Nonparametric Bayesian techniques have been developed recently to extend the sophistication of factor models, allowing one to infer the number of appropriate factors from the observed data. We consider such techniques for sparse factor analysis, with application to gene-expression data from three virus challenge studies. Particular attention is placed on employing the Beta Process (BP), the Indian Buffet Process (IBP), and related sparseness-promoting techniques to infer a proper number of factors. The posterior density function on the model parameters is computed using Gibbs sampling and variational Bayesian (VB) analysis.Time-evolving gene-expression data are considered for respiratory syncytial virus (RSV), Rhino virus, and influenza, using blood samples from healthy human subjects. These data were acquired in three challenge studies, each executed after receiving institutional review board (IRB) approval from Duke University. Comparisons are made between several alternative means of per-forming nonparametric factor analysis on these data, with comparisons as well to sparse-PCA and Penalized Matrix Decomposition (PMD), closely related non-Bayesian approaches.Applying the Beta Process to the factor scores, or to the singular values of a pseudo-SVD construction, the proposed algorithms infer the number of factors in gene-expression data. For real data the "true" number of factors is unknown; in our simulations we consider a range of noise variances, and the proposed Bayesian models inferred the number of factors accurately relative to other methods in the literature, such as sparse-PCA and PMD. We have also identified a "pan-viral" factor of importance for each of the three viruses considered in this study. We have identified a set of genes associated with this pan-viral factor, of interest for early detection of such viruses based upon the host response, as quantified via gene-expression data. X ∈ ℝp×n, where each column corresponds to one of n samples, quantifying the associated gene-expression values for all p genes under investigation. We typically must address the "large p, small n" problem , o, oA reprtudent-t ) priors.tudent-t  and the tudent-t .i.e., to infer r). The non-Bayesian approaches are often sequential, and one may infer r by monitoring the error ||F E||as a function of iteration number  to  to 9] toss (IBP) , 20]. I. Ir (in ss (IBP) , 22]..r , computational efficiency is of major practical importance. In previous use of nonparametric Bayesian methods to this problem, a Gibbs sampler has typically been employed .In practice ak from the Beta-Bernoulli construction above defines the kth column of the factor-loading matrix A. The factor-score matrix S utilizes the binary vectors zj = T defined in (5), for j ∈ {1, . . . , n}. Specifically, we define the jth column of S as S are sparse.To complete the model specifications, note that S, as above, they may alternatively be used to define A, with factor scores designed as in traditional factor models. This approach was considered in [p draws from the IBP to construct A, and typically p is very large for the gene-expression problems of interest. When presenting results in Section II-B, we discuss our experience with this model on small-scale problems, although this approach was found computationally intractable for the motivating virus studies considered in Section II-D.In the above discussion the Beta-Bernoulli and IBP processes were presented for a specific construction of the factor-analysis model, with the goal of making the connection to the model explicit and hence clearer. However, there are alternative ways of utilizing the IBP for design of factor models. Specifically, rather than using the binary vectors to construct dered in , using ar is closely related to the non-Bayesian sparse-PCA [ak}k = 1,K as before, using a sparseness-promoting prior like that discussed in Section IV-A. Further, the factor scores ξk for factor loading k is drawn ξk ~ In), for kth row of S, and we consider K such rows, for large K (relative to the anticipated r). Finally, the vector of pseudo singular values λ =  is generatedThe final Bayesian construction considered for inferring arse-PCA  and penaarse-PCA  models. λ select the columns of A used across all columns of X. As discussed in Section IV, the number of non-zero components of λ is drawn Binomial)), and the posterior on the number of such components provides desired information on the number of factors r. Note that this construction is like the Beta-Bernoulli process discussed above, in that it utilizes k π~ Beta) and the Bernoulli distribution; however, it only draws the binary vector z once, and therefore there is not the idea of multiple "customers", as in the two IBP-related formulations discussed above.The matrix product AS in (1) is now constituted as r. Running on a typical modern PC, the computation times are summarized in Table The MCMC results presented here correspond to using 5000 collection samples, after a burn-in of 2000 iterations. However, with 2000 burn-in iterations and 500 collection samples, the average results of the factor scores and factor loadings are almost identical to those found with 5000. For all MCMC results, we employed a singular value decomposition (SVD) of the data matrix to initialize the factor loading and factor score matrix in the FA model, as well as the right-and left-singular matrix in the matrix decomposition model. For each iteration of the Gibbs sampler a particular number of factors r are employed, and based upon all collection samples one may infer an approximate posterior distribution for α = 1, β = 1, c = 1, and d = g = h = e = f = 10−6. These parameters were not optimized, and were set in the same manner for all experiments. Although the PMD model is a non-Bayesian method, it also has parameter settings that must be addressed carefully; two hyper-parameters need adjusting: the sparseness threshold and the stop condition [To be explicit, we provide detailed hyper-parameter settings for the model in (7)-(14); the other models are set similarly. Specifically, ondition . In all All calculations were performed on PCs with Intel Pentium Dual E2200 processors and 2.00 GB memory, and all software was written in Matlab. For the large-scale analysis performed on the real data discussed above, MCMC required approximately 4 hours of CPU, while VB required 3 hours .The following authors performed the statistical analysis: BC, MC, JP, AH, JL, DD and LC. The following authors executed the three viral challenge studies, and performed all biological interpretation of the ressults: AZ, CW and GSG. All authors read and contributed to writing this paper.We here provide a concise summary of the inference methods applied to one of the Bayesian FA models discussed above, with this representative of the analysis applied to the rest. Specifically, we consider the model discussed in Section IV-B, in which the BP is applied within the factor-score matrix. The complete model may be expressed asi = 1, . . . , n, j = 1, . . . , p and k = 1, . . . , K.where The full likelihood of the model isThe sequential update equations of the Gibbs sampler are as follows.kiz. The probability of ki z= 1 is expressed as• Sample each entry of the binary matrix, k πfrom kp(π|-) = Beta where • Sample jk Afrom jkp(A|−) = jkA; jkμ, Σjk) where • Sample each entry of factor loading matrix, si, from sip(|−) = is; ξi, Λi) where K-dimensional vector,iz, repeated p times, 1 ≤ i ≤ n.• Sample each column of factor score matrix, j ψfrom • Sample jk from p(γjk|-) = Gamma  where c' = c+1/2 and • Sample γδ from p(δ|-) = Gamma  where e' = e + nK/2 and jk|p(γ−) represent the probability of jk γconditioned on all other parameters.• Sample X), where in Θ ≡{A,S,Z,α,π,ψ,γ,δ} Our objective is to optimize the parameters Γ in the approximation Q. Toward that end, consider the variational expressionWe seek a distribution Q to approximate the exact posterior p(Θ|p(X) is a constant with respect to Γ, and therefore KL||p(Θ|X)) is minimized. However, we cannot explicitly compute the KL divergence, since p(Θ|X) is unknown. However, the denominator term in p(X)p(Θ|X) = p(X|Θ)p(Θ), and the prior p(Θ) and likelihood function p(X|Θ) are available. To make computation of Q(Θ|Γ) has a factorized form Q = ΠiQi. With appropriate choice of iQ, the variational expression Note that the term ki zwe have ki z= 1. We consider the following two conditions:• For discussion below, the symbol < • > represents the expectation of the argument.k πwe have • For jk Awe have • For , where i swe have K-dimensional vector of all zi repeated p times. In order to exactly calculate the expectation, • For B are k ≤ K, 1 ≤ j ≤ p and 1 ≤ i ≤ n., we have to consider it as two parts. Specifically, the off-diagonal elements of j ψwe have • For jk γwe have • For δ we have Q(δ) Gamma , where e' = e + Kn/2 and • For The research reported here was supported under the DARPA PHD program. The research and results presented here are the contributions of the authors, and the results were not influenced in any way by the sponsor.


