Distinguishing short and long FermiFermi gamma-ray bursts

Two classes of gamma-ray bursts (GRBs), short and long, have been determined without any doubts, and are usually ascribed to different progenitors, yet these classes overlap for a variety of descriptive parameters. A subsample of 46 long and 22 short $Fermi$ GRBs with estimated Hurst Exponents (HEs), complemented by minimum variability time-scales (MVTS) and durations ($T_{90}$) is used to perform a supervised Machine Learning (ML) and Monte Carlo (MC) simulation using a Support Vector Machine (SVM) algorithm. It is found that while $T_{90}$ itself performs very well in distinguishing short and long GRBs, the overall success ratio is higher when the training set is complemented by MVTS and HE. These results may allow to introduce a new (non-linear) parameter that might provide less ambiguous classification of GRBs.Comment: 9 pages, 6 figures; matches the before-proof version accepted for publication in MNRA

Analysis of the observed and intrinsic durations of SwiftSwift/BAT gamma-ray bursts

The duration distribution of 947 GRBs observed by $Swift$/BAT, as well as its subsample of 347 events with measured redshift, allowing to examine the durations in both the observer and rest frames, are examined. Using a maximum log-likelihood method, mixtures of two and three standard Gaussians are fitted to each sample, and the adequate model is chosen based on the value of the difference in the log-likelihoods, Akaike information criterion and Bayesian information criterion. It is found that a two-Gaussian is a better description than a three-Gaussian, and that the presumed intermediate-duration class is unlikely to be present in the $Swift$ duration data.Comment: 7 pages, 5 figures; matches the published version. arXiv admin note: text overlap with arXiv:1506.0780

Analysis of gamma-ray burst duration distribution using mixtures of skewed distributions

Two classes of GRBs have been confidently identified thus far and are prescribed to different physical scenarios -- NS-NS or NS-BH mergers, and collapse of massive stars, for short and long GRBs, respectively. A third, intermediate in duration class, was suggested to be present in previous catalogs, such as BATSE and $Swift$, based on statistical tests regarding a mixture of two or three log-normal distributions of $T_{90}$. However, this might possibly not be an adequate model. This paper investigates whether the distributions of $\log T_{90}$ from BATSE, $Swift$, and $Fermi$ are described better by a mixture of skewed distributions rather than standard Gaussians. Mixtures of standard normal, skew-normal, sinh-arcsinh and alpha-skew-normal distributions are fitted using a maximum likelihood method. The preferred model is chosen based on the Akaike information criterion. It is found that mixtures of two skew-normal or two sinh-arcsinh distributions are more likely to describe the observed duration distribution of $Fermi$ than a mixture of three standard Gaussians, and that mixtures of two sinh-arcsinh or two skew-normal distributions are models competing with the conventional three-Gaussian in the case of BATSE and $Swift$. Based on statistical reasoning, existence of a third (intermediate) class of GRBs in $Fermi$ data is rejected, and it is shown that other phenomenological models may describe the observed $Fermi$, BATSE, and $Swift$ duration distributions at least as well as a mixture of standard normal distributions.Comment: 9 pages, 7 figures, 3 tables; matches the version accepted by MNRAS. arXiv admin note: text overlap with arXiv:1602.0236

On the fractal dimension of the Duffing attractor

The box counting dimension $d_C$ and the correlation dimension $d_G$ change with the number of numerically generated points forming the attractor. At a sufficiently large number of points the fractal dimension tends to a finite value. The obtained values are $d_C\approx 1.43$ and $d_G\approx 1.38$.Comment: 10 pages, 5 figures; template changed, introduction and discussion enlarged, references and appendices added; accepted version published in the Romanian Reports of Physic

Analysis of the observed and intrinsic durations of gamma-ray bursts with known redshift

