Detection of Large-Scale Noisy Multi-Periodic Patterns with Discrete Double Fourier Transform. II. Study of Correlations Between Patterns

2020 ◽  
pp. 2150003
Author(s):  
V. R. Chechetkin ◽  
V. V. Lobzin
Keyword(s):  
Fourier Transform ◽  
Correlation Function ◽  
Geomagnetic Activity ◽  
Large Scale ◽  
Statistical Significance ◽  
Periodic Patterns ◽  
Pattern Correlation ◽  
Double Fourier Transform ◽  
The Fourier Transform ◽  
Combined Technique

The discrete double Fourier transform (DDFT) was developed to search for large-scale multi-periodic patterns in the presence of noise and is based on detection of the equidistant series of harmonics generated by the periodic patterns in the discrete Fourier transform (DFT) spectra. As DDFT retains all generic features of the Fourier transform, the corresponding pattern correlation function (PCF) related to DDFT can be introduced similarly to the data correlation function (DCF) related to DFT on the basis of the Wiener–Khinchin relationship. Peaks in PCF indicate the number of periodic patterns in a dataset under analysis and have direct correspondence with the counterpart peaks in the DFT spectrum. The close correspondence between positions of the peaks in the PCF and DFT spectra strongly enhances statistical significance of detected periodicities. Similar PCFs can also be defined for the cepstrum transform. The combined DFT–DCF and DDFT–PCF technique was applied to the detection of cycles in geomagnetic activity using disturbance storm-time (Dst) index. In addition to the known 27-day, semiannual and 11-year cycles of geomagnetic activity, we have also found the annual cycle of activity. The results were compared with those obtained by the cepstrum transform. A multiple cross-check makes the combined technique much more efficient and robust in comparison with the detection based on a unique particular method.

scholarly journals Detection of Large-Scale Noisy Multi-Periodic Patterns with Discrete Double Fourier Transform

2019 ◽  
Vol 19 (02) ◽  
pp. 2050019 ◽  
Author(s):  
V. R. Chechetkin ◽  
V. V. Lobzin
Keyword(s):  
Fourier Transform ◽  
Dna Sequences ◽  
Large Scale ◽  
Solar Rotation ◽  
Statistical Significance ◽  
Solar Wind Speed ◽  
Periodic Patterns ◽  
Long Range Correlations ◽  
Practical Applications ◽  
Double Fourier Transform

In many processes, the variations in underlying characteristics can be approximated by noisy multi-periodic patterns. If large-scale patterns are superimposed by a noise with long-range correlations, the detection of multi-periodic patterns becomes especially challenging. To solve this problem, we developed a discrete double Fourier transform (DDFT). DDFT is based on the equidistance property of harmonics generated by multi-periodic patterns in the discrete Fourier transform (DFT) spectra. As the large-scale patterns generate long enough equidistant series, they can be detected by the iteration of the primary DFT. DDFT is defined as Fourier transform of intensity spectral harmonics or of their functions. It comprises widely used cepstrum transform as a particular case. We present also the relevant analytical criteria for the assessment of the statistical significance of peak harmonics in DDFT spectra in the presence of noise. DDFT technique was tested by extensive numerical simulations. The practical applications of the DDFT technique are illustrated by the analysis of variations in solar wind speed related to solar rotation and by the study of large-scale multi-periodic patterns in DNA sequences. The latter application can be considered as a generic example for the general spectral analysis of symbolic sequences. The results are compared with those obtained by the cepstrum transform. The mutual combination of DFT and DDFT provides an efficient technique to search for noisy large-scale multi-periodic patterns.

scholarly journals FastMMD: Ensemble of Circular Discrepancy for Efficient Two-Sample Test

2015 ◽  
Vol 27 (6) ◽  
pp. 1345-1372 ◽  
Author(s):  
Ji Zhao ◽  
Deyu Meng
Keyword(s):  
Fourier Transform ◽  
Time Complexity ◽  
Large Scale ◽  
Test Statistic ◽  
Approximation Accuracy ◽  
Sample Test ◽  
Geometric Explanation ◽  
Core Idea ◽  
The Fourier Transform ◽  
Biased Estimates

