Some properties of Hermite polynomials.
\r\n\t
",isbn:"978-1-83881-922-4",printIsbn:"978-1-83881-921-7",pdfIsbn:"978-1-83881-923-1",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"dcfc52d92f694b0848977a3c11c13d00",bookSignature:"Dr. Fiaz Ahmad and Prof. Muhammad Sultan",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10454.jpg",keywords:"Agricultural Engineering, Technologies, Application, Sustainable Agriculture, Information Technology in Agriculture, Food Security, Renewable Energies, Precision Farming, Smart Agriculture, Farm Mechanization, Robotics, Post Harvest Technologies",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"November 25th 2020",dateEndSecondStepPublish:"December 23rd 2020",dateEndThirdStepPublish:"February 21st 2021",dateEndFourthStepPublish:"May 12th 2021",dateEndFifthStepPublish:"July 11th 2021",remainingDaysToSecondStep:"2 months",secondStepPassed:!0,currentStepOfPublishingProcess:4,editedByType:null,kuFlag:!1,biosketch:"Dr. Ahmad is a researcher in the field of agricultural mechanization and agricultural equipment engineering, in-charge of Farm Machinery Design Laboratory at Bahauddin Zakariya University, with expertise in modeling and simulation. He applied for two patents at the national level.",coeditorOneBiosketch:"Renowned researcher with a focus on developing energy-efficient heat- and/or water-driven temperature and humidity control systems for agricultural storage, greenhouse, agricultural livestock and poultry applications including HVAC, desiccant air-conditioning, adsorption, Maisotsenko cycle (M-cycle), and adsorption desalination.",coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"338219",title:"Dr.",name:"Fiaz",middleName:null,surname:"Ahmad",slug:"fiaz-ahmad",fullName:"Fiaz Ahmad",profilePictureURL:"https://mts.intechopen.com/storage/users/338219/images/system/338219.jpg",biography:"Fiaz Ahmad obtained his Ph.D. (2015) from Nanjing Agriculture University China in the field of Agricultural Bioenvironmental and Energy Engineering and Postdoc (2020) from Jiangsu University China in the field of Plant protection Engineering. He got the Higher Education Commission, Pakistan Scholarship for Ph.D. studies, and Post-Doctoral Fellowship from Jiangsu Government, China. During postdoctoral studies, he worked on the application of unmanned aerial vehicle sprayers for agrochemical applications to control pests and weeds. He passed the B.S. and M.S. degrees in agricultural engineering from the University of Agriculture Faisalabad, Pakistan in 2007. From 2007 to 2008, he was a Lecturer in the Department of Agricultural Engineering, Bahauddin Zakariya University, Multan-Pakistan. Since 2009, he has been an Assistant Professor in the Department of Agricultural Engineering, BZ University Multan, Pakistan. He is the author of 33 journal articles. He also supervised 6 master students and is currently supervising 5 master and 2 Ph.D. students. In addition, Dr. Ahmad completed three university-funded projects. His research interests include the design of agricultural machinery, artificial intelligence, and plant protection environment.",institutionString:"Bahauddin Zakariya University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"0",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"Bahauddin Zakariya University",institutionURL:null,country:{name:"Pakistan"}}}],coeditorOne:{id:"199381",title:"Prof.",name:"Muhammad",middleName:null,surname:"Sultan",slug:"muhammad-sultan",fullName:"Muhammad Sultan",profilePictureURL:"https://mts.intechopen.com/storage/users/199381/images/system/199381.jpeg",biography:"Muhammad Sultan completed his Ph.D. (2015) and Postdoc (2017) from Kyushu University (Japan) in the field of Energy and Environmental Engineering. He was an awardee of MEXT and JASSO fellowships (from the Japanese Government) during Ph.D. and Postdoc studies, respectively. In 2019, he did Postdoc as a Canadian Queen Elizabeth Advanced Scholar at Simon Fraser University (Canada) in the field of Mechatronic Systems Engineering. He received his Master\\'s in Environmental Engineering (2010) and Bachelor in Agricultural Engineering (2008) with distinctions, from the University of Agriculture, Faisalabad. He worked for Kyushu University International Institute for Carbon-Neutral Energy Research (WPI-I2CNER) for two years. Currently, he is working as an Assistant Professor at the Department of Agricultural Engineering, Bahauddin Zakariya University (Pakistan). He has supervised 10+ M.Eng./Ph.D. students so far and 10+ M.Eng./Ph.D. students are currently working under his supervision. He has published more than 70+ journal articles, 70+ conference articles, and a few magazine articles, with