Classification of local stellar populations: the improved MEMPHIS algorithm - Part II

Discontinuities of the local velocity distribution which are associated with stellar populations are studied from the improved statistical method MEMPHIS (Maximum Entropy of the Mixture Probability from HIerarchical Segregation), by combining a sampling parameter, optimisation of the mixture approach, and maximum partition entropy of populations composing the stellar sample. The sampling parameter is associated with isolating integrals of the star motion and it is used to build a hierarchical family of subsamples. An accurate characterisation of the entropy graph is given where a local maximum of entropy takes place simultaneously with a local minimum 2 error. By working from different sampling parameters the method is applied to samples from HIPPARCOS and Geneva-Copenhagen survey (GCS) to obtain kinematic parameters and mixture proportions of thin disk, thick disk and halo. The sampling parameter P = |(U, V,W)|, absolute heliocentric velocity, allows to build an optimal subsample containing thin and thick disk stars, by leaving aside most of the halo population. The sampling parameter P = |W|, absolute perpendicular velocity, is able to build an optimal subsample containing a mixture of total disk and halo stars, although it does not allow an optimal segregation of thin and thick disks. Other sampling parameters like P = |(U,W)| or P = |V | are found to be less population informative. By comparing both samples, HIPPARCOS provides more accurate estimates for thick disk and halo, while GCS does for the total disk. In particular, the radial velocity dispersion of the halo fits perfectly into the empirical Titius-Bode like law U = 6.6 ( 4 3 )3n+2, which was previously proposed for discrete kinematic components, where the values n = 0, 1, 2, 3 stands for early-type stars, thin disk, thick disk, and halo populations. Population statistics are used to segregate thin disk, thick disk, and halo, and to obtain a more accurate bayesian estimation of the population fractions.Preprin

Mixture parameters of a trivariate normal superposition from sample cumulants: application to the local stellar velocity distribution

The velocity distribution of nearby stars can be studied as a mixture of two main population components. In order to determine the mixing proportions and the population parameters a combined geometric-statistical method has been developed. The overall distribution is approximated from a superposition of two trivariate normal velocity density functions. The peculiar velocity is projected on a plane containing the global centroid (mean of the distribution), which is ortogonal to the direction D through both population subcentroids, obtaining two linear independent projected velocities. The statistical moments of these new variables are computed from second, third and fourth-order sample cumulants. The symmetric behaviour of the distribution around the direction D allows to determine it working only from third cumulants. Finally the overall set of projected peculiar velocity moments is used to determine the population covariance matrices, population means, and mixture proportions. The method does not require any extra hypotheses such as those concerning to prior population parameters, or specific symmetries of the distribution.Postprint (published version

Three kinematical methods to identify local galactic structures

Method one: by combining a sampling parameter related to an isolating integral of the stellar motion, an optimisation of the mixture approach, and a maximisation of the partition entropy for the constituent populations of the stellar sample. Method two: by segregating into different kinematical components in terms of the stellar orbital parameters. Method three: by approaching a maximum entropy velocity distribution to samples selected in terms of stellar eccentricity layers. Working samples: HIPPARCOS and Geneva-Copenhagen survey catalog. Results: kinematical characterisation of large-scale structures, such as thin disc, thick disc and halo, and identification of small-scale structures, such as moving groups in the solar neighbourhood. Consequences: confirmation of the Titius-Bode-like law for radial velocity dispersions and explanation of the apparent vertex deviation of the disc from the swinging of two major kinematic groups around the LSR, by predicting a continuously changing orientation of the disc pseudo ellipsoid.Postprint (published version

Contribución al Estudio de la Dinámica Galáctica: Superposición de Sistemas Estelares

