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Bivariate goodness-of-fit test, similar to Kolmogoroff-Smirnoff univariate test for estimating error and erasure probability of indicator channel - information processing
Statistical method for evaluating goodness of fit between data set and theoretical distribution of random variables
New methods are presented that utilize the optimization of goodness-of-fit statistics in order to estimate Weibull parameters from failure data. It is assumed that the underlying population is characterized by a three-parameter Weibull distribution. Goodness-of-fit tests are based on the empirical distribution function (EDF). The EDF is a step function, calculated using failure data, and represents an approximation of the cumulative distribution function for the underlying population. Statistics (such as the Kolmogorov-Smirnov statistic and the Anderson-Darling statistic) measure the discrepancy between the EDF and the cumulative distribution function (CDF). These statistics are minimized with respect to the three Weibull parameters. Due to nonlinearities encountered in the minimization process, Powell's numerical optimization procedure is applied to obtain the optimum value of the EDF. Numerical examples show the applicability of these new estimation methods. The results are compared to the estimates obtained with Cooper's nonlinear regression algorithm.
Frequency function for testing small samples for goodness of fit to normal populations
This paper presents solutions to the equation of transfer for Thomson scattering in a constant magnetic field of arbitrary orientation. Results from several atmospheres are combined to give the flux from a dipole star. The results are compared to the polarization data of the magnetic white dwarf Grw + 70 deg 8247. The fit is good, though it implies a very large polarization in the ultraviolet. Thomson scattering is not thought to be an important opacity source in white dwarfs, so the good fit is either fortuitous or is perhaps explained by assuming the magnetic field affects the polarization processes in all opacities similarly.
Probability density function of chi-square statistics applied to goodness of fit tests
Autocorrelated normal random variates were generated via computer and the effects of various levels of correlation on goodness of fit problems were studied. The results are useful in determining the distribution or estimating the parameters of populations that correlate observations such as wind speeds and temperature. The model used to generate the autocorrelated data is an autoregressive process of order 1. The Kolmogorov-Smirnov and chi-square statistics are used in the analysis. It was observed in the simulation that high positive correlations tend to shift the sample mean away from the population mean and negative correlations tend to shift the sample mean towards the population mean. In many cases, it was observed that positive and negative correlations tend to decrease the standard deviation. However, since this did not occur in all cases, no definite conclusion can be made regarding the standard deviation. Since the autoregressive process is a linear transformation, it is not surprising that normality was preserved. However, a possible extension of this problem could be to generate non-normal data and observe how the distribution is affected by correlation. Another extension might utilize another model such as autoregressive of order k or a moving average process of order k.
The origin of the Martian dichotomy, which divides highlands from lowlands, is unknown. We examine a section of the dichotomy ( 50 - 90E) defined by steep scarps and normal faults. Stratigraphy and age relationships preclude the formation of the 2.5 km high boundary via erosion. The abrupt disappearance of topographic knobs similar to 300 - 500 km to the northeast is interpreted as a buried fault. Alignment of the buried fault with grabens, stratigraphy, and age determinations using crater counts indicate that the lowland bench is down faulted highlands crust. The estimated local strain (3.5%) and fault pattern are broadly consistent with gravitational relaxation of a plateau boundary. Magnetic and gravity anomalies occur on either side of the buried fault. Admittance analysis indicates isostatic compensation. Although nonunique, a model with a 10 km thick intracrustal block under the lowland bench, a 20 km thick block under the plains, and an excess density of 200 kg/m(3) provides a good fit to the isostatic anomaly. A good fit to a profile of the magnetic field perpendicular to the dichotomy is produced using uniformly polarized intracrustal blocks 10 - 20 km thick, an intensity of 6 Am/m, a field inclination of -30 degrees, and gaps aligned with the isostatic anomalies. One interpretation is that high-density intrusions demagnetized the crust after dynamo cessation and that low-lying magnetized areas could be down faulted highlands crust. Another model (inclination of 30 degrees) has magnetized crust beneath the isostatic anomalies, separated by gaps. The gaps could result from hydrothermal alteration of the crust along fault zones.
The X square statistic is a useful measure of the discrepancy between the actual distribution of a set of data points and the theoretical distribution of a random variable of which the data points supposedly are values. Thus the X square statistic is frequently used in goodness of fit tests.
