Signal Processing for Remote Sensing - Chapter 3 docx

3 A Universal Neural Network–Based Infrasound Event Classifier Fredri c M. Ham and Ranjan Acha ryya CONTE NTS 3.1 Over view of Infrasound and Why Cl assify Infrasou nd Ev ents? 31 3.2 Neu ral Netw orks for Infrasound Class ification 32 3.3 Detail s of the Approa ch 33 3.3.1 Infraso und Data Collect ed for Trainin g and Testin g 34 3.3.2 Radial Ba sis Functio n Neu ral Ne tworks 34 3.4 Data Pr eprocessin g 38 3.4.1 Noi se Filterin g 38 3.4.2 Feat ure Extractio n Pro cess 38 3.4.3 Useful Definition s 42 3.4.4 Sel ection Proce ss for the Optima l Numbe r of Feature Vect or Compo nents 44 3.4.5 Optima l Outp ut Thres hold Value s and 3-D ROC Curves 44 3.5 Simul ation Results 47 3.6 Conc lusions 51 Ackno wledg ments 51 Refere nces 51 3.1 Overview of I nfrasound and Why C las sify Inf rasound Events? Infrasound is a longitudinal pressure wave [1–4]. The characteristics of these waves are similar to audible acoustic waves but the frequency range is far below what the human ear can detect. The typical frequenc y ran ge is from 0.01 to 10 Hz (Figure 3 .1). Natur e is an incredible creator of infrasonic signals that can emanate from sources such as volcano eruptions, earthquakes, severe weather, tsunamis, meteors (bolides), gravity waves, microbaroms (infrasound radiated from ocean waves), surf, mountain ranges (mountain associated waves), avalanches, and auroral waves to name a few. Infrasound can also result from man-made events such as mining blasts, the space shuttle, high-speed aircraft, artillery fire, rockets, vehicles, and nuclear events. Because of relatively low atmospheric absorption at low frequencies, infrasound waves can travel long distances in the Earth’s atmosphere and can be detected with sensitive ground-based sensors. An integral part of the comprehensive nuclear test ban treaty (CTBT) international monitoring syst em (IMS) is an infrasound network system [3]. The goal is to have 60 ß 2007 by Taylor & Francis Group, LLC. infrasound arrays operational worldwide over the next several years. The main objective of the infrasound monitoring system is the detection and verification, localization, and classification of nuclear explosions as well as other infrasonic signals-of-interest (SOI). Detection refers to the problem of detecting an SOI in the presence of all other unwanted sources and noises. Localization deals with finding the origin of a source, and classifica- tion deals with the discrimination of different infrasound events of interest. This chapter concentrates on the classification part only. 3.2 Neural Networks for Infrasound Classification Humans excel at the task of classifying patterns. We all perform this task on a daily basis. Do we wear the checkered or the striped shirt today? For example, we will probably select from a group of checkered shirts versus a group of striped shirts. The grouping process is carried out (probably at a near subconscious level) by our ability to discriminate among all shirts in our closet and we group the striped ones in the striped class and the checkered ones in the checkered class (that is, without physically moving them around in the closet, only in our minds). However, if the closet is dimly lit, this creates a potential problem and diminishes our ability to make the right selection (that is, we are working in a ‘‘noisy’’ environment). In the case of using an artificial neural network for classification of patterns (or various ‘‘events’’) the same problem exists with noise. Noise is everywhere. In general, a common problem associated with event classification (or detection and localization for that matter) is environmental noise. In the in frasound problem, many times the distance between the source and the sensors is relatively large (as opposed to region infrasonic phenomena). Increases in the distance between sources and sensors heighten the environmental dependence of the signals. For example, the signal of an infrasonic event that takes place near an ocean may have significantly different charac- teristics as compared to the same event that occurs in a desert. A major contributor of noise for the signal near an ocean is microbaroms. As mentioned above, microbaroms are generated in the air from larg e ocean waves. One important characteristic of neural networks is their noise rejection capability [5]. This, and several other attributes, makes them highly desirable to use as classifiers. FIGURE 3.1 Infrasound spectrum. 