The abnormal event detection problem is an important subject in real-time video surveillance. In this paper, we propose a novel online one-class classification algorithm, online least squares one-class support vector machine (online LS-OC-SVM), combined with its sparsified version (sparse online LS-OC-SVM). LS-OC-SVM extracts a hyperplane as an optimal description of training objects in a regularized least squares sense. The online LS-OC-SVM learns a training set with a limited number of samples to provide a basic normal model, then updates the model through remaining data. In the sparse online scheme, the model complexity is controlled by the coherence criterion. The online LS-OC-SVM is adopted to handle the abnormal event detection problem. Each frame of the video is characterized by the covariance matrix descriptor encoding the moving information, then is classified into a normal or an abnormal frame. Experiments are conducted, on a two-dimensional synthetic distribution dataset and a benchmark video surveillance dataset, to demonstrate the promising results of the proposed online LS-OC-SVM method. After recording events by a visual sensor, such as a camera, obtaining detailed information of individual or crowd behavior is a challenging object in this area; automatic abnormal event detection is required to provide convenience, safety and an efficient lifestyle for humanity . An abnoIn –4, abnorOn the other hand, low-level motion features were employed. In , the autTrajectory information is also adopted to detect abnormal events. In ,11, the We consider the model-free approach, which does not require an explicit statistical model. To be accurate, the support vector machine (SVM) classification method is relied on in this paper. Inspired by the satisfactory performance of a covariance feature descriptor representing object in a tracking problem, a covariance descriptor characterizes the moving information of a global frame. In a tracking problem, the covariance descriptor is constructed of the blob intensity or color for template matching. In this paper, covariance encodes the optical flow of the global frame.The rest of the paper is organized as follows. In Section 2, related works are briefly reviewed. In Section 3, the online least squares one-class support vector machine (online LS-OC-SVM) classification method is originally derived. In Section 4, a covariance matrix descriptor is described to provide feature vectors for the classification algorithm. In Section 5, we propose abnormal detection methods based on the online LS-OC-SVM. In Section 6, we present the results on synthetic data and real-world video scenes. Finally, Section 7 concludes the paper.2.i.e., all training data are given a priori and are learned together. If additional training data arrive afterward, the SVM must be retrained from scratch T at the time step, n, could be calculated as:A, B, C and D with suitable sizes T becomes:Instead of i.e.,After providing these relations with the dictionary, we now discuss the dictionary construction. The coherence criterion is adopted to characterize a dictionary in sparse approximation problems. It provides an elegant model reduction criterion with a less computationally-demanding procedure ,40,41. Txjw is the element in the dictionary, x. Presetting a threshold, μ0, the new arrival sample, xt, at the time step, t, is tested with the coherence criterion to judge whether the dictionary remains unchanged or is incremented by including the new element. For n training samples, the subset, which includes m (1 ≤ m ≪ n) samples, is considered the initial dictionary. Then, each remaining sample is tested with tϵ ≤ μ0, it will be included into the dictionary. Concretely, the algorithm is performed with two cases described herein below.In the online case, the coherence between a new datum and the current dictionary is calculated by:First case:tϵ > μ0n + 1, the new data, xn+1, is not included into the dictionary. The Gram matrix, K, with the entries, κ, i, j ∈ {1, 2, …, D}, is unchanged. When a new sample, x, arrives, we need to compute:n + 1, κ is the column vector with entries κ, j ∈ {1, 2, …, D}. K (x) is the matrix with the -th entry κ, i ∈ {1, 2, …, n}, j ∈ {1, 2, …, D}.In this case, at time step Second case:ϵt ≤ μ0nx+1, is added into the dictionary, x. Then, the Gram matrix should be changed by:In this case, the new data, where K̅ is the Gram matrix of the dictionary, including the new arrival dictionary sample, xn+1, and K is the Gram matrix of the dictionary at the last time step, n. Let x = {xw1, xw2, …, xDw } denote the dictionary at time step n; d is the column vector with entries j= κd, j ∈ {1, 2, …, D}, and d = κ. By adopting the matrix inverse identity K(x) and also S denote the updatedn + 1; we have:n + 1, q is the column vector with entries iq = κ, i ∈ {1, 2, …, n}, and xD+1 is the new arrival datum xn+1, which is included into the dictionary. The matrix inverse in A, U, C and V, such that U and V should be chosen as two vectors, and A should be chosen as a scaler. Thus, the inverse, (C−1 + V A−1U), is a scaler; K(x)bq⊤), we regard two vectors, (K(x)b) and q⊤, as vector U and V, respectively, while C in with proper choices of matrices S, using κ with entries κ, j ∈ {1, 2, …,D, D + 1}, xjw is an element of the dictionary.Once knowing 4.I is the intensity of the image, xI, yI and tI are the derivatives of the image intensity value along the x, y and time t dimension, u and υ are the components of the optical flow in horizontal and vertical direction and γ represents the weight of the regularization term.The optical flow is chosen as the basic low-level feature to represent the movement direction and amplitude. We apply the Horn-Schunck (HS) method to compute optical flow in this paper. The optical flow can provide important information about the spatial arrangement of the object and the change rate of this arrangement . The optI is the image , hue-saturation-value (HSV), hue-lightness-saturation (HLS), etc), iϕ is a mapping relating the image with the i – th feature, F is the W × H × d dimension feature, W is the image width, H is the image height and d is the number of the chosen features. For each frame, the feature, F, can be represented as the d × d covariance matrix:n is the number of the pixels sampled in the frame, μ is the mean of n feature vectors of the selected points and zk is the feature vector of the k – th point. C is the covariance matrix of the feature vector, F. The covariance matrix descriptor proposes a way to merge multiple features. Different choices of feature vectors are shown in u and υ are horizontal and vertical components of optical flow, xu and xυ are the first derivatives of horizontal and vertical optical flow in the x direction, respectively, yu and yυ are the first derivatives of the corresponding feature in the y direction, xxu and xxυ are the second derivatives in x direction and yyu and yyυ are the second derivatives in y direction. The flowchart of covariance matrix descriptor computation is shown in The covariance feature descriptor was originally proposed by Tuzel  for pattXi and Xj are matrices in Lie group G; the parameter σ determines the scale at which the data is probed.If proper parameters are given, classical kernels, such as Gaussian, polynomial and sigmoidal kernels, have similar performances . In our fined as ,46:(45)5.I1, I2,…, nI} (the positive class), describing the normal behavior is obtained. The abnormal detection strategies relative to the online algorithms proposed in Section 3.3 and Section 3.4 are introduced below.In an abnormal event detection problem, it is assumed that a set of training frames, {5.1.The general architecture of the abnormal event detection method via online least squares one-class SVM (online LS-OC-SVM) proposed in Section 3.3 is summarized in Algorithm 1; the flowchart is shown in Algorithm 1: Visual abnormal event detection via online least squares one-class support vector machine (LS-OC-SVM) and sparse online LS-OC-SVM.Requiren training frames {Ii}i=1nand the corresponding optical flow {OPi}i=1n.(a)Online strategy: Applying LS-OC-SVM on the small subset of training samples to calculate the coefficient matrix.(b)Sparse online strategy: Applying LS-OC-SVM to train the initial dictionary, C, offline.(a)Online strategy: Applying online LS-OC-SVM on the remaining samples to calculate the coefficient matrix.(b)Sparse online strategy: Applying sparse online LS-OC-SVM on the remaining samples to calculate the coefficient matrix and to update the dictionary.Each frame Cn+l is classified via LS-OC-SVM.Compute the covariance matrix of each frame.Step 1: The first step consists of calculating the covariance matrix descriptor of the training frames. The features could be chosen as any form shown in OP1,OP2,…, nOP} are the image optical flows of the 1st to n – th frames; {C1,C2,…, Cn} are the covariance matrix descriptors.where {Step 2: The second step is applying LS-OC-SVM on a small subset of the training samples to calculate the coefficient parameters, α and ρ, in m ≪ n of data selected from the training set m frames are chosen. These m samples are trained offline. This step can be described in the following equation:K ] and [α − ρ ]T are defined in Step 3: After learning the first m samples, the coefficient matrices, K and [α − ρ ]T, are obtained. The online LS-OC-SVM method (Section 3.3) is applied to learn the remaining n – m samples {Cm+1,Cm+2n…C}. This step can be expressed as:Step 4: Based on the coefficient matrix, [α − ρ ]T, the distance of the training samples Cn+l, with respect to the decision plane is computed. By comparing the distances of the samples, an abnormal event is detected:Cn+l is the covariance matrix descriptor of the (n + l) – th frame needed to be classified, and iC is the sample of the training data. “1” corresponds to an abnormal frame; “ − 1” corresponds to a normal frame. disT is the threshold of the distance, it is the maximum distance of the training samples to the hyperplane.where 5.2.C, making a sparse representation of the training data. The initial dictionary, C, is learned offline. Each remaining training sample is learned one-by-one online. Meanwhile, it is checked to be included, or not, into the dictionary. The test datum is classified based on the dictionary. The feature extraction step (Step 1) and the detection step (Step 4) are the same as the ones presented in Section 5.1. Owing to the dictionary, the training steps are different.The abnormal event detection via sparse online least squares one-class SVM (sparse online LS-OC-SVM) is introduced below. A subset of the samples is chosen to form the dictionary, Step 2-sparse: The second step is applying LS-OC-SVM to train the initial dictionary offline. The first m samples are the initial dictionary denoted as C. This step can be generalized as:Step 3-sparse: After learning the initial dictionary, C, including the first m(1 ≤ m ≪n) samples, the remaining training samples, {Cm+1, Cm+2, …, Cn}, are learned via sparse online LS-OC-SVM described in Section 3.4. This step can be described in the following equations:C is the dictionary and Ck is a new incoming remaining sample in the training dataset. According to the coherence criterion introduced in Section 3.4, if the new sample, Ck, satisfies the dictionary updated condition, it will be included into the dictionary, C.6.This section presents the results of experiments conducted to illustrate the performance of the two proposed classification algorithms, online least square one-class SVM (online LS-OC-SVM) and sparse online least square one-class SVM (sparse online LS-OC-SVM). The two-dimensional synthetic distribution dataset and the University of Minnesota (UMN)  dataset 6.1.Two synthetic data, “square” and “ring-line-square” , are useThe first sample is used for initializing the online LS-OC-SVM proposed in Section 3.3; the 399 remaining samples in “square” and 849 remaining samples in “ring-ling-square” are learned in the online manner.Via the sparse online LS-OC-SVM method proposed in Section 3.4, the first sample is trained offline, and this sample is considered the initial dictionary. Then, each arrival sample in 399 remaining samples in “square” and 849 remaining samples in “ring-ling-square” are checked by the coherence criterion to determine whether the dictionary should be retained or updated by including the new element.σ = 0.065. The preset threshold of the coherence criterion is μ0 = 0.08. The detection results obtained by these two online training algorithms are the same as the ones when training data were learned in a batch model.The distances are shown in contours illustrating the boundary. The contours of “square” and “ring-line-square” are shown in 6.2.σ, in the Gaussian function and the penalty factor, C, are chosen to form the receiver operating characteristic (ROC) curve. In the indoor scene, time lags of the frame labels lead to the lower area under the ROC curve (AUC) value. In the last few frames, labeled as abnormal of abnormal sequences, there are no people, while, in the training samples, there are no people in the upper half of the image. The covariance of the training frame is similar to the covariance of the abnormal frame without people. Our covariance feature descriptor-based classification method cannot distinguish between these two situations. However, this issue can be resolved by utilizing the foreground information. For example, if there are no moving objects in the frame, this frame is immediately classified as abnormal. The results of these three scenes show that the covariance descriptor can distinguish between normal and abnormal events. The performance of online LS-OC-SVM is almost the same as that of the offline method.UMN dataset detection results via online LS-OC-SVM proposed in Section 3.3 are shown below. The UMN dataset consists of eleven sequences of crowded panic escape events, which are recorded in a lawn, an indoor and a plaza scene. A frame where the people are walking in different directions is considered as a normal sample for training or for normal testing. A scene where the people are running is taken as an abnormal sample for testing. The detection results of the lawn scene, the indoor scene and the plaza scene are shown in 6.3.UMN dataset abnormal event detection results via sparse online LS-OC-SVM proposed in Section 3.4 are presented. Taking the lawn scene as an example, the first normal covariance matrix descriptor from the training samples is included into the dictionary firstly; then, the remaining training covariance descriptors are learned online by the sparse online LS-OC-SVM method. The ROC curve of the detection results of the lawn scene, the indoor scene and the plaza scene are shown in The resulting performances when all training samples are learned offline via one-class SVM (OC-SVM), learned via least squares one-class SVM (LS-OC-SVM), learned via online least squares one-class SVM (online LS-OC-SVM) and learned via sparse online least squares one-class SVM (sparse LS-OC-SVM), are shown in The resulting performances of the covariance matrix descriptor-based online least squares one-class SVM method, and of state-of-the-art methods, are shown in 7.In this paper, we proposed a method to detect abnormal events via online least squares one-class SVM (online LS-OC-SVM) and sparse online least squares one-class SVM (sparse online LS-OC-SVM). Online LS-OC-SVM learns training samples sequentially; sparse online LS-OC-SVM incorporates the coherence criterion to form the dictionary for a sparse representation of the detector. The covariance matrix descriptor encodes the movement feature of the frame to distinguish between normal and abnormal events. The proposed detection algorithms have been tested on a synthetic dataset and a real-world video dataset yielding successful results in detecting abnormal events.


