron structure. Moreover, to show the efficiency of the proposed scheme in solving real-world classification problems, the Iris data classification problem is solved using an MLP whose neurons are equipped by nonmonotonic activation functions, and the result is compared with two well-known monotonic activation functions.     

Interpretation and knowledge discovery from the multilayer perceptron network: Opening the black box

This paper interprets the outputs from the multilayer perceptron (MLP) network by finding the input data features at the input layer of the network which activate the hidden layer feature detectors. This leads directly to the deduction of the significant data inputs, the inputs that the network actually uses to perform the input/output mapping for a classification task, and the discovery of the most significant of these data inputs. The analysis presents a method for providing explanations for the network outputs and for representing the knowledge learned by the network in the form of significant input data relationships. During network development the explanation facilities and data relationships can be used for network validation and verification, and after development, for rule induction and data mining where this method provides a potential tool for knowledge discovery in databases (KDD).     

Computation of madalines' sensitivity to input and weight perturbations

The sensitivity of a neural network's output to its input and weight perturbations is an important measure for evaluating the network's performance. In this letter, we propose an approach to quantify the sensitivity of Madalines. The sensitivity is defined as the probability of output deviation due to input and weight perturbations with respect to overall input patterns. Based on the structural characteristics of Madalines, a bottom-up strategy is followed, along which the sensitivity of single neurons, that is, Adalines, is considered first and then the sensitivity of the entire Madaline network. By means of probability theory, an analytical formula is derived for the calculation of Adalines' sensitivity, and an algorithm is designed for the computation of Madalines' sensitivity. Computer simulations are run to verify the effectiveness of the formula and algorithm. The simulation results are in good agreement with the theoretical results.     

Evaluation of unsupervised feature extraction neural networks for face recognition

In this paper, new appearances based on neural networks (NN) algorithms are presented for face recognition. Face recognition is subdivided into two main stages: feature extraction and classifier. The suggested NN algorithms are the unsupervised Sanger principal component neural network (Sanger PCNN) and the self-organizing feature map (SOFM), which will be applied for features extraction of the frontal view of a face image. It is of interest to compare the unsupervised network with the traditional Eigenfaces technique. This paper presents an experimental comparison of the statistical Eigenfaces method for feature extraction and the unsupervised neural networks in order to evaluate the classification accuracies as comparison criteria. The classifier is done by the multilayer perceptron (MLP) neural network. Overcoming of the problem of the finite number of training samples per person is discussed. Experimental results are implemented on the Olivetti Research Laboratory database that contains variability in expression, pose, and facial details. The results show that the proposed method SOFM/MLP neural network is more efficient and robust than the Sanger PCNN/MLP and the Eigenfaces/MLP, when used a few number of training samples per person. As a result, it would be more applicable to utilize the SOFM/MLP NN in order to accomplish a higher level of accuracy within a recognition system.     

Sensitivity Analysis for Decision Boundaries

A novel approach is presented to visualize and analyze decision boundaries for feedforward neural networks. First order sensitivity analysis of the neural network output function with respect to input perturbations is used to visualize the position of decision boundaries over input space. Similarly, sensitivity analysis of each hidden unit activation function reveals which boundary is implemented by which hidden unit. The paper shows how these sensitivity analysis models can be used to better understand the data being modelled, and to visually identify irrelevant input and hidden units.     

A Novel Structure for Radial Basis Function Networks—WRBF

A novel structure for radial basis function networks is proposed. In this structure, unlike traditional RBF, we set some weights between input and hidden layer. These weights, which take values around unity, are multiplication factors for input vector and perform a linear mapping. Doing this, we increase free parameters of the network, but since these weights are trainable, the overall performance of the network is improved significantly. According to the new weight vector, we called this structure Weighted RBF or WRBF. Weight adjustment formula is provided by applying the gradient descent algorithm. Two classification problems used to evaluate performance of the new RBF network: letter classification using UCI dataset with 16 features, a difficult problem, and digit recognition using HODA dataset with 64 features, an easy problem. WRBF is compared with classic RBF and MLP network, and our experiments show that WRBF outperforms both significantly. For example, in the case of 200 hidden neurons, WRBF achieved recognition rate of 92.78% on UCI dataset while RBF and MLP achieved 83.13 and 89.25% respectively. On HODA dataset, WRBF reached 97.94% recognition rate whereas RBF achieved 97.14%, and MLP accomplished 97.63%.     