0.01 0.03 0.1 0.2 1.0 10.0 1 Megaton yield Volcano events Micro- baroms 1 Kiloton yield Impulsive events Gravity waves Mountain associated waves Bolide 0.01 0.03 0.1 0.2 1.0 10.0 Hz Hz ß 2007 by Taylor & Francis Group, LLC. 3.3 Details of the Approach Our approach of classifying infrasound events is based on a parallel bank neural network structure [6–10]. The basic architecture is shown in Figure 3.2. There are several reasons for using such an architecture; however, one very important advantage of dedicating one module to perform the classification of one event class is that the architecture is fault tolerant (i.e., if one module fails, the rest of the individual classifiers will continue to function). However, the overall performance of the classifier is enhanced when the parallel bank neural network classifier (PBNNC) architecture is used. Individual banks (or mod- ules) within the classifier architecture are radial basis function neural networks (RBF NNs) [5]. Also, each classifier has its own dedicated preprocessor. Customized feature vectors are computed optimally for each classifier and are based on cepstral coefficients and a subset of their associated derivatives (differences) [11]. This will be explained in detail later. The different neural modules are trained to classify one and only one class; however, for the requisite module responsible for one of the classes, it is also trained not to recognize all other classes (negative reinforcement). During the training process, the output is set to a ‘‘1’’ for a correct class and a ‘‘0’’ for all the other signals associated with all the other classes. When the training process is complete the final output thresholds will be set to an optimal value based on a three-dimensional receiver operating characteristic (3-D ROC) curve for each one of the neural modules (see Figure 3.2). Infrasound signal Pre-processor 1 Pre-processor 2 Pre-processor 3 Pre-processor 4 Infrasound class 1 neural network Infrasound class 2 neural network Infrasound class 3 neural network Infrasound class 4 neural network Infrasound class 5 neural network Infrasound class 6 neural network Pre-processor 5 Pre-processor 6 Optimum threshold set by ROC curve Optimum threshold set by ROC curve Optimum threshold set by ROC curve Optimum threshold set by ROC curve Optimum threshold set by ROC curve Optimum threshold set by ROC curve 1 0 1 0 1 0 1 0 1 0 1 0 FIGURE 3.2 Basic parallel bank neural network classifier (PBNNC) architecture. ß 2007 by Taylor & Francis Group, LLC. 3.3. 1 Infras ound Data Colle cted fo r Traini ng and Testing The data used for train ing and testing the individual network s are obt ained from mult iple infr asound arr ays locate d in differen t geogr aphi cal regions with differen t geome tries. The six infr asound classes used in this study are shown in Table 3.1, and the vari ous arr ay geomet ries are shown in Figure 3.3(a) through Figu re 3.3(e) [12,13 ]. Table 3.2 sho ws the vario us classes, along with the arr ay numbe rs where the data were collected , and the ass ociated sa mpling freque ncies. 3.3. 