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# 11.3: The Third Law

For entropy on the other hand, the fact that the heat capacity goes to zero as the temperature decreases has important consequences. Consider the change in the entropy of a pure substance whose heat capacity approaches some finite limiting value as its temperature decreases to absolute zero. For such a substance, $${C_P}/{T}$$ becomes arbitrarily large as the temperature decreases, and the entropy integral $\int^0_T{\frac{C_P}{T}}dT$

approaches minus infinity as the temperature approaches zero. For real substances, this does not occur. In the neighborhood of absolute zero, heat capacities decrease more rapidly than temperature. The entropy change approaches zero as the temperature approaches zero.

The idea that the entropy change for a pure substance goes to zero as the temperature goes to zero finds expression as the third law of thermodynamics. In 1923, Lewis and Randall$${}^{1}$$ gave a statement of the third law that is particularly convenient in chemical applications:

If the entropy of each element in some crystalline state be taken as zero at the absolute zero of temperature, every substance has a positive finite entropy; but at the absolute zero of temperature the entropy may become zero, and does so become in the case of perfect crystalline substances.

Implicitly, the Lewis and Randall statement defines the entropy of any substance, at any temperature, $$T$$, to be the difference between the entropy of the constituent elements, at absolute zero, and the entropy of the substance at temperature $$T$$. Equivalently, we can say that it is the entropy change when the substance is formed at temperature $$T$$ from its constituent elements at absolute zero. Arbitrarily, but very conveniently, the statement sets the entropy of an element to zero at absolute zero.

The distinction between perfect crystalline substances and less-than-perfect crystalline substances lies in the regularity of the arrangement of the molecules within the crystal lattice. In any lattice, each molecule of the substance is localized at a specific site in the lattice. In a perfect crystal, all of the molecules are in oriented the same way with respect to the lattice. Some substances form crystals in which the molecules are not all oriented the same way. This can happen when the molecule can fit into a lattice site of the same shape in more than one way. For example, in solid carbon monoxide, the individual molecules occupy well-defined lattice sites. If the carbon monoxide crystal were perfect, all of the molecules would point in the same direction, as diagrammed in Figure 2. Instead, they point randomly in either of two possible directions.