To evaluate if white matter lesions (WML) on MRI can be a potential marker for onabotulinum toxin A (Botox®) treatment success in migraine, given the limited response rate and high costs per treatment.Retrospective data base and MRI analysis of 529 migraineurs who received Botox® between 2002 and 2009. Responders were defined as patients who underwent three or more treatments, whereas non-responders had only one or two treatments. MRIs were analysed on axial T2 and coronar FLAIR (fluid attenuated inversion recovery) sequences for the presence of WML. Statistical analysis was done with the Chi-Square-Test and the Mann–Whitney-U-Test.Of 529 Botox® treated migraineurs, 111 patients had a MRI. Of these 111 patients, 47 were responders, 64 non-responders to Botox®. Response rate to Botox® in migraineurs with WML was 55.3%, in migraineurs without WML 44.7%. In the investigated items “age”, “age at onset”, “gender”, “attack duration”, “frequency”, “aura”, “WML”, “size of WML”, we found no statistical significant difference between the two groups. 55% of the responders and 50% of the non-responders showed WML. All WML were located supratentorially, anteriorly, mostly of small size (3–5 mm).WML on MRIs cannot serve as a marker to predict a positive response to Botox®. Migraine is a primary headache disorder. According to the WHO, the lifetime prevalence of migraine in Europe and North America is 6% in men and 15-18% in women for one year  mediates its postulated mechanism of action in migraine by inhibiting the release of nociceptive agents, such as glutamate, substance P, calcitonin gene-related peptide and acetylcholine , medium (6–9 mm) or large (>10 mm). Total number of lesions was recorded. The distribution was classified in supratentorial or infratentorial. If supratentorial, in anterior or posterior, with cut at the middle of corpus callosum.Each WML analysis was performed independently by two neurologists, each blinded to the history of the patient. In case of disagreement between the two readers, a consensus was achieved by discussion.Statistical analysis followed the statistical program SPSS (Superior Performing Software System). The Chi-Square-Test was applied for the items gender, duration, frequency (episodic vs. chronic), aura and non-parametric values such as age, age at onset, WML, were analyzed by the Mann–Whitney-U-Test.In the current group of 111 Botox® treated patients, 47 have been responders and 64 non-responders. We found in none of the investigated parameters a statistical significance to characterize or distinct responders from non-responders, details are shown in Table The analyzed WML in the MRIs followed no pattern to permit a conclusion for a positive response to a Botox® treatment. WML were absent in 45% of the responders, present in 55%. In the non-responders, WML were absent in 50%, present in 50%. Mean lesion load of small-size- WML in responders was 2.3 per person, in non-responders 2.9 per person. Mean lesion load of medium-size- WML was 0.2 in both groups per person and of large-size- WML was 0.02 per person in responders and 0.03 per person in non-responders. Figure The aim of our study was to find a marker of response to Botox®, in order to optimize treatment of migraine patients in clinical practice.The response rate to Botox® in the treatment of migraine is generally in a range of 30% to 50%  was demonstrated in a study over three years  could be “cured” of migraine for a certain time. Second, patients with a very long treatment interval were included in the study, ending in 2009. Third, patients, corresponding to a treatment, but unable to pay for further treatments. Some false results in the responder group (≥ three treatments) could arise, first, from non-responders, having tried several times Botox®. However, more than three treatments without any sort of response are very unlikely. Second, an initial responder becomes a non-responder.Improvements could be obtained by carrying on the study in a prospective design and by realizing more MRIs in our clinic.WML on MRI scans cannot serve as a marker to predict a positive response to Botox®. The meaning of the WML in the migraine population remains unclear, being probably not of clinical importance. But they are often a sign for migraine chronification and longer lifetime history of migraine. They can be seen as well in other clinical conditions like cerebrovascular diseases, different types of dementia, inflammatory diseases and bipolar depression, which can be important comorbidities to migraine. They have to be considered while having a look at white matter lesions in the context of migraine.


Photoplethysmography is one of the optical techniques has been developed for experimental use in vasculardisease. It has several advantages over other traditional experimental approaches. Because of its non-invasive, safe, costeffectiveand easy-to-use properties, it is considered as a useful diagnostic tool. The further developments in the Photoplethysmographmay replace it among other tools used in the assessment of vascular diseases such as blood test and ultrasound.This overview discusses the different terminologies used for the photoplethysmograph and reveals the researchdiscontinuity among different disciplines. Moreover, it suggests standard terminologies as a resolution for a confusionpersisted for more than 50 years. It is an optical measurement technique uses an invisible infrared light send in the tissue and the amount of the backscattered light corresponds with the variation of the blood volume [Photoplethysmography reflects the blood movement in the vessel, which goes from the heart to the fingertips through the blood vessels in a wave-like motion , as showd volume .The photoplethysmography (PPG), introduced by Hertzman in 1938 [1b). Therefore, the second derivative of the photoplethysmogram was developed as a method allowing more accurate recognition of the inflection points and easier interpretation of the original plethysmogram wave.In 1978, Ozawa reported (in Japanese)  that the1a). The second derivative of the original photoplethysmogram waveform includes four systolic waves and one diastolic wave, namely a-wave (early systolic positive wave), b-wave (early systolic negative wave), c-wave (late systolic reincreasing wave), d-wave (late systolic redecreasing wave) and e-wave (early diastolic positive wave), as shown Fig. (1b). The original photoplethysmogram wave form consists of two waves: systolic and diastolic waves, as shown Fig. , as well as the values of sanguineous pressure, whereas abPWV = Lab / ΔT , and Lab measured distance between ankle-brachial location. In 2006, Padilla et al.  used the et al. [In 2007, Alty et al.  used theet al. [In 1999, Mishima et al. [In 2005, Yashima et al.  used theet al. [In 2007, Kageyama et al.  used theet al. [In 2008, Abe et al.  used theet al. [In 2008, Cox et al.  used theet al. [In 2009, Gil et al.  used theSome of the literature review to explain the different usage of different terminologies for the second derivative plethysmogram:et al. [b/a ratio reflects increased arterial stiffness, hence the b/a ratio increases with age. Moreover, they found the (b-c-d-e)/a ratio may be useful for evaluation of vascular aging and for screening of arteriosclerotic disease. The waves a, b, c, d, and e are demonstrated in Fig. (1b).In 1998, Takazawa et al.  used theet al. [b/a of the APG is related to the distensibility of the peripheral artery, and suggest that magnitude of b/a is a useful non-invasive index of atherosclerosis and altered arterial distensibility.In 1998, Imanaga et al.  used theet al. [b/a ratio and the aging index (SDPTG-AI) increased with age, and the c/a, d/a, and e/a ratios decreased with age. The informal aging index (b-e)/a has been suggested as a replacement for the known formula (b-c-d-e)/a.In 2007, Baek et al.  used theet al. [In 2007, Kimura et al.  used theet al. [In 1994, Katsuki et al.  used theet al. [In 1996, Takada In 2000, Bortolotto [et al. [(b - c - d)/a upon Sho treatment, which includes Kamishoyosan. Sho is one of the main concepts of Kampo medicine, and corresponds to the holistic tailored treatment suitable for the individual patient’s symptoms. In 2005,Ushiroyama et al.  used the et al. [ et al. [ et al. [In 2006, Nousou et al.  used the[ et al.  used the[ et al.  used the et al. in 2003 [To date, two papers have been found using the SDDVP. The first by Millasseau in 2003  when the et al. in 2007, [The second paper by Rivas-Vilchisin 2007,  when the et al. [In 2007, Taniguchi et al.  used theStandardization of Photoplethysmogram terminologies is highly desirable.Each terminology system should be as simple as possible and has been used the most in literature.The symbols should indicate the nature of the Photoplethysmogram signal type.A symbol representing a given charge should not be easily confused with any of those representing the other biomedical signals types. Several general preferences regarding standard terminologies for Photoplethysmogram signals have been introduced as following:Photoplethysmograph, Photoplethysmogram,and Digital Pulse Volume are the same terminologies with different a symbol PTG, PPG, and DVP.Second Derivative Photoplethysmograph, Acceleration Photoplethysmograph are the same terminologies with different symbols SDPTG, and APG.Various sets of symbols were proposed and problems associated with each set were noted as following:The data collected from PubMed database has been used for English literature and ScienceLinksJapan for Japanese literature. The reason behind choosing the PubMed database, it comprises more than 19 million citations for biomedical articles from MEDLINE and life science journals back to 1948.PubMed includes links to full-text articles which may be found in other databases such as PubMed Central or at publisher web sites.The total number of references for “Photoplethysmography” terminology was 1024, “Photoplethysmogram” was 91, “PPG”was 232, “PTG”was 13, “DVP” was 10, “APG” was 4, and “SDPTG” was 15 in PUBMED database. On the other hand, the total number of references for “Photoplethysmography” terminology was 23, “Photoplethysmogram” was 24, “PPG”was 2, “PTG”was 4, “DVP” was 8, “APG” was 28, and “SDPTG” was 8 in SCIENCELINKS Japanese database.2a); the number of publications for “PPG” is increasing with years more than “PTG” and “DVP”. In Fig. (2b), the cumulative sum of PPG clearly shows an exponential increase through years.As shown in Fig. (3a); the number of publications for“APG” is increasing with years more than “SDPTG”. In Fig.(3b), the cumulative sum of APG clearly shows an exponentialincrease through years.As shown in Fig. (4a) that the usage of “PPG “,”PTG”, and “DVP” terminologies is more in Engineering than science “SDPTG”. In Fig. (4b), the usage of “APG” and “SDPTG” vanishes in Engineering compared to Science.By categorizing the number of publication by its discipline, it is interesting to notice in Fig. (5) shows that all Photoplethysmography terminologies are coherent between PubMed and ScienceLinksJapan in all terminologies except APG. APG has been used by Japanese scientist more than western scientist. Therefore, the recommended as standard naming for the discussed terminologies is: The significant result in Fig.  and APG (for the second derivative photoplethysmogram) terminologies in oral presentations and written publications. The usage of different terminologies for the same signal can lead to a disconnect between different groups of engineers and scientists. These standard terminologies will save the researcher’s efforts and overcome generating redundant results. It will also speed up the scientific progress in this multidisciplinary field. This paper is an attempt to resolve the difficulties and confusion regarding plethysmography terminologies which have persisted for more than 50 years.