the addition of 2 book chapters and 2 edited/co-edited books. Dr. Sultan is serving as a Leading Guest Editor of a special issue in the Sustainability (MDPI) journal (IF 2.58). In addition, he is appointed as a Regional Editor for the Evergreen Journal of Kyushu University. His research is focused on developing energy-efficient heat- and/or water-driven temperature and humidity control systems for agricultural storage, greenhouse, livestock, and poultry applications. His research keywords include HVAC, desiccant air-conditioning, evaporative cooling, adsorption cooling, energy recovery ventilator, adsorption heat pump, Maisotsenko cycle (M-cycle), wastewater, energy recovery ventilators; adsorption desalination; and agricultural, poultry and livestock applications.",institutionString:"Bahauddin Zakariya University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"Bahauddin Zakariya University",institutionURL:null,country:{name:"Pakistan"}}},coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"8",title:"Chemistry",slug:"chemistry"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"252211",firstName:"Sara",lastName:"Debeuc",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/252211/images/7239_n.png",email:"sara.d@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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\nwhere the terms are in the usual meanings, namely, \n
where \n
The QHO is a very good approximation in solving systems of diatomic molecules vibrating under the spring constant [1, 2, 5] and finds various modern physics applications such as in [8, 9, 10] as stated in a famous quotation: “the career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction by Sidney Coleman” [10, 11]. Here, U(r) is central potential which can be given in Cartesian coordinates (\n
JWKB method is known to give exact eigenenergies for the QHO, but eigenfunctions fail at and around the classical turning points (CTPs) where \n
The QHO in (3) is a bound-state problem which can be written in 1D for the potential function in 1D (\n
or, simply,
\nwhose solution by various conventional approaches (such as analytical, algebraic, approximation, etc.) is given in any fundamental textbooks, that is, [1, 2, 3, 5] and whose results can be summarized as follows [14]:
\n\n
TISE for the QHO in 1D in dimensionless form:
\n
Note that here \n
Exact eigenenergies:
Graphs of \n\nf\n\n\nλ\n≥\n0\n\ny\n\n\n and \n\ng\n\n\nλ\n≥\n0\n\ny\n\n=\n0\n\n.
\n
Exact eigenfunctions (wave functions) in \n
\n
By using (6), we have the wave functions in x:
\n
We used two different symbols (\n
Rodriguez formula: \n\n | \n
Generating function: \n\n | \n
Some of the Hermite polynomials: \n\n \n\n | \n
Recurrence relations: \n\n \n\n | \n
Evenness and oddity \n\n \n\n | \n
Orthogonality: \n\n | \n
Some properties of Hermite polynomials.
is given along with the related Mathematica codes in [14].
\n2D plot of Figure 1 is schematically given in Figure 2 for the QHO under study (in the dimensionless form) from which we have the following outcomes [14]:
\nSchematic 2D sketch of \n\nf\n\n\n\nλ\nn\n\n\n\nE\nn\n\n\n≥\n0\n\ny\n\n≕\n\nk\n2\n\n\n\n\nλ\nn\n\n\n\n\nE\nn\n\n≥\n0\n\n\n\ny\n\n\n for a given \n\n\nλ\nn\n\n\n.
JWKB eigenenergies can be found by applying the Bohr-Sommerfeld quantization formula given by [1, 2, 3, 4, 5, 6, 7, 14]:
\nas follows:
\nwhich is already the exact solution given in (8) [1, 2, 3, 4, 6, 7, 14]. Results are given along with the MAF solutions in Table 2 for comparison. Note that we use the following notation for the symmetrical (or even parity (EP)) and antisymmetrical (or odd parity (OP)) solutions:
\nIndex (=MAF index) | \nEP | \nOP | \n||||||||
---|---|---|---|---|---|---|---|---|---|---|
MAF | \nJWKB* | \nMAF | \nJWKB* | \n|||||||
\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n
0 | \n1.20348 | \n0.0603317 | \n\n\n | \n0 | \n\n\n | \n2.33811 | \n1.01735 | \n\n\n | \n1 | \n\n\n | \n
1 | \n3.27162 | \n2.0115 | \n\n\n | \n2 | \n\n\n | \n4.08795 | \n3.0079 | \n\n\n | \n3 | \n\n\n | \n
2 | \n4.83082 | \n4.0063 | \n\n\n | \n4 | \n\n\n | \n5.52056 | \n5.00508 | \n\n\n | \n5 | \n\n\n | \n
3 | \n6.16988 | \n6.00435 | \n\n\n | \n6 | \n\n\n | \n6.78671 | \n7.00374 | \n\n\n | \n7 | \n\n\n | \n
4 | \n7.37677 | \n8.00332 | \n8.50332 \n | \n8 | \n\n\n | \n7.94413 | \n9.00295 | \n\n\n | \n9 | \n\n\n | \n
5 | \n8.49195 | \n10.0027 | \n\n\n | \n10 | \n\n\n | \n9.02265 | \n11.0024 | \n\n\n | \n11 | \n11.5 \n | \n
6 | \n9.5382 | \n12.0023 | \n\n\n | \n12 | \n\n\n | \n10.0402 | \n13.0021 | \n\n\n | \n13 | \n\n\n | \n
7 | \n10.5299 | \n14.0019 | \n\n\n | \n14 | \n\n\n | \n11.0085 | \n15.0018 | \n\n\n | \n15 | \n\n\n | \n
JWKB and MAF eigenenergies.