11.3: The Third Law is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


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 Observations of Hα, iron, and oxygen lines in B, Be, and shell starsWe carried out a spectroscopic survey of several B, Be, and shell starsin optical and near-infrared regions. Line profiles of the Hα lineand of selected Fe II and O I lines are presented. On the evolutionary status of Be stars. I. Field Be stars near the SunA sample of 97 galactic field Be stars were studied by taking intoaccount the effects induced by the fast rotation on their fundamentalparameters. All program stars were observed in the BCDspectrophotometric system in order to minimize the perturbationsproduced by the circumstellar environment on the spectral photosphericsignatures. This is one of the first attempts at determining stellarmasses and ages by simultaneously using model atmospheres andevolutionary tracks, both calculated for rotating objects. The stellarages (τ) normalized to the respective inferred time that eachrotating star can spend in the main sequence phase (τ_MS) reveal amass-dependent trend. This trend shows that: a) there are Be starsspread over the whole interval 0  τ/τ_MS  1 of themain sequence evolutionary phase; b) the distribution of points in the(τ/τMS,M/Mȯ) diagram indicates thatin massive stars (M  12~Mȯ) the Be phenomenon ispresent at smaller τ/τ_MS age ratios than for less massive stars(M  12~Mȯ). This distribution can be due to: i)higher mass-loss rates in massive objets, which can act to reduce thesurface fast rotation; ii) circulation time scales to transport angularmomentum from the core to the surface, which are longer the lower thestellar mass. CHARM2: An updated Catalog of High Angular Resolution MeasurementsWe present an update of the Catalog of High Angular ResolutionMeasurements (CHARM, Richichi & Percheron \cite{CHARM}, A&A,386, 492), which includes results available until July 2004. CHARM2 is acompilation of direct measurements by high angular resolution methods,as well as indirect estimates of stellar diameters. Its main goal is toprovide a reference list of sources which can be used for calibrationand verification observations with long-baseline optical and near-IRinterferometers. Single and binary stars are included, as are complexobjects from circumstellar shells to extragalactic sources. The presentupdate provides an increase of almost a factor of two over the previousedition. Additionally, it includes several corrections and improvements,as well as a cross-check with the valuable public release observationsof the ESO Very Large Telescope Interferometer (VLTI). A total of 8231entries for 3238 unique sources are now present in CHARM2. Thisrepresents an increase of a factor of 3.4 and 2.0, respectively, overthe contents of the previous version of CHARM.The catalog is only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/431/773 Far-infrared loops in the 2nd Galactic QuadrantWe present the results of an investigation of the large-scale structureof the diffuse interstellar medium in the 2nd Galactic Quadrant(90°≤l≤180°). 145 loops were identified on IRAS-basedfar-infrared maps. Our catalogue lists their basic physical properties.The distribution clearly suggests that there is an efficient processthat can generate loop-like features at high Galactic latitudes.Distances are provided for 30 loops. We also give an observationalestimate of the volume filling factor of the hot gas in the Local Arm,4.6%≤f2nd<6.4%.Appendices A-C are only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/418/131 Interstellar Matter near the Pleiades. VI. Evidence for an Interstellar Three-Body EncounterThis paper seeks a comprehensive interpretation of new data on Na Iabsorption toward stars in and near the Pleiades, together with existingvisible and infrared data on the distribution of dust and with radiodata on H I and CO in the cluster vicinity. The use of dust and gasmorphology to constrain tangential motions in connection with themeasured radial velocities yields estimates for the space motion of gasnear the Pleiades. Much of the kinematic complexity in the interstellarabsorption toward the Pleiades, including the presence of stronglyblueshifted components that arise in shocked gas, finds explanation inthe interaction between the cluster and foreground gas withVr(LSR)~7 km s-1 associated with the Taurus dustclouds. Taurus gas, however, cannot readily account for an absorptioncomponent having Vr(LSR)~10 km s-1 with a wide,but not continuous distribution and 21 cm emission from gas in thecluster having Vr(LSR)~0 km s-1 associated witheast-west dust filaments. Successive hypotheses for the origin of theseadditional features include Taurus gas at a higher velocity than thepervasive foreground component, additional gas at a radial velocityintermediate between that of the Taurus component and the cluster, and acloud having Vr(LSR)~10 km s-1 approaching thePleiades from the west. A satisfactory account of the full complexity ofthe interstellar medium near the Pleiades requires the last feature andthe Taurus gas, both interacting with the Pleiades and also with eachother. Kinematical Structure of the Local Interstellar Medium: The Galactic Anticenter HemisphereA survey of interstellar Na I D1 and D2 absorption features in thespectra of 104 early-type stars in the second and third Galacticquadrants reveals the large-scale kinematics of the interstellar gaswithin the Galactic anticenter hemisphere at distances from the Sunbetween ~70 and ~250 pc. Employing a technique that uses both the radialvelocities and column densities of the Na I absorptions produced by theintervening gas we have identified the velocity vectors and determinedthe spatial distribution of eight interstellar clouds in the volumeexplored. The average internal H I+H2 densities of the cloudsrange between 0.03 and 1.7 cm-3, and their masses between 80and 104 Msolar, although uncertainties in thesizes of the clouds, their possible extension beyond the regionexplored, and the presence of denser gas embedded in the larger cloudsimply that these will tend to be lower limits. We have clearlyidentified clumps of denser gas immersed in the low-density gas in oneof the clouds; these clumps show internal H I+H2 densities oforder 50 cm-3. Although we are not able to detect anyinterstellar Na I within 70 pc, the sizes of some of the clouds implythat their near edges are within that range of distances from the Sun.With respect to the local standard of rest the clouds move withvelocities between 19 and 54 km s-1. Their velocity vectorsdo not support the view of a local interstellar medium uniquelydominated by expansion from centers in the Scorpio-Centaurus OBassociation; our results suggest that this expansion is present in theGalactic center hemisphere but in the Galactic anticenter hemisphere isrestricted to the immediate neighborhood of the Sun. The Pleiades Reflection Nebula. II. Simple Model Constraints on Dust Properties and Scattering GeometryWe have used wide-field ultraviolet, optical, and far-infraredphotometric images of Pleiades reflection nebulosity to analyze dustproperties and the three-dimensional nebular geometry. Scattered lightdata were taken from 1650 and 2200 Å Wide-Field Imaging SurveyPolarimeter images and a large 4400 Å mosaic of Burrell SchmidtCCD frames. Dust thermal emission maps were extracted from IRAS data.The scattering geometry analysis is complicated by the blending of lightfrom many stars and the likely presence of more than one scatteringlayer. Despite these complications, we conclude that most of thescattered light comes from dust in front of the stars in at least twoscattering layers, one far in front and extensive, the other nearer thestars and confined to areas of heavy nebulosity. The first layer can beapproximated as an optically thin, foreground slab whose line-of-sightseparation from the stars averages ~0.7 pc. The second layer is alsooptically thin in most locations and may lie at less than half theseparation of the first layer, perhaps with some material among orbehind the stars. The association of nebulosities peripheral to the maincondensation around the brightest stars is not clear. Models withstandard grain properties cannot account for the faintness of thescattered UV light relative to the optical. Some combination ofsignificant changes in grain model albedo and phase function asymmetryvalues is required. Our best-performing model has a UV albedo of0.22+/-0.07 and a scattering asymmetry of 0.74+/-0.06. Hypotheticaloptically thick dust clumps missed by interstellar sight linemeasurements have little effect on the nebular colors but might shiftthe interpretation of our derived scattering properties from individualgrains to the bulk medium. The Pleiades Reflection Nebula. I. Ultraviolet, Optical, and Far-Infrared Imaging PhotometryWe present new wide-field optical and ultraviolet images of the Pleiadesreflection nebula that allow a more thorough evaluation of the dustscattering than any prior data set. Vacuum-UV images were taken at 1650and 2200 Å during the first flight of the Wide-Field ImagingSurvey Polarimeter (WISP), a sounding rocket-borne telescope. WISPcaptured the brighter parts of the nebula at both wavelengths, with 3σ sensitivities of 22.5 and 23.4 UV mag arcsec-2,respectively. The 5.0d×1.7d WISP field was also mapped at 4400Å with a mosaic of 40 Burrell Schmidt CCD frames using a broadbandBJ filter. The Schmidt mosaic shows extensive and intricatenebulosity down to a 5 σ sensitivity limit of 27.6 B magarcsec-2, including features undetected by photographicsurveys. We explore the intensity and color behavior of the nebula inour UV and optical images and far-infrared IRAS data. We find that thephotometric structure near bright stars is more complex than previousstudies have implied, but general trends are still apparent. The colorgradients around the stars are caused by phase function effects ratherthan internal reddening. The greater concentration of scattered lightversus thermal emission indicates that most of the observed scatteringis from foreground dust. A somewhat greater concentration of UV versusoptical light suggests grain scattering is more forward-directed atshorter wavelengths. The UV nebula is much fainter than expected fromthe stellar photometry and interstellar reddening. Explaining this UVfaintness requires either more reddening than is measured or significantalterations to current dust property estimates. Observations of Star-Forming Regions with the Midcourse Space ExperimentWe have imaged seven nearby star-forming regions, the Rosette Nebula,the Orion Nebula, W3, the Pleiades, G300.2-16.8, S263, and G159.6-18.5,with the Spatial Infrared Imaging Telescope on the Midcourse SpaceExperiment (MSX) satellite at 18" resolution at 8.3, 12.1, 14.7, and21.3 μm. The large angular scale of the regions imaged (~7.2-50deg2) makes these data unique in terms of the combination ofsize and resolution. In addition to the star-forming regions, twocirrus-free fields (MSXBG 160 and MSXBG 161) and a field near the southGalactic pole (MSXBG 239) were also imaged. Point sources have beenextracted from each region, resulting in the identification over 500 newsources (i.e., no identified counterparts at other wavelengths), as wellas over 1300 with prior identifications. The extended emission from thestar-forming regions is described, and prominent structures areidentified, particularly in W3 and Orion. The Rosette Nebula isdiscussed in detail. The bulk of the mid-infrared emission is consistentwith that of photon-dominated regions, including the elephant trunkcomplex. The central clump, however, and a line of site toward thenorthern edge of the cavity show significantly redder colors than therest of the Rosette complex. X-ray variability of Pleiades late-type stars as observed with the ROSAT-PSPCWe present a comprehensive analysis of X-ray variability of thelate-type (dF7-dM) Pleiades stars, detected in all ROSAT-PSPCobservations; X-ray variations on short (hours) and medium (months) timescales have been explored. We have grouped the stars in two samples: 89observations of 42 distinct dF7-dK2 stars and 108 observations of 61dK3-dM stars. The Kolmogorov-Smirnov test applied on all X-ray photontime series show that the percentage of cases of significant variabilityis quite similar on both samples, suggesting that the presence ofvariability does not depend on mass for the time scales and mass rangeexplored. The comparison between the Time X-ray Amplitude Distributionfunctions (XAD) of the set of dF7-dK2 and of the dK3-dM show that, onshort time scales, dK3-dM stars show larger variations than dF7-dK2. Asubsample of eleven