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For a unimodal distribution, negative skew commonly indicates that the tail is on the left side of the distribution, and positive skew indicates that the tail is on the right. In cases where one tail is long but the other tail is fat, skewness does not obey a simple rule. For example, a zero value means that the tails on both sides of the mean balance out overall; this is the case for a symmetric distribution, but can also be true for an asymmetric distribution where one tail is long and thin, and the other is short but fat.

Consider the two distributions in the figure just below. Within each graph, the values on the right side of the distribution taper differently from the values on the left side. These tapering sides are called tailsand they provide a visual means to determine which of the two kinds of skewness a distribution has:.

Skewness in a data series may sometimes be observed not only graphically but by simple inspection of the values. For instance, consider the numeric sequence 49, 50, 51whose values are evenly distributed around a central value of We can transform this sequence into a negatively skewed distribution by adding a value far below the mean, which is probably a negative outliere.

Therefore, the mean of the sequence becomes Similarly, we can make the sequence positively skewed by adding a value far above the mean, which is probably a positive outlier, e.

Table of contents for issues of Supplement to the Journal of the Royal Statistical Society

As mentioned earlier, a unimodal distribution with zero value of skewness does not imply that this distribution is symmetric necessarily. However, a symmetric unimodal or multimodal distribution always has zero skewness. The skewness is not directly related to the relationship between the mean and median: a distribution with negative skew can have its mean greater than or less than the median, and likewise for positive skew.

However, the modern definition of skewness and the traditional nonparametric definition do not always have the same sign: while they agree for some families of distributions, they differ in some of the cases, and conflating them is misleading. If the distribution is symmetricthen the mean is equal to the median, and the distribution has zero skewness.

This is the case of a coin toss or the series 1,2,3,4, Note, however, that the converse is not true in general, i. A journal article points out: [2]. Many textbooks teach a rule of thumb stating that the mean is right of the median under right skew, and left of the median under left skew. This rule fails with surprising frequency. It can fail in multimodal distributionsor in distributions where one tail is long but the other is heavy.

Most commonly, though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal. Such distributions not only contradict the textbook relationship between mean, median, and skew, they also contradict the textbook interpretation of the median.

For example, in the distribution of adult residents across US households, the skew is to the right. However, due to the majority of cases is less or equal to the mode, which is also the median, the mean sits in the heavier left tail. As a result, the rule of thumb that the mean is right of the median under right skew failed. It is sometimes referred to as Pearson's moment coefficient of skewness[5] or simply the moment coefficient of skewness[4] but should not be confused with Pearson's other skewness statistics see below.

This is analogous to the definition of kurtosis as the fourth cumulant normalized by the square of the second cumulant. The skewness is also sometimes denoted Skew[ X ].

For a sample of n values, a natural method of moments estimator of the population skewness is [6].The MGF network uses the network structure of two layers. First layer which is input layer constitutes the adaptive feature extraction part and second layer constitutes the signal classification part. The simulation results in the form confusion matrix show that proposed modified modulation classification algorithm has high classification accuracy at low signal to noise ratio SNR.

The performance comparison with state-of-the-art existing techniques shows the significant performance improvement of proposed MGF based classifier. Automatic Modulation Classification AMC is an approach which classifies the modulation format of the received signal at the receiver side.

AMC has found extensive importance in the field of electronic surveillance, military domain, electronic counter measures, civil domain, software defined radios, and lately cognitive radios. For example, in military domains, it may be employed for monitoring and interference recognitions, while in civil domain it includes interference confirmation, spectrum management, and signal confirmation.

The most important applications in civil domain are intelligent modems, software defined radios, and cognitive radios. Due to incremental technologies such as cognitive radios, the recent research has been focussed to identify and then classify these types of signal as discussed by Haykin [ 1 ].

To accomplish AMC, there are two approaches, decision theoretic approach, which is based upon likelihood function of the received signal, and pattern recognition approach, which is based upon features extraction from the received signal [ 2 ].

Interplay between Graph Topology and Correlations of Third Order in Spiking Neuronal Networks

The likelihood function based decision theoretic approach is optimal, but computationally complex. The classifier based upon decision theoretic approach is proposed in [ 3 ]. In [ 4 ], author gives survey of the decision theoretic approach and the comparison of proposed classifier performances in the literature.