JWKB solution is exact.
where the subscripts “J, ns” represent J, JWKB, and ns, symmetrical indices (ns = even), and similarly, “\n
Conventional first-order JWKB solution of the QHO given in the normal form in (4) or (7) is as follows:
\nwhere \n
However, we are fortunately not helpless: since \n
where \n
where the superscripts (par.m.) and (asy.m.) represent parity matched and asymptotically matched JWKB solutions, respectively. Eqs. (18) and (19) tells that we will take \n
so that they exhibit the following asymptotic behaviors:
\nWhen initial values at \n
is applied to the JWKB solution in (15), we find the following:
\nwhere the second complementary solution (in the sine form) has been canceled and calculation of the integral in the cosine term can be calculated by the similar change of variable as in (13) whose result will give \n
and by using (16), we have.
\nNow, by applying the JWKB connection formula with a small phase term \n
and the asymptotically matched (modified) wave function in region III via (20) and (21) of [3, 4, 7, 14] gives:
\nAbbreviations we use for the EP JWKB solutions here (and also for the OP solutions in the next subsection) are as follows [14]:
\nSince we have already calculated \n
Exact and JWKB solutions of EP wave functions (for \n\np\n=\n1\n\n).
Similarly, by using the boundary conditions for the OP case in (17), namely (by using (16)),
\nand starting with region II.
\nconnecting to region III in the first quadrant (\n
whose asymptotic matching gives.
\nAgain, since we have already obtained \n
If we follow the QHO in dimensionless form given in (7), we have the following properties in MAF theories [3, 18, 19, 20, 21, 22, 23]:
\nFormal MAF method suggests a solution to the TISE in (7) in terms of Airy functions as follows:
\nwhere \n
Now, with the choice of the last term in (35) as zero, we find the following:
\nHere, the property of the Airy functions, \n
where \n
Starting from the second term, we have.
\nwhere b1 is some constant, and finally, making the first term in (35) zero (which is the only assumption in the MAF method), we have the following:
\nOr more correctly in two-variable form in our eigenvalue system.
\ncan be thought as a measure of the accuracy of the MAF solution, namely, \n
The same results would also be obtained if we had chosen the other linearly independent solution, \n
or more correctly in two variables here in our study.
\nwhere \n
For a symmetrical f as in Figure 2, we have even-parity (EP) and odd-parity (OP) MAF wave functions just as in JWKB method, but now it leads to two different MAF quantization formulas with two different MAF universal constants regarding EP and OP solutions as given in [3] and as we study in this section. We again use the symbolism in (9) (\n
where the denominator can be written in the following form [3]:
\nEven-parity (EP) eigenenergies: if we apply the EP formulas of the exact solution in (17), by using (16), to the MAF wave functions, we have the following:
where we used \n
where the last term vanishes as \n
whose substitution in (48) gives.
\nNow, by the substitution of \n
where the subscripts sn stand for s, symmetrical solution (EP), and n, quantization order (nth quantization), and Zsn is the nth solution of the differential equation in (52) regarding the symmetrical solution. Now, by using the results in (13), we find the MAF quantization formula regarding the symmetrical solution:
\nwhere \n
Odd-parity (OP) eigenenergies: similarly, if we apply the OP formulas of the exact solution in (17), by using (16), to the MAF wave functions, we have the following:
where, similarly, the subscripts an stand for a, antisymmetrical (OP), and n, quantization order (nth quantization), and Zan is the nth solution of the equation in (55) regarding the asymmetrical solutions. Similarly, by using the results in (13), we find the MAF quantization formula regarding the antisymmetrical solution:
\nwhere \n
By using a tentative boundary condition with q = 1 for the EP solutions, we have found the result in (47), and we said that we would extend it by considering the parity matching for \n
or
\nSimilarly, for the antisymmetric parity wave functions, we have.
\nwhere constant coefficients \n
Again, we use the schematic sketch given in Figure 2 for the QHO under study.
\nSince we have tactically used (53)–(57) to resemble the MAF quantization formula to the JWKB quantization formula given in (12), by using the result of calculation of the same integral in (13), we have the following results regarding the MAF eigenenergies of the QHO:
\nMAF eigenenergies are given in Table 2 along with the JWKB solutions (which are already exact) for some n values in comparison. (Note again that we used ns and na to represent the symmetrical (EP) and antisymmetrical MAF indices in Table 2, respectively).
\nFor the regions IIb and III, we have the following definitions:
\nCalculation of \n
where \n
Similarly, calculation of the constant coefficient in (60) for the antisymmetric boundary values given in (54) with q = 1 gives.