The interaction of Io with its plasma torus and the Jovian magnetic field is described and examined in the context of several currently popular models. Three specific matters are addressed. First, features implied by sub-Alfvenic flow which must be common to all models are discussed. Next, the magnetic signature observed near Io by the Goddard Space Flight Center Voyager 1 magnetometer is examined and it is pointed out that the preliminary estimate of 5,000,000 A current may be an overestimate. Good fits are obtained with alternative current distributions which yield currents as small as 700,000 A through Io. The best fits are obtained for an Alfven Mach number of 0.15, but good fits are also found for Alfven Mach numbers between 0.1 and 0.25. Lastly, the crucial role of charged particle data for probing the near Io interaction is pointed out. Published data from the low energy charged particle (LECP) detector are made to suggest that Io has an intrinsic magnetic field of magnitude comparable with earlier estimates. Predictions are made which can test this picture as further data become available.
The first results of a major survey of globular cluster core structure are reported. Surface brightness profiles for three clusters are presented: one with a normal flat core profile (NGC 6388), and two with central cusps (NGC 6624 and M15). The profiles are fitted with both seeing-convolved King models and seeing-convolved power laws. A King model provides a reasonable fit to the entire profile of NGC 6388, but there are no acceptable King profiles for NGC 6624 and M15. For these two clusters, a seeing-convolved power law provides a good fit to the inner part of the profile, while a King model gives a good fit to the outer part of the profile. The interpretation of the measured power-law slopes of the central surface brightness cusps of NGC 6624 and M15 are discussed in terms of models for core collapse in a multicomponent cluster.
A collisionless distribution of stars with an anisotropic velocity distribution is considered as a model for the cores of spherical galaxies. The first moment of the corresponding Boltzmann equation is integrated to give the resulting density and velocity dispersion profiles. These distributions plus a central point source of light give a good fit to the data from recent observations of M87 without requiring a supermassive central object. This model requires one less fitting parameter than published models incorporating a central black hole. The same model with no point source of light also provides a good fit to the normal E1 galaxy NGC 3379. The success of this model in describing the cores of both M87 and NGC 3379 suggests that a supermassive black hole is not necessary in either.
The observed sharp drop in the geometric albedo of Titan shortward of 6000 A (also observed for Jupiter and Saturn) has been tentatively attributed to the presence of small absorbing particles referred to as Axel dust. The properties of Axel dust particles are studied, with the effect of scattering included through Mie theory, in an attempt to obtain a physically plausible set of parameters that properly characterizes the dust. Quantitative characteristics of the dust are described, scattering and absorption properties of various particle size distributions are calculated, and fits are made to observations of Titan and Saturn. It is found that: (1) a flat distribution of particle radii up to 0.1 micron and an imaginary part of the refractive index that varies as the -2.5 power of wavelength produce a good fit for Titan provided the extinction optical depth of the satellite's atmosphere is about 10 at 5000 A; (2) a clear layer of about 7 km-am hydrogen is required above the dust for a good fit to Saturn with the same dust properties; and (3) the inclusion of dust significantly increases the computed abundance of methane over the value given by reflecting-layer models.