The modulation classification in decision theoretic approach is viewed as multiple hypothesis test or may be sequence of pairwise multiple hypothesis test. Once the likelihood function is set up, average likelihood ratio test ALRTgeneralized likelihood ratio test GLRThybrid likelihood ratio test HLRTand combinations of these tests are to be used to determine the modulation format of the received signal [ 5 ].

Due to phase errors, channel effects, timing jitter, and frequency offset, the decision theoretic approach is not robust to model mismatch [ 6 ]. Maximum likelihood method is used in classification of digital modulations in [ 7 ]. The author shows that ML classifier is capable of classifying any finite set constellations with zero error rate when the number of available data symbols goes to infinity.

The modulation classification algorithm proposed for identification of software defined radio modulation schemes without pilot symbols between transmitter and receiver in [ 8 ]. The classifier based upon likelihood ratio test loads the values of test function for likelihood ratio test; the proposed algorithm converts unknown signal symbol to the address of lookup table.

The feature extraction based pattern recognition approach PRA is robust to model mismatch, but not optimal with less computational complexity as compared to decision theoretic approach. The PRA is divided into two modules. In the first module, distinct features are extracted from the received signal, which undergo channel effects such as fading and also channel noise such as additive white Gaussian noise AWGN. After the successful extraction of these features, second module is classifier which decides about the modulation format of the received signal [ 9 ].

The previous techniques employed in literature for feature extraction based modulation classification are discussed below. In [ 10 ], authors considered seven modulation formats for classification using genetic algorithm GA based clustering.

The features extracted are spectral features from the received signal and reduced set of parameters is derived from these coefficients and input to GA based clustering technique. The modulation classification based upon combination of 2nd, 4th, 6th, and 8th order cumulants and spectral features are proposed in [ 11 ]. Hierarchical support vector machine SVM is used as classifier. The optimization Bee algorithm is used to improve the overall performance of proposed classifier.

Spectral features, statistical features, and wavelet based features are used to classify the modulation formats in [ 12 ] and performance is evaluated on AWGN channel. The authors proposed a classifier based upon SVM and optimization of algorithm is done using particle swarm optimization PSO. The modulation formats are recognized using artificial neural network ANN and resilient back propagation in [ 13 ]. The GA is used to select the best feature subset from the combined spectral features and statistical features.Handbook of Statistics.

Recommend Documents. Handbook of Statistics Jun 14, - Jeffrey C. Miecznikowski, Dan Wang, David L. Gold and Song Liu. Gorilla gorilla. Homo sapiens. Pan paniscus. Pan troglodytes. Computational Statistics Handbook with Aug 22, - a probability mass function in the case of discrete random variables. These functions for calculating descriptive statistics are available in the European Champion Clubs' Cup History Statistics and Statistical Thinking.

South Beach. Cognitive Computing: Theory and Applications. Edited by. Venkat N. Vijay V.

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Venu Govindaraju. London: Sage.Conceived and designed the experiments: SJ SR. Performed the experiments: SJ. Analyzed the data: SJ.

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Wrote the paper: SJ SR. The study of processes evolving on networks has recently become a very popular research field, not only because of the rich mathematical theory that underpins it, but also because of its many possible applications, a number of them in the field of biology.

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Indeed, molecular signaling pathways, gene regulation, predator-prey interactions and the communication between neurons in the brain can be seen as examples of networks with complex dynamics. The properties of such dynamics depend largely on the topology of the underlying network graph. In this work, we want to answer the following question: Knowing network connectivity, what can be said about the level of third-order correlations that will characterize the network dynamics?

We consider a linear point process as a model for pulse-coded, or spiking activity in a neuronal network. Using recent results from theory of such processes, we study third-order correlations between spike trains in such a system and explain which features of the network graph i.

This, however, ceases to be the case in networks with a geometric out-degree distribution, where topological specificities have a strong impact on average correlations. Many biological phenomena can be viewed as dynamical processes on a graph. Understanding coordinated activity of nodes in such a network is of some importance, as it helps to characterize the behavior of the complex system.