\nSince the MAF solutions of both EP and OP solutions are very close to the exact solutions given in Figures 3 and 4, their absolute and relative error graphs with respect to the exact solution are given in Figures 5 and 6. We can also see that there are no discontinuities at the CTPs in the MAF solutions when compared with the JWKB solutions given in Figures 3 and 4.
\nExact and JWKB solutions of OP wave functions (for \n\nq\n=\n1\n\n).
Relative and absolute error of EP MAF solutions.
Relative and absolute error of OP MAF solutions.
Here we studied the fundamental outcomes of the two conventional semiclassical approximation methods, namely, JWKB and MAF methods pedagogically, and obtained the solutions of the QHO by these semiclassical methods by using the parity conditions of the expected solutions by using the dimensionless form of the QHO system. We applied the asymptotic matching and parity matching procedure to obtain the correct form of semiclassical solutions. As expected, JWKB solutions diverge at and around the CTPs, whereas MAF solutions do not. As also expected (since being typical), JWKB eigenenergies are exact, whereas MAF eigenenergies are unfortunately not but very accurate as expected from an approximation method. In the MAF method, function p in (40) or in (43) is assumed zero. Indeed, it is very close to zero to give approximate results, and function P in (40) or in (43) can be used as an approximation criterion for the MAF method [3, 18]. However, improved MAF methods (IMAF) or perturbation corrections concerning the nonzero P function seem straightforward to improve the accuracy of the MAF solutions as in [3, 20, 22]. Normally, for an even potential function in the TISE, EP and OP initial values are as given in (17), but due to the conversion factor β in (11) or (16), for the QHO in the dimensionless form (in \n
Author acknowledges special thanks to the IntechOpen for the financial support in the publishment of this chapter.
\nEpileptic seizure detection deals with the process of detecting a seizure when it occurs. The need of the day is to take forward this work to eventually predict a seizure much before it is detected as it the very nature of the seizure that it is random. This chapter discusses various methods to do the same.
\nThe cause of disorder will remain unexplained unless a complete cure is possible and available. Two practical engineering approaches are used to research in epilepsy. The first approach involves monitoring the brain activity on multiple scales which gives us a base to understand the generation of seizures. The second approach is to model the natural properties of the brain network and manipulate these for the modulation of seizure generation.
\nThis work mainly concentrates on amalgamation of the above approaches towards developing a closed loop device which has a feedback of brain signals to the device so that it can control interventions that stop seizures.
\nThe main objective in this chapter is a search for a precursor for seizure prediction mainly in the preictal phase as shown in the Figure 1. This may have form of an identifiable, significant pattern, feature or a pattern to extract the feature.
\nSeizure prediction methodology.
Five techniques are used to achieve this objective. They are:
\nUsing Lyapunov exponents.
\nUsing Cross wavelets [1].
\nFourier Bessel function [2].
\nWavelets [3].
\nEMD [4].
\nA seizure prediction method to predict the transitions between Inter ictal and pre ictal states using cross wavelet and Lyapunov exponent features and neural network for binary classification had been proposed [1]. The CHB-MIT database was used.
\nThe cross wavelet transform (XWT) of two time series xn and yn is defined as WXY = WXWY∗, where * denotes complex conjugation. We further define the cross wavelet power as \n
A mathematical function which detects chaos is the Lyapunov exponents. Lyapunov exponents are the average exponential rates of divergence or convergence of nearby orbits in phase space.
\nWhere \n
The data is divided into Preictal and interictal as per the information of expert. Three types of preictal data is considered for experimentation. The methods adopted for prediction system are as shown in the block diagram below (Figures 2 and 3):
\nBlock diagram of epilepsy prediction system using cross wavelets, Lyapunov exponents and neural networks.
Block diagram showing flow of seizure prediction using wavelet.
The data is having 23 channels. The channels are selected as per standard bipolar montage, electrode placement and channel information is provided in Table 1 in which channels are divided as 11 pairs to calculate cross wavelet coefficients.
\nPair Number | \nLeft side Electrodes | \nChannel Number | \nRight Side electrodes | \nChannel Number | \n
---|---|---|---|---|
1 | \nFp1 -F7 | \n1 | \nFp2-F8 | \n13 | \n
2 | \nFp1-F3 | \n5 | \nFp2-F4 | \n9 | \n
3 | \nT7-P7 | \n3 | \nT8-P8 | \n15 | \n
4 | \nC3-P3 | \n7 | \nC4-P4 | \n11 | \n
5 | \nP3-O1 | \n8 | \nP4-O2 | \n12 | \n
6 | \nP7-O1 | \n4 | \nT8-O2 | \n16 | \n
Division of channels into 11 pairs to calculate cross wavelet coefficients.
where F:Frontal P:Posterior T:Temporal C:Central O:Occipital.
Cross wavelet features are extracted from 11 channel pairs which are applied to Feed forward Back propagation neural network having two layers with 11 input neurons as input layer and one output neuron as one output layer. +1 is assigned as target for pre ictal features and − 1 for inter ictal features. The network trained and tested for various feature vectors and the results are tabulated in Table 2.