In November 2014 Uranus was observed with the Wide Field Camera 3 (WFC3) instrument of the Hubble Space Telescope as part of the Hubble 2020: Outer Planet Atmospheres Legacy program, OPAL. OPAL annually maps Jupiter, Uranus and Neptune (and will also map Saturn from 2018) in several visible near- infrared wavelength filters. The Uranus 2014 OPAL observations were made on the 89th November at a time when a huge cloud complex, first observed by de Pater et al. (2015) and subsequently tracked by professional and amateur astronomers (Sayanagi et al., 2016), was present at 30-40deg N. We imaged the entire visible atmosphere, including the storm system, in seven filters spanning 467924 nm, capturing variations in the coloration of Uranus clouds and also vertical distribution due to wavelength dependent changes in Rayleigh scattering and methane absorption optical depth. Here we analyse these new HST observations with the NEMESIS radiative-transfer and retrieval code in multiple-scattering mode to determine the vertical cloud structure in and around the storm cloud system. The same storm system was also observed in the H-band (1.4-1.8 micrometers) with the SINFONI Integral Field Unit Spectrometer on the Very Large Telescope (VLT) on 31st October and 11th November, reported by Irwin et al. (2016, 10.1016j.icarus.2015.09.010). To constrain better the cloud particle sizes and scattering properties over a wide wavelength range we also conducted a limb-darkening analysis of the background cloud structure in the 30-40deg N latitude band by simultaneously fitting: a) these HSTOPAL observations at a range of zenith angles; b) the VLTSINFONI observations at a range of zenith angles; and c) IRTFSpeX observations of this latitude band made in 2009 at a single zenith angle of 23deg, spanning the wavelength range 0.8-1.8 micrometers (Irwin et al., 2015, 10.1016j.icarus.2014.12.020). We find that the HST observations, and the combined HSTVLTIRTF observations at all locations are well modelled with a three-component cloud comprised of: 1) a vertically thin, but optically thick deep tropospheric cloud at a pressure of approximately 2 bars; 2) a methane-ice cloud based at the methane-condensation level of approximately 1.23 bar, with variable vertical extent; and 3) a vertically extended tropospheric haze, also based at the methane-condensation level of 1.23 bar. We find that modelling both haze and tropospheric cloud with particles having an effective radius of approximately 0.1 micron provides a good fit the observations, although for the tropospheric cloud, particles with an effective radius as large as 1.0 micron provide a similarly good fit. We find that the particles in both the tropospheric cloud and haze are more scattering at short wave- lengths, giving them a blue color, but are more absorbing at longer wavelengths, especially for the tropospheric haze. We find that the spectra of the storm clouds are well modelled by localized thickening and vertical extension of the methane-ice cloud. For the particles in the storm clouds, which we assume to be composed of methane ice particles, we find that their mean radii must lie somewhere in the range 0. 1 1. 0 m. We find that the high clouds have low integrated opacity, and that streamers reminiscent of convective thunderstorm anvils are confined to levels deeper than 1 bar. These results argue against vigorous moist convective origins for the cloud features.
Laboratory band-model absorption coefficients of CH4 were used to calculate the Uranus spectrum from 5400 to 10,400 A. A good fit for both strong and weak bands for the Uranus spectrum over the entire wavelength interval was achieved; three atmospheric models were employed: (1) a reflecting layer model; (2) a homogeneous scattering layer model; and (3) a clear atmosphere sandwiched between two scattering layers. The spectrum for the reflecting layer model shows serious discrepancies but indicates that large amounts of CH4 are necessary to reproduce the Uranus spectrum. Both scattering models gave reasonably good fits; the homogeneous model requires a particle scattering albedo of at least 0.998 and an abundance per scattering mean free path of about 1 km-am. For the sandwich model, a continuum single scattering albedo of 0.995 was derived for the upper scattering layer, and the scattering optical depth variable wavelength was consistent with Rayleigh scattering.
Laboratory infrared emission and absorption spectra have been taken of terrestrial silicates, meteorites, and lunar soils in the form of micrometer and submicrometer grains. The emission spectra were taken in a way that imitates telescopic observations. The purpose was to see which materials best simulate the 10-micron astrophysical feature. The emission spectra of dunite, fayalite, and Allende give a good fit to the 10-micron broadband emission feature of comets Bennett and Kohoutek. A study of the effect of grain size on the presence of the 10-micron emission feature of dunite shows that for particles larger than 37 microns no feature is seen. The emission spectrum of the Murray meteorite, a Type 2 carbonaceous chrondrite, is quite similar to the intermediate-resolution spectrum of comet Kohoutek in the 10-micron region. Hydrous silicates or amorphous magnesium silicates in combination with high-temperature condensates, such as olivine or anorthite, would yield spectra that match the intermediate-resolution spectrum of comet Kohoutek in the 10-micron region. Glassy olivine and glassy anorthite in approximately equal proportions would also give a spectrum that is a good fit to the cometary 10-micron feature.
A criterion for characterizing an iteratively trained classifier is presented. The criterion is based on an information theoretic measure that is developed from modeling classifier training iterations as a set of cascaded channels. The criterion is formulated as a figure of merit and as a performance index to check the appropriateness of application of the characterized classifier to an unknown data base and for implementing classifier updates and data selection respectively.
A criterion for characterizing an iteratively trained classifier is presented. The criterion is based on an information theoretic measure that is developed from modeling classifier training iterations as a set of cascaded channels. The criterion is formulated as a figure of merit and as a performance index to check the appropriateness of application of the characterized classifier to an unknown data base and for implementing classifier updates and data selection, respectively.