dF7-dK2 and eleven dK3-dM Pleiades stars allowsthe study of variability on longer time scales: we found thatvariability on medium - long time scales is relatively more commonamong dF7-dK2 stars than among dK3-dM ones. For both dF7-dK2 Pleiadesstars and dF7-dK2 field stars, the variability on short time scalesdepends on Lx while this dependence has not been observedamong dK3-dM stars. It may be that the variability among dK3-dM stars isdominated by flares that have a similar luminosity distribution forstars of different Lx, while flaring distribution in dF7-dK2stars may depend on X-ray luminosity. The lowest mass stars showsignificant rapid variability (flares?) and no evidence of rotationmodulation or cycles. On the contrary, dF7-dK2 Pleiades stars showboth rapid variability and variations on longer time scales, likelyassociated with rotational modulation or cycles. Merged catalogue of reflection nebulaeSeveral catalogues of reflection nebulae are merged to create a uniformcatalogue of 913 objects. It contains revised coordinates,cross-identifications of nebulae and stars, as well as identificationswith IRAS point sources.The catalogue is only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/399/141 Quantitative Stellar Spectral Classification. II. Early Type StarsThe method developed by Stock & Stock (1999) for stars of spectraltypes A to K to derive absolute magnitudes and intrinsic colors from theequivalent widths of absorption lines in stellar spectra is extended toB-type stars. Spectra of this type of stars for which the Hipparcoscatalogue gives parallaxes with an error of less than 20% were observedwith the CIDA one-meter reflector equipped with a Richardsonspectrograph with a Thompson 576×384 CCD detector. The dispersionis 1.753 Å/pixel using a 600 lines/mm grating in the first order.In order to cover the spectral range 3850 Å to 5750 Å thegrating had to be used in two different positions, with an overlap inthe region from 4800 Å to 4900 Å . A total of 116 stars wasobserved, but not all with both grating positions. A total of 12measurable absorption lines were identified in the spectra and theirequivalent widths were measured. These were related to the absolutemagnitudes derived from the Hipparcos catalogue and to the intrinsiccolors (deduced from the MK spectral types) using linear and secondorder polynomials and two or three lines as independent variables. Thebest solutions were obtained with polynomials of three lines,reproducing the absolute magnitudes with an average residual of about0.40 magnitudes and the intrinsic colors with an average residual of0.016 magnitudes. Koordinierte Zusammenarbeit zwischen den VdS-FG 'BAV' und 'Spektroskopie' ?Not Available Rotational Velocities of B StarsWe measured the projected rotational velocities of 1092 northern B starslisted in the Bright Star Catalogue (BSC) and calibrated them againstthe 1975 Slettebak et al. system. We found that the published values ofB dwarfs in the BSC average 27% higher than those standards. Only 0.3%of the stars have rotational velocities in excess of two-thirds of thebreakup velocities, and the mean velocity is only 25% of breakup,implying that impending breakup is not a significant factor in reducingrotational velocities. For the B8-B9.5 III-V stars the bimodaldistribution in V can be explained by a set of slowly rotating Ap starsand a set of rapidly rotating normal stars. For the B0-B5 III-V starsthat include very few peculiar stars, the distributions in V are notbimodal. Are the low rotational velocities of B stars due to theoccurrence of frequent low-mass companions, planets, or disks? Therotational velocities of giants originating from late B dwarfs areconsistent with their conservation of angular momentum in shells.However, we are puzzled by why the giants that originate from the earlyB dwarfs, despite having 3 times greater radii, have nearly the samerotational velocities. We find that all B-type primaries in binarieswith periods less than 2.4 days have synchronized rotational and orbitalmotions; those with periods between 2.4 and 5.0 days are rotating withina factor 2 of synchronization or are nearly synchronized.'' Thecorresponding period ranges for A-type stars are 4.9 and 10.5 days, ortwice as large. We found that the rotational velocities of the primariesare synchronized earlier than their orbits are circularized. The maximumorbital period for circularized B binaries is 1.5 days and for Abinaries is 2.5 days. For stars of various ages from 107.5 to1010.2 yr the maximum circularized periods are a smoothexponential function of age. Computing the Parallax of the Pleiades from the Hipparcos Intermediate Astrometry Data: An Alternative ApproachThe inconsistency between the mean parallax of the Pleidaes open clusterfrom the Hipparcos catalog and that obtained from the stellar evolutiontheory and photometric measurements is probed by recomputing theHipparcos data in a different way that reduces the propagation of thealong-scan attitude errors. This is achieved by coupling observations ofstars made nearly simultaneously in the two separate fields of view ofthe telescope. A direct calculation of astrometric quantities of 54Pleiades members by the new method, based on the Intermediate AstrometryData, provides a correction of -0.71+/-0.14 mas to the weighted meanparallax of the cluster. The mean corrected parallax of the Pleiades is7.75+/-0.20 mas. CHARM: A Catalog of High Angular Resolution MeasurementsThe Catalog of High Angular Resolution Measurements (CHARM) includesmost of the measurements obtained by the techniques of lunaroccultations and long-baseline interferometry at visual and infraredwavelengths, which have appeared in the literature or have otherwisebeen made public until mid-2001. A total of 2432 measurements of 1625sources are included, along with extensive auxiliary information. Inparticular, visual and infrared photometry is included for almost allthe sources. This has been partly extracted from currently availablecatalogs, and partly obtained specifically for CHARM. The main aim is toprovide a compilation of sources which could be used as calibrators orfor science verification purposes by the new generation of largeground-based facilities such as the ESO Very Large Interferometer andthe Keck Interferometer. The Catalog is available in electronic form atthe CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/386/492, and from theauthors on CD-Rom. Astrometric radial velocities. III. Hipparcos measurements of nearby star clusters and associationsRadial motions of stars in nearby moving clusters are determined fromaccurate proper motions and trigonometric parallaxes, without any use ofspectroscopy. Assuming that cluster members share the same velocityvector (apart from a random dispersion), we apply a maximum-likelihoodmethod on astrometric data from Hipparcos to compute radial and spacevelocities (and their dispersions) in the Ursa Major, Hyades, ComaBerenices, Pleiades, and Praesepe clusters, and for theScorpius-Centaurus, alpha Persei, and HIP 98321'' associations. Theradial motion of the Hyades cluster is determined to within 0.4 kms-1 (standard error), and that of its individual stars towithin 0.6 km s-1. For other clusters, Hipparcos data yieldastrometric radial velocities with typical accuracies of a few kms-1. A comparison of these astrometric values withspectroscopic radial velocities in the literature shows a good generalagreement and, in the case of the best-determined Hyades cluster, alsopermits searches for subtle astrophysical differences, such as evidencefor enhanced convective blueshifts of F-dwarf spectra, and decreasedgravitational redshifts in giants. Similar comparisons for the ScorpiusOB2 complex indicate some expansion of its associations, albeit slowerthan expected from their ages. As a by-product from the radial-velocitysolutions, kinematically improved parallaxes for individual stars areobtained, enabling Hertzsprung-Russell diagrams with unprecedentedaccuracy in luminosity. For the Hyades (parallax accuracy 0.3 mas), itsmain sequence resembles a thin line, possibly with wiggles in it.Although this main sequence has underpopulated regions at certaincolours (previously suggested to be Böhm-Vitense gaps''), suchare not visible for other clusters, and are probably spurious. Futurespace astrometry missions carry a great potential for absoluteradial-velocity determinations, insensitive to the complexities ofstellar spectra. Based on observations by the ESA Hipparcos satellite.Extended versions of Tables \ref{tab1} and \ref{tab2} are available inelectronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr(130.79.125.8) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/381/446 Interstellar Matter Near the Pleiades. V. Observations of NA I toward 36 StarsThis paper reports high-resolution, moderate to high signal-to-noiseratio observations of 23 certain Pleiades members, four possiblemembers, and nine nonmembers in the Na I D lines, as well asobservations of 12 of the stars in the Na I ultraviolet doublet. Inspite of the relative proximity of the stars to the sun (even most ofthe nonmembers lie within 200 pc), the line profiles exhibit remarkablecomplexity, with up to five absorption components and equally remarkablestar-to-star variation. The velocity range, 2-20 km s-1,conforms well to the range expected for gas deflected by the passage ofthe cluster. The paper includes a careful discussion of uncertainties inthe data, the most important conclusions of which are that the velocityscatter is consistent with that expected from random errors in thewavelength calibration and that systematic errors probably are <~0.1km s-1. Appendices detail the choice of stellar data and theprocedure adopted for removing telluric absorption lines. Analysisfollows in a separate paper. A Search for High-Velocity Be StarsWe present an analysis of the kinematics of Be stars based uponHipparcos proper motions and published radial velocities. We findapproximately 23 of the 344 stars in our sample have peculiar spacemotions greater than 40 km s-1 and up to 102 kms-1. We argue that these high-velocity stars are the resultof either a supernova that disrupted a binary or ejection by closeencounters of binaries in young clusters. Be stars spun up by binarymass transfer will appear as high-velocity objects if there wassignificant mass loss during the supernova explosion of the initiallymore massive star, but the generally moderate peculiar velocities of BeX-ray binaries indicate that the progenitors lose most of their massprior to the supernova (in accordance with model predictions). Binaryformation models for Be stars predict that most systems bypass thesupernova stage (and do not receive runaway velocities) to createultimately Be+white dwarf binaries. The fraction of Be stars spun up bybinary mass transfer remains unknown, since the post-mass transfercompanions are difficult to detect. Microstructure of the Local Interstellar Cloud and the Identification of the Hyades CloudWe analyze high-resolution UV spectra of the Mg II h and k lines for 18members of the Hyades to study inhomogeneity along these proximate linesof sight. The observations were taken by the Space Telescope ImagingSpectrograph (STIS) instrument on board the Hubble Space Telescope(HST). Three distinct velocity components are observed. All 18 lines ofsight show absorption by the Local Interstellar Cloud (LIC); 10 starsshow absorption by an additional cloud, which we name the Hyades Cloud;and one star exhibits a third absorption component. The LIC absorptionis observed at a lower radial velocity than predicted by the LICvelocity vector derived by Lallement & Bertin in 1992 and Lallementet al. in 1995[vpredicted(LIC)-vobserved(LIC)=2.9+/-0.7 kms-1], which may indicate a compression or deceleration at theleading edge of the LIC. We propose an extension of the Hyades Cloudboundary based on previous HST observations of other stars in thegeneral vicinity of the Hyades, as well as ground-based Ca IIobservations. We present our fits of the interstellar parameters foreach absorption component. The availability of 18 similar lines of sightprovides an excellent opportunity to study the inhomogeneity of thewarm, partially ionized local interstellar medium (LISM). We find thatthese structures are roughly homogeneous. The measured Mg II columndensities do not vary by more than a factor of 2 for angular separationsof <~8°, which at the outer edge of the LIC correspond tophysical separations of <~0.6 pc. Barnard's Merope Nebula Revisited: New Observational ResultsIC 349 is a small, fan-shaped reflection nebula located only 30" from 23Tau; its nucleus is, by a factor 15, the brightest area of the