[spa] Se analizan diversos aspectos del comportamiento cinemático de las poblaciones estelaresen el entorno solar. Para ello se pone a punto un método numérico de superposición desistemas estelares que permite aproximar una muestra global de estrellas por dos o máscomponentes cada una de ellas con distribución normal de velocidades. Se parte de losprincipios y métodos de la Dinámica Galáctica. Se plantea el modelo estadístico de lasuperposición de n funciones de distribución de Schwarzschild generalizadas. Se desarrolla un método de cálculo numérico para el caso particular de dos poblaciones que finalmente se aplica a muestras estelares locales. Los resultados que se obtienen de la aplicación delmétodo se encajan dentro de los modelos dinámicos que se presentan al principio. Adicionalmente se desarrolla un método de selección de la muestra de estrellas por máximaentropía de la probabilidad de mezcla. Se resumen varios modelos dinámicos de sistemas estelares de Chandrasekhar que utilizandiferentes hipótesis de simetrías de la distribución de velocidades: Simetría cilíndrica en estado estacionario, en estado no estacionario con y sin simetría respecto del planogaláctico y simetría axial no cilíndrica. Se utiliza el principio de superposición depoblaciones para obtener grupos de estrellas que se ajustan a modelos dinámicos sencillosaun cuando el conjunto global de estrellas no pueda interpretarse de acuerdo con lasmismas simplificaciones. Se describen las diferentes interpretaciones del fenómeno de ladesviación del vértex resumiendo sus posibles causas. Se introduce el concepto depoblaciones estelares de acuerdo con el criterio de que cinemática y distribución espacialhacen referencia a componentes mientras que edad y metalicidad se refieren a poblacionesdentro de una componente. Se presenta el desarrollo estadístico que da lugar al algoritmode cálculo y se deducen las expresiones de los momentos de orden n de una superposiciónarbitraria dep poblaciones. Tales expresiones dan lugar a un sistema de ecuaciones cuya resolución para el caso p=2 constituye el método numérico de separación de poblaciones. Se optimiza así un desarrollo analítico previo utilizando el mínimo número de grupos estelares que expliquen los parámetros característicos de la muestra. Se mejoran los resultados mediante propagación estadística de errores y resolución de sistemas de ecuaciones por mínimos cuadrados ponderados. Para entrenar el método numérico se utilizan muestras sintéticas. Estas muestras, permiten introducir estrellas de comportamiento cinemático extremo y sugieren el criterio de selección de la muestra. Además, las muestras sintéticas permiten aplicar el método numérico de forma recurrente previa extracción de la población más dispersa y así obtener más de dos poblaciones gausianas a partir de una muestra global. Se define un criterio de selección de estrellas de la muestra para excluir las que presentan características cinemáticas más extremas. Este criterio puede asociarse con la idea de máxima entropía para obtener la aproximación general representativa del máximo número de estrellas. Finalmente se aplica todo lo planteado a muestras del entorno solar: CNS3 e HIPPARCOS. La aplicación del método a estas muestras permite deducir interesantes conclusiones sobre la cinemática local. Se aportan nuevos valores para las velocidades radiales, desviación del vértex y proporciones de mezcla de poblaciones en el entorno solar. En la muestra procedente del CNS3 se aprecian los denominados discos joven y viejo siendo esta última componente compatible con un modelo dinámico de simetría cilíndrica. En la procedente de HIPPARCOS se aprecia además el disco grueso presentándose desviación del vértexpara todas las componentes. Adicionalmente, desde un punto de vista metodológico se aporta la optimización de un método numérico, su tratamiento de errores y la forma de seleccionar la muestra.[eng] In order to study the kinematic behaviour of local stellar populations, it has been developeda statistical method which allows approximating a local stellar sample as superposition oftwo or more stellar systems each one with normal velocity distribution. The partialcomponents are supposed large enough as to be represented by gaussian functions lookingfor macroscopic properties, so that they may be associated with stellar populations.Some Galactic dynamic models developed by different authors are reviewed. These modelsare based in the Chandrasekhar's approximation for the velocity distribution function. Depending on the symmetry hypothesis taken for describing the model some conclusionsabout the values of the moments, mean velocities and vertex deviation are obtained.Then, a statistical model based on the use of the moments of second, third and fourth orderis developed. The velocity density function is approximated by the superposition of twotrivariate normal distributions leading to an equation system optimized for minimizing theerrors. A local stellar sample is drawn from neighbour star catalogues by using a non-informativefiltering method looking for the maximum entropy of the mixture probability. Thus, we areable to apply the method recursively in order to identify more than two groups in thesample. The population covariance matrices are determined as well as the mean velocities,the vertex deviation and the mixture proportions. The method has been applied to CNS3and HIPPARCOS. The most remarkable conclusions deduced from the kinematic parameters are: In CNS3 twoclear components are clearly detected. The one corresponding to old disk stars iscompatible with dynamic models accepting axial symmetry and shows no vertex deviation.In HIPPARCOS, three components are shown which are associable to young, old and thickdisks. The component of CNS3 associable with young as well as these three ones show nonegligible vertex deviation and require a point axial symmetry model in order to explain itskinematics. For both samples, a radial differential movement between young and old diskcomponents is also detected