\nData | \nTrue positive (TP) | \nFalse positive(FP) | \nSensitivity (%) | \nSpecificity (%) | \n
---|---|---|---|---|
Preictal (1 min) | \n152 | \n28 | \n8x.4 | \n— | \n
Preictal (2 min) | \n295 | \n65 | \n81.9 | \n— | \n
Preicta (5 min) | \n634 | \n86 | \n88.05 | \n— | \n
\n | TN | \nFN | \n\n | \n |
Inter Ictal | \n902 | \n34 | \n\n | 96.36 | \n
\n | Over all accuracy (%) | \n90.3 | \n
Prediction performance of neural network with cross wavelet features.
The above table can be interpreted as follows:
\nFor the consideration of interictal period, it is the TN and FN values which are taken into consideration as we need to minimize false alerts. It can be seen that the TN and FN values were 902 and 34 respectively with 96.36% specificity. The preictal data on the other hand had 88.05 sensitivity for 5 minutes data.
\nThe lyapunov exponent is calculated from 23 channels, the extracted features are given to Feed forward back propagation neural network. 23 input nodes and one output node. The network is trained with preictal and interictal features the training performance is evaluated and results are tabulated in Table 3.
\nData | \nTrue positive (TP) | \nFalse positive(FP) | \nSensitivity (%) | \nSpecificity (%) | \n
---|---|---|---|---|
Preictal | \n180 | \n0 | \n100 | \n— | \n
Inter Ictal | \n297-TN | \n3-FN | \n— | \n99 | \n
\n | Overall accuracy (%) | \n99.37 | \n
Prediction performance of neural network with lyapunov features.
From the above Table 3, we can notice that the number of TP values for preictal period is 180 whereas there were no FP and 100% sensitivity when prediction was done with lyapunov features. In comparison, the inter ictal period had shown 287 TN and 3 FN with 99% specificity. The overall accuracy was 99.37%.
\nFeature extraction is done using DWT. EEG signals contain all the useful information below 30 Hz and for this reason 4 decomposition levels D1-D4 and one final approximation, A4 are chosen [3].
\nBased on EEG Ictal period marking of experts selected preictal and interictal periods. These data is decomposed using discrete wavelet transform [3]. Out of 7 sub bands selected three sub bands D2, D3, Dx. These decomposition details are mentioned in Table 4.
\nDECOMPOSED SIGNAL | \nFREQUENCY BANDS(HZ) | \nDECOMPOSITION LEVEL | \n
---|---|---|
D1 | \n128—256 | \n1(NOISES) | \n
D2 | \n64–128 | \n2(HIGHGAMA) | \n
D3 | \n32----64 | \n3(GAMA) | \n
D4 | \n16---32 | \n4(BETA) | \n
D5 | \n8-----16 | \n5ALPHA) | \n
D6 | \n4 -----8 | \n6(THEETA) | \n
A6 | \n0---4 | \n6(DELTA) | \n
Frequency bands and corresponding decomposition levels.
From these sub bands 4 features power, covariance, inter Quartile Range (IQR) and median absolute deviation (MAD) are extracted from 23 channels of pre ictal and interictal EEG data. Three channels are selected and the feature vector size is Equal to 36 = 3 (channels) x 3 (sub bands D2, D3, D4) x4 (features-power, covariance, IQR, and MAD) from each epochs of preictal and Interictal EEG data. These features are applied to feed forward back propagation neural network as shown in Figure 4. Two layers are used hidden layer 36 neurons and output layer having 36 neurons. It is binary classification target +1 is assigned for preictal (Epiliptic) data and − 1 is assigned to Inter Ictal (normal). Total 1588 epochs (1 second) are used for classification 800 for training and 788 used for testing. The performance is evaluated in terms of sensitivity, Specificity and Overall accuracy.
\nTwo types of data is chosen. First data has a time horizon of around 5 minutes for the pre-ictal period while the second has the time horizon for 10 minutes. The inter-ictal period is considered to be around 2 hours in order to nullify the post-ictal or seizure effects.
For comparison of performance, Elman Back propagation neural network is used. The performance of Elman Network is tabulated in Table 5. Sensitivity in Elman network is high, specificity and overall accuracy are low. By comparisons of two types of neural networks feed forward network having better overall performance as the overall accuracy is about 88.71% compared to 85.9% of Elman back propagation.
\n(TP) | \n(FP) | \nSensitivity(%) | \n(TN) | \n(FN) | \nSpecificity (%) | \nOverall accuracy (%) | \n
---|---|---|---|---|---|---|
296 | \n4 | \n98.6 | \n381 | \n107 | \n78.1 | \n85.9 | \n
Elman back propagation neural network performance.
(TP) | \n(FP) | \nSensitivity (%) | \n(TN) | \n(FN) | \nSpecificity (%) | \nOverall accuracy(%) | \n
---|---|---|---|---|---|---|
273 | \n27 | \n91 | \n462 | \n62 | \n87 | \n88.71 | \n
Feed forward neural network performance.
