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Sparse identification of nonlinear dynamics (SINDy) is a data-driven algorithm for obtaining dynamical systems from data. [1] Given a series of snapshots of a dynamical system and its corresponding time derivatives, SINDy performs a sparsity-promoting regression (such as LASSO) on a library of nonlinear candidate functions of the snapshots against the derivatives to find the governing equations.
Ideally, unevenly spaced time series are analyzed in their unaltered form. However, most of the basic theory for time series analysis was developed at a time when limitations in computing resources favored an analysis of equally spaced data, since in this case efficient linear algebra routines can be used and many problems have an explicit ...
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation [1] [2] is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting ...
By comparison, if principal component analysis, which is a linear dimensionality reduction algorithm, is used to reduce this same dataset into two dimensions, the resulting values are not so well organized. This demonstrates that the high-dimensional vectors (each representing a letter 'A') that sample this manifold vary in a non-linear manner.
Here x ≥ 0 means that each component of the vector x should be non-negative, and ‖·‖ 2 denotes the Euclidean norm. Non-negative least squares problems turn up as subproblems in matrix decomposition, e.g. in algorithms for PARAFAC [2] and non-negative matrix/tensor factorization. [3] [4] The latter can be considered a generalization of ...
Lur'e problem block diagram. An early nonlinear feedback system analysis problem was formulated by A. I. Lur'e.Control systems described by the Lur'e problem have a forward path that is linear and time-invariant, and a feedback path that contains a memory-less, possibly time-varying, static nonlinearity.
In stochastic processes, chaos theory and time series analysis, detrended fluctuation analysis (DFA) is a method for determining the statistical self-affinity of a signal. It is useful for analysing time series that appear to be long-memory processes (diverging correlation time, e.g. power-law decaying autocorrelation function) or 1/f noise.
Its subject matter includes: [1]: 1–2 generalizations of calculus to Banach spaces; implicit function theorems; fixed-point theorems (Brouwer fixed point theorem, Fixed point theorems in infinite-dimensional spaces, topological degree theory, Jordan separation theorem, Lefschetz fixed-point theorem)
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