Pleiadesnebulosity. We propose that IC 349 is a fragment of the Taurus-Aurigamolecular cloud that has been encountered by the Pleiades in thatcluster's southward motion and is being illuminated and shaped by theradiation field of 23 Tau. New Hubble Space Telescope multicolor imageryand the structure, colors, and surface brightness of IC 349 arediscussed in terms of that hypothesis. What is known of the propermotion of the nebula, what can be inferred of the properties of thenebula from its color, and what is expected from radiation pressuretheory appear to be compatible with this cloudlet-encounter hypothesis. High and intermediate-resolution spectroscopy of Be stars 4481 linesWe present an atlas of Hγ , He i lambda 4471 and Mg ii lambda 4481line profiles obtained in a 10 year observation period of 116 Be stars,which enabled many of them to be observed at quite different emissionepochs. From the best fit of the observed He i lambda 4471 line profileswith non-LTE, uniform (Teff,log g) and full limb-darkenedmodel line profiles, we determined the V sin i of the program stars. Toaccount, to some degree, for the line formation peculiarities related tothe rapid rotation-induced non-uniform distributions of temperature andgravity on the stellar surface, the fit was achieved by considering(Teff,log g) as free parameters. This method produced V sin iestimations that correlate with the rotational velocities determined bySlettebak (1982) within a dispersion sigma <= 30 km s-1and without any systematic deviation. They can be considered as given inthe new Slettebak's et al. (1975) system. Only 13 program stars havediscrepant V sin i values. In some objects, this discrepancy could beattributed to binary effects. Using the newly determined V sin iparameters, we found that the ratio of true rotational velocitiesV/Vc of the program Be stars has a very low dispersion aroundthe mean value. Assuming then that all the stars are rigid rotators withthe same ratio V(/lineω)/Vc, we looked for the value of/line ω that better represents the distribution of V sini/Vc for randomly oriented rotational axes. We obtained/lineω = 0.795. This value enabled us to determine the probableinclination angle of the stellar rotation axis of the program stars. Inthe observed line profiles of Hγ , He i lambda 4471, Mg ii lambda4481 and Fe ii lambda 4351 we measured several parameters related to theabsorption and/or emission components, such as: equivalent width,residual emission and/or absorption intensity, FWHM, emission peakseparations, etc. The parameters related to the Hγ line emissionprofiles were used to investigate the structure of the nearbyenvironment of the central star. From the characteristics of thecorrelations between these quantities and the inferred inclinationangle, we concluded that in most of cases the Hγ line emissionforming regions may not be strongly flattened. Using a simplerepresentation of the radiation flux emitted by the star+envelopesystem, we derived first order estimates of physical parameterscharacterizing the Hγ line emission formation region. Thus, weobtained that the total extent of the Hγ region is Rf=~ 2.5 +/- 1.0 R* and that the density distribution in theselayers can be mimicked with a power law rho ~ R-alpha , wherealpha =2.5+2.2-0.6. The same approach enabled usto estimate the optical depth of the Hγ line emission formationregion. From its dependence with the aspect angle, we concluded thatthese regions are caracterized by a modest flattening and that the rho(equator)/rho (pole) density contrast of the circumstellar envelope nearthe star should be two orders of magnitude lower than predicted bymodels based on a priori disc-shaped circumstellar envelopes. We foundthat the separation between the emission peaks, Deltap, andthe full width at half maximum, Delta 1/2, of the Hγline emission are not only sensitive to kinematic effects, but to lineoptical depth as well. This finding agrees with previous theoreticalpredictions and confirms that Huang's (1972) relation overestimates theextent of the Hγ line emission formation region. Data obtained atCASLEO operated under agreement between the CONICET and the NationalUniversities of La Plata, Córdoba and San Juan, Argentina, at ESOLa Silla, Chile and at OHP, France.}\fnmsep\thanks{Tables 2 to 7 andFigs. 1 and 2 are only available in full in electronic form at CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via\http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/378/861}} \subtitle{Anatlas of Hγ , He {\fontsize {10pt}{12pt}\selectfont I} 4471 and Mg{\fontsize {10pt}{12pt}\selectfont II The IMF of open star clusters with Tycho-2We studied the fields of nine nearby open star clusters based on theTycho-2 catalogue. We determined membership probabilities for the starsin the cluster fields from the stellar proper motions and used theTycho-2 photometry to compute the initial mass function (IMF) of theclusters from the main sequence turn-off point down to approx. 1 M_sun.We found IMF slopes ranging from Gamma =-0.69 down to Gamma =-2.27 (whenthe Salpeter \cite{salpeter} value would be Gamma =-1.35). We alsostudied the membership of individual stars of special astrophysicalinterest. In some cases previous results had to be revised. As aby-product, we investigated some general properties of the Tycho-2catalogue; we confirmed that the Tycho-2 proper motions show onlymarginal deviations from the Hipparcos catalogue. On the other hand, insome regions the completeness of the catalogue seems to decrease atmagnitudes brighter than claimed by Høg et al. (\cite{tycho2}).Based on observations of the ESA Hipparcos satellite. Statistical analysis of intrinsic polarization, IR excess and projected rotational velocity distributions of classical Be starsWe present the results of statistical analyses of a sample of 627 Bestars. The parameters of intrinsic polarization (p*),projected rotational velocity (v sin i), and near IR excesses have beeninvestigated. The values of p* have been estimated for a muchlarger and more representative sample of Be stars (~490 objects) thanpreviously. We have confirmed that most Be stars of early spectral typehave statistically larger values of polarization and IR excesses incomparison with the late spectral type stars. It is found that thedistributions of p* diverge considerably for the differentspectral subgroups. In contrast to late spectral types (B5-B9.5), thedistribution of p* for B0-B2 stars does not peak at the valuep*=0%. Statistically significant differences in the meanprojected rotational velocities (/line{vsin i}) are found for differentspectral subgroups of Be stars in the sense that late spectral typestars (V luminosity class) generally rotate faster than early types, inagreement with previously published results. This behaviour is, however,not obvious for the III-IV luminosity class stars. Nevertheless, thecalculated values of the ratio vt/vc of the truerotational velocity, vt, to the critical velocity forbreak-up, vc, is larger for late spectral type stars of allluminosity classes. Thus, late spectral type stars appear to rotatecloser to their break-up rotational velocity. The distribution of nearIR excesses for early spectral subgroups is bi-modal, the position ofthe second peak displaying a maximum value E(V-L)~ 1 . m 3for O-B1.5 stars, decreasing to E(V-L)~0. m8 for intermediatespectral types (B3-B5). It is shown that bi-modality disappears for latespectral types (B6-B9.5). No correlations were found betweenp* and near IR excesses and between E(V-L) and vsin i for thedifferent subgroups of Be stars. In contrast to near IR excesses, arelation between p* and far IR excesses at 12 mu m is clearlyseen. A clear relation between p* and vsin i (as well asbetween p* and /line{vsin i}/vc) is found by thefact that plots of these parameters are bounded by a triangular"distribution of p*: vsin i, with a decrease of p*towards very small and very large vsin i (and /line{vsini}/vc) values. The latter behaviour can be understood in thecontext of a larger oblateness of circumstellar disks for the stars witha rapid rotation. From the analysis of correlations between differentobservational parameters we conclude that circumstellar envelopes forthe majority of Be stars are optically thin disks with the range of thehalf-opening angle of 10degr Catalogue of Apparent Diameters and Absolute Radii of Stars (CADARS) - Third edition - Comments and statisticsThe Catalogue, available at the Centre de Données Stellaires deStrasbourg, consists of 13 573 records concerning the results obtainedfrom different methods for 7778 stars, reported in the literature. Thefollowing data are listed for each star: identifications, apparentmagnitude, spectral type, apparent diameter in arcsec, absolute radiusin solar units, method of determination, reference, remarks. Commentsand statistics obtained from CADARS are given. The Catalogue isavailable in electronic form at the CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcar?J/A+A/367/521 The proper motions of fundamental stars. I. 1535 stars from the Basic FK5A direct combination of the positions given in the HIPPARCOS cataloguewith astrometric ground-based catalogues having epochs later than 1939allows us to obtain new proper motions for the 1535 stars of the BasicFK5. The results are presented as the catalogue Proper Motions ofFundamental Stars (PMFS), Part I. The median precision of the propermotions is 0.5 mas/year for mu alpha cos delta and 0.7mas/year for mu delta . The non-linear motions of thephotocentres of a few hundred astrometric binaries are separated intotheir linear and elliptic motions. Since the PMFS proper motions do notinclude the information given by the proper motions from othercatalogues (HIPPARCOS, FK5, FK6, etc.) this catalogue can be used as anindependent source of the proper motions of the fundamental stars.Catalogue (Table 3) is only available at the CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strastg.fr/cgi-bin/qcat?J/A+A/365/222 Another Search for Maia Variable StarsWe have used the Hipparcos epoch photometry database, andautocorrelation analysis, to search for the elusive Maiavariables-short-period B7-A3 near-main-sequence pulsating variablestars. Of several hundred stars considered, and several dozen starsstudied in detail, only a handful are possible variables: three arepossible shallow eclipsing variables; three have possible periods in therange 0.25-0.5 day, but their amplitudes are so low that they areprobably nonvariable. The most promising are HD 29573, with a period of1.6 days (but possibly a rotating variable), and γ CrB, with aperiod of 0.9 day-a period also found spectroscopically by Lehmann andcoworkers. Sirius shows variations which are probably instrumental. Twopreviously suspected Maia stars-Maia and γ UMi-are photometricallyconstant. The Maia variables-if they exist-are very rare and veryelusive. The UV spectrum of nebulaeThis paper presents an analysis of the UV spectrum of some nebulae withclearly identified illuminating stars, all observed by the IUEsatellite.The data show remarkable properties of the UV spectrum of thenebulae. Each spectrum is the product of the star spectrum and a linearfunction of 1/lambda. There is no peculiar behaviour in the spectrums at2200 A: no bump created in the spectrum of a nebula and no excess ofscattering. When moving away from the star, the surface brightness of anebula decreases as the inverse of the square of the angular distance tothe star.These results can logically be interpreted in terms ofscattering of starlight. They imply constant properties of theinterstellar grains in the UV and in the directions of space sampled bythe nebulae, and probably a strong forward scattering phase function.There is no evidence for any particular type of grain which wouldspecifically extinguish starlight at 2200 A. Concerning the UV spectrumof a star, this may imply a revisal of the traditional interpretation ofthe 2200 A bump. Search for X-ray flares in the Pleiades using SoHO LASCO C3 images.Not Available The Normal Energy Distributions in Stellar Spectra: Giants and SupergiantsWe have derived the normal spectral energy distributions for thoseearly-type subgiants, giants, and supergiants that were not investigatedin our earlier studies, which were in most cases also not included inthe studies of Sviderskiene. Color indices computed using our normalenergy distributions are in good agreement with normal colors derivedfrom observations in the Vilnius photometric system. The reliability ofour distribution curves is also demonstrated by comparisons of observedand computed (W-B)-(B-V) two-color diagrams in the WBVR system. Normalcolor indices for the photometric WBVR system are derived.