The maximum mean discrepancy (MMD) is a recently proposed test statistic for the two-sample test. Its quadratic time complexity, however, greatly hampers its availability to large-scale applications. To accelerate the MMD calculation, in this study we propose an efficient method called FastMMD. The core idea of FastMMD is to equivalently transform the MMD with shift-invariant kernels into the amplitude expectation of a linear combination of sinusoid components based on Bochner’s theorem and Fourier transform (Rahimi & Recht, 2007 ). Taking advantage of sampling the Fourier transform, FastMMD decreases the time complexity for MMD calculation from [Formula: see text] to [Formula: see text], where N and d are the size and dimension of the sample set, respectively. Here, L is the number of basis functions for approximating kernels that determines the approximation accuracy. For kernels that are spherically invariant, the computation can be further accelerated to [Formula: see text] by using the Fastfood technique (Le, Sarlós, & Smola, 2013 ). The uniform convergence of our method has also been theoretically proved in both unbiased and biased estimates. We also provide a geometric explanation for our method, ensemble of circular discrepancy, which helps us understand the insight of MMD and we hope will lead to more extensive metrics for assessing the two-sample test task. Experimental results substantiate that the accuracy of FastMMD is similar to that of MMD and with faster computation and lower variance than existing MMD approximation methods.

scholarly journals On the Importance of Statistical Homogeneity to the Scaling of Rain

2019 ◽  
Vol 36 (6) ◽  
pp. 1063-1078 ◽  
Author(s):  
A. R. Jameson
Keyword(s):  
Fourier Transform ◽  
Correlation Function ◽  
Power Spectrum ◽  
Wide Sense ◽  
Time Data ◽  
Statistical Homogeneity ◽  
Statistical Heterogeneity ◽  
Cross Correlations ◽  
The Fourier Transform ◽  
Universal Scaling Function

AbstractScaling studies of rainfall are important for the conversion of observations and numerical model outputs among all the various scales. Two common approaches for determining scaling relations are the Fourier transform of observations and the Fourier transform of a correlation function using the Wiener–Khintchine (WK) theorem. In both methods, the observations must be wide-sense statistically stationary (WSS) in time or wide-sense statistically spatially homogeneous (WSSH) in space so that the correlation function and power spectrum form a Fourier transform pair. The focus here is on developing an explicit understanding for the requirement. Statistically heterogeneous (either in space or time) data can produce serious scaling errors. This work shows that the effects of statistical heterogeneity appear as contributions from cross correlations among all of the distinct contributing rainfall components using either method so that the correlation function and its FFT do not form a transform pair. Moreover, the transform then also depends upon the time and location of the observations so that the “observed” power spectrum no longer represents a “universal” scaling function beyond the observations. An index of statistical heterogeneity (IXH) defined in previous work provides a way of determining whether or not a set of rain data may be considered to be WSS or WSSH. The greater IXH exceeds the null, the more likely the derived power spectrum should not be used for general scaling. Numerical simulations and some observations are used to demonstrate all of these findings.

On-line electron-optical correlation computing in the CTEM

1978 ◽  
Vol 36 (1) ◽  
pp. 88-89 ◽  
Author(s):  
R. Guckenberger ◽  
W. Hoppe
Keyword(s):  
Fourier Transform ◽  
Correlation Function ◽  
Transfer Function ◽  
Inverse Fourier Transform ◽  
Main Peak ◽  
Optical Correlation ◽  
Auto Correlation ◽  
Auto Correlation Function ◽  
On Line ◽  
The Fourier Transform

Light diffractograms of electron micrographs are frequently used to study the transfer function of the microscope. In order to utilize diffractograms for control operations in the microscope, several attempts have been undertaken to obtain on-line diffractograms /1 - 3/. Alternatively correlation functions (CF) may be used /4-8/. In this paper we describe an electron-optical device for the computation of such CF and its on-line operation in a microscope.The auto-correlation function (ACF) is the inverse Fourier transform of the squared modulus of the Fourier transform (diffractogram) of an image. Therefore it also contains the transfer function. It is its zero peak (main peak) which is of particular interest. In noisy images the main ACF-peak of the noise contributes in an unwanted way to the main ACF-peak of the image. This can be avoided if the ACF will be computed of two images which are identical except for noise /9/ (noise-reduced ACF= NRACF).