Any signal can be represented in terms of Fourier Bessel series due to its decaying nature. An EEG signal is expanded into a Fourier Bessel series [2]. In this way, an EEG signal can be segmented and periods interictal and ictal are classified to predict the occurrence of seizure.
\nA 1–1 mapping exists between the frequencies and the coefficients. \n
All the above Figures 5–7 show the segmented bands of a seizure signal.
\nFirst plot shows original signal followed by segmented EEG seizure signal of ictal period.
First plot shows original signal followed by segmented EEG seizure signal of inter ictal period.
First plot shows original signal followed by segmented EEG seizure signal of pre-ictal period.
The five features energy in each sub band, fmean, IQR and MAD are extracted from each sub band.
\nThe Figure 8 shows the sum of all Bessel coefficients the preictal and interictal features are discriminating.
\nAbsolute sum of Bessel coefficients with red being Preictal and blue being Interictal EEG signals.
From the Figure 9 it can be observed that the feature, Median absolute deviation of Fourier Bessel coefficients for the Interictal and preictal are discriminating.
\nMAD of coefficients with red being Preictal and blue being Interictal EEG signals.
The inter ictal and pre ictal data is prepared as per the information in Table 7. The calculated Fourier-Bessel Coefficients from inter ictal and pre ictal data is given to Neural Network with 64 input neurons, one output neuron and one hidden layer. The Feed Forward Back propagation algorithm was used as shown in Figure 10. The network is trained −1 as target for inter -ictal and + 1 for pre-ictal.
\nEEG Sub Band | \nFrequency Range (Hz) | \nFourier-Bessel Coefficient(m) | \n
---|---|---|
DELTA | \n0–4 | \n0–4 | \n
THEETA | \n4–7 | \n4–7 | \n
ALPHA | \n7–13 | \n7–13 | \n
LOW BETA | \n13–15 | \n13–15 | \n
HIGH BETA | \n15–30 | \n15–30 | \n
LOW GAMA | \n30–65 | \n30–65 | \n
HIGH GAMA | \n65–120 | \n65–120 | \n
Mapping of frequencies to the Fourier-Bessel coefficients.
The neural network architecture used above contains three layers: 64 neuron input layer, 1 neuron output layer and a hidden layer in the middle which also has 64 neurons.
The trained network is simulated with Inter-ictal and Pre-ictal data. There was one epoch as false negative and zero epochs as false positives. The simulation results had garnered 150 epochs of inter -ictal and 150 epochs of pre-ictal data. Inter ictal period is used to study sensitivity where as the pre ictal data is used for specificity.
\nThe number of false negative values should be low so that it should have high sensitivity. The specificity must be high with lower false positive values. From Table 8, it is observed that sensitivity, specificity and accuracy of the proposed method is superior and the seizure is predicted before 5 minutes for subject 1.
\nFile Name | \nFile Start Time | \nFile End Time | \nNumber of Seizures | \nSeizure start seconds | \nSeizure End seconds | \n
---|---|---|---|---|---|
chb01_01 | \n11:42:54 | \n12:42:54 | \n0 | \n— | \n— | \n
chb01_03 | \n13:43:04 | \n14:43:04 | \n1 | \n2996 | \n3036 | \n
chb01_15 | \n01:44:44 | \n2:44:44 | \n1 | \n1732 | \n1772 | \n
Seizure information of Subject-1 with timing in seconds.
TP | \nFN | \nSensitivity | \nTN | \nFP | \nSpecificity | \nTCA | \n
---|---|---|---|---|---|---|
149 | \n01 | \n99.33 | \n150 | \n0 | \n100% | \n99.6% | \n
Sensitivity, specificity and classification accuracy.
The inter-ictal and pre ictal data is prepared as per the information in Table 10. The trained network is simulated with inter-ictal and pre-ictal data. There were zero epochs as false negative and zero epochs as false positives.
\nFile Name | \nNumber of Seizures | \nSeizure Start(seconds) | \nSeizure End(seconds) | \n
---|---|---|---|
chb24_13 | \n1 | \n3288 | \n3304 | \n
chb24_14 | \n1 | \n1939 | \n1966 | \n
chb24_15 | \n1 | \n3552 | \n3569 | \n
Seizure information of Subject-2 with timing in seconds.
The simulation results of 150 epochs of inter-ictal and 150 epochs of pre-ictal data have been tabulated as below in Table 11.
\nTP | \nFN | \nSE | \nTN | \nFP | \nSP | \nTCA | \n
---|---|---|---|---|---|---|
150 | \n0 | \n100% | \n150 | \n0 | \n100% | \n100% | \n
Sensitivity, specificity and classification accuracy.
The number of false negative and false positive values was minimum due to the fact that the testing was done for shorter periods.