Of course, the topology of a network plays a pivotal role in determining the level of coordination among its different vertices. In particular, correlations between triplets of events here: action potentials generated by neurons have recently garnered some interest in the theoretical neuroscience community.

Moments of Distributions

In this paper, we present a decomposition of an average measure of third-order coordinated activity of neurons in a spiking neuronal network in terms of the relevant topological motifs present in the underlying graph. We study different network topologies and show, in particular, that the presence of certain tree motifs in the synaptic connectivity graph greatly affects the strength of third-order correlations between spike trains of different neurons.

Analyzing networks of interacting elements has become the tool of choice in many areas of biology. In recent years, network models have been used to study the interactions between predator and prey [ 1 ], gene interactions [ 2 ] and neural network dynamics [ 34 ].

A fundamental question in the study of complex networks is how the topology of the graph on which a dynamic process evolves influences its activity. A particularly interesting issue is the emergence of synchronized, or correlated patterns of events. While it is obvious that the presence or absence of such patterns of activity depends largely on how individual nodes in the network are connected, it is by no means a trivial task to explain exactly how this happens. In theoretical neuroscience, the connection between network topology and correlated activity continues to be an important topic of study.

Not only are correlations between neuronal spike trains believed to have an important function in information processing [ 56 ] and coincidence detection [ 7 ], but they are also believed to be tied to expectation and attention see [ 7 ] for details. In addition, it been shown that nerve cells can be extremely sensitive to synchronous input from large groups of neurons [ 8 ].

While there has been much work on elucidating the causes and effects of pairwise correlations between spike trains [ 3 ], it seems that correlations beyond second order also have a role to play in the brain. Higher-order correlations have also been reported in the rat somatosensory cortex and the visual cortex of the behaving macaque [ 10 ]. Indeed, it has been suggested that these correlations are inherent properties of cortical dynamics in many species [ 1112 ]. As a result, neural data has recently been intensively investigated for signs of higher-order synchrony using classical means such as maximum entropy models [ 13 — 18 ].

In addition, new methods are being developed in order to shed more light on what seems to be a very important property of networks in the brain [ 19 — 21 ]. In this work, we study the relation between the topology i. Our aim was to show how triplet correlations depend on topological motifs in a network with known connectivity. We hope our results can be used to facilitate thought experiments to relate hypothetical connectivity to third-order correlations by, for example, assuming specific network topologies and then computing how these assumptions affect the dynamics.

While this might be a point of contention, in previous work, it was clearly shown that that a mapping between synaptically coupled spiking networks e. In addition, it has been demonstrated that synaptic connectivity can be reconstructed from simulated spike trains with very high fidelity, provided the network has a connectivity which is not too dense and not too sparse [ 23 ].

However, we would also like to point out that knowing the true connectivity in an experimental setting is close to impossible. In addition, the inference of connectivity from neural data is confounded by undersampling. One can typically only record from a tiny fraction of all neurons that constitute the network, while most of the population remains effectively hidden to the experimenter.

Similar work, pertaining to the influence of connectivity on correlations of second order has already been published [ 324 — 26 ]. In it, the authors dissect the contribution of specific structural motifs to the emergence of pairwise correlations in a recurrent network of interconnected point processes, meant to represent neurons communicating via spikes.The dependence of multiproton correlation functions and cumulants on the acceptance in rapidity and transverse momentum is studied.

Here, we found that the preliminary data of various cumulant ratios are consistent, within errors, with rapidity and transverse momentum-independent correlation functions.

But, rapidity correlations which moderately increase with rapidity separation between protons are slightly favored.

We propose to further explore the rapidity dependence of multiparticle correlation functions by measuring the dependence of the integrated reduced correlation functions as a function of the size of the rapidity window.

Works referenced in this record:. GOV collections:. Title: Rapidity dependence of proton cumulants and correlation functions. Abstract The dependence of multiproton correlation functions and cumulants on the acceptance in rapidity and transverse momentum is studied.

Bzdak, Adam, and Koch, Volker. Rapidity dependence of proton cumulants and correlation functions. United States: N.

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Copy to clipboard. United States. Free Publicly Available Full Text. Accepted Manuscript Publisher. Accepted Manuscript DOE. Copyright Statement. Other availability. Search WorldCat to find libraries that may hold this journal. Cited by: 2 works. Citation information provided by Web of Science. LinkedIn Pinterest Tumblr. Physical Review C, Vol. Physical Review Letters, Vol. Nuclear Physics A, Vol.