2 Radial Basis Fu nction Neur al Networ ks As previousl y mentioned , eac h of the neural netw ork modu les in Figure 3.2 is an RBF NN. A brief overview of RBF NNs will be given here. Thi s is not meant to be an exhaustive discourse on the subject, but only an introduction to the subject. More details can be found in Refs. [5,14]. Earlier work on the RBF NN was carried out for handling multivariate interpolation problems [15,16]. However, more recently they have been used for probability density estimation [17–19] and approximations of smooth multivariate functions [20]. In prin- ciple, the RBF NN makes adjustments of its weights so that the error between the actual and the desired responses is minimized relative to an optimization criterion through a defined learning algorithm [5]. Once trained, the networ k performs the interp olation in the output vector space, thus the generalization property. Radial basis functions are one type of positive-definite kernels that are extensively used for multivariate interpolation and approximation. Radial basis functions can be used for problems of any dimension, and the smoothness of the interpolants can be achieved to any desirable extent. Moreover, the structures of the interpolants are very simple. How- ever, there are several challenges that go alon g with the aforementioned attributes of RBF NNs. For example, many times an ill-conditioned linear system must be solved, and the complexity of both time and space increases with the number of interpolation points. But these types of problems can be overcome. The interpolation problem may be formulated as follows. Assume M distinct data points X ¼ {x 1 , ,x M }. Also assume the data set is bounded in a region V (for a specific class). Each observed data point x 2 R u (u corresponds to the dimension of the input space) may correspond to some function of x. Mathematically, the interpolation problem may be stated as follows. Given a set of M points, i.e., {x i 2 R u ji ¼ 1, 2, . . . , M} and a corresponding set of M real numbers {d i 2 R ji ¼ 1, 2, . . . , M} (desired outputs or the targets), find a function F:R M ! R that satisfies the interpolation condition F(x i ) ¼ d i , i ¼ 1, 2, , M (3:1) TABLE 3.1 Infrasound Classes Used for Training and Testing Class Number Event No. SOI (n ¼574) No. SOI Used for Training (n ¼351) No. SOI Used for Testing (n ¼223) 1 Vehicle 8 4 4 2 Artillery fire (ARTY) 264 132 132 3 Jet 12 8 4 4 Missile 24 16 8 5 Rocket 70 45 25 6 Shuttle 196 146 50 ß 2007 by Taylor & Francis Group, LLC. −20 YDIS, m XDIS, m 30 20 10 −10 −10 10 20 30−20−30−40 −20 −30 Sensor 1 (0.0, 0.0) Sensor 2 (−18.6, −7.5) Sensor 4 (15.3, −12.7) Array BP1 Sensor 3 (3.1, 19.9) YDIS, m XDIS, m 30 20 10 −10 −10 10 20 30 −20−30−40 −20 −30 Sensor 1 (0.0, 0.0) Sensor 5 (−19.8, 0.0) Sensor 2 (−1.3, −19.9) Sensor 4 (19.8, −0.1) Array BP2 Sensor 3 (0.7, 20.1) YDIS, m XDIS, m 30 20 10 −10 −10 10 20 30 40−20−30−40 −20 −30 Sensor 1 (0.0, 0.0) Sensor 2 (−22.0, 10.0) Sensor 4 (45.0, −8.0) Array K8201 Sensor 3 (28.0, −21.0) YDIS, m XDIS, m 30 20 10 −10 −10 10 20 30−30−40 −20 −30 Sensor 1 (0.0, 0.0) Sensor 2 (−20.1, 0.0) Sensor 4 (20.3, 0.5) Array K8202 Sensor 3 (0.0, 20.2) YDIS, m XDIS, m 30 20 10 −10 −10 10 20 30 −20−30−40 −20 −30 Sensor 1 (0.0, 0.0) Sensor 2 (−12.3, −15.8) Sensor 4 (14.1, −14.1) Arra y K8203 Sensor 3 (1.1, 20.0) (a) (b) (c) (e) (d) FIGURE 3.3 Five different array geometries. ß 2007 by Taylor & Francis Group, LLC. Thus , all the point s must pass thr ough the interpol ating sur face. A radial basis func tion may be a special inte rpolating func tion of the form F( x)¼ X M i¼ 1 w i f i ( xÀ x i kk 2 )(3:2) wh ere f ( . ) is kno wn as the radial basis functi on andk . k 2 deno tes the Euc lidean norm . In gene ral, the data point s x i are the center s of the radial ba sis func tions and are frequ ently writt en as c i . One of the problem s encounter ed when attempti ng to fit a func tion to da ta point s is over -fitting of the data, that is, the value of M is too large. Howeve r, general ly speaking, this is less a problem the RBF NN that it is with , for example , a multi-l ayer per ceptron train ed by backpro pagation [5]. The RBF NN is attemptin g to constru ct the hype rspace for a particul ar pro blem wh en given a limited number of da ta point s. Let