\nFrom Table 11 it is observed that for shorter periods under consideration seizure is predicted before 5 minutes for subject 2 with 100% accuracy.
\nThe selection of data was done a bit different from the previous works. Care has been taken to reduce the effects of post seizure by taking a minimum gap of 2 hours in the inter-ictal period.
\nUsing the EEG data as compiled from above, IMF’s are extracted using the EMD technique. Using these IMF’s, features such as Kurtosis, Inter-quartile range and Median Absolute Deviation are extracted. The following Figure 11 shows the steps involved in the study for prediction. The extracted features are used for training the Neural network and the results are tabulated.
\nSteps involved in epileptic seizure prediction using epileptic zone. It is divided into three parts. 1) the first part extracted the IMF’s while in the second part 2) features are extracted from these IMF’s. These features are given as 3) input to the neural network in the third part.
For patient 8, source has been localized as discussed in the topic of source localization. It has been observed that 4 channels 6,8,20 and 21 have been the most significant channels. These channels are decomposed into 4 IMF’s out of which 3 significant features are extracted thus a total of 4x4x3 = 48 features are extracted.
\n600 preictal and interictal epochs of 2 second duration are considered respectively, which means 1200 epochs (600 + 600 = 1200) with 48 features add up to a total input vector of 1200x48 to the neural network. This is tabulated as shown below in Table 12.
\nFEATURE | \nVECTOR LENGTH | \n
---|---|
CHANNELS | \n4 (6,8,20 and 21) | \n
INTRINSIC MODE FUNCTIONS | \n4 levels | \n
FEATURES | \n3 (MAD, IQR, Kurtosis) | \n
TOTAL FEATURE VECTOR | \n4 X 4 X 3 = 48 | \n
PRE-ICTAL EPOCHS [2 SECOND] | \n600 | \n
INTER-ICTAL EPOCHS [2 SECOND] | \n600 | \n
TOTAL INPUT VECTOR TO NN | \n(1200) X 48 | \n
An overview of the input vector to neural network.
The following results were obtained in this method:
\nThe concept is extended to all the patients whose source has been localized as shown in below Table 14.
\nThe prediction method is run on the entire channels localized from the source as derived from Table 14. The results are as shown in the Table 13. The above results are obtained for data of short intervals. A testing has been run for continues data whose results are as shown in the figures below:
\n\n | True Positive(TN) | \nFalse Negative(FP) | \nSensitivity (%) | \nTrue Negative(TN) | \nFalse Positive(FP) | \nSpecificity (%) | \nOver all accuracy | \n
---|---|---|---|---|---|---|---|
[5 Min] | \n289 | \n11 | \n96.33 | \n290 | \n10 | \n96.67 | \n96.5 | \n
[10 Min] | \n300 | \n— | \n100 | \n295 | \n5 | \n98.33 | \n99.16 | \n
Sensitivity, specificity and classification accuracy using epileptic zone for prediction.
When a seizure free data is considered, there is a chance for false alarm. Consider the Figure 12 where the result of testing of continuous seizure free data is shown.
\nThe testing for continuous seizure data where seizure is predicted 30.4 min before onset.
This false positive problem in seizure free data cannot be taken as a chance for seizure. Thus a false alarm avoidance methodology should be used (Figures 13 and 14).
\nA continuous seizure free data is used for testing. Since it is seizure free no transition should occur. There can be some spikes observed from the above zoomed in figure.
Continuous seizure data with false positive values.
A continuous occurrence of around 10 can be ignored so that no false alarm is triggered. In the above Figures 9 and 10 continuous occurrences happen. Thus, it can be ignored.
\nA new method is proposed for generalization of prediction. There are a few limitations using generalization of epileptic seizure prediction. One of the limitations is the variation issue. Focal seizures are particular to the part of the brain.
\nGeneralization of seizure prediction is possible with the help of epileptic source localized perfectly with clinical support using PET, FMRI, etc. For this work, the results of source localization are used. Table 14 shows the results obtained from source localization. The data of these six patients are considered and a generalization is applied by averaging of the each level. The results obtained are as tabulated below in Table 16.
\nPatient | \nChannels | \n
---|---|
1 | \n1, 5, 9, 13, 14, 15 and 21 | \n
2 | \n1,12,15 and 9 | \n
3 | \n1,4,6,8,14,20 and 21 | \n
5 | \n2,3,9,15,19 and 23 | \n
8 | \n6,8,20 and 21 | \n
24 | \n5,6,20 and 21 | \n
Source localization results.