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Physical Review D, Vol. Journal of High Energy Physics, Vol. Nuclear Physics B, Vol.In this paper, the topic of coherent two-dimensional direction of arrival 2D-DOA estimation is investigated. Compared to the traditional PARALIND decomposition, the proposed algorithm owns lower computational complexity and smaller data storage capacity due to the process of compression.

Besides, the proposed algorithm can obtain autopaired azimuth angles and elevation angles and can achieve the same estimation performance as the traditional PARALIND, which outperforms some familiar algorithms presented for coherent sources such as the forward backward spatial smoothing-estimating signal parameters via rotational invariance techniques FBSS-ESPRIT and forward backward spatial smoothing-propagator method FBSS-PM.

Array signal processing has aroused considerable concerns in recent decades owing to its extensive engineering application in satellite communication, radar, sonar, and some other fields [ 1 — 4 ]. In array signal processing, spectrum estimation, also known as direction of arrival DOA estimation, is a crucial issue.

Till now, there are already many neoteric algorithms [ 5 — 8 ] proposed for DOA estimation with linear array, which include estimating signal parameters via rotational invariance techniques ESPRIT algorithm [ 56 ], multiple signal classification MUSIC algorithm [ 7 ], and propagator method PM [ 8 ].

Compared with linear array, rectangular array can measure both azimuth angle and elevation angle, and hence 2D-DOA estimation with rectangular array has motivated enormous investigations. Some other algorithms such as angle estimation with generalized coprime planar array [ 13 ] and 2D-DOA estimation with nested subarrays [ 14 ] have better performance for 2D-DOA estimation as well.

However, in many practical situations, the complex propagation environment usually leads to the presence of coherent signals, which makes it complicated to obtain the valid DOA estimation. Therefore, the research on coherent angle estimation has gained great significance. The algorithms mentioned above are only applicable to noncoherent sources and the coherent sources will lead to severe invalidity for these methods.

Some other coherent estimation methods including traditional forward spatial smoothing FSS or forward backward spatial smoothing FBSS [ 1516 ] algorithm have good estimation performance. Exploiting coprime multiple-input multiple-output MIMO radar, [ 17 ] has proposed the DOA estimation method for mixed coherent and uncorrelated targets and [ 18 ] obtained the DOA estimation of coherent sources via fourth-order cumulants.

The trilinear decomposition, namely, parallel factor PARAFAC technique [ 19 — 21 ], has been widely employed to resolve the problem of angular estimation with rectangular array [ 22 ]. Compressed sensing CS [ 2728 ] has aroused considerable concern, which is introduced to areas including channel estimation, image, beamforming, and radar [ 29 — 32 ]. Specifically, the angular information of sources can be structured as a sparse vector and hence the CS technique can be directly utilized [ 3334 ].

By applying CS theory, many novel parameter estimation algorithms have been proposed for different scenarios. Finally, to acquire the 2D-DOA estimation, we formulate a sparse recovery problem which can be solved by the orthogonal matching pursuit OMP method [ 37 ]. Note that the compression process only compresses the directional matrix and the source matrix while the coherent matrix remains the same. Therefore, the coherent structure of impinging signals is not destroyed after compression.

Comparing to [ 36 ], we summarize the main contributions of our research in this paper:. In this paper, the proposed algorithm can obtain autopaired 2D-DOA estimation of coherent signals. In addition, the corresponding correlated matrix can also be obtained. In addition, due to the process of compression, the proposed algorithm consumes lower computational burden and requires limited storage capacity in practical application.

A series of simulation results verify the effectiveness of our approach. The remainder of our paper is organized as follows: Section 2 presents the received data model of coherent signals with uniform rectangular array. Numerical simulations are exhibited in Section 4and we conclude this paper in Section 5. The Frobenius norm and l 0 —norm are denoted by and.

Assume that there are far-field narrow-band signals with DOA impinging on a uniform rectangular array consisting of sensors, where is the azimuth angle of the - th signal, represents the elevation angle, and the distance between any two adjacent elements is.


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