Classification of local stellar populations: the improved MEMPHIS algorithm - Part II

Discontinuities of the local velocity distribution which are associated with stellar populations are studied from the improved statistical method MEMPHIS (Maximum Entropy of the Mixture Probability from HIerarchical Segregation), by combining a sampling parameter, optimisation of the mixture approach, and maximum partition entropy of populations composing the stellar sample. The sampling parameter is associated with isolating integrals of the star motion and it is used to build a hierarchical family of subsamples. An accurate characterisation of the entropy graph is given where a local maximum of entropy takes place simultaneously with a local minimum 2 error. By working from different sampling parameters the method is applied to samples from HIPPARCOS and Geneva-Copenhagen survey (GCS) to obtain kinematic parameters and mixture proportions of thin disk, thick disk and halo. The sampling parameter P = |(U, V,W)|, absolute heliocentric velocity, allows to build an optimal subsample containing thin and thick disk stars, by leaving aside most of the halo population. The sampling parameter P = |W|, absolute perpendicular velocity, is able to build an optimal subsample containing a mixture of total disk and halo stars, although it does not allow an optimal segregation of thin and thick disks. Other sampling parameters like P = |(U,W)| or P = |V | are found to be less population informative. By comparing both samples, HIPPARCOS provides more accurate estimates for thick disk and halo, while GCS does for the total disk. In particular, the radial velocity dispersion of the halo fits perfectly into the empirical Titius-Bode like law U = 6.6 ( 4 3 )3n+2, which was previously proposed for discrete kinematic components, where the values n = 0, 1, 2, 3 stands for early-type stars, thin disk, thick disk, and halo populations. Population statistics are used to segregate thin disk, thick disk, and halo, and to obtain a more accurate bayesian estimation of the population fractions

Mixture parameters of a trivariate normal superposition from sample cumulants: application to the local stellar velocity distribution

The velocity distribution of nearby stars can be studied as a mixture of two main population components. In order to determine the mixing proportions and the population parameters a combined geometric-statistical method has been developed. The overall distribution is approximated from a superposition of two trivariate normal velocity density functions. The peculiar velocity is projected on a plane containing the global centroid (mean of the distribution), which is ortogonal to the direction D through both population subcentroids, obtaining two linear independent projected velocities. The statistical moments of these new variables are computed from second, third and fourth-order sample cumulants. The symmetric behaviour of the distribution around the direction D allows to determine it working only from third cumulants. Finally the overall set of projected peculiar velocity moments is used to determine the population covariance matrices, population means, and mixture proportions. The method does not require any extra hypotheses such as those concerning to prior population parameters, or specific symmetries of the distribution

