scholarly journals Analysis of Natural and Artificial Phenomena Using Signal Processing and Fractional Calculus

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
J. A. Tenreiro Machado ◽  
António M. Lopes
Keyword(s):  
Time Series ◽  
Fourier Transform ◽  
Vector Field ◽  
System Dynamics ◽  
Power Law ◽  
System Behavior ◽  
Amplitude Spectra ◽  
The Fourier Transform ◽  
Complex Phenomena ◽  
Power Law Parameters

AbstractIn this paper we study several natural and man-made complex phenomena in the perspective of dynamical systems. For each class of phenomena, the system outputs are time-series records obtained in identical conditions. The time-series are viewed as manifestations of the system behavior and are processed for analyzing the system dynamics. First, we use the Fourier transform to process the data and we approximate the amplitude spectra by means of power law functions. We interpret the power law parameters as a phenomenological signature of the system dynamics. Second, we adopt the techniques of non-hierarchical clustering and multidimensional scaling to visualize hidden relationships between the complex phenomena. Third, we propose a vector field based analogy to interpret the patterns unveiled by the PL parameters.

An analysis of the spectrum of a gravity anomaly over an inclined dike

Geophysics ◽  
1985 ◽  
Vol 50 (9) ◽  
pp. 1500-1501
Author(s):  
B. N. P. Agarwal ◽  
D. Sita Ramaiah
Keyword(s):  
Fourier Transform ◽  
Gravity Anomaly ◽  
Fourier Spectrum ◽  
Fourier Transforms ◽  
Weighted Average ◽  
Window Function ◽  
Average Spectrum ◽  
Amplitude Spectra ◽  
Distance Data ◽  
The Fourier Transform

Bhimasankaram et al. (1977) used Fourier spectrum analysis for a direct approach to the interpretation of gravity anomaly over a finite inclined dike. They derived several equations from the real and imaginary components and from the amplitude and phase spectra to relate various parameters of the dike. Because the width 2b of the dike (Figure 1) appears only in sin (ωb) term—ω being the angular frequency—they determined its value from the minima/zeroes of the amplitude spectra. The theoretical Fourier spectrum uses gravity field data over an infinite distance (length), whereas field observations are available only for a limited distance. Thus, a set of observational data is viewed as a product of infinite‐distance data with an appropriate window function. Usually, a rectangular window of appropriate distance (width) and of unit magnitude is chosen for this purpose. The Fourier transform of the finite‐distance and discrete data is thus represented by convolution operations between Fourier transforms of the infinite‐distance data, the window function, and the comb function. The combined effect gives a smooth, weighted average spectrum. Thus, the Fourier transform of actual observed data may differ substantially from theoretic data. The differences are apparent for low‐ and high‐frequency ranges. As a result, the minima of the amplitude spectra may change considerably, thereby rendering the estimate of the width of the dike unreliable from the roots of the equation sin (ωb) = 0.

Testing for nonlinearity in time series without the Fourier transform

2005 ◽  
Vol 72 (5) ◽  
Author(s):  
Tomomichi Nakamura ◽  
Xiaodong Luo ◽  
Michael Small
Keyword(s):  
Time Series ◽  
Fourier Transform ◽  
The Fourier Transform

scholarly journals NONPARAMETRIC ANALYSES OF LOG-PERIODIC PRECURSORS TO FINANCIAL CRASHES

2003 ◽  
Vol 14 (08) ◽  
pp. 1107-1125 ◽  
Author(s):  
WEI-XING ZHOU ◽  
DIDIER SORNETTE
Keyword(s):  
Time Series ◽  
Fourier Transform ◽  
Power Law ◽  
Hilbert Transform ◽  
Strong Evidence ◽  
Unpublished Report ◽  
Dow Jones Industrial Average ◽  
The Hilbert Transform ◽  
Financial Indices ◽  
Good Agreement