us take another point of view concer ning how an RBF NN per form s its constr uction of a hype rsurface . Regul arization theo ry [5,14] is applied to the constr uction of the hype rsurface . A geomet rical explanati on follo ws. Consi der a set of input data obt ained from sever al events from a single cl ass. The inp ut data may be from temp oral sign als or defined features obt ained from thes e sign als usin g an appropriate transformation. The input data would be transformed by a nonlinear function in the hidd en layer of the RBF NN. Each event would then correspond to a point in the featu re spac e. Figure 3.4 depicts a two- dimensi onal (2-D) feature set, that is, the dimension of the output of the hidden layer in the RBF NN is two. In Figure 3.4, ‘‘(a)’’, ‘‘(b)’’, and ‘‘(c)’’ correspond to three separate events. The purpose here is to construct a surface (shown by the dotted line in Figure 3.4) such that the dotted region encompasses events of the same class. If the RBF network is to classify four different classes, there must be four different regions (four dotted contours), one for each class. Ideally, each of these regions should be separate with no overlap. However, because there is always a limited amount of observed data, perfect reconstruction of the hyperspace is not possible and it is inevitable that overlap will occur. To overcome this problem it is necessary to incorporate global information from V (i.e., the class space) in approximating the unknown hypersp ace. One choice is to introduce a smoothness constraint on the targets. Mathematical details will not be given here, but for an in-depth development see Refs. [5,14]. Let us now turn our attention to the actual RBF NN architecture and how the network is trained. In its basic form, the RBF NN has three layers: an input layer, one hidden TABLE 3.2 Array Numbers Associated with the Event Classes and the Sampling Frequencies Used to Collect the Data Class Number Event Array Sampling Frequency, Hz 1 Vehicle K8201 100 2 Artillery fire (ARTY) (K8201: Sites 1 and 2) K8201; K8203 (K8201: Sites 1 and 100; Sites 2 and 50); 50 3 Jet K8201 50 4 Missile K8201; K8203 50; 50 5 Rocket BP1; BP2 100; 100 6 Shuttle BP2; BP103 a 100; 50 a Array geometry not available. ß 2007 by Taylor & Francis Group, LLC. layer, and one output layer. Referring to Figure 3.5, the source nodes (or the input components) make up the input layer. The hidden layer performs a nonlinear trans- formation (i.e., the radial basis functions residing in the hidden layer perform this transformation) of the input to the network and is generally of a higher dimension than the input. This nonlinear transformation of the input in the hidden layer may be viewed as a basis for the construction of the input in the transformed space. Thus, the term radial basis function. In Figure 3.5, the output of the RBF NN (i.e., at the output layer) is calculated according to y i ¼ f i (x) ¼ X N k¼1 w ik f k (x,c k ) ¼ X N k¼1 w ik f k ( x À c k kk 2 ), i ¼ 1, 2, , m (no: outputs) (3:3) where x 2 R uÂ1 is the input vector, f k ( . ) is a (RBF) function that maps R þ (set of all positive real numbers) to R (field of real numbers), k . k 2 denotes the Euclidean norm, w ik are the weights in the output layer, N is the number of neurons in the hidden layer, and c k 2 R uÂ1 are the RBF centers that are selected based on the input vector space. The Euclidean distance between the center of each neuron in the hidden layer and the input to the network is computed. The output of the neuron in a hidden layer is a nonlinear (a) (b) (c) Feature 1 Feature 2 FIGURE 3.4 Example of a two-dimensional feature set. f 1 f 2 f N W T ∈ ℜ m × N Σ Σ y 1 y m x 1 x 2 x 3 x u Input layer Hidden layer Output layer FIGURE 3.5 RBF NN architecture. ß 2007 by Taylor & Francis Group, LLC. func tion of this dista nce, and the output of the network is compu ted as a wei ghted sum of the hidde n layer outpu ts. The func tional form of