High-speed face recognition using self-adaptive radial basis function neural networks

In this work, we have proposed a self-adaptive radial basis function neural network (RBFNN)-based method for high-speed recognition of human faces. It has been seen that the variations between the images of a person, under varying pose, facial expressions, illumination, etc., are quite high. Therefore, in face recognition problem to achieve high recognition rate, it is necessary to consider the structural information lying within these images in the classification process. In the present study, it has been realized by modeling each of the training images as a hidden layer neuron in the proposed RBFNN. Now, to classify a facial image, a confidence measure has been imposed on the outputs of the hidden layer neurons to reduce the influences of the images belonging to other classes. This process makes the RBFNN as self-adaptive for choosing a subset of the hidden layer neurons, which are in close neighborhood of the input image, to be considered for classifying the input image. The process reduces the computation time at the output layer of the RBFNN by neglecting the ineffective radial basis functions and makes the proposed method to recognize face images in high speed and also in interframe period of video. The performance of the proposed method has been evaluated on the basis of sensitivity and specificity on two popular face recognition databases, the ORL and the UMIST face databases. On the ORL database, the best average sensitivity (recognition) and specificity rates are found to be 97.30 and 99.94%, respectively using five samples per person in the training set. Whereas, on the UMIST database, the above quantities are found to be 96.36 and 99.81%, respectively using eight samples per person in the training set. The experimental results indicate that the proposed method outperforms some of the face recognition approaches.     

An on-line wastewater quality predication system based on a time-delay neural network

The biological treatment process in a wastewater treatment system is a very complex process. The efficiency of the treatment is usually measured by laboratory tests, which typically take five days. In this paper, a time-delay neural network (TDNN) modeling method is proposed for predicting the treatment results. As the first step, a sensitivity analysis performed on a multi-layer perceptron (MLP) network model is used to reduce the input dimensions of the model. Then a TDNN model is further used to improve the performance of the original MLP network model. Subsequently, an on-line prediction and model-updating strategy is proposed and implemented. Simulations using industrial process data show that the prediction accuracy can be improved by the on-line model updating.     

Damage detection in Timoshenko beam structures by multilayer perceptron and radial basis function networks

This study investigates the efficiency of artificial neural networks (ANNs) in health monitoring of pristine and damaged beam-like structures. Beam modeling is based on Timoshenko theory. Two commonly used network models, multilayer perceptron (MLP) and radial basis neural network (RBNN), are used. Beam material and geometrical properties, beam end conditions and dynamically obtained data are used as input to the neural networks. The combinations of these parameters yield umpteenth input data. Therefore, to examine the effectiveness of ANNs, the frequency of intact beams is first tried to be determined by the network models, given the material and geometrical characteristics of beam elements and support conditions. The methodology to compute the vibrational data utilized in training the networks is provided. Showing the robustness of network models, the second stage of the study is carried out. At this stage, the crack parameters (e.g. the location and severity of crack) are estimated by the ANNs using the beam properties, beam end conditions and vibrational data, which consist of natural frequencies and mode shape rotation values. Despite the multiplexed input data, no data reduction schemes or multistage computations are executed in training and validation of neural network models. As a result of analysis runs, the optimal MLP and RBNN models are determined. Comparison of these models shows that the optimal RBNN algorithm performs better. The effectiveness of optimal ANN models in the presence of noise is also presented. As a conclusion, the trained network can be used as a diagnosis method in structural health monitoring of beam-like structures.     

Efficient detection of spurious inputs for improving the robustness of MLP networks in practical applications

The problem of the rejection of patterns not belonging to identified training classes is investigated with respect to Multilayer Perceptron Networks (MLP). The reason for the inherent unreliability of the standard MLP in this respect is explained, and some mechanisms for the enhancement of its rejection performance are considered. Two network configurations are presented as candidates for a more reliable structure, and are compared to the so-called negative training approach. The first configuration is an MLP which uses a Gaussian as its activation function, and the second is an MLP with direct connections from the input to the output layer of the network. The networks are examined and evaluated both through the technique of network inversion, and through practical experiments in a pattern classification application. Finally, the model of Radial Basis Function (RBF) networks is also considered in this respect, and its performance is compared to that obtained with the other networks described.     

基于函数逼近的多层前馈神经网络灵敏度分析

神经网络灵敏度分析对网络结构设计、硬件实现等具有重要的指导意义,已有的灵敏度计算公式对权值和输入扰动有一定限制或者计算误差较大。基于Piché的随机模型,通过使用两个逼近函数对神经网络一类Sigmoid激活函数进行高精度逼近,获得了新的神经网络灵敏度计算公式,公式取消了对权值扰动和输入扰动的限制,与其他方法相比提高了计算精度,实验证明了公式的正确性和精确性。     

Rough knowledge-based network, fuzziness and classification

A method of integrating rough sets and fuzzy multilayer perceptron (MLP) for designing a knowledge-based network for pattern recognition problems is described. Rough set theory is used to extract crude knowledge from the input domain in the form of rules. The syntax of these rules automatically determines the optimal number of hidden nodes while the dependency factors are used in the initial weight encoding. Results on classification of speech data demonstrate the superiority of the system over the fuzzy and conventional versions of the MLP.