We apply two nonparametric methods to further test the hypothesis that log-periodicity characterizes the detrended price trajectory of large financial indices prior to financial crashes or strong corrections. The term "parametric" refers here to the use of the log-periodic power law formula to fit the data; in contrast, "nonparametric" refers to the use of general tools such as Fourier transform, and in the present case the Hilbert transform and the so-called (H, q)-analysis. The analysis using the (H, q)-derivative is applied to seven time series ending with the October 1987 crash, the October 1997 correction and the April 2000 crash of the Dow Jones Industrial Average (DJIA), the Standard & Poor 500 and Nasdaq indices. The Hilbert transform is applied to two detrended price time series in terms of the ln (tc-t) variable, where tcis the time of the crash. Taking all results together, we find strong evidence for a universal fundamental log-frequency f=1.02±0.05 corresponding to the scaling ratio λ=2.67±0.12. These values are in very good agreement with those obtained in earlier works with different parametric techniques. This note is extracted from a long unpublished report with 58 figures available at , which extensively describes the evidence we have accumulated on these seven time series, in particular by presenting all relevant details so that the reader can judge for himself or herself the validity and robustness of the results.

scholarly journals Time series forecasting using amplitude-frequency analysis of STL components

2021 ◽  
Vol 2094 (3) ◽  
pp. 032019
Author(s):  
D G Chkalova
Keyword(s):  
Time Series ◽  
Fourier Transform ◽  
Frequency Analysis ◽  
Software Implementation ◽  
Phase Operator ◽  
Economic Time Series ◽  
Economic Time ◽  
Harmonic Components ◽  
The Fourier Transform ◽  
Forecast Quality

Abstract The problem of economic time series analysis and forecasting using amplitude-frequency analysis of STL decomposition is considered. An amplitude-phase operator was chosen as an apparatus for extraction the series harmonic components, the advantages of which (compared to the Fourier transform) are: calculations speed, result accuracy, simplicity and interpretability of software implementation. The forecast quality was carried out using the MAPE metric. Significantly higher prediction quality was achieved compared to Facebook Prophet forecasting package.

scholarly journals Time-series Imputation Algorithm

2021 ◽  
Author(s):  
David Howe
Keyword(s):  
Time Series ◽  
Fourier Transform ◽  
Time Series Data ◽  
Series Data ◽  
Allan Deviation ◽  
Original Time Series ◽  
Power Spectral ◽  
Power Spectral Densities ◽  
The Fourier Transform ◽  
Original Time

Statistical imputation is a field of study that attempts to fill missing data. It is commonly applied to population statistics whose data have no correlation with running time. For a time series, data is typically analyzed using the autocorrelation function (ACF), the Fourier transform to estimate power spectral densities (PSD), the Allan deviation (ADEV), trend extensions, and basically any analysis that depends on uniform time indexes. We explain the rationale for an imputation algorithm that fills gaps in a time series by applying a backward, inverted replica of adjacent live data. To illustrate, four intentional massive gaps that exceed 100% of the original time series are recovered. The L(f) PSD with imputation applied to the gaps is nearly indistinguishable from the original. Also, the confidence of ADEV with imputation falls within 90% of the original ADEV with mixtures of power-law noises. The algorithm in Python is included for those wishing to try it.

scholarly journals Solar 5-minute Oscillations at 2.23 μm

1994 ◽  
Vol 154 ◽  
pp. 271-276
Author(s):  
Torben Leifsen
Keyword(s):  
Time Series ◽  
Fourier Transform ◽  
Near Infrared ◽  
Broad Band ◽  
Solar Observatory ◽  
Fourier Transform Spectrometer ◽  
Large Intensity ◽  
National Solar Observatory ◽  
Long Run ◽  
The Fourier Transform

Large amplitude solar 5-min intensity oscillations have recently been detected at 2.23 μm using broad band (650 Å FWHM) photometry (Leifsen and Maltby, 1990). Large intensity amplitudes in a broad range in the near infrared was unexpected, and several questions concerning the source of the high amplitudes were raised. In an attempt to study the nature of these oscillations, time series of spectra have been obtained with the Fourier Transform Spectrometer (FTS) of the McMath telescope at National Solar Observatory at Kitt Peak. We present preliminary results from a 10 day long run in May 1991 in support for the suggestion that the results may be useful in both helio- and asteroseismological investigations.

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