the radial ba sis func tion, f k ( . ), can be any of the follow ing: . Line ar func tion: f( x)¼ x . Cubi c appr oximatio n: f (x)¼ x 3 . Thin- plate-s pline function: f (x)¼ x 2 ln(x) . Gaus sian function : f (x)¼ exp(À x 2 /s 2 ) . Mult i-quadrat ic functi on: f ( x)¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þs 2 p . Inv erse multi-qu adratic function : f (x)¼ 1=( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þs 2 p ) The parameter s contr ols the ‘‘width ’’ of the RBF and is common ly referr ed to as the spre ad par ameter. In man y pr actical appl ications the Gaus sian RBF is used. The center s, c k , of the Gau ssian func tions are points used to per form a sampl ing of the inp ut vector spac e. In general , the center s form a subset of the inp ut data. 3. 4 D ata P reproc essi ng 3.4. 1 Noise Filtering Microb aroms, a s pre viousl y def ined, are a persisten tly pres ent sour ce of noise that resides in most collected infr asound sign als [21–23 ]. Mi crobarom s are a clas s of infr asonic sign als charact eriz ed by narrow-b and, nearly sinu soidal wavef orms, with a period bet ween 6 and 8 sec . These signal s can be gene rated by marin e sto rms through a non- linear inte raction of surfac e waves [24]. The freque ncy conten t of the microbar oms often coinci des with that of small-yi eld nuclea r ex plosions. This co uld be bothersome in man y appli cations; howev er, simple band-pass filtering can allevi ate the pro blem in man y case s. The refore, a band-pass filter with a pass band betwee n 1 and 49 Hz (for signal s samp led at 100 Hz) is used here to elimi nate the effects of the microb aroms. Figure 3.6 shows how band-pass filteri ng can be used to elimina te the mi crobarom s problem . 3.4. 2 Featur e E xtractio n Process Dep icted in eac h of the six graphs in Figure 3.7 is a collectio n of eight sign als from each cl ass, that is, y ij (t ) for i¼ 1, 2, . . . , 6 (classe s) and j¼ 1, 2, . . . , 8 (num ber of sign als) (see Ta ble 3.1 for total number of signals in each class). A feature extra ction proces s is desired that wil l captu re the salient features of the sign als in each class and at the same time be invarian t relative to the arr ay geome try, the geog raphica l locat ion of the array , the samp ling frequenc y, and the leng th of the time window . The overall per formance of the cl assifier is co ntingen t on the data that is used to train the neural netw ork in each of the six modu les shown in Figure 3.2. Moreo ver, the neu ral netw ork’s ability to distinguish between the various events (presented the neural networks as feature vectors) is the distinctiveness of the features between the classes. However, within each class it is desirable to have the feature vectors as similar to each other as possible. There are two major questions to be answered: (1) What will cause the signals in one class to have markedly different characteristics? (2) What can be done to minimize these ß 2007 by Taylor & Francis Group, LLC. differen ces and achieve uni formit y with in a class and distin ctive ly dif ferent featu re vector charact eristic s between class es? The answer to the first questi on is quite simple—n oise. This can be noise as sociated with the sensors, the data acquis ition equip ment, or other unwa nted sign als that are not of interest. The answ er to the sec ond question is also quite sim ple (once you know the answ er)—using a feature extra ction process ba sed on compu ted cepst ral co efficients and a subset of thei r assoc iated der ivatives (di fferences ) [10,11 ,25 –28]. As me ntioned in Secti on 3.3, each cl assifier has its own dedicat ed pre process or (see Figure 3.2). Customi zed feature vec tors are compute d optim ally for each clas sifier (or neural module) and are based on the aforem entio ned cepst ral coefficie nts and a subset of their associa ted deriva tives (or differen ces). The pr eprocessin g proce dure is as follow s. Each