Gamma (30–80 Hz) rhythms in hippocampus and neocortex resulting from the interaction of excitatory and inhibitory cells (E- and I-cells), called Pyramidal-Interneuronal Network Gamma (PING), require that the I-cells respond to the E-cells, but don't fire on their own. In idealized models, there is a sharp boundary between a parameter regime where the I-cells have weak-enough drive for PING, and one where they have so much drive that they fire without being prompted by the E-cells. In the latter regime, they often de-synchronize and suppress the E-cells; the boundary was therefore called the “suppression boundary” by Börgers and Kopell . The mod Gamma-frequency (30–80 Hz) oscillations in hippocampus and neocortex are known to result, in many instances, from the interaction of excitatory pyramidal cells (E-cells) and fast-spiking inhibitory interneurons (I-cells)  relation is said to be of type 1 if it has no discontinuity, i.e., if arbitrarily slow spiking is possible for drives sufficiently close to (but above) threshold, and of type 2 if it has a discontinuity at spike onset. , and mav = −20 and dv/dt < 0. If the firing period is T > 0, the frequency is f = 1000/T. The factor of 1000 arises because we measure time in ms, but frequencies not in reciprocal ms, but in reciprocal s, namely, in Hz.We define the “firing times” of the neurons as times at which f - I) relation of the Erisir model neuron as follows. We begin with a simulation for I = 6, starting at  = . The computed trajectory converges to a fixed point which appears to be globally attracting. We then raise I, in steps of 0.05, from 6.0 to 7.5, starting each new simulation at the point in phase space at which the previous simulation ended. As soon as I rises above 7.0 (approximately), periodic spiking begins, at an onset frequency of approximately 60 Hz. The dots in Figure f as a function of I. We then lower I, in steps of 0.05, from 7.5 to 6, again starting each new simulation at the point in phase space at which the previous simulation ended. Periodic spiking continues as I falls below 7, and ceases only when I falls below 6.5. The circles in Figure f as a function of I, as I is gradually lowered. For I approximately between 6.5 and 7, there is bi-stability: both rest and periodic spiking are possible and stable in this range.We compute the frequency-current . Denote the firing period by T. Suppose that  =  is the uniquely determined point on the limit cycle with v = −20 and dv/dt < 0. At time t = 0, we start a simulation at this point. At time φT, with 0 < φ < 1, we abruptly increase v by 1 mV; this corresponds to an instantaneous charge injection at time φT. Denote by v = −20, dv/dt < 0. The phase advance produced by the input isWe also compute a phase response curve for the Erisir interneuron, defined as follows. Assume that g, for I = 7.2. It is of type 2: Excitatory input received early in the cycle delays the next spike instead of advancing it. Since in our network simulations, the I-cells inhibit each other, we also compute the phase response curve for the Erisir interneuron with an inhibitory autapse, modeled as described in Section 2.4, with maximal conductance equal to 0.2. The self-inhibition makes the type 2 character of the phase response more pronounced; see Figure Figure I ≤ 7.4. For I = 6.2, there is a single stable fixed point. Figure I rises.I, with blue indicating stability, and red instability. At a value of I near 6.5, a stable limit cycle arises; Figure I very slightly above 7.Figure I in Figure To confirm that the fixed point loses its stability in a Hopf bifurcation, we plot, in Figure f - I relation and the phase response curve, without self-inhibition (panel B) and with self-inhibition (panel C). The f - I relation now indicates no region of bi-stability, and is of type 1, with a square-root-like appearance. The phase response is of type 1 as well, positive throughout.We compare networks in which the I-cells are the Erisir neurons presented in the previous section with networks in which they are Wang–Buzsáki (WB) neurons  = (1 + tanh(v/4))/2, v denotes the presynaptic membrane potential, and τR and τD are the rise and decay time constants, respectively. To model the synaptic input from neuron i to neuron j, we add to the right-hand side of the equation governing the membrane potential vj of neuron j a term of the form gijsi(t) (vrev − vj) where gij denotes the maximal conductance associated with the synapse, si denotes the gating variable associated with neuron i, and vrev denotes the synaptic reversal potential. For excitatory, AMPA-receptor-mediated synapses, we use τR = 0.1, τD = 3, and vrev = 0; for inhibitory, GABAA-receptor-mediated synapses, τR = 0.3, τD = 9, and vrev = −80.where ρ denotes a smoothed Heaviside function, ρ of neurons i and j. For instance, we denote by gEI the value of gij when the i-th neuron is an E-cell, and the j-th neuron is an I-cell. Parameters gIE, gII, and gEE are defined similarly. We scale these parameters with network size: gEI = ĝEI/NE, gIE = ĝIE/NI, gII = ĝII/NI, gEE = ĝEE/NE. We choose parameters similar to those in Börgers and Kopell (EI is so strong that a population spike volley of the E-cells promptly triggers one of the I-cells (but not much stronger), and ĝIE is significantly stronger: ĝEI = 0.2, ĝIE = 0.8, ĝII = 0.2, ĝEE = 0. The value of ĝIE is varied in Figure Connectivity is all-to-all in our networks.  The value of d Kopell : ĝEI is i-th and j-th I-cells are gap-junctionally connected with probability 1/5. If vi and vj are the membrane potentials of the two cells, we add to the right-hand side of the equation for vi the term ggap(vj − vi), and to the right-hand side of the equation for vj the term ggap(vi − vj), with ggap = 0.8; this ensures that the I-cells remain synchronous when there are gap junctions.In some of our networks, the I-cells are also gap-junctionally coupled. The Each figure in this paper was generated by a stand-alone Matlab code, available from the first author upon request.II, increases linearly with time. The drive to the I-cells is heterogeneous, here and in all of our large network simulations; actual drive to the j-th I-cell isFigure II varies from 0 to 2 for WB neurons, and from 6 to 8 for Erisir neurons. The horizontal axis shows II. Panel (A) of the figure illustrates the “suppression boundary” as described in Börgers and Kopell  through (D) of Figure B and C). When both changes are made at the same time, however, one returns to an abrupt suppression transition (panel D). In the presence of gap junctions  through (C) of Figure (D) of Figure II. We discuss these two points in greater detail in sections 3.2 and 3.3.Panels Our main goal in this section is to explain why the suppression transition is gradual in Figure II, to the I-cell for which the E-cells skip every second cycle; see Figure If one couples an RTM neuron with a WB neuron, there is typically a fairly broad range of drives, II = 7.27 shown in the figure, there is no cycle-skipping, and the network settles into an oscillation with a period of approximately 26 ms. If II is raised from 7.27 to 7.28 (not shown in the figure), the E-cell is suppressed altogether, and the period of the I-cell falls to approximately 23 ms. Because the I-cell is of type 2, the input from the E-cell, by lasting for a few milliseconds beyond the spike of the I-cell, does not advance the next I-cell spike; it delays it. Thus if the I-cell is unable to suppress the E-cell on a given cycle, it will be delayed on the next cycle, and is therefore even less able to suppress the E-cell on the next cycle. As a result, the E-cell is either suppressed on all cycles, or on none.Figures fE, of E-cell spikes in 1000 ms  as a function of II, with gIE = 0.5 (blue), 0.8 (black), and 1.1 (red). The figure confirms that there is an abrupt suppression transition when the I-cell is an Erisir neuron (panel B), but not when it is a WB neuron (panel A).In Figure II typically causes either an abrupt drop in fE, or a slight increase. The reason for the increase in fE with increasing II is that the enhanced drive to the I-cells makes them fire earlier on each cycle, thereby allowing the E-cell to fire earlier on the next cycle, unless there is a change in entrainment pattern, and the E-cell is suppressed on more cycles than before. When there is a change in entrainment pattern as a result of an increase in II, the frequency fE drops abruptly.In Figure II, to the I-cells in the transition regime. For illustration, Figure (C) of Figure There are seemingly irregular sequences of strong and weak E-cell spike volleys in Figure sk the number of E-cells that fire on the k-th gamma cycle in Figure sk, sk + 1) for the simulation of Figure sk < 160, then sk + 1 can be deduced from sk, approximately at least:g has exactly one fixed point s*, which is unstable: g′(x*) < −1. If sk = 160, then sk + 1 is not determined by sk. Instead, sk + 1 then depends on the precise placement of the inhibitory spike volley in the k-th cycle, relatively to the excitatory one. If the inhibitory spike volley comes early, then the following I-cell spike volley comes so early that it largely or completely suppresses the E-cell spike volley; see Figure To understand this irregularity, we reduce the network dynamics to a one-dimensional map, examining how the strength of a given E-cell spike volley depends on the strength of the previous volley. We denote by sk, sk + 1). However, we describe now a method for constructing an idealized version of Figure To construct a figure analogous to Figure t = 0, in a point, X0, in phase space chosen so that a full population spike volley of the E-cells is imminent within a few milliseconds. Exactly how X0 is defined is largely irrelevant. We obtain it from the simulation of Figure t = t0 > 0, we re-set the phase space variables associated with the I-cell population to a point Y0 to force an immediate population spike volley of the I-cells. Again, it is largely irrelevant exactly how Y0 is defined. We obtain it from the simulation of Figure We start the simulation of the network, at time t0, the first E-cell spike volley may be suppressed partially or completely. The number, s1, of E-cells firing soon after time t = 0, 0 ≤ s1 ≤ 160, is a function of t0 ≥ 0; see Figure s2, of E-cells firing in the second E-cell spike volley, and plot it as a function of t0 , where g = g(s) is a smooth function defined for 0 ≤ s ≤ 160 with 0 ≤ g(s) ≤ 160, may not have a stable fixed point when g is decreasing, but it does have at least one stable fixed point when g is increasing. So the mapping from s1 to s2 has an attracting fixed point when the interneurons are of type 2, but not necessarily, and in particular not in the example shown in Figure An iteration of the form II denotes the mean drive to the I-cells. With increasing II, increasingly many I-cells have enough drive to fire without being prompted by the E-cells. Once enough I-cells are in this group, the E-cells are suppressed altogether. In a network of cells coupled by inhibitory chemical synapses, but not by gap junctions, heterogeneity of drives typically prevents synchronization (White et al., Simulations similar to that of Figure II passes through it rapidly.This discussion suggests that even in Figure f - I relations of WB and Erisir neurons.Figure broader suppression transition than with WB interneurons. This is opposite to what is seen in the presence of gap junctions. We have no explanation of this effect. In particular, our arguments based on the phase response of the I-cells do not apply when the I-cells are not kept synchronous by gap junctions: once the I-cells de-synchronize, they receive excitatory input from the E-cells at all phases, not just at the early phases at which type 2 I-cells are delayed by such inputs.Figures The fast-spiking inhibitory basket cells believed to be central in the formation of gamma rhythms are in fact gap-junctionally connected (Beierlein et al., In this paper, we have discussed the loss of gamma rhythms due to too much excitation of the I-cells. By contrast, a recent paper of Börgers et al.  investigAll numerical experiments in this paper have been for “strong PING,” that is, for PING oscillations in which participating E-cells fire at or near gamma frequency. By contrast, “weak PING” oscillations are noise-driven, and individual participating E-cells fire on a small, randomly selected fraction of cycles only. In Börgers et al. , it was Our discussion suggests that the competition among E-cells associated with PING oscillations [see for instance (Olufsen et al., In summary, we have found that in the presence of gap junctions, when the I-cells are of type 2, the suppression transition tends to be both narrower and more orderly than when the I-cells are of type 1. It is tempting to speculate that fast-spiking inhibitory basket cells might have evolved to have type 2 phase response curves precisely because that leads to clean suppression transitions, reducing the amount of modulation of local recurrent inhibition needed to turn gamma frequency cell assemblies off or on, an operation that seems likely to be crucial in brain function.