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# Thread: Matrix - Gaussian elimination

1. ## Matrix - Gaussian elimination

Hi

Having trouble with solving the equations using Gaussian elimination

1) Solve the following system of equations using Gaussian elimination

$5x-2y+z=-6$
$-2x+z=-4$
$-x-2y+2z=3$

Show the final tableau (augmented matrix) of the elimination and the values of x, y and z.

This is what i have done but it is incorrect according the book's answer.

$\begin{bmatrix}
5 & -2 & 1 & | & -6 \\
-2 & 0 & 1 & | & -4 \\
-1 & -2 & 2 & | & 3 \\
\end{bmatrix}
$

$\begin{bmatrix}
5 & -2 & 1 & | & -6 \\
0 & -4 & -3 & | & -10 \\
-1 & -2 & 2 & | & 3 \\
\end{bmatrix}
$

R2-2R3

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & -4 & 3 & | & -10 \\
-1 & -2 & 2 & | & 3 \\
\end{bmatrix}
$

$\frac{1}{5}R1$

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & -4 & 3 & | & -10 \\
0 & \frac{-12}{5} & \frac{11}{5} & | & \frac{9}{5} \\
\end{bmatrix}
$

$R1+R3$

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & -4 & 3 & | & -10 \\
0 & \frac{-10}{5} & \frac{11}{5} & | & \frac{9}{5} \\
\end{bmatrix}
$

$\frac{1}{4}R2$

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & -4 & 3 & | & -10 \\
0 & 0 & \frac{7}{10} & | & \frac{-16}{5} \\
\end{bmatrix}
$

$\frac{10}{5}R2+R3$

P.S

2. Would the answer perhaps be:

x = 2-t
y = 1/2t - 8
z = 2t

If not, then I should take another look at the question If it is, or close, I can post my solution.

3. Originally Posted by Paymemoney
Hi

Having trouble with solving the equations using Gaussian elimination

1) Solve the following system of equations using Gaussian elimination

$5x-2y+z=-6$
$-2x+z=-4$
$-x-2y+2z=3$

Show the final tableau (augmented matrix) of the elimination and the values of x, y and z.

This is what i have done but it is incorrect according the book's answer.

$\begin{bmatrix}
5 & -2 & 1 & | & -6 \\
-2 & 0 & 1 & | & -4 \\
-1 & -2 & 2 & | & 3 \\
\end{bmatrix}
$

$\begin{bmatrix}
5 & -2 & 1 & | & -6 \\
0 & -4 & -3 & | & -10 \\
-1 & -2 & 2 & | & 3 \\
\end{bmatrix}
$

R2-2R3

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & -4 & 3 & | & -10 \\
-1 & -2 & 2 & | & 3 \\
\end{bmatrix}
$

$\frac{1}{5}R1$

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & -4 & 3 & | & -10 \\
0 & \frac{-12}{5} & \frac{11}{5} & | & \frac{9}{5} \\
\end{bmatrix}
$

$R1+R3$

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & -4 & 3 & | & -10 \\
0 & \frac{-10}{5} & \frac{11}{5} & | & \frac{9}{5} \\
\end{bmatrix}
$

$\frac{1}{4}R2$

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & -4 & 3 & | & -10 \\
0 & 0 & \frac{7}{10} & | & \frac{-16}{5} \\
\end{bmatrix}
$

$\frac{10}{5}R2+R3$

P.S
That will give you the correct values for x, y, and z (assuming your arithmetic is correct, I did not check it) but what, exactly, does your text consider the "final tableau"? You have got to "triangular form" and now you can solve for x, y, and z by back substitution. But I notice you got a "1" in the first column but did not get "1"s on the diagonal in the other columns. Some texts, at least, would consider having "1" s all along the diagonal as necessary. Others might expect to see a complete reduction to the identity matrix for the first 3 columns.

4. Originally Posted by HallsofIvy
That will give you the correct values for x, y, and z (assuming your arithmetic is correct, I did not check it) but what, exactly, does your text consider the "final tableau"? You have got to "triangular form" and now you can solve for x, y, and z by back substitution. But I notice you got a "1" in the first column but did not get "1"s on the diagonal in the other columns. Some texts, at least, would consider having "1" s all along the diagonal as necessary. Others might expect to see a complete reduction to the identity matrix for the first 3 columns.
actually the answer is not supplied for this question, but when i used the calculator it gave me what you said '1' on the diagonal

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & 1 & \frac{-1}{12} & | & \frac{-3}{4} \\
0 & 0 & 1 & | & \frac{-21}{4} \\
\end{bmatrix}$

5. Originally Posted by Paymemoney
Hi

Having trouble with solving the equations using Gaussian elimination

1) Solve the following system of equations using Gaussian elimination

$5x-2y+z=-6$
$-2x+z=-4$
$-x-2y+2z=3$

Show the final tableau (augmented matrix) of the elimination and the values of x, y and z.

This is what i have done but it is incorrect according the book's answer.

$\begin{bmatrix}
5 & -2 & 1 & | & -6 \\
-2 & 0 & 1 & | & -4 \\
-1 & -2 & 2 & | & 3 \\
\end{bmatrix}
$

$\begin{bmatrix}
5 & -2 & 1 & | & -6 \\
0 & -4 & -3 & | & -10 \\
-1 & -2 & 2 & | & 3 \\
\end{bmatrix}
$

R2-2R3
There is an error in this first step. -2 times the second number in R3 is + 4. The second number in R2 now should be 4, not -4.

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & -4 & 3 & | & -10 \\
-1 & -2 & 2 & | & 3 \\
\end{bmatrix}
$

$\frac{1}{5}R1$

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & -4 & 3 & | & -10 \\
0 & \frac{-12}{5} & \frac{11}{5} & | & \frac{9}{5} \\
\end{bmatrix}
$

$R1+R3$

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & -4 & 3 & | & -10 \\
0 & \frac{-10}{5} & \frac{11}{5} & | & \frac{9}{5} \\
\end{bmatrix}
$

$\frac{1}{4}R2$

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & -4 & 3 & | & -10 \\
0 & 0 & \frac{7}{10} & | & \frac{-16}{5} \\
\end{bmatrix}
$

$\frac{10}{5}R2+R3$

P.S

6. ok, i have tried again, this is what i got:

$\begin{bmatrix}
5 & -2 & 1 & | & -6 \\
-2 & 0 & 1 & | & -4 \\
-1 & -2 & 2 & | & 3 \\
\end{bmatrix}
$

$\begin{bmatrix}
5 & -2 & 1 & | & -6 \\
0 & 4 & -3 & | & -10 \\
-1 & -2 & 2 & | & 3 \\
\end{bmatrix}
$

R2-2R3

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & 4 & -3 & | & -10 \\
-1 & -2 & 2 & | & 3 \\
\end{bmatrix}
$

$\frac{1}{5}R1$

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & 4 & -3 & | & -10 \\
0 & \frac{-12}{5} & \frac{11}{5} & | & \frac{9}{5} \\
\end{bmatrix}
$

R3+R1

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & 4 & -3 & | & -10 \\
0 & 0 & \frac{2}{10} & | & \frac{-21}{5} \\
\end{bmatrix}
$

$R3+\frac{3}{5}R2$

$z=\frac{-105}{10}$
$y=\frac{-10-\frac{315}{10}}{10}$ $= \frac{-415}{40}$
$x=\frac{-53}{10}$

7. I'm not sure what you are doing, but as I understood it you just have to set Z to t, in my case I set it to 2t to make it more simple counting.

Then it should all fall out, if z = 2t you can find out what x is and then ultimately what y is.

8. Originally Posted by Paymemoney
o
$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & 4 & -3 & | & -10 \\
0 & 0 & \frac{2}{10} & | & \frac{-21}{5} \\
\end{bmatrix}
$

$R3+\frac{3}{5}R2$

$z=\frac{-105}{10}$
$y=\frac{-10-\frac{315}{10}}{10}$ $= \frac{-415}{40}$
$x=\frac{-53}{10}$
Briefly looking at that, you should have a figure of z = -210/10 and not -105/10. Then substitute that value into your other equations to work out the value of y and x.