\n | True Positive (TP) | \nFalse Negative (FN) | \nSensitivity (%) | \nTrue Negative (TN) | \nFalse Positive (FP) | \nSpecificity (%) | \nOver all accuracy | \n
---|---|---|---|---|---|---|---|
[chb01] | \n290 | \n10 | \n96.66 | \n277 | \n33 | \n89 | \n92.8 | \n
Chb02 | \n282 | \n18 | \n94 | \n290 | \n10 | \n96.66 | \n95.3 | \n
Chb03 | \n284 | \n16 | \n9x.66 | \n288 | \n12 | \n96 | \n95.3 | \n
Chb05 | \n270 | \n30 | \n90 | \n264 | \n36 | \n88 | \n89 | \n
Chb24 | \n288 | \n12 | \n96 | \n286 | \n14 | \n95.33 | \n95.6 | \n
Sensitivity, specificity and classification accuracy using epileptic zone for prediction for all patients from source localization in Table 14.
Subjects | \nSensitivity (%) | \nSpecificity (%) | \nOver all accuracy (%) | \n
---|---|---|---|
Multiple(6) | \n81.7 | \n76.2 | \n79.75 | \n
Sensitivity, specificity and over all accuracy obtained for generalization of prediction.
S No | \nAuthor | \nyear | \nData Base | \nAlgorithm | \nPrediction Time | \nSpecificity | \nSensitivity | \nAccuracy | \n
---|---|---|---|---|---|---|---|---|
1 | \nHaddad, T [5] | \n2014 | \nEEG | \ngraph theory | \n30 min | \n— | \n— | \n72% | \n
2 | \nNai-Fu Chang [6] | \n2012 | \nCHB-MIT | \nwavelet coherence | \n— | \n\n | \n | 70% | \n
3 | \nChristopher J. James [7] | \n2009 | \n— | \nICA, Phase Synchronization | \n35 min | \n65–80% | \n65–100% | \n— | \n
4 | \nMaryann D’Alessandro [8] | \n2003 | \nEEG | \nintelligent genetic search process | \n\n | 90.47% | \n62.5% | \n\n |
5 | \nLeon D. Iasemidis [9] | \n2003 | \nEEG | \nLyapunov exponents | \n71.7 min | \n\n | \n | \n |
6 | \nPiotr Mirowski [10] | \n2009 | \nEEG | \ncross correlation | \n\n | \n | 71% | \n\n |
7 | \nChisci [11] | \n2010 | \nFreiburg ECOG | \nSVM classifier based on the Kalman filter, | \n— | \n100% | \n100% | \n— | \n
8 | \nDorai, Arvind [12] | \n2010 | \nEEG | \nLyapunov exponents | \n65 seconds | \n\n | \n | 8x.17% | \n
9 | \nYang Zheng[13] | \n2014 | \nEEG | \nbivariate empirical mode decomposition | \n— | \n— | \n\n | \n |
10 | \nPeyvand Ghaderyan [14] | \n2014 | \nFreiburg EEG | \nKNN-SVM | \n— | \n86.1% | \n91.11% | \n— | \n
11 | \npresent work | \n2013 | \nCHB-MIT | \nLyapunov exponents | \n2 min | \n99% | \n100% | \n99.37% | \n
12 | \npresent work | \n2013 | \nCHB-MIT | \nWavelets | \n5 min | \n100% | \n91% | \n88.71% | \n
13 | \npresent work | \n2014 | \nCHB-MIT | \nFourier Bessel | \n5 min | \n100% | \n99.33% | \n99.6% | \n
14 | \npresent work | \n2014 | \nCHB-MIT | \nLocalization-EMD-ANN | \n5 min | \n96.67% | \n96.33% | \n96.5% | \n
15 | \npresent work | \n2014 | \nCHB-MIT | \nLocalization-EMD-ANN | \n10 min/30 min | \n98.33% | \n100% | \n99.16% | \n
Comparison of prediction results.
From the above table it can be noticed that the sensitivity obtained by generalization is 81.7%, while the specificity is 76.2%. The overall prediction accuracy stands at 79.75%.
\nEMD proves to be a good technique for seizure prediction. The main distinguishing attribute of this work is that it has been able to forecast the seizure about 30 minutes in advance. This might be a result obtained due to the preictal period being much longer and the effects being nullified. The other existing prediction works were capable of only a few minutes. This gives the work much weight in the field of medicine as an alarm can be raised much well in advance and the life of a patient can be saved by alerting either the doctors or the patient himself to take necessary precautions. The concept of generalization can be improved with the help of other existing source localization techniques which make use of PET, FMRI, etc.
\nThe existing works for prediction using Lyapunov exponents as seen in S.no “5” had a prediction time of 71.7 minutes. The present work done using Lyapunov exponents was able to achieve a staggering result of 2 minutes prediction time with 99% specificity, 100% sensitivity and an overall classification accuracy of 99.97%.
\nS.no “2” had got a classification accuracy of 70% using wavelet coherence. The present work achieved a classification accuracy of 88.71% with 100% specificity and 91% sensitivity. The present works using Fourier Bessel as well as the EMD techniques have got good results.
The above table is an indicator that progressive improvement has taken place in both the prediction time and prediction accuracy after the employment of localization and selecting only certain electrodes of interest. Most of the previous literature is incomplete and this work aimed to bridge the gap. There has been significant success achieved in this segment.
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