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Analogously, the consciousness makes the mental phenomenon fundamental to any voluntary motor operation as seen from the psychological perspective.The graphical presentation of Bernstein five-level movements' construction system has been presented in Figure A-level—intrinsic stimuli,B-level—extrinsic contact stimuli,C-level—extrinsic remote stimuli,D-level—verbal code,E-level—symbolic code  A, B and C levels are “driven” by physical stimuli, whereas the D and E levels are “powered” by purely mental representations of environment, where any connection with reality is possible, indeed, but not necessary. The “common denominator” of Bernstein's model and ML is the same, systemic structure. The final product of the information processing system is a movement, even if merely that of jaws and tongue.Symptomatically enough, also Rochat (who did not refer to Bernstein) developed a five-level model of self-awareness Rochat, . Though Such differentiation of information processing modalities assigned to particular levels remains in keeping with the scales' conformity premise by Morawski , p. 162.In physics one of the especially important laws is the “correspondence rule” formulated by Jammer . It stat


In this paper, we explore a new resource called multi-target diversity to optimize the performance of multiple input multiple output (MIMO) radar with widely separated antennas for detecting multiple targets. In particular, we allocate antennas of the MIMO radar to probe different targets simultaneously in a flexible manner based on the performance metric of relative entropy. Two antenna allocation schemes are proposed. In the first scheme, each antenna is allocated to illuminate a proper target over the entire illumination time, so that the detection performance of each target is guaranteed. The problem is formulated as a minimum makespan scheduling problem in the combinatorial optimization framework. Antenna allocation is implemented through a branch-and-bound algorithm and an enhanced factor 2 algorithm. In the second scheme, called antenna-time allocation, each antenna is allocated to illuminate different targets with different illumination time. Both antenna allocation and time allocation are optimized based on illumination probabilities. Over a large range of transmitted power, target fluctuations and target numbers, both of the proposed antenna allocation schemes outperform the scheme without antenna allocation. Moreover, the antenna-time allocation scheme achieves a more robust detection performance than branch-and-bound algorithm and the enhanced factor 2 algorithm when the target number changes. Unlike a standard phased-array radar, where different antennas transmit the scaled and phase-shifted version of the same waveform, MIMO radar can transmit different waveforms at different antennas . In geneThere are many resource optimization problems we could consider to improve the performance of MIMO radar. Waveform design is one of the most interesting resource optimization problems –17. Optii.e., relative entropy, was used to study the detection performance. It originated from Stein's lemma , where the subscript k1 ∈ k denotes the first transmit antenna in k steered towards target k. The transmitted waveform is assumed to be narrowband. Therefore, the waveform that arrives at the n-th receiver can be modeled as T is a kM × 1 vector, which denotes the k-th target scattering coefficients for receiver n, and each element mh,k,n in hk,n denotes the scattering coefficient from the m-th transmitter to the n-th receiver for the k-th target. Let Rk denote the matrix that collects rk,n at different receivers  as its columns: Rk = . Thus, Rk can be written as:Hk =  is the kM × N target scattering matrix. The total received signal at all receivers is the superposition of signals from the K target cell, which can be written as follows:W =  is the noise matrix, and wn is the noise vector at the n-th receiver.Assume that the entire search area of the space is divided into = sk1, sk, ⋯, skMkdeled as :(1)rkHk can be ignored T. Based on Assumptions 2 and 3, the autocorrelation matrix of the waveform transmitted by the k-th antenna subset and the autocorrelation matrix of the k-th target scattering coefficients are both diagonal matrices, which can be written as Assuming that the transmit waveform i.e., the next allocation is determined while the MIMO radar system is in operation and in view of the results obtained from prior radar measurements in prior epochs 30], the -complete in the strong sense [The minimum makespan scheduling problems are classified as ng sense . We use 3.2.1.This is a classical algorithm for finding optimal solutions of various optimization problems, which was first proposed by Land and Doig in . If compThe branch and bound algorithm also consists of an enumeration of all candidate antenna allocation solutions. The search process for all antenna combinations is a tree structure, whose nodes are the antenna subsets of all combinations. The step that splits an antenna set into two or more smaller antenna sets is called branching. The step that computes upper and lower bounds for the relative entropy value within a given antenna set is called bounding. A tree node will be discarded, if its upper bound is less than some other node's lower bound. This step is called pruning -36. Usin edit 1 To apply the branch-and-bound algorithm in our problem, the problem  can be wTherefore, a branch-and-bound algorithm can be used for this integer programming problem. We need to be aware that branch-and-bound algorithm has exponential worst case complexity, but luckily, it can work with much less effort in some cases. We obtain the optimal antenna allocation result through the branch-and-bound algorithm as a performance bound and compare it with the performance of the proposed enhanced factor 2 algorithm in the experiments.3.2.2.i.e., the antenna that can contribute more to the detection performance must have priority. The heuristic scheme is summarized as Algorithm 1.The enhanced factor 2 algorithm is proposed based on the factor 2 algorithm in . The prok = 0, k ∈ . For each kQ,m corresponding to target k and transmit antenna m, calculate m to target k. Out of k ∈  and m ∈ , select a pair  with the maximal kr,m, and ℚk′ = ℚk′ + kQ′,m′. Meanwhile, the antenna is discarded from the remaining antenna set,  = \m′. Next, we consider the targets, except k′, and thus, we set kr′,m = − 1(∀m ∈ ). Repeat this step till all targets have been selected once. After that, out of k ∈ , select the one with the minimal ℚk. For this cell k″, select a transmit antenna m″ from the remaining antenna set with the maximal kQ″,m, and ℚk″ = ℚk″ + kQ″,m″. Next, transmit antenna m″ is also discarded. Repeat this step till all transmit antennas have been selected, and max min kD (p0 ‖ p1) = min ℚk.The relative entropy of each target cell is initialized ℚDk(p0 ‖ p1) is achieved, there can be some antenna m̃ illuminating the target k̃ that is not assigned the least relative entropy. Thus, we could reduce the power of these antennas, i.e., kD (p0‖p1), but it is useful in realistic applications.Though an exhaustive search algorithm can always achieve a solution if it exists, it is implemented with exponential complexity. The branch-and-bound algorithm has exponential worst case complexity. However, the proposed heuristic enhanced factor 2 algorithm offers significantly reduced complexity, and the performance can be seen in the experiments. Here, we need to note that after max min Algorithm 1 Enhanced factor 2 algorithm.1k = 0, k ∈ .Initialize ℚ2k ∈ , m ∈ .Calculate 3fori = 1 : Kk′ ∈  and m′∈  s.t. max kr,m;  select k′ = ℚk′ + kQ′,m′;  ℚkr′,m = −1 (∀m ∈ ),  =  \ m′;  end.4while ≠ nullk″, s.t. min ℚk;  select m″ ∈ , s.t. max kQ″,m;  select k″ = ℚk″ + kQ″,m″,  = \ m″;  ℚend.5kD (p0‖p1) = min ℚk.max min 3.3.Other valid approximation algorithms with high precision for the minimum makespan scheduling problem can be exploited, but we do not focus on these in this paper. We consider that if each target has an opportunity to be illuminated by all transmit antennas, a new appropriate antenna allocation result can be explored. Therefore, we propose an antenna-time allocation scheme to solve the problem from another perspective. Using the antenna-time allocation scheme, antennas are allocated to illuminate different targets with different illumination times.In order to achieve a proper antenna-time allocation, we propose a method to partition the illumination time based on the illumination probabilities. In particular, the optimal illumination probabilities are obtained first; next, the illumination probabilities can be converted to the illumination time. identical units of time. Denote the illumination from antenna m to target k at the l-th unit time by:l = 1,2,…, . Over each unit time, each transmit antenna selects one of all target cells to illuminate at random with a certain probability. Next, our objective is to obtain the optimal probabilities. We introduce a variable kp,m(l), which denotes the probability of kI,m(l) = 1. For any unit time l, it is obvious that Assume that the total illumination time is divided into kp,m(l) can be obtained by solving the following linear optimization problem: edit 1 The optimal solution of kp,m(l). Observing the problem kp,m(l) remains constant for l = 1, 2,…, , since kQ,m is a constant for given k and m. In fact, we assume the target features remain the same over the entire illumination time. Therefore, kp,m(l) can be written as:l = 1, 2,…, . So far, transmit antenna m has been arranged to illuminate target k at random with a probability kp,m per unit time. Next, we consider the entire illumination process from antenna m to target k, which consists of a sequence of illumination units: kI,m(l), kI,m(2), …, kI,m(L). This illumination process can be regarded as a stochastic process, which is denoted by kI,m(l). According to l = 1, 2,…, . Next, in order to convert the illumination probabilities to the illumination time, we introduce a lemma.Through linear programming, we can easily obtain the optimal numerical solution of m to target k is a stochastic process with ergodicity:The entire illumination process from antenna kI,m(l) is the same as the ensemble average. Denote the illumination time from antenna m to target k by kt,m. According to Lemma 1, kt,m can be obtained:unitt and totalt are the unit of illumination time and the total illumination time, respectively. Here, we need to be aware that kQ,m in kQ,m does not increase linearly with the unit time, the unit time should not be too short. Though this method is based on approximation analysis, it is not a relaxed solution to the previous -hard allocation problem. In fact, it is an implementable scheme for antenna-time allocation, and the effectiveness of this method can be demonstrated in the experiments.Therefore, when the number of unit time is sufficiently large, the time average of the sequence Next, in order to show an intuitive explanation of the antenna-time allocation scheme, we compare the antenna-time allocation scheme with the antenna-only allocation scheme proposed before. Uniform allocation and time allocation are also