Partition entropy and chi-squared error: the improved MEMPHIS algorithm - Part I

The entropy of the population partition is studied as a function of the sampling parameter, so that within a particular interval of its graph, the plateau region, it is possible to get a stable estimation of the mixture parameters. The optimal estimation is associated with a local maximum of entropy. Alter natively, the $\chi^2$ error of the mixture approach may also be used to obtain an optimal segregation. The relationship between the fitting error and the population entropy has been analysed in detail. We have proved that, by using an appropriate sampling parameter, within a plateau region of the entropy graph, a local entropy maximum takes place simultaneously with a local minimum of the $\chi^2$ error. Therefore, the combined statistical method provides the best approximation mixture, as well as the less informative partiti on, to estimate the kinematic parameters of populations.Preprin

Partition entropy and chi-squared error: the improved MEMPHIS algorithm - Part I

The entropy of the population partition is studied as a function of the sampling parameter, so that within a particular interval of its graph, the plateau region, it is possible to get a stable estimation of the mixture parameters. The optimal estimation is associated with a local maximum of entropy. Alter natively, the $\chi^2$ error of the mixture approach may also be used to obtain an optimal segregation. The relationship between the fitting error and the population entropy has been analysed in detail. We have proved that, by using an appropriate sampling parameter, within a plateau region of the entropy graph, a local entropy maximum takes place simultaneously with a local minimum of the $\chi^2$ error. Therefore, the combined statistical method provides the best approximation mixture, as well as the less informative partiti on, to estimate the kinematic parameters of populations

Programas del algoritmo de clasificacion de poblaciones normales trivariantes a partir de la entropia de mezcla

The complete FORTRAN codified programs of the segregation algorithm described in Alcobé (2001) are described. The classification algorithm is applied to study the trivariate velocity distribution of stars from the star catalogs, which can be locally approximated by a superposition of two or more normal components. An auxiliar sampling parameter P (such as the velocity module referred to a specific point, the absolute value of one peculiar velocity component alone, the distance to the galactic plane, etc.) is introduced in order to define the sample boundaries. The sampling paramenter must induce a hyerarchical incorporation of stars to the population components, in the sense that the greater the P value, the greater the number of stars in each component. For a fixed P, a sample S(P) is drawn from the global catalog. Depending on the sampling parameter the population entropy H(P) of a two-component mixture is computed from the mixing proportions. The purpose is to find the optimal P value in order to maximize H(P). Then the algorithm is used recursively in order to segregate a global sample in more than two populations. Moreover, for each subsample S(P), the goodness of the superposition approximation is evaluated by reconstructing the sample central moments up to fourth-order from the population parameters. A chi-square test, taking into account the sampling distribution moments, is evaluated to measure the fitting error. For subsamples S(P) a total accordance between the minimum chi-square and the maximum population entropy H(P) is produced.Preprin

Classification of local stellar populations: the improved MEMPHIS algorithm - Part II

Discontinuities of the local velocity distribution which are associated with stellar populations are studied from the improved statistical method MEMPHIS (Maximum Entropy of the Mixture Probability from HIerarchical Segregation), by combining a sampling parameter, optimisation of the mixture approach, and maximum partition entropy of populations composing the stellar sample. The sampling parameter is associated with isolating integrals of the star motion and it is used to build a hierarchical family of subsamples. An accurate characterisation of the entropy graph is given where a local maximum of entropy takes place simultaneously with a local minimum 2 error. By working from different sampling parameters the method is applied to samples from HIPPARCOS and Geneva-Copenhagen survey (GCS) to obtain kinematic parameters and mixture proportions of thin disk, thick disk and halo. The sampling parameter P = |(U, V,W)|, absolute heliocentric velocity, allows to build an optimal subsample containing thin and thick disk stars, by leaving aside most of the halo population. The sampling parameter P = |W|, absolute perpendicular velocity, is able to build an optimal subsample containing a mixture of total disk and halo stars, although it does not allow an optimal segregation of thin and thick disks. Other sampling parameters like P = |(U,W)| or P = |V | are found to be less population informative. By comparing both samples, HIPPARCOS provides more accurate estimates for thick disk and halo, while GCS does for the