|
Multitaper
In signal processing, multitaper is a spectral density estimation technique developed by David J. Thomson. It can estimate the power spectrum ''S''''X'' of a stationary ergodic finite-variance random process ''X'', given a finite contiguous realization of ''X'' as data. Motivation The multitaper method overcomes some of the limitations of non-parametric Fourier analysis. When applying the Fourier transform to extract spectral information from a signal, we assume that each Fourier coefficient is a reliable representation of the amplitude and relative phase of the corresponding component frequency. This assumption, however, is not generally valid for empirical data. For instance, a single trial represents only one noisy realization of the underlying process of interest. A comparable situation arises in statistics when estimating measures of central tendency i.e., it is bad practice to estimate qualities of a population using individuals or very small samples. Likewise, a sing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MultiTaper Plot
In signal processing, multitaper is a spectral density estimation technique developed by David J. Thomson. It can estimate the power spectrum ''S''''X'' of a stationary ergodic finite-variance random process ''X'', given a finite contiguous realization of ''X'' as data. Motivation The multitaper method overcomes some of the limitations of non-parametric Fourier analysis. When applying the Fourier transform to extract spectral information from a signal, we assume that each Fourier coefficient is a reliable representation of the amplitude and relative phase of the corresponding component frequency. This assumption, however, is not generally valid for empirical data. For instance, a single trial represents only one noisy realization of the underlying process of interest. A comparable situation arises in statistics when estimating measures of central tendency i.e., it is bad practice to estimate qualities of a population using individuals or very small samples. Likewise, a sing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chronux
Chronux is an open-source software package developed for the loading, visualization and analysis of a variety of modalities / formats of neurobiological time series data. Usage of this tool enables neuroscientists to perform a variety of analysis on multichannel electrophysiological data such as LFP (local field potentials), EEG, MEG, Neuronal spike times and also on spatiotemporal data such as FMRI and dynamic optical imaging data. The software consists of a set of MATLAB routines interfaced with C libraries that can be used to perform the tasks that constitute a typical study of neurobiological data. These include local regression and smoothing, spike sorting and spectral analysis - including multitaper spectral analysis, a powerful nonparametric method to estimate power spectrum. The package also includes some GUIs for time series visualization and analysis. Chronux is GNU GPL v2 licensedhttp://chronux.org/downloads/chronux/chronux/manual.pdf (and MATLAB is proprietary). The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Window Function
In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions. The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra. The duration of the segments is determined in ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Concentration Problem
The spectral concentration problem in Fourier analysis refers to finding a time sequence of a given length whose discrete Fourier transform is maximally localized on a given frequency interval, as measured by the spectral concentration. Spectral concentration The discrete-time Fourier transform (DTFT) ''U''(''f'') of a finite series w_t, t = 1,2,3,4,...,T is defined as :U(f) = \sum_^w_t e^. In the following, the sampling interval will be taken as Δ''t'' = 1, and hence the frequency interval as ''f'' ∈ ½,½ ''U''(''f'') is a periodic function with a period 1. For a given frequency ''W'' such that 0\lambda_), then the eigenvector corresponding to \lambda_ is called ''nth''–order Slepian sequence (DPSS) (0≤''n''≤''N''-1). This ''nth''–order taper also offers the best sidelobe suppression and is pairwise orthogonal to the Slepian sequences of previous orders (0,1,2,3....,n-1). These lower order Slepian sequences form the basis for spectral estimation by multitaper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Slepian
David S. Slepian (June 30, 1923 – November 29, 2007) was an American mathematician. He is best known for his work with algebraic coding theory, probability theory, and distributed source coding. He was colleagues with Claude Shannon and Richard Hamming at Bell Labs. Life and work Born in Pittsburgh, Pennsylvania, he gained a B.Sc. at University of Michigan before joining the US Army in World War II, as a sonic deception officer in the Ghost army. He received his Ph.D. from Harvard University in 1949, writing his dissertation in physics. After post-doctoral work at the University of Cambridge and University of Sorbonne, he worked at the Mathematics Research Center at Bell Telephone Laboratories, where he pioneered work in algebraic coding theory on group codes, first published in the paper ''A Class of Binary Signaling Alphabets''. Here, he also worked along with other information theory giants such as Claude Shannon and Richard Hamming. He also proved the possibility of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Concentration Problem
The spectral concentration problem in Fourier analysis refers to finding a time sequence of a given length whose discrete Fourier transform is maximally localized on a given frequency interval, as measured by the spectral concentration. Spectral concentration The discrete-time Fourier transform (DTFT) ''U''(''f'') of a finite series w_t, t = 1,2,3,4,...,T is defined as :U(f) = \sum_^w_t e^. In the following, the sampling interval will be taken as Δ''t'' = 1, and hence the frequency interval as ''f'' ∈ ½,½ ''U''(''f'') is a periodic function with a period 1. For a given frequency ''W'' such that 0\lambda_), then the eigenvector corresponding to \lambda_ is called ''nth''–order Slepian sequence (DPSS) (0≤''n''≤''N''-1). This ''nth''–order taper also offers the best sidelobe suppression and is pairwise orthogonal to the Slepian sequences of previous orders (0,1,2,3....,n-1). These lower order Slepian sequences form the basis for spectral estimation by multitaper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Window Function
In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions. The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra. The duration of the segments is determined in ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neuroscience
Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, developmental biology, cytology, psychology, physics, computer science, chemistry, medicine, statistics, and Mathematical Modeling, mathematical modeling to understand the fundamental and emergent properties of neurons, glia and neural circuits. The understanding of the biological basis of learning, memory, behavior, perception, and consciousness has been described by Eric Kandel as the "epic challenge" of the Biology, biological sciences. The scope of neuroscience has broadened over time to include different approaches used to study the nervous system at different scales. The techniques used by neuroscientists have expanded enormously, from molecular biology, molecular and cell biology, cellular studies of individual neurons to neuroimaging, imaging ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequency Spectrum
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum. When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density. More commonly used is the power spectral density (or simply power spectrum), which applies to signals existing over ''all'' time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The power spectral density (PSD) then refers to the spectral energy distribution that would b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shannon Number
The Shannon number, named after the American mathematician Claude Shannon, is a conservative lower bound of the game-tree complexity of chess of 10120, based on an average of about 103 possibilities for a pair of moves consisting of a move for White followed by a move for Black, and a typical game lasting about 40 such pairs of moves. Shannon's calculation Shannon showed a calculation for the lower bound of the game-tree complexity of chess, resulting in about 10120 possible games, to demonstrate the impracticality of solving chess by brute force, in his 1950 paper "Programming a Computer for Playing Chess". (This influential paper introduced the field of computer chess.) Shannon also estimated the number of possible positions, "of the general order of \scriptstyle \frac, or roughly 1043". This includes some illegal positions (e.g., pawns on the first rank, both kings in check) and excludes legal positions following captures and promotions. After each player has mov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