used for comparison, both of which employ all antennas to sequentially detect targets one by one. Time allocation optimizes the division of the entire time, while uniform allocation divides the entire time into equal parts for all targets without any optimization.4.M > K. However, the number of targets can be very large in the entire surveillance region. Therefore, we have to detect several times to cover the entire region. For each detection, the number of targets to be illuminated simultaneously should not exceed the number of transmitters, i.e., M > K.In this section, we set scenarios to investigate the detection performance via four different illumination schemes, including a uniform allocation scheme, a time allocation scheme, an antenna-only allocation scheme and an antenna-time allocation scheme. Moreover, the antenna-only allocation scheme contains the branch-and-bound algorithm and the enhanced factor 2 algorithm. Therefore, we use five methods in total to make a comparative analysis. Before simulations, we need to note that in order to allocate transmit antennas for each target cell, we need 4.1.M = 4, N = 4 and target number K = 2. Consider a typical scenario in which RH,1 = diag, RH,2 = diag, which means some antennas are more suitable to illuminate a certain target than the others. Particularly, Target 1 is more sensitive to the illumination from Antennas 1 and 4, while Target 2 is more sensitive to the illumination from Antennas 2 and 3. On the whole, the target scattering intensity of Target 1 is higher than that of Target 2. The different sensitivities can be caused by different path losses, angular views, wave lengths and polarizations. versus 0–25 dB transmitted power via different schemes. It can be observed that the relative entropy of the schemes that we propose outperforms that of uniform allocation and time allocation. It is obvious that uniform allocation is the worst, since it does not use any optimization. When the transmitted power grows, the relative entropy of the antenna-time allocation scheme gradually outperforms that of the antenna-only allocation scheme, i.e., branch-and-bound algorithm and enhanced factor 2 algorithm. In addition, we can find that branch-and-bound algorithm and enhanced factor 2 algorithm have the same relative entropy value for each transmitted power.We set In fact, the branch-and-bound algorithm and enhanced factor 2 algorithm lead to the same antenna allocation result for each point on the curves in versus transmitted power via different schemes. The parameters used for simulation in fP = 0.0001. The transmitted power per antenna ranges from 0 dB to 25 dB. In order to observe the detection performance, 104 Monte Carlo trials are executed to achieve the detection probability of each point on the curves. We can find that the branch-and-bound algorithm, the enhanced factor 2 algorithm and the antenna-time allocation scheme almost have the same detection probability for each transmitted power point. All of them can improve detection performance significantly compared with time allocation and uniform allocation.versus -12-12 dB radar cross-section (RCS) fluctuations of targets. The statistics of RCS fluctuations can be given by the log-normal distribution model. The variance is from -12 dB to 12 dB. In 6 Monte Carlo trials are executed to achieve the detection probability for each RCS fluctuation variance. It can be observed that two antenna-only allocation algorithms, i.e., the branch-and-bound algorithm and the enhanced factor 2 algorithm, outperform the time allocation scheme.In order to investigate the influence of different targets on the proposed schemes, we use all illumination schemes to detect different targets in different scenarios. Over a large range of RCS fluctuations, the antenna-time allocation scheme is the best, while the uniform allocation is the worst.M = 9, N = 9. The target number varies from two to nine. versus target number via different schemes. It can be observed that the proposed antenna-only allocation scheme and antenna-time allocation scheme still have advantages over time allocation and uniform allocation with different target numbers. Moreover, the antenna-time allocation scheme achieves the most robust detection performance when the number of targets is larger.Next, we investigate the influence of the target number on the proposed schemes. We set more antennas: 4.2.In the experiments above, we have found that the idea of antenna allocation has advantages over the idea without antenna allocation for multi-target detection. In this part, we will show some theoretical analysis.i.e., SNR. Let kL is the pulse number for the k-th target. To obtain an intuitive explanation of the relative kρ,m's for different k and m are equal, which means all targets are the same isotropic scatterers, and all antennas are the same transmitters with the same path losses, angular views, wave lengths, polarizations, etc. Therefore, we drop the subscript of kρ,m and obtain:In the relative entropy kD (p0‖p1) increases linearly with the number of receive antennas N. Therefore, when the other parameters are fixed, the number of receivers should be as large as possible to increase kD (p0 ‖ p1).Observing N to study the influence of kM and kL. Applying the Taylor expansion of log(·) in the small ρ region, ρ region, Next, we fix K denotes the target number. In the scheme without antenna allocation, i.e., time allocation or uniform allocation, we employ all antennas to illuminate targets sequentially Thus, kM = M and kL = L/K, where M and L are the total number of transmit antennas and the total number of signal pulses, respectively Therefore, we have:For multi-target detection, K targets simultaneously Thus, the average value of kM is M/K, and kL = L. We still obtain C is a constant for a given K. Substituting kL in the small ρ region. kD (p0‖p1) in ρ region, we should select a scheme with larger pulse number kL, i.e., longer illumination time for a given pulse repetition frequency. Considering kM × kL is a constant, we should increase kL and decrease kM. Thus, for multi-target detection, we will adopt antenna allocation to illuminate multiple targets simultaneously to lengthen illumination time in low SNR region.In antenna allocation scheme, we partition all antennas to illuminate kD (p0‖p1) in kM × kL is a constant, we can conclude that we should decrease kL and increase kM when ρ is relatively large. In Substituting 5.In this paper, a multi-target detection problem for MIMO radar with widely separated antennas is investigated based on relative entropy. In order to explore multi-target diversity, we propose the antenna-only allocation scheme and the antenna-time allocation scheme to implement multi-target detection simultaneously. In the antenna-only allocation scheme, we use a branch-and-bound algorithm and an enhanced factor 2 algorithm, respectively. In contrast to sequential illumination without antenna allocation, simultaneous illumination via antenna allocation can increase illumination time at the expense of decreasing the number of illumination antennas for each target. It is shown that proper antenna allocation outperforms time allocation and uniform allocation over a large range of transmitted power, RCS fluctuations and target numbers. Moreover, the antenna-time allocation scheme can achieve a more robust performance than branch-and-bound algorithm and enhanced factor 2 algorithm for more targets.


E = b + skN, where E: expended energy, b: baseline rate, k: cost per spike, and sN: the number of spikes fired. Figure s(t) as a spike-train. Given an energy constraint E, the goal is to obtain the best possible reconstruction r(t) of s(t), with a simple low-pass filter h(t) = A exp(-t/τ) (mimicking a post-synaptic cell membrane) such that average squared reconstruction error e(t) = s(t) - r(t)  and firing level γ. Here, we show that this encoder can be interpreted as an instantaneous rate encoder with rate i(t) = (s(t)/τ + s'(t))/A. The function closely approximates the PSTH. We tested the instantaneous rate coder by predicting: i) spike-times . For signals with little variability, this predicts a rate-code of signal intensity. For signals with high variability, the spike-rate is driven by the changes in the signal . We conclude that optimal encoder can optimally time spikes while maintaining high coding fidelity, which can be interpreted as an instantaneous rate code.An action potential (spike) is metabolically expensive to generate, and it is likely that selective pressure has been exerted on the nervous system to generate energy-efficient neural codes . An addi) Figure  is minims Figure ee rasters Figure op panel,s Figure  for sing


The zero field equation of state (EOS) is hard sphere without any re-scaling of particle size, and the hydrodynamic corrections to the long-time self-diffusion coefficient are quantitatively consistent with the expected value for hard spheres. Care is taken to ensure that both the dimensionless gravitational energy, which is equivalent to a Peclet number Peg, and dipolar strength Λ are of order unity. In the presence of an external electric field, anisotropic chain-chain clusters form; this cluster formation manifests itself with the appearance of a plateau in the diffusion coefficient when the dimensionless dipolar strength Λ ~ 1. The structure and dynamics of this chain-chain cluster state is examined for a monodisperse system for two particle sizes.In this work, we use structure and dynamics in sedimentation equilibrium, in the presence of gravity, to examine,  Electric-field-induced dipolar colloids are a simple two-component system where both cluster formation and network formation123453710kBT).Computer simulations of colloidal hard spheres with an imposed pair-wise electric polarization interaction that neglect contributions of higher order than the point-dipolar term seem to capture the essence of the behaviours261251415Peg scales as a4, and colloids of the size and particle-solvent density mismatch that is typical in many microscopy experiments have Peg ~ O(1), and thus gravity can often not be neglected.One can minimize the effects of gravity by density matching, by ensuring that the gravitational length, defined by equilibrium colloidal systems with competing interactions on different lengthscales4Other forms of aggregation have been studied extensively in colloidal systemsa = 0.8 and 1.0 μm-diameter in a nearly density-matched solvent mixture, our system, with Peg ~ O(1) is as strongly Brownian as possible while still allowing the particles to be just large enough for microscopy. In our experiments, there is a 20% difference in the two particle sizes used, which is significant given the a4 dependence of the Peclet number, and much larger than the (less than) 5% polydispersity in particle size. Sedimentation in the presence of gravity has been used to good effect in many colloidal systems to extract information about equations of state; an early example is from Hachisu and TakanoIn this work, we employ a colloidal system to study dipolar clusters. Our system is easily index matched and density matched for confocal microscopy experiments, and we show it to be an excellent candidate for true hard-sphere-plus-dipolar interactions. With colloidal spheres of diameter 2μm-diameter spheres, as well as a bidisperse system, at several values of a uniaxial AC electric field (see Methods). Snapshots of colloidal sediments at zero field and some selected field strengths are shown in z, i.e. parallel and anti-parallel to the direction of gravity (which points to the left). μm diameter PMMA colloids in 70:30 decalin/TCE. As the field increases from zero to 1667 V/mm, the sediment goes from fluid-like to fully-formed chains. The middle panel in μm diameter PMMA colloids in 60:40 decalin/TCE. The field strengths chosen are ones for comparable values of the dipolar strengthExperiments were carried out for two monodisperse colloidal systems with 0.8 and 1 εp and εf are the particle and fluid dielectric constant respectively, a is the particle radius, and E0 is the amplitude of the sinusoidal AC electric field; see E0 and Λ). For comparison, μm and 1.0 μm diameter