New Probability Distributions in Astrophysics: XII. Truncation for the Gompertz Distribution

Analytical functions which fit the probability distributions of stars and galaxies can provide insight into how these distributions originate. In order to introduce a truncated version of the Gompertz distribution, we derive its probability density function, its distribution function, its average value, its second moment about the origin, its median, its random generation of values and a maximum likelihood estimator for its two unknown parameters. The astrophysical applications of the Gompertz distribution are the initial mass function for stars, the luminosity function for the galaxies of the Sloan Digital Sky Survey, the photometric maximum of galaxies visible in the GLADE+ catalog and a model for the mean absolute magnitude in the GLADE+ catalog as a function of the redshift.

Advances in statistical mechanics of rock masses and its engineering applications

To efficiently link the continuum mechanics for rocks with the structural statistics of rock masses,a theoretical and methodological system called the statistical mechanics of rock masses(SMRM)was developed in the past three decades.In SMRM,equivalent continuum models of stressestrain relationship,strength and failure probability for jointed rock masses were established,which were based on the geometric probability models characterising the rock mass structure.This follows the statistical physics,the continuum mechanics,the fracture mechanics and the weakest link hypothesis.A general constitutive model and complete stressestrain models under compressive and shear conditions were also developed as the derivatives of the SMRM theory.An SMRM calculation system was then developed to provide fast and precise solutions for parameter estimations of rock masses,such as full-direction rock quality designation(RQD),elastic modulus,Coulomb compressive strength,rock mass quality rating,and Poisson’s ratio and shear strength.The constitutive equations involved in SMRM were integrated into a FLAC3D based numerical module to apply for engineering rock masses.It is also capable of analysing the complete deformation of rock masses and active reinforcement of engineering rock masses.Examples of engineering applications of SMRM were presented,including a rock mass at QBT hydropower station in northwestern China,a dam slope of Zongo II hydropower station in D.R.Congo,an open-pit mine in Dexing,China,an underground powerhouse of Jinping I hydropower station in southwestern China,and a typical circular tunnel in Lanzhou-Chongqing railway,China.These applications verified the reliability of the SMRM and demonstrated its applicability to broad engineering issues associated with jointed rock masses.

Ensemble prediction model of solar proton events associated with solar flares and coronal mass ejections

An ensemble prediction model of solar proton events （SPEs）, combining the information of solar flares and coronal mass ejections （CMEs）, is built. In this model, solar flares are parameterized by the peak flux, the duration and the longitude. In addition, CMEs are parameterized by the width, the speed and the measurement position angle. The importance of each parameter for the occurrence of SPEs is estimated by the information gain ratio. We find that the CME width and speed are more informative than the flare’s peak flux and duration. As the physical mechanism of SPEs is not very clear, a hidden naive Bayes approach, which is a probability-based calculation method from the field of machine learning, is used to build the prediction model from the observational data. As is known, SPEs originate from solar flares and/or shock waves associated with CMEs. Hence, we first build two base prediction models using the properties of solar flares and CMEs, respectively. Then the outputs of these models are combined to generate the ensemble prediction model of SPEs. The ensemble prediction model incorporating the complementary information of solar flares and CMEs achieves better performance than each base prediction model taken separately.

The inversion of the real kinematic properties of coronal mass ejections by forward modeling

Kinematic properties of coronal mass ejections （CMEs） suffer from projection effects,and it is expected that the real velocity should be larger and the real angular width should be smaller than the apparent values.Several attempts have been taken to correct the projection effects,which however led to an inflated average velocity probably due to the biased choice of CME events.In order to estimate the overall influence of the projection effects on the kinematic properties of the CMEs,we perform a forward modeling of real distributions of CME properties,such as the velocity,the angular width,and the latitude,by requiring their projected distributions to best match observations.Such a matching is conducted by Monte Carlo simulations.According to the derived real distributions,we found that （1） the average real velocity of all non-full-halo CMEs is about 514 km s-1,and the average real angular width is about 33°,in contrast to the corresponding apparent values of 418 km s-1 and 42.7° in observations;（2） For the CMEs with the angular width in the range of 20°-120°,the average real velocity is 510 km s-1 and the average real angular width is 43.4°,in contrast to the corresponding apparent values of 392 km s-1 and 52° in observations.

New Probability Distributions in Astrophysics: X. Truncation and Mass-Luminosity Relationship for the Frèchet Distribution

The Frèchet distribution has aided the modelling of scientific data in many contexts. We demonstrate how it can be adapted to model astrophysical data. We analyze the truncated version of the Frèchet distribution deriving the probability density function (PDF), the distribution function, the average value, the rth moment about the origin, the median, the random generation of values and the maximum likelihood estimator, which allows us to derive the two unknown parameters. This first PDF in the regular and truncated version is then applied to model the mass of the stars. A canonical transformation from the mass to the luminosity allows us to derive a new PDF, which is derived in its regular and truncated version. Finally, we apply this new PDF model on the distribution in luminosity of NGC 2362.

New Probability Distributions in Astrophysics: XI. Left Truncation for the Topp-Leone Distribution

The Topp-Leone (T-L) distribution has aided the modeling of scientific data in many contexts. We demonstrate how it can be adapted to model astrophysical data. We analyse the left truncated version of the T-L distribution, deriving its probability density function (PDF), distribution function, average value, rth moment about the origin, median, the random generation of its values, and its maximum likelihood estimator, which allows us to derive the two unknown parameters. The T-L distribution, in its regular and truncated versions, is then applied to model the initial mass function for the stars. A comparison is made with specific clusters and between proposed functions for the IMF. The Topp-Leone distribution can provide an excellent fit in some cases.

A quick and effective multivariate statistical strategy for imaging mass spectrometry

A new multivariate statistical strategy for analyzing large datasets that are produced by imaging mass spectrometry（IMS） techniques is reported.The strategy divides the whole datacube of the sample into several subsets and analyses them one by one to obtain the results.Instead of analyzing the whole datacube at one time,the strategy makes the analysis easier and decreases the computation time greatly.In this report,the IMS data are produced by the air flow-assisted ionization IMS（AFAI-IMS）.The strategy can be used in combination with most multivariate statistical analysis methods.In this paper,the strategy was combined with the principal component analysis（PCA） and partial least square analysis（PLS）.It was proven to be effective by analyzing the handwriting sample.By using the strategy,the m/z corresponding to the specific lipids in rat brain tissue were distinguished successfully.Moreover the analysis time grew linearly instead of exponentially as the size of sample increased.The strategy developed in this study has enormous potential for searching for the mjz of potential biomarkers quickly and effectively.

New Probability Distributions in Astrophysics: XIII. Truncation for the Benini Distribution

In order to introduce a right truncated version of the Benini distribution, we derive its probability density function, its distribution function, its average value, its kth moment about the origin, its median, how to randomly generate its values, and the maximum likelihood estimator for its three unknown parameters. The astrophysical application of the Benini distribution and its right truncated version is to the initial mass function for stars.