time-do main signal is first norm alized and then its mean value is co mpute d and remove d. Next , the power spect ral dens ity (P SD) is calcu lated for each signal, whic h is a mixtur e of the desire d comp onent and possi bly other unwa nted signal s and noise. Therefor e, when the PSDs are comp uted for a set of signal s in a defin ed class there will be spect ral compo nents ass ociate d wi th noise and ot her unwa nted signal s that need to be suppres sed. This can be systemati cally accomp lished by first compu ting the av erage PSD (i.e., PSD avg ) over the suite of PSDs for a particul ar class . The spect ral co mponent s are define d as m i for i¼ 1, 2, . . . for PSD avg . The max imum spect ral compo nent, m max ,of PSD avg is then deter mined. This is consi dered the dominan t spectral comp onent wi thin a particul ar cl ass and its value is used to supp ress selected comp onents in the res ident PSDs for any particu lar cl ass according to the follo wing: if m i >« 1 m max (typ ically « 1 ¼ 0: 001) then m i m i else « 2 m i (typically « 2 ¼ 0:00001) 1.5 0.5 0 0 500 Time samples Before filtering Amplitude 1000 –0.5 1 1 0.0 –0.5 0 500 Time samples After filtering Amplitude 1000 –1 0.5 Before filtering 8 6 4 2 0 –50 0 Frequency (Hz)Frequency (Hz) After filtering Magnitude 50 400 300 200 100 0 –50 0 Magnitude 50 FIGURE 3.6 Results of band-pass filtering to eliminate the effects of microbaroms (an artillery signal). ß 2007 by Taylor & Francis Group, LLC. To some extent, this will minimize the effects of any unwanted components that may reside in the signals and at the same time minimize the effects of noise. However, another step can be taken to further minimize the effects of any unwanted signals and noise that may reside in the data. This is based on a minimum variance criterion applied to the spectral components of the PSDs in a particular class after the previously described step is completed. The second step is carried out by taking the first 90% of the spectral compon- ents that are rank-ordered according to the smallest variance. The rest of the components Raw time-domain 8 signals for vehicle Amplitude 0.8 0.6 0.4 0.2 –0.2 –0.4 –0.6 0 100 200 300 400 Time (sec) 500 600 700 0 Raw time-domain 8 signals for missile Amplitude 2 1.5 1 0.5 –0.5 –1 –1.5 –2 –2.5 –3 0 1000 2000 3000 4000 Time (sec) 5000 6000 7000 8000 0 Raw time-domain 8 signals for artillery Amplitude 1.8 1.6 1.4 1.2 0.6 0.8 0.4 0.2 0 0 50 100 150 200 Time (sec) 250 300 350 1 Raw time-domain 8 signals for rocket Amplitude 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 200 400 600 800 Time (sec) 1000 1200 1400 1600 Vehicle class Missile class Artillery class Rocket class (a) (b) (d) (c) Raw time-domain 8 signals for jet Amplitude 1.4 1.2 1 0.8 0.4 0.2 0 –0.4 –0.2 0 500 1000 1500 25002000 Time (sec) 3000 3500 4000 4500 0.6 Raw time-domain 8 signals for shuttle Amplitude 1 0.9 0.8 0.7 0.5 0.4 0.3 0.2 0.1 0 100 200 300 400 Time (sec) 500 600 700 800 900 1000 0.6 Jet class Shuttle class (e) (f) FIGURE 3.7 (See color insert following page 178.) Infrasound signals for six classes. ß 2007 by Taylor & Francis Group, LLC. [...]... (3: 12) 10 (a) Feature set for vehicle 10 8 Amplitude 9 8 Amplitude 9 7 6 7 6 5 5 4 4 3 5 10 15 20 25 Feature number 3 30 5 10 15 20 25 Feature number 30 (d) Feature set for rocket 10 (c) Feature set for artillery fire 10 9 8 8 Amplitude 9 Amplitude (b) Feature set for missile 7 6 7 6 5 5 4 4 3 5 10 15 20 25 Feature number 3 30 5 (e) Feature set for jet 10 15 20 25 Feature number 30 (f) Feature set for. .. 70 Performance mean 65 60 34 Features 55 50 45 40 10 15 20 25 Feature number 30 35 40 Feature number 45 Mean 66.9596 69.2 735 60 50 55 60 47.95 53 40 55 Variance 34 50 122. 834 500 450 400 Performance variance 35 0 30 0 250 34 Features 200 150 100 50 0 10 15 20 25 30 35 40 Feature number 45 FIGURE 3. 10 Performance means and performance variances versus feature number used to determine the optimal length... 