particles  in 50:50 ratio by volume fraction. The particle columns in the bidisperse system are always thicker: this is even clearly the case when only half the particles (by volume fraction) are visible.where z, for all fields for the 0.8 μm-diameter colloids (mean volume fraction Φ = 0.017), obtained from the 3-dimensional image stacks. We see in z) in thin slices dz at height z, parallel to and above the bottom substrate and the local osmotic pressure μm PMMA particles shows excellent agreement with what is expected for a hard-sphere EOS from the Carnahan-Starling relation, and is in agreement with careful experiments in true bulk (non-microscopy) systems24even at low concentrations. For example, in the work of Li et al.apparent osmotic pressure Π. It is nevertheless a useful quantity because the intra-chain structure is the result of both gravitational force and free thermal exchange of particles when Λ is O(1). At non-zero fields (i.e. with increasing Λ), the key observation is that Π at any given packing fraction Φ is larger than the hard-sphere value, and increases with the field. This is in contrast with experiments , the probability of unassociated two-particle clusters is not zero. On the other hand, as the field is increased, the probability of size 2, 3, and larger chain clusters increases systematically above the zero-field value; this represents real in-plane clustering  of chains .D  as a function of distance z from the substrate. At zero field, the diffusion coefficient D saturates to the bulk value D0 ~ 0.44 μm2/s when z is greater than 50 μm. We can utilize the diffusion profiles in D = D0(1+KLΦ) with KL = −2.80 ± 0.05. This is comparable to the range −2.8 to −2.9 obtained in experiments by Blees et al.et al.28Evidence of clustering can also be obtained from “sedimentation-diffusion” profiles where we measure the lateral (in-plane) self-diffusion coefficient z = 30 μm in D = 0.42 μm2/s to 0.27μm2/s as the field increases from zero to E = 1667 V/mm. At high fields, there is a plateau for z = 28 μm (position B) where sharp clusters are seen at the highest fields, and in time averages of the movies (bottom panel), where the particle dynamics smear the intensities at lowest fields, but the clusters are still visible at highest fields, suggesting particles stay confined to their respective clusters. At intermediate fields, the time average  is more diffuse than the snapshot , but the clusters are still visible, indicating that the dynamics is also intermediate between free “particle” and cluster “particle”  occurs at a dipolar strength Λ that is close to 1 for both particle sizes . This again suggests that a pure-dielectric dipolar interaction is a dominant contributor to the interactions.Finally, we can assess the magnitude of the dipolar strength at the transition from single particles to clusters. For the chain-cluster (or “string fluid”) structure, the hard-sphere-plus-dipolar modelμm-scale colloids in a solvent mixture of decahydronapthalene and tetrachloroethylene are hard-sphere-like, density- and refractive-index-matchable, and yet have a strong enough dipolar inter-particle interaction in the presence of an external field.In this work, we have examined Brownian colloidal systems where the equation of state is shown to be in quantitative agreement with the hard-sphere equation of state, even in a thin sediment, and with no use of “effective” particle sizes. In addition, we find that First, we used sedimentation-diffusion profiles, in zero field, to obtain self-diffusion coefficients as a function of volume fraction Φ. The colloidal systems exhibited a decrease in self-diffusion coefficient with increasing Φ that was consistent with bulk experiments for hard-sphere colloids.apparent osmotic pressure at any given volume fraction Φ is found to be greater than the hard-sphere value. It should be noted that, in the presence of clustering, the true local volume fraction is larger than the mean value at a particular height, so the EOS must be taken with a pinch of salt; however, we reiterate the observation in the raw sedimentation profiles that the profiles are more extended with higher fields.Next, we applied external electric fields. The “hard-sphere-plus-dipolar” equation of state (EOS) is obtained: the We characterized the onset of field-induced interactions by sedimentation-diffusion profiles. The deviation from zero field profiles occurs at Λ ~ 1 see . Stable μm-diameter particles than for 0.8 μm-diameter particles, as can be seen even by a visual inspection of The lateral extent of chain-chain clusters increases with the dipolar strength. This is unsurprising, but the nature of the clusters is interesting: the clusters are thicker for 1 For context, two close-to-Brownian colloidal systems used in optical microscopy in which dipolar colloidal interactions have been previously employed are either at Peclet numbers of order 10, or not true hard-sphere systems. In the first system, silica microspheres in aqueous suspensions610Finally, an interesting aspect of the cluster state might have relevance to the low-density networking forming “void” phase, which are seen in experiments51415a = 10 and 0.8 μm), the solvent mixture was adjusted to volume ratios of 60:40 and 70:30 of decalin to TCE respectively /tetrachloroethylene (TCE) solvent mixture; in this case the gravitational length, lg was significantly larger than the sample thickness. The 2a = 10 and 0.8 μm particles are dyed with 4-methylaminoethylmethacrylate-7-nitrobenzo-2-oxa-1,3-diazol (NBD) and 1,1-dioctadecyl-3,3,3,3-tetramethylindocarbocyanine perchlorate (DiIC18) respectively, and excited by 490 nm (blue) and 561 nm (green) laser lines. In the binary mixture, while the emission from the green excitation comes purely from the 0.8 μm particles, the emission from the blue excitation also has a small contribution from the 1.0 and 0.8 μm particles. Shown in The system used is a colloidal suspension of PMMA colloids in a cis-trans-decahydronaphthalene  and tetrachloroethylene (TCE) solvent. Two particle diameters were used in the experiments. For both particle diameters  at 29 frames per second (fps) using a Visitech VT Eye confocal scanner. Slices in z are acquired in 0.15 μm z-increments. Standard particle-tracking and counting methodsa being identified as connected.For all the electric-field cells, the top and bottom plates were 22 mm square cover slips coated with a transparent conducting layer of indium tin oxide (ITO). These cover slides were glued together  with two sets of 50 fps or 56 fps. From these time series mean-squared displacements versus time are obtained, and fitted to obtain diffusion coefficients. The time of seconds, in colloids with reasonably high local densities, is in the long-time limit.Time series are obtained at several depths How to cite this article: Newman, H. D. and Yethiraj, A. Clusters in sedimentation equilibrium for an experimental hard-sphere-plus-dipolar Brownian colloidal system. Sci. Rep.5, 13572; doi: 10.1038/srep13572 (2015).


Resonance is an important phenomenon in vibrating systems and, in systems of nonlinear coupled oscillators, resonant interactions can occur between constituent parts of the system. In this paper, out-of-unison resonance is defined as a solution in which components of the response are 90° out-of-phase, in contrast to the in-unison responses that are normally considered. A well-known physical example of this is whirling, which can occur in a taut cable. Here, we use a normal form technique to obtain time-independent functions known as backbone curves. Considering a model of a cable, this approach is used to identify out-of-unison resonance and it is demonstrated that this corresponds to whirling. We then show how out-of-unison resonance can occur in other two degree-of-freedom nonlinear oscillators. Specifically, an in-line oscillator consisting of two masses connected by nonlinear springs—a type of system where out-of-unison resonance has not previously been identified—is shown to have specific parameter regions where out-of-unison resonance can occur. Finally, we demonstrate how the backbone curve analysis can be used to predict the responses of forced systems. A large class of resonant interactions may be described as a vibration-in-unison, meaning that the physical displacement coordinates of the system reach their extrema and pass through zero simultaneously. The vibration-in-unison concept was famously used by Rosenberg to give the first definition of a nonlinear normal mode for a conservative system –4. In moHere we define out-of-unison resonance as a resonant interaction between two displacement coordinates of a conservative system, where one coordinate reaches an extrema as the other passes through zero, such that they are 90° out-of-phase. The best-known physical example of this behaviour is whirling which can occur, for example, in a taut cable . As we sPrevious authors have applied approximate analytical techniques to problems involving weakly nonlinear oscillators e.g. –15). The. The15])In this paper, the second-order normal form technique  is used A method for the application of the second-order normal form technique is introduced in §2. Then, in §3, we consider whirling motion in a taut, horizontal, undamped cable, which is a well-known physical example of out-of-unison resonance. In §4, we consider an in-line, conservative, two degree-of-freedom (2-d.f.) oscillator to demonstrate how out-of-unison motion may exist in other nonlinear systems. Furthermore, we demonstrate how, from the analytical descriptions developed, we may determine the values of the physical parameters that yield out-of-unison resonance. Finally, we demonstrate how the out-of-unison motion, seen in the conservative systems, translates to a similar behaviour when the in-line system is forced and damped (and so is non-conservative).2.In this paper, we use the second-order normal form technique to transform the equations of motion of weakly nonlinear systems into sets of time-invariant equations, describing the approximate dynamics. The technique uses the assumption that the nonlinear terms are small; therefore, as the amplitude decreases the results converge to the true solution, but they diverge at higher amplitudes. Here, we consider responses in amplitude and frequency ranges where the nonlinear terms are small, and it is confirmed that the resulting errors are negligible via comparison with results found using numerical continuation. Methods for directly estimating the validity of the normal form results may also be used (e.g. ). The deN d.f. may be writtenM, C and K are the {N×N} mass, damping and stiffness matrices respectively, Nx and Px are {N×1} vectors of nonlinear terms and forcing amplitudes, respectively, and Ωf is the forcing frequency. To find the backbone curves of this system, we consider the unforced and undamped equivalent of equation , where u describes the fundamental response, and h  contains the harmonic contents of q. From this, we can write the transformed equation of motion asun  to be sinusoidal, such that it may be writtenUn, ωrn and ϕn are the amplitude, response frequency and the phase of un, respectively. Note that ωrn and ωnn are distinct and represent the fundamental response frequency and linear natural frequency of qn, respectively. The subscripts p and m denote the positive and negative (plus and minus) signs of the complex exponents, respectively.Next we apply the nonlinear near-identity transform Nq is small in relation to other terms—denoted as order ɛ1, where ɛ is a bookkeeping parameter denoting smallness. Furthermore, as the harmonics are small we may also state that h is of order ɛ1. Therefore, we may write ɛNq(u+ɛh)≈ɛNq(u) whilst maintaining order ɛ1 accuracy. Substituting equation (Nq(u) allows us to write Nq(u)=[nq]u* where [nq] is an {N×L} matrix of coefficients and u* is an {L×1} vector of all the unique combinations of unp and unm that exist in Nq(u). The ℓth element of u* can be written as follows:skpℓ and skmℓ are the exponents of ukp and ukm in As we are assuming the system to be weakly nonlinear, we assume that equation  into Nq by defiℓ=0 , the cable exhibits a small sag, breaking