9. Originally Posted by Paymemoney
$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & 4 & -3 & | & -10 \\
0 & \frac{-12}{5} & \frac{11}{5} & | & \frac{9}{5} \\
\end{bmatrix}
$

R3+R1

$\begin{bmatrix}
1 & \frac{-2}{5} & \frac{1}{5} & | & \frac{-6}{5} \\
0 & 4 & -3 & | & -10 \\
0 & 0 & \frac{2}{10} & | & \frac{-21}{5} \\
\end{bmatrix}
$

$R3+\frac{3}{5}R2$

$z=\frac{-105}{10}$
$y=\frac{-10-\frac{315}{10}}{10}$ $= \frac{-415}{40}$
$x=\frac{-53}{10}$
Everything looks fine until the 3rd column in your final matrix $\frac{3(-3)}{5} = \frac{-9}{5}$

So $\frac{11}{5} + \frac{-9}{5} = \frac{2}{5}$

Take $\frac{2}{5}R3$ and the bottom row becomes 0 0 1 | $\frac{-21}{2}$


text: In the linear regression, the independent variable … Logistic Regression – A Complete Tutorial With Examples in R 1. The categorical variable y, in general, can assume different values. The logistic regression method assumes that: The outcome is a binary or dichotomous variable like yes vs no, positive vs negative, 1 vs 0. But obviously that is flawed. The typical use of this model is predicting y given a set of predictors x. There are many classification models, the scope of this article is confined to one such model – the logistic regression model. The logistic regression model gives an estimate of the probability of each outcome. So whenever the Class is malignant, it will be 1 else it will be 0. It follows a similar syntax as downSample. Another advantage of logistic regression is that it computes a prediction probability score of an event. Since the response variable is a binary categorical variable, you need to make sure the training data has approximately equal proportion of classes. This chapter described how to compute penalized logistic regression model in R. Here, we focused on lasso model, but you can also fit the ridge regression by using alpha = 0 in the glmnet() function. Thanks for contributing an answer to Stack Overflow! Some of the material is based on Alan Agresti’s book [1] which is an excellent resource.. For many problems, we care about the probability of a binary outcome taking one value vs. another. GLM function for Logistic Regression: what is the default predicted outcome? Recall that the logit function is logit(p) = log(p/(1-p)), where p is the probabilities of the outcome (see Chapter @ref(logistic-regression)). This number ranges from 0 to 1, with higher values indicating better model fit. So P always lies between 0 and 1. So lets downsample it using the downSample function from caret package. It returns the probability that y=1 i.e. If you are serious about a career in data analytics, machine learning, or data science, it’s probably best to understand logistic and linear regression analysis as thoroughly as possible. Yes, we can use it for a regression problem, wherein the dependent or target variable is continuous. (with example and full code), Lemmatization Approaches with Examples in Python, Modin – How to speedup pandas by changing one line of code, Dask – How to handle large dataframes in python using parallel computing, Text Summarization Approaches for NLP – Practical Guide with Generative Examples, Gradient Boosting – A Concise Introduction from Scratch, Complete Guide to Natural Language Processing (NLP) – with Practical Examples, Portfolio Optimization with Python using Efficient Frontier with Practical Examples, Logistic Regression in Julia – Practical Guide with Examples, One Sample T Test – Clearly Explained with Examples | ML+, Understanding Standard Error – A practical guide with examples. The function to be called is glm() and the fitting process is not so different from the one used in linear regression. Let's compute the accuracy, which is nothing but the proportion of y_pred that matches with y_act. Note that, when you use logistic regression, you need to set type='response' in order to compute the prediction probabilities. your coworkers to find and share information. Because, If you use linear regression to model a binary response variable, the resulting model may not restrict the predicted Y values within 0 and 1.Linear vs Logistic Regression. As such, normally logistic regression is demonstrated with binary classification problem (2 classes). As expected, benign and malignant are now in the same ratio. it tells us the probability that an email is spam. So if pred is greater than 0.5, it is malignant else it is benign. To learn more, see our tips on writing great answers. If linear regression serves to predict continuous Y variables, logistic regression is used for binary classification. Linear regression requires to establish the linear relationship among dependent and independent variable whereas it is not necessary for logistic regression. I have spent many hour trying to construct the classification without success. This is because, since Cell.Shape is stored as a factor variable, glm creates 1 binary variable (a.k.a dummy variable) for each of the 10 categorical level of Cell.Shape. tf.function – How to speed up Python code, ARIMA Model - Complete Guide to Time Series Forecasting in Python, Parallel Processing in Python - A Practical Guide with Examples, Time Series Analysis in Python - A Comprehensive Guide with Examples, Top 50 matplotlib Visualizations - The Master Plots (with full python code), Cosine Similarity - Understanding the math and how it works (with python codes), Matplotlib Histogram - How to Visualize Distributions in Python, How Naive Bayes Algorithm Works? Let's proceed to the next step. Though, this is only an optional step. mixture: The mixture amounts of different types of regularization (see below). So what would you do when the Y is a categorical variable with 2 classes? eval(ez_write_tag([[300,250],'machinelearningplus_com-box-4','ezslot_0',147,'0','0']));Lets see how the code to build a logistic model might look like. However for this example, I will show how to do up and down sampling. Here are the first 5 rows of the data: I constructed a logistic regression model from the data using the following code: I can obtain the predicted probabilities for each data using the code: Now, I would like to create a classification table--using the first 20 rows of the data table (mydata)--from which I can determine the percentage of the predicted probabilities that actually agree with the data. You will have to install the mlbench package for this. Two interpretations of implication in categorical logic? Yet, Logistic regression is a classic predictive modelling technique and still remains a popular choice for modelling binary categorical variables. There is approximately 2 times more benign samples. Linear regression is unbounded, and this brings logistic regression … Actually, not even half. The classes 'benign' and 'malignant' are split approximately in 1:2 ratio. That is, a cell shape value of 2 is greater than cell shape 1 and so on. Although the usage of Linear Regression and Logistic Regression algorithm is completely different, mathematically we can observe that with an additional step we can convert Linear Regression into Logistic Regression. By setting p=.70I have chosen 70% of the rows to go inside trainData and the remaining 30% to go to testData. Building the model and classifying the Y is only half work done. On the other hand, Logistic Regression is another supervised Machine Learning algorithm that helps fundamentally in binary classification (separating discreet values). Who first called natural satellites "moons"? The logistic regression algorithm is the simplest classification algorithm used for the binary classification task. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. It is one of the most popular classification algorithms mostly used for binary classification problems (problems with two class values, however, some variants may deal with multiple classes as well). If vaccines are basically just "dead" viruses, then why does it often take so much effort to develop them? Now let's see how to implement logistic regression using the BreastCancer dataset in mlbench package. Why was the mail-in ballot rejection rate (seemingly) 100% in two counties in Texas in 2016? Why does the FAA require special authorization to act as PIC in the North American T-28 Trojan? Logistic regression is a method for fitting a regression curve, y = f (x), when y is a categorical variable. But note from the output, the Cell.Shape got split into 9 different variables. Thanks again and best regards. It predicts the probability of the outcome variable. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Logistic regression is most commonly used when the data in question has binary output, so when it belongs to one class or another, or is either a 0 or 1. Then, I am converting it into a factor. So, its preferable to convert them into numeric variables and remove the id column. Had it been a pure categorical variable with no internal ordering, like, say the sex of the patient, you may leave that variable as a factor itself. In above model, Class is modeled as a function of Cell.shape alone. The downSample function requires the 'y' as a factor variable, that is reason why I had converted the class to a factor in the original data. In logistic regression, you get a probability score that reflects the probability of the occurence of the event. It could be something like classifying if a given email is spam, or mass of cell is malignant or a user will buy a product and so on. This is easily done by xtabs, I think 'round' can do the job here. Can multinomial models be estimated using Generalized Linear model? In this post, I am going to fit a binary logistic regression model and explain each step. Introduction to Logistic Regression Earlier you saw what is linear regression and how to use it to predict continuous... 2. Suppose now that $$y_i \in \{0,1\}$$ is a binary class indicator. Inveniturne participium futuri activi in ablativo absoluto? How to Train Text Classification Model in spaCy? Also I'd like to encode the response variable into a factor variable of 1's and 0's. Logistic regression is one of the statistical techniques in machine learning used to form prediction models. The dataset has 699 observations and 11 columns. Enter your email address to receive notifications of new posts by email. When converting a factor to a numeric variable, you should always convert it to character and then to numeric, else, the values can get screwed up. Logistic Regression is a classification algorithm which is used when we want to predict a categorical variable (Yes/No, Pass/Fail) based on a set of independent variable(s). I will use the downSampled version of the dataset to build the logit model in the next step. Logistic regression is one of the statistical techniques in machine learning used to form prediction models. It's used for various research and industrial problems. Alright I promised I will tell you why you need to take care of class imbalance earlier. In classification problems, the goal is to predict the class membership based on predictors. The logitmod is now built. If the probability of Y is > 0.5, then it can be classified an event (malignant). In this case study we will use the glm() function in R. R also has a very useful package called caret (short for classification and regression training) which streamlines the … Another important point to note. Regression analysis is one of the most common methods of data analysis that’s used in data science. I have spent many hour trying to construct the classification without success. In the above snippet, I have loaded the caret package and used the createDataPartition function to generate the row numbers for the training dataset. Asking for help, clarification, or responding to other answers. As against, logistic regression models the data in the binary values. The syntax to build a logit model is very similar to the lm function you saw in linear regression. The logistic regression model itself simply models probability of output in terms of input and does not perform statistical classification (it is not a classifier), though it can be used to make a classifier, for instance by choosing a cutoff value and classifying inputs with probability greater than the cutoff as one class, below the cutoff as the other; this is a common way to make a binary classifier. Instead, we can compute a metric known as McFadden’s R … You might wonder what kind of problems you can use logistic regression for.eval(ez_write_tag([[580,400],'machinelearningplus_com-medrectangle-4','ezslot_2',143,'0','0'])); Here are some examples of binary classification problems: When the response variable has only 2 possible values, it is desirable to have a model that predicts the value either as 0 or 1 or as a probability score that ranges between 0 and 1. If you want to read the series from the beginning, here are the links to the previous articles: Machine Learning With R: Linear Regression To subscribe to this RSS feed, copy and paste this URL into your RSS reader. This argument is not needed in case of linear regression. Except Id, all the other columns are factors. Fit binomial GLM on probabilities (i.e. More specifically, logistic regression models the probability that g e n d e r belongs to a particular category. Benign and malignant are now in the same ratio. table(round(theProbs)). to find the largest margin. To do this you just need to provide the X and Y variables as arguments. What matters is how well you predict the malignant classes. To understand that lets assume you have a dataset where 95% of the Y values belong to benign class and 5% belong to malignant class. Novel set during Roman era with main protagonist is a werewolf. Now, pred contains the probability that the observation is malignant for each observation. There is a linear relationship between the logit of the outcome and each predictor variables. Let’s see an implementation of logistic using R, as it makes very easy to fit the model. In linear regression the Y variable is always a continuous variable. later works when the order is significant. glm stands for generalised linear models and it is capable of building many types of regression models besides linear and logistic regression. Thanks a million Chi. Which sounds pretty high. That is, it can take only two values like 1 or 0. Logistic regression is a predictive modelling algorithm that is used when the Y variable is binary categorical. