Bounds on the truncation error of periodic signals

Giardina, Charles R.; Chirlian, Paul M.

doi:10.1109/tct.1972.1083433

Cited by 13 publications

(9 citation statements)

References 2 publications

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“…We then consider the Fourier expansion errors. According to Theorem 2 in [16], the truncation error of the Fourier expansion, F can be bounded as follows…”

Section: Proof Of Theorem 26mentioning

confidence: 99%

On the Numerical Rank of Radial Basis Function Kernels in High Dimensions

Wang¹,

Li²,

Darve³

2018

SIAM J. Matrix Anal. Appl.

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Low-rank approximations are popular methods to reduce the high computational cost of algorithms involving large-scale kernel matrices. The success of low-rank methods hinges on the matrix rank of the kernel matrix, and in practice, these methods are effective even for highdimensional datasets. Their practical success motivates our analysis of the function rank, an upper bound of the matrix rank. In this paper, we consider radial basis functions (RBF), approximate the RBF kernel with a low-rank representation that is a finite sum of separate products and provide explicit upper bounds on the function rank and the L∞ error for such approximations. Our three main results are as follows. First, for a fixed precision, the function rank of RBFs, in the worst case, grows polynomially with the data dimension. Second, precise error bounds for the low-rank approximations in the L∞ norm are derived in terms of the function smoothness and the domain diameters. And last, a group pattern in the magnitude of singular values for RBF kernel matrices is observed and analyzed, and is explained by a grouping of the expansion terms in the kernel's low-rank representation. Empirical results verify the theoretical results.

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“…We then consider the Fourier expansion errors. According to Theorem 2 in [16], the truncation error of the Fourier expansion, F can be bounded as follows…”

Section: Proof Of Theorem 26mentioning

confidence: 99%

On the Numerical Rank of Radial Basis Function Kernels in High Dimensions

Wang¹,

Li²,

Darve³

2018

SIAM J. Matrix Anal. Appl.

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“…as a function of the number of terms included in the truncated series approximation. While this may be done for the error given by the L 2 norm [25], ||U −U N || 2 , we are not able to do so using the error given by the pointwise difference |U −U N |. In fact, Lemma 10.2 from [30] shows that the pointwise convergence of a Fourier series may be arbitrarily slow.…”

Section: Fourier Series Expansion Of a Periodic Potentialmentioning

confidence: 99%

Approximation of the Mean Escape Time for a Tilted Periodic Potential

Tamra¹,

Davis²,

Gedeon³

2019

CICP

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We present a formula approximating the mean escape time (MST) of a particle from a tilted multi-periodic potential well. The potential function consists of a weighted sum of a finite number of component functions, each of which is periodic. For this particular case, the least period of the potential function is a common period amongst all of its component functions. An approximation of the MST for the potential function is derived, and this approximation takes the form of a product of the MSTs for each of the individual periodic component functions. Our first example illustrates the computational advantages of using the approximation for model validation and parameter tuning in the context of the biological application of DNA transcription. We also use this formula to approximate the MST for an arbitrary tilted periodic potential by the product of MSTs of a finite number of its Fourier modes. Two examples using truncated Fourier series are presented and analyzed.

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“…For the describing function case, we need to estimate

ϵ_{1} = ∥ f - f_{1} ∥ = \int_{- π}^{π} falsefalse| f - f_{1} |^{2} normald t,

which is basically a scaled version of the error defined in (6). For Fourier series, an upper bound of this error can be obtained using the principle of bounded variation [23].…”

Section: Analytical Error Boundmentioning

confidence: 99%

“…Then

f \in V false[a, b false]

if and only if

f \in V false[a, c false]

and

f \in V false[c, b false]

. Furthermore,

V_{a}^{b} false(f false) = V_{a}^{c} false(f false) + V_{c}^{b} false(f false)

. Theorem [23]: If f ( t ) is periodic with frequency

ω_{0}

and the total variation over one period is bounded by V , then the mean‐square error in approximating up to

m th

harmonic is

ϵ_{m}^{2} \leq \frac{V^{2}}{π ω_{0} m} .

We use this result to develop an error bound for the describing function approximation. We apply this to classically known static nonlinearities and then illustrate how this can be applied to the dynamic nonlinearities such as the biomolecular systems discussed in Section 2.…”

Section: Analytical Error Boundmentioning

confidence: 99%

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Describing function‐based approximations of biomolecular systems

Dey

Sen

2017

IET syst. biol.

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Mathematical methods provide useful framework for the analysis and design of complex systems. In newer contexts such as biology, however, there is a need to both adapt existing methods as well as to develop new ones. Using a combination of analytical and computational approaches, the authors adapt and develop the method of describing functions to represent the input-output responses of biomolecular signalling systems. They approximate representative systems exhibiting various saturating and hysteretic dynamics in a way that is better than the standard linearisation. Furthermore, they develop analytical upper bounds for the computational error estimates. Finally, they use these error estimates to augment the limit cycle analysis with a simple and quick way to bound the predicted oscillation amplitude. These results provide system approximations that can add more insight into the local behaviour of these systems than standard linearisation, compute responses to other periodic inputs and to analyse limit cycles.

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Bounds on the truncation error of periodic signals

Cited by 13 publications

References 2 publications

On the Numerical Rank of Radial Basis Function Kernels in High Dimensions

On the Numerical Rank of Radial Basis Function Kernels in High Dimensions

Approximation of the Mean Escape Time for a Tilted Periodic Potential

Describing function‐based approximations of biomolecular systems

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