0.2 0 (c) ROC 3- D plot for artillery 0.2 0.4 0.6 0 8 2 1 Tru 0.5 osi tive ep 0.2 0 0 (d) ROC 3- D plot for rocket False positive 0.4 0.6 1 0.8 False positive Misclassification 5 4 3 Artillery 2 1 0 1 0.5 ep osit ive Tru 0 0 0.2 0.4 0.6 0.8 4 Rocket 3 2 1 0 1 1 Tru e p 0.5 osit ive False positive (e) ROC 3- D plot for jet 0 0 0.2 0.4 0.6 0.8 1 False positive (f) ROC 3- D plot for shuttle 5 5 4 3 Misclassification... value for 40 elements in the feature vector is (slightly) larger than that for 34 elements, the variance for 40 is nearly three times that for 34 elements Therefore, a length of 34 elements for the feature vectors is the best choice 3. 4.5 Optimal Output Threshold Values and 3- D ROC Curves At the output of the RBF NN for each of the six neural modules, there is a single output neuron with hard-limiting... this approach TABLE 3. 6 Six-Class Classification Result Using a Threshold Value for Each Network Performance Type CCR ACC Binary Outputs (%) Bi-Polar Outputs (%) 90.1 94.6 88 .38 92.8 TABLE 3. 7 Six-Class Classification Result Using ‘‘Winner-Takes-All’’ Performance Type CCR ACC ß 2007 by Taylor & Francis Group, LLC Binary Method (%) Bi-Polar Method (%) 93. 7 93. 7 92.4 92.4 3. 6 Conclusions Radial basis... classifications for one event The accuracy (ACC) is given by ACC ¼ ß 2007 by Taylor & Francis Group, LLC No: correct predictions pþs ¼ No predictions pþqþrþs (3: 13) TABLE 3. 3 Confusion Matrix for a Two-Class Classifier Predicted Value Actual Value Positive Negative Positive p r Negative q s As seen from Equation 3. 12 and Equation 3. 13, if multiple classifications occur, the CCR is a more conservative performance... and the output threshold for each neural module was determined one by one by fixing the spread parameter, i.e., s, for all other neural modules to 0 .3, and holding the threshold value at 0.5 Once the first neural module’s spread ß 2007 by Taylor & Francis Group, LLC (a) ROC 3- D plot for vehicle (b) ROC 3- D plot for missile 5 Misclassification Misclassification 4 3. 5 Vehicle 3 2.5 2 1.5 1 0.8 Tru e... of Signals in Noise, 2nd ed., Academic Press, San Diego, CA, 1995 31 Smith, S.W., The Scientist and Engineer’s Guide to Digital Signal Processing, California Technical Publishing, San Diego, CA, 1997 32 Hanley, J.A and McNeil, B.J., The meaning and use of the area under a receiver operating characteristic (ROC) curve, Radiology, 1 43( 1), 29 36 , 1982 33 Demuth, H and Beale, M., Neural Network Toolbox for. .. fixed at 0 .3 and 0.5, respectively Table 3. 4 gives the final values of the spread parameter and the output threshold for the global classifier Figure 3. 13 shows the classifier architecture with the final values indicated for the RBF NN spread parameters and the output thresholds Table 3. 5 shows the confusion matrix for the six-classifier Concentrating on the 6 Â 6 portion of the matrix for each of... set by ROC curve (0 .31 44) 1 0 s2 = 2.2 Pre-processor 2 Infrasound class 2 neural network 1 0 Optimum threshold set by ROC curve (0.6921) s3 = 0 .3 Pre-processor 3 Infrasound signal Infrasound class 3 neural network 1 0 Optimum threshold set by ROC curve (0.9221) s4 = 1.8 Pre-processor 4 Infrasound class 4 neural network 1 0 Optimum threshold set by ROC curve (0.4446) s5 = 0.2 Pre-processor 5 Infrasound . to 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.5 2 2.5 3 3.5 4 False positive (a) ROC 3- D plot for vehicle (c) ROC 3- D plot for artillery (e) ROC 3- D plot for jet (f) ROC 3- D plot for shuttle (d) ROC 3- D plot for rocket (b) ROC 3- D plot for missile True. orks for Infrasound Class ification 32 3. 3 Detail s of the Approa ch 33 3. 3.1 Infraso und Data Collect ed for Trainin g and Testin g 34 3. 3.2 Radial Ba sis Functio n Neu ral Ne tworks 34 3. 4 Data. three-dimensional receiver operating characteristic ( 3- D ROC) curve for each one of the neural modules (see Figure 3. 2). Infrasound signal Pre-processor 1 Pre-processor 2 Pre-processor 3 Pre-processor