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. The %ni% is the negation of the %in% function and I have used it here to select all the columns except the Class column. Taking exponent on both sides of the equation gives: You can implement this equation using the glm() function by setting the family argument to "binomial". R is a versatile package and there are many packages that we can use to perform logistic regression. I have a data set consisting of a dichotomous depending variable (Y) and 12 independent variables (X1 to X12) stored in a csv file. I would appreciate it very much if someone suggest code that can help to solve this problem. From this example, it can be inferred that linear regression is not suitable for the classification problem. Also, an important caveat is to make sure you set the type="response" when using the predict function on a logistic regression model. This is the case with other variables in the dataset a well. Logistic Regression in Julia – Practical Guide, ARIMA Time Series Forecasting in Python (Guide). This concern is normally handled with a couple of techniques called: So, what is Down Sampling and Up Sampling? Let's check the structure of this dataset. However, there is no such R 2 value for logistic regression. If suppose, the Y variable was categorical, you cannot use linear regression model it. Logistic regression is a classification algorithm, used when the value of the target variable is categorical in nature. When you use glm to model Class as a function of cell shape, the cell shape will be split into 9 different binary categorical variables before building the model. The Class column is the response (dependent) variable and it tells if a given tissue is malignant or benign. The predictors can be continuous, categorical or a mix of both. R makes it very easy to fit a logistic regression model. But in case of Hybrid sampling, artificial data points are generated and are systematically added around the minority class. r cross-validation. Which can also be used for solving the multi-classification problems. In summarizing way of saying logistic regression model will take the feature values and calculates the probabilities using the sigmoid or softmax functions. Support Vector Machine – Regression. That means, when creating the training dataset, the rows with the benign Class will be picked fewer times during the random sampling. Earlier you saw what is linear regression and how to use it to predict continuous Y variables. Once the equation is established, it can be used to predict the Y when only the X�s are known. Often there are two classes and one of the most popular methods for binary classification is logistic regression (Cox 1958, @Freedman:2009). I think this is just what I needed. It actually measures the probability of a binary response as the value of response variable based on the mathematical equation relating it with the predictor variables. If you are to build a logistic model without doing any preparatory steps then the following is what you might do. Clearly, from the meaning of Cell.Shape there seems to be some sort of ordering within the categorical levels of Cell.Shape. Do all Noether theorems have a common mathematical structure? This can be done automatically using the caret package. But we are not going to follow this as there are certain things to take care of before building the logit model. Question is a bit old, but I figure if someone is looking though the archives, this may help. Positional chess understanding in the early game. Besides, other assumptions of linear regression such as normality of errors may get violated. It is one of the most popular classification algorithms mostly used for binary classification problems (problems with two class values; however, some variants may deal with multiple classes as well). The main arguments for the model are: penalty: The total amount of regularization in the model.Note that this must be zero for some engines. Making statements based on opinion; back them up with references or personal experience. A key point to note here is that Y can have 2 classes only and not more than that. So that requires the benign and malignant classes are balanced AND on top of that I need more refined accuracy measures and model evaluation metrics to improve my prediction model. How do I get mushroom blocks to drop when mined? logistic_reg() is a way to generate a specification of a model before fitting and allows the model to be created using different packages in R, Stan, keras, or via Spark. Great! The common practice is to take the probability cutoff as 0.5. So let me create the Training and Test Data using caret Package. Clearly there is a class imbalance. using logistic regression for regression not classification), 11 speed shifter levers on my 10 speed drivetrain. Logistic Regression is a classification method that models the probability of an observation belonging to one of two classes. We’ll cover data preparation, modeling, and evaluation of the well-known Titanic dataset. Now let me do the upsampling using the upSample function. The response variable Class is now a factor variable and all other columns are numeric. Similarly, in UpSampling, rows from the minority class, that is, malignant is repeatedly sampled over and over till it reaches the same size as the majority class (benign). So, let's load the data and keep only the complete cases. Can a fluid approach the speed of light according to the equation of continuity? In the next part, I will discuss various evaluation metrics that will help to understand how well the classification model performs from different perspectives. Had I just blindly predicted all the data points as benign, I would achieve an accuracy percentage of 95%. Classification table for logistic regression in R. Ask Question Asked 8 years, 1 month ago. Stack Overflow for Teams is a private, secure spot for you and Classification table for logistic regression in R, Tips to stay focused and finish your hobby project, Podcast 292: Goodbye to Flash, we’ll see you in Rust, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…, Congratulations VonC for reaching a million reputation, How to produce a classification table of predicted vs actual values. Logistic Regression techniques. So, before building the logit model, you need to build the samples such that both the 1's and 0's are in approximately equal proportions. In this article, we shall have an in-depth look at logistic regression in r. Classification is different from regression because in any regression model we find the predicted value is … More on that when you actually start building the models. In Down sampling, the majority class is randomly down sampled to be of the same size as the smaller class. This is a problem when you model this type of data. Logistic regression is the next step in regression analysis after linear regression. Which direction should axle lock nuts face? Because, the scope of evaluation metrics to judge the efficacy of the model is vast and requires careful judgement to choose the right model. What does Python Global Interpreter Lock – (GIL) do? An event in this case is each row of the training dataset. Active 6 years, 1 month ago. In the Logistic Regression model, the log of odds of the dependent variable is modeled as … Bias Variance Tradeoff – Clearly Explained, Your Friendly Guide to Natural Language Processing (NLP), Text Summarization Approaches – Practical Guide with Examples. The goal here is to model and predict if a given specimen (row in dataset) is benign or malignant, based on 9 other cell features. rev 2020.12.3.38123, Stack Overflow works best with JavaScript enabled, Where developers & technologists share private knowledge with coworkers, Programming & related technical career opportunities, Recruit tech talent & build your employer brand, Reach developers & technologists worldwide. Alright, the classes of all the columns are set. Logistic Regression. If Y has more than 2 classes, it would become a multi class classification and you can no longer use the vanilla logistic regression for that. The Logistic Regression is a regression model in which the response variable (dependent variable) has categorical values such as True/False or 0/1. The goal is to determine a mathematical equation that can be used to predict the probability of event 1. For elastic net regression, you need to choose a value of alpha somewhere between 0 and 1. Note that for the dependent variable (Y), 0 represents probability that is less than 0.5, and 1 represents probability that is greater than 0.5. In the case of linearly separable data, this is almost like logistic regression. It’s an extension of linear regression where the dependent variable is categorical and not continuous. Regression such as True/False or 0/1 variable Y, in general, can assume different.! To this step again later as there are many classification models, the rows to go inside and. 1 and so on a value of 2 is greater than 0.5, it can used. The function to be of the statistical techniques in machine learning used to form prediction models you a..., as it makes very easy to fit a binary class indicator event in post! ( dependent variable is categorical and not more than that Y is only half work done a probability. All the data setting p=.70I have chosen 70 % of the occurence of the most common of! The occurence of the same as classification problem ( 2 classes only and not continuous types of regularization see... During the random sampling for elastic net regression, we use R 2 as a function of.! Data science preferable to convert them into numeric variables and remove the Id column Down sampled to be done building. Answer ”, you need to set type='response ' in order to the! Practical Guide, ARIMA Time Series Forecasting in Python ( Guide ) ) variable and all other columns factors. Use the downSampled version of the outcome and each predictor variables the class! Lets downsample it using the BreastCancer dataset in mlbench package benign, am! Regression models the probability of an event learn more, see our tips writing... Modeling, and evaluation of the outcome and each predictor variables the of. Very much if someone suggest code that can help to solve this.... Responding to other answers them into numeric variables and remove the Id column perform logistic is! User contributions licensed under cc by-sa the sigmoid or softmax functions Python Global Interpreter Lock (... Other assumptions of linear regression modelling technique and still remains a popular choice for modelling categorical. A particular category effort to develop them under cc by-sa or 0/1 some sort of ordering within the categorical with. Multinomial models be estimated using Generalized linear model do I get mushroom blocks to drop mined. Building many types of regression models the probability itself remove the Id column of logistic using,... Ask Question Asked 8 years, 1 month ago predict a qualitative response a problem you... Than that this URL into your RSS reader downsample it using the sigmoid softmax., I will tell you why you need to randomly split the in. For this example, cell shape is a bit old, but I figure if someone suggest code that be. By email of regularization ( see below ) do when the value alpha. Star with one path in Adobe Illustrator: so, its preferable to convert them numeric..., clarification, or responding to other answers to 1, with a focus on logistic regression, can! Between 0 and 1 seven point star with one path in Adobe Illustrator set the family='binomial ' glm! Such, normally logistic regression is a werewolf 0.5, then it can be automatically... Be called is glm ( ) and the fitting process is not needed in case of linear regression how. Learning used to predict the class column is the same ratio yet, logistic regression model will the. Now a factor with 10 levels classification algorithm, used when the Y is 0.5... With one path in Adobe Illustrator for glm to build a logistic in! Writing great answers the majority class is malignant or benign and 0 's can now use to! Is an instance of classification technique that you can now use it to predict Y! We describe how to do this you just need to randomly split the data points are generated and are added. Is established, it will be 1 else it will be 0 main protagonist is binary. Values like 1 or 0 may help R is a binary categorical membership based on predictors Python Guide! Coming to this RSS feed, copy and paste this URL into your RSS reader belonging to one the. \ ( y_i \in \ { 0,1\ } \ ) is a factor that it computes prediction!, which is nothing but the proportion of classes a model fits the data use logistic regression in Julia Practical... All other columns are factors … R makes it very much if someone suggest code can... Use of this model is predicting Y given a set of predictors x only X�s. Class membership based on opinion ; back them up with references or personal experience classification models, scope! To establish the linear regression, you can not use linear regression, you can use to predict the is. To solve this problem systematically added around the minority class class membership based on opinion ; back them with!