The duration distribution of 408 GRBs with measured both duration $T_{90}$ and redshift $z$ is examined. Mixtures of a number of distributions (standard normal, skew-normal, sinh-arcsinh, and alpha-skew-normal) are fitted to the observed and intrinsic durations using the maximum log-likelihood method. The best fit is chosen via the Akaike information critetion. The aim of this work is to assess the presence of the presumed intermediate GRB class, and to provide a phenomenological model more appropriate than the common mixture of standard Gaussians. While $\log T^{obs}_{90}$ are well described by a truly trimodal fit, after moving to the rest frame the statistically most significant fit is unimodal. To trace the source of this discrepancy, 334 GRBs observed only by $Swift$/BAT are examined in the same way. In the observer frame, this results in a number of statistically plausible descriptions, being uni- and bimodal, and with the number of components ranging from one to three. After moving to the rest frame, no unambiguous conclusions may be put forward. It is concluded that the size of the sample is not big enough to infer reliably GRB properties based on a univariate statistical reasoning only.Comment: 12 pages, 10 figures; accepted in Astrophysics and Space Scienc

Analysis of Fermi gamma-ray burst duration distribution

Two classes of GRBs, short and long, have been determined without any doubts, and are usually prescribed to different physical scenarios. A third class, intermediate in $T_{90}$ durations, has been reported to be present in the datasets of BATSE, Swift, RHESSI and possibly BeppoSAX. The latest release of $>1500$ GRBs observed by Fermi gives an opportunity to further investigate the duration distribution. The aim of this paper is to investigate whether a third class is present in the $\log T_{90}$ distribution, or is it described by a bimodal distribution. A standard $\chi^2$ fitting of a mixture of Gaussians is applied to 25 histograms with different binnings. Different binnings give various values of the fitting parameters, as well as the shape of the fitted curve. Among five statistically significant fits none is trimodal. Locations of the Gaussian components are in agreement with previous works. However, a trimodal distribution, understood in the sense of having three separated peaks, is not found for any binning. It is concluded that the duration distribution in Fermi data is well described by a mixture of three log-normal distributions, but it is intrinsically bimodal, hence no third class is present in the $T_{90}$ data of Fermi. It is suggested that the log-normal fit may not be an adequate model.Comment: 6 pages, 3 figures; matches the version to be publishe

On the relationship between the Hurst exponent, the ratio of the mean square successive difference to the variance, and the number of turning points

The long range dependence of the fractional Brownian motion (fBm), fractional Gaussian noise (fGn), and differentiated fGn (DfGn) is described by the Hurst exponent $H$. Considering the realisations of these three processes as time series, they might be described by their statistical features, such as half of the ratio of the mean square successive difference to the variance, $\mathcal{A}$, and the number of turning points, $T$. This paper investigates the relationships between $\mathcal{A}$ and $H$, and between $T$ and $H$. It is found numerically that the formulae $\mathcal{A}(H)=a{\rm e}^{bH}$ in case of fBm, and $\mathcal{A}(H)=a+bH^c$ for fGn and DfGn, describe well the $\mathcal{A}(H)$ relationship. When $T(H)$ is considered, no simple formula is found, and it is empirically found that among polynomials, the fourth and second order description applies best. The most relevant finding is that when plotted in the space of $(\mathcal{A},T)$, the three process types form separate branches. Hence, it is examined whether $\mathcal{A}$ and $T$ may serve as Hurst exponent indicators. Some real world data (stock market indices, sunspot numbers, chaotic time series) are analyzed for this purpose, and it is found that the $H$'s estimated using the $H(\mathcal{A})$ relations (expressed as inverted $\mathcal{A}(H)$ functions) are consistent with the $H$'s extracted with the well known wavelet approach. This allows to efficiently estimate the Hurst exponent based on fast and easy to compute $\mathcal{A}$ and $T$, given that the process type: fBm, fGn or DfGn, is correctly classified beforehand. Finally, it is suggested that the $\mathcal{A}(H)$ relation for fGn and DfGn might be an exact (shifted) $3/2$ power-law.Comment: 20 pages in one-column format, 7 figures; matches the version accepted for publicatio