Classical and Nonclassical Optical Diffusion

1990 ◽  
Vol 195 ◽  
Author(s):  
J. M. Drake ◽  
A. Z. Genack
Keyword(s):  
Fourier Transform ◽  
Diffusion Coefficient ◽  
Correlation Function ◽  
Optical Transmission ◽  
Optical Field ◽  
Photon Absorption ◽  
Absorption Time ◽  
Transmission Data ◽  
The Fourier Transform ◽  
Field Correlation

ABSTRACTWe present a brief summary of our recent work on classical and nonclassical optical diffusion in disorder metal oxides. Results on the optical transmission through a slab of titania spheres is presented and discussed in some detail illustrating the experimental method for obtaining the optical diffusion coefficient and photon absorption time from the transmission data. We conclude that the optical field-field correlation function with frequency is the Fourier transform of the time of flight distribution.

ON THE FOURIER TRANSFORMS OF THE SPINOR COMPONENTS OF HYDROGEN-LIKE ATOM RELATIVISTIC EIGENFUNCTIONS

2006 ◽  
Vol 20 (09) ◽  
pp. 1123-1139 ◽  
Author(s):  
A. V. SOLDATOV ◽  
J. SEKE ◽  
G. ADAM
Keyword(s):  
Fourier Transform ◽  
Large Scale ◽  
Fourier Transforms ◽  
Singular Part ◽  
Generalized Function ◽  
Momentum Representation ◽  
One Dimensional ◽  
Multiple Products ◽  
The Fourier Transform ◽  
Dimensional Integral

An approach to calculate both the Fourier transforms of singular spinor components of relativistic eigenfunctions of hydrogen-like atoms and those of their multiple products is developed. The method simplifies calculations of the momentum representation for the eigenfunctions belonging to the discrete spectrum. Besides, it reduces the procedure of the Fourier transform for eigenfunctions of the continuous spectrum to the evaluation of only one one-dimensional integral, allowing the explicit separation of its singular part representing a generalized function. The proposed algorithm is especially convenient for large-scale numerical calculations.

Dietary Patterns: A New Therapeutic Approach for Depression?

2019 ◽  
Vol 20 (2) ◽  
pp. 123-129 ◽  
Author(s):  
Mariana Jesus ◽  
Tânia Silva ◽  
César Cagigal ◽  
Vera Martins ◽  
Carla Silva
Keyword(s):  
Depressive Symptoms ◽  
Dietary Patterns ◽  
Large Scale ◽  
Randomized Clinical Trials ◽  
Dietary Intervention ◽  
Common Mental Disorder ◽  
Controlled Trial ◽  
Statistical Significance ◽  
Inverse Association ◽  
The Impact

Introduction: The field of nutritional psychiatry is a fast-growing one. Although initially, it focused on the effects of vitamins and micronutrients in mental health, in the last decade, its focus also extended to the dietary patterns. The possibility of a dietary cost-effective intervention in the most common mental disorder, depression, cannot be overlooked due to its potential large-scale impact. Method: A classic review of the literature was conducted, and studies published between 2010 and 2018 focusing on the impact of dietary patterns in depression and depressive symptoms were included. Results: We found 10 studies that matched our criteria. Most studies showed an inverse association between healthy dietary patterns, rich in fruits, vegetables, lean meats, nuts and whole grains, and with low intake of processed and sugary foods, and depression and depressive symptoms throughout an array of age groups, although some authors reported statistical significance only in women. While most studies were of cross-sectional design, making it difficult to infer causality, a randomized controlled trial presented similar results. Discussion: he association between dietary patterns and depression is now well-established, although the exact etiological pathways are still unknown. Dietary intervention, with the implementation of healthier dietary patterns, closer to the traditional ones, can play an important role in the prevention and adjunctive therapy of depression and depressive symptoms. Conclusion: More large-scale randomized clinical trials need to be conducted, in order to confirm the association between high-quality dietary patterns and lower risk of depression and depressive symptoms.

Sign in / Sign up

Export Citation Format

Share Document