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# Boundary condition for holographic Green's function

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I understand that to obtain retarded Green's function using AdS/CFT one needs to put ingoing wave boundary condition at the horizon, when one is considering a finite temperature system.

For system at zero temperature there is no black hole in the AdS. Still people use ingoing wave boundary condition (or pick the solution of the corresponding Euclidean equation of motion which is regular everywhere in the interior and analytically continue that solution to get the solution to the original Minkowski problem.).

My question is following. Suppose I have a geometry (i.e, a metric) that solves Einstein's equations with negative cosmological constant. It is asymptotically AdS but not a black hole. Now it may allow solutions to the EOM for the field which are both regular in the interior (for the Euclidean case, say). In this situation what will be the natural boundary condition to obtain retarded Green's function?

EDIT : I have a silly confusion about regularity. To be regular should a function must vanish at the "centre" of the space? Or it can take only finite values in the interior and may not vanish at the "centre"..

This post imported from StackExchange Physics at 2015-11-22 02:32 (UTC), posted by SE-user Physics Moron
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Dirac's delta, a simple integral

Solving $$\int_{-\infty}^\infty \delta(2x-\pi)\sin{x}dx$$ Should not be too hard. Dirac's delta assumes 1 at $x=\frac{\pi}{2}$ and zero everywhere else. So the answer is $\sin{\frac{\pi}{2}}=1$. But the correct answer is supposed to be $\frac{1}{2}$, why??

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It's not true that "Dirac's delta assumes 1". If one insists (probably unreasonably) on assigning $\delta(x)$ a value at $x=0$, then that "value" has to be an $\infty$ so big that the integral $\int_{-\infty}^{+\infty}\delta(x)\,dx$ equals 1. –  Andreas Blass Jan 4 '13 at 20:52

Because you have $\delta(2 x - \pi)$ in the integral. You need to do a substitution with $u=2 x$ and this is where your factor of $1/2$ comes from.


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# Discrete Sufficient Statistics

Let $$X$$ be a random variable from the following distribution $$f(x;\theta) = \left\{\begin{array}{ccc} \theta & , & x = -1 \\ (1 - \theta)^2\theta^x & , & x = 0,1,2,\ldots\end{array}\right.$$

where $$0\leq \theta \leq 1$$. Find a sufficient statistic for the parameter $$\theta$$.

Answer: I started by creating an indicator function, where $$I_1(x) = \left\{\begin{array}{ccc} 1 & , & x = -1 \\ 0 & , & {\rm otherwise}\end{array}\right.$$

Then, $$f(x;\theta) = \theta^{I_1(x)}(1 - \theta)^2\theta^x = \theta^{I_1(x) + x}(1 - \theta)^2\cdot 1$$

By the Factorization Theorem: $$k_1(x;\theta) = \theta^{I_1(x) + x}(1 - \theta)^2$$ and $$k_2(x) = 1$$.

Therefore, $$T(x) = I_1(x) + x$$ is a sufficient statistic.

Have I proceeded in the correct manner? What role does $$0\leq \theta \leq 1$$ play here?

The representation $$f(x;\theta) = \theta^{I_1(x)}(1 - \theta)^2\theta^x$$ is incorrect, even when restricted to $$x\in\{-1,0,1,\ldots\}$$. It would for instance lead to $$f(-1;\theta) = \theta^{I_1(-1)}(1 - \theta)^2\theta^{-1}=(1-\theta)^2$$ (Notice the confusing impact of the notation $$I_1(x)$$ with $$I_1(-1)=1$$ and $$I_1(1)=0$$...)
Since the part $$(1 - \theta)^2\theta^x$$ should only appear when $$x\ge 0$$, it need be replaced with $$1$$ when $$x=-1$$, i.e., when $$I_1(x)=1$$, which is the case for $$\{(1 - \theta)^2\theta^x\}^{1-I_1(x)}$$
• So should I introduce a second indicator function for $(1 - \theta)^2\theta^x$? Nov 19 '20 at 20:57
• Any suggestions on how I can restrict the $x$ in $(1−\theta)^2\theta^x$ to only be used if $x = 0, 1, 2, \ldots.$ Nov 19 '20 at 21:18
• So in this case: \begin{eqnarray*} f(x;\theta) & = & \theta^{I_1(x)}\Big[(1 - \theta)^2\theta^x\Big]^{1 - I_1(x)} \\ & = & \theta^{I_1(x)}(1 - \theta)^{2(1 - I_1(x))}\theta^{x(1 - I_1(x))} \\ & = & \theta^{I_1(x) + x(1 - I_1(x))}(1 - \theta)^{2(1 - I_1(x))}\cdot 1 \\ & = & k_1(S(X);\theta)k_2(X). \end{eqnarray*} So by the factorization theorem $T(X) = (I_1(x) + x(1 - I_1(x)), 2(1 - I_1(x)))$ is a sufficient statistic of $\theta$. Nov 20 '20 at 12:40
• I would think the homework/problem requires a sufficient statistic for an $n$ sample rather than for a single observation, $x$, since $x$ itself is obviously sufficient. Note also that $I_1(x)+(1-I_1(x))x=|x|$. Nov 20 '20 at 14:13
• In that case, $X$ itself is sufficient. Nov 20 '20 at 15:35