the rotational symmetry about x. This leads to a difference in the dynamic behaviour of the corresponding linear modes in the y-direction and in the z-direction—most notably, a difference in the linear natural frequencies  (see equations (2.10)), and hence Nu, using Nu=[nu]u*. We may then calculate Ω=ωr1=ωr2. The real parts of equations (3.4) then lead top=±1 where p=+1 corresponds to responses where u1 and u2 are in-phase or anti-phase, and p=−1 corresponds to u1 and u2 being ±90° out-of-phase.From equations , we may equation ) as3.3aquations  and  can be found by setting U1≠0 and U2=0, and U1=0 and U2≠0. These correspond to sets of responses of the system, denoted S1 and S2, which are defined byU1≠0 and U2≠0, equations (3.5) can be written as follows:ωn2≠ωn1 (due to the influence of gravity), p=+1 cannot be a valid solution.Besides the trivial solution where p=−1 two additional sets of responses, denoted S3+ and S3− (or S3± when referring to both), are produced. S3± have the phase differences (ϕ1−ϕ2)=±π/2. Substituting p=−1 into equation  at a bifurcation, the point on S3± where U2=0 is the point at which the bifurcation occurs. From equations (3.10), this point is described byS1, which can be seen from equation −3, a Young's Modulus of 2×1011 Pa and a static tension of 200 N. The equations derived by Warnitchai et al. =. All other system parameters are those used in §4c. Owing to the antisymmetric shape of the forcing, only the second underlying linear mode, q2, experiences direct modal forcing. We therefore expect the forced responses to only follow backbone curves with a q2 component (i.e. the responses will not follow S1). Ω , against the maximum displacement amplitude of the first physical coordinate, X1. Light-blue lines represent S1, S2 and S3± whilst dark-blue lines represent S4± . Thin-black and dashed-red lines show the stable and unstable forced responses, respectively. The forced responses that are of interest here (following S4±) are highlighted with a thick-black line. Other responses in We now compare the backbone curves to the forced and damped responses of this system. The damping and forcing parameters used here are S2. This branch of the forced response . Reducing the forcing and damping would result in the secondary branches collapsing onto S4− and a decrease in the distortion of the responses.Four sets of points—labelled Ab shows that, aside from the distortion, the linear modal coordinates are out-of-unison for responses on the secondary branches, i.e. as one coordinate reaches an extrema the other passes through zero. Figure 9d shows that, in responses B1,2 and D1,2, the physical coordinates are neither in-unison nor out-of-unison, i.e. as one coordinate reaches an extrema or zero the other does not. In response C1,2, however, the physical coordinates are out-of-unison (aside from the distortion). As discussed in §4c, this corresponds to the point on the backbone curves where the amplitudes of q1 and q2 are equal (neglecting harmonics). The similarity between the corresponding responses in the forced system and in the backbone curves shows that, for lightly damped systems, the forced response is governed by the backbone curves.Figure 95.In this paper, we have investigated the phenomenon of out-of-unison resonance for systems of weakly nonlinear, coupled oscillators. This phenomenon has physical manifestations, such as whirling in a taut cable, and we have shown how this phenomenon can be modelled in order to describe the dynamic behaviour of the system. A cable was modelled using two sets of modes, one in the vertical and the other in the transverse direction, where the natural frequencies of the corresponding modes are close. The proximity of these natural frequencies allows for resonant interactions between the modes at a 1 : 1 ratio and leading to out-of-unison resonance. We then investigated an in-line nonlinear oscillator, with two close linear natural frequencies. Using a backbone curve analysis, we demonstrated that there were specific parameters for this system that led to out-of-unison resonance. In addition to this, we considered the case where the system was forced and lightly damped using continuation methods to track the steady-state periodic motion. As would be expected for a lightly damped system, the forced responses were shown to be governed by the backbone curves defined for the underlying conservative system. Therefore, for this particular system, we were able to show examples of resonances in the forced responses that have similar features to those obtained from studying the underlying conservative system.


Biomedical researchers often encounter the large-p-small-n situations—a great number of variables are measured/recorded for only a few subjects. The authors propose a fuzzy permutation method to address the multiple testing problem for small sample size studies. The method introduces fuzziness into standard permutation analysis to produce randomized p-values, which are then converted into q-values for false discovery rate controls. Simple algebra shows that the fuzzy permutation method is at least as powerful as the standard permutation method under any alternative. Monte-Carlo simulations show that the proposed method has desirable statistical properties whether the study variables are normally or non-normally distributed. A real dataset is analyzed to illustrate its use. The proposed fuzzy permutation method is recommended for use in the large-p-small-n settings. Biomedical studies nowadays often involve a large number of variables for the study subjects, such as gene expression data analyses and genome-wide association studies, etc.123Benjamini and Hochbergn < 107n < 10 again, are repeatedly tested in different batchesIn this paper, we are concerned with situations where only very few subjects say, <10) are recruited but a great number of variables  are measured/recorded for each person—the ‘large-p-small-n’ situations.  are measured for each and every subject in the study. The algorithm of the fuzzy permutation procedure is described below:pj, for j = 1, …, m.Perform a two-sample test, such as Student t test or Wilcoxon rank sum test, and calculate the p-value, j = 1, … m, where U is a uniform  distribution, and ε is a very ‘small’ number (explained later).Add a fuzzy term to the p-value: j = 1, … m, based on the permuted data.Permute case and control status, and calculate the p-value j = 1, … m.Add a fuzzy term to the p-value: k times.Repeat Steps 3 and 4 for a total of k permutations, for j = 1, … m.Calculate a permutation-based p-value, Convert the permutation-based p-values in Step 6 to q-values.Assume that a case-control study recruits a total of ε → 0, so that they will have absolutely no effects on the ranking of the p values when they are not tied. But ε should not be exactly zero, otherwise ties will not be resolved. In practice, we can let ε to be slightly larger than the precision of a computer as allowed by a programming language. The limit of resolution for the widely used R code is 10−16. Thus one can safely use ε = 10−12, which shall be small enough for all data size. S1 Exhibit presents an alternative but equivalent algorithm based on fractional counting.Note that the fuzzy terms to be added are very small numbers. Ideally one should let Theoretical comparisons of fuzzy and standardA function (fuzz.perm) written in R codep1 = 0.744222017 (using two-sample Student t test). After adding a fuzzy term  to the p-value p1, we obtain p1 = 0.744222017 = We perform Monte-Carlo simulations to study the performances of the fuzzy permutation procedure. We assume the study variables to follow respectively, the normal distribution, the gamma distribution, the truncated normal distribution and the beta distribution. The total number of variables is set at 100000. For each of the four types of distributions, we consider four scenarios: HS, HW, LS and LW, respectively . The parameters for all these scenarios are detailed in S4 Exhibit. We assume a total sample size of 10 (5 cases and 5 controls).Using the algorithm described above, we calculate the permutation-based p-values using the fuzzy permutation method with Student t test and with Wilcoxon rank sum test, respectively. A total of 5000 permutations were performed for each of the HS, HW, LS and LW alternative hypotheses. [We also simulate the null hypothesis when all the variables are noises and find that the fuzzy permutation p-values are indeed distributed according to a uniform  distribution under the null (results not shown).] For comparison, we also calculate the p-values, using the Student t test (without permutation), the Wilcoxon rank sum test (without permutation), as well as the standard permutation methodWe use Package fdrtool in RThe results when the variables are normally distributed are shown in We reduced the total number of variables to 20000, and the results (S5–S8 Exhibits) are essentially the same as the above The gene expression data of Biswas and PoidingerWe use the six methods in the previous section to analyze this example. A total of 10000 permutations were performed. With FDR controlled at a higher value of 0.09, the fuzzy permutation method with Student t test can now detect a total of 245 significant genes. When FDR controlled at 0.10, the fuzzy permutation method with Student t test and with Wilcoxon rank sum test can even detect more significant genes than the Student t test without permutation.For this example, the three discrete tests are extremely conservative in making significant calls. Not until FDR control value being raised to 0.20 that the standard permutation method with Student t test is beginning to detect 2 significant genes. When the FDR control value is raised to 0.40, the standard permutation method with Student t test can detect 18 significant genes, the Wilcoxon rank sum test without permutation, a total of 10523 significant genes, and the standard permutation method with Wilcoxon rank sum test, still zero significant gene. By contrast, with the same high FDR control value of 0.40, the fuzzy permutation method with Student t test can detect a total of 22745 significant genes, and the fuzzy permutation method with Wilcoxon rank sum test, a total of 22135 significant genes.A study with a very small sample size (such as this example) should best be viewed as a screening study. Any statistically significant gene found in the study is to be subject to further confirmatory testing in studies with larger sample sizes. In consideration of this, many researchers may not opt for more significant genes in the initial screening study; due to limited budget they may work with only a fixed number of genes in subsequent follow-up large sample sized studies. Under this constraint, the fuzzy permutation method is still a better choice; the FDR control value can be set more stringent than other methods to obtain the same requested number of genes. For example, if one wishes to select for further study around 10000 genes from among the total 47322 genes in this example, the FDR control value for Wilcoxon rank sum test without permutation has to be set at 0.40, producing <10000 × 0.4 = 4000 false positives (this test was found to be conservative in the simulation study). But for the two fuzzy permutation methods, the FDR control value can be at 0.10, producing ~10000 × 0.1 = 1000 false positives (these two tests were found to be well controlled in the simulation study). If one wishes to select no more than 300 genes, the FDR control value for the standard permutation with Student t test will need to be set higher than 0.40 , but for the fuzzy permutation with Student t test, it can be at 0.09 .Previously, Kulinskaya and Lewin also used randomized p-values for FDR controlThe standard permutation methodIn summary, the proposed method introduces fuzziness into standard permutation analysis to produce randomized p-values, which are then converted into q-values for FDR controls. It is a distribution-free method and is recommended for use in the large-p-small-n settings.How to cite this article: Yang, Y.-H. et al. A Fuzzy Permutation Method for False Discovery Rate Control. Sci. Rep.6, 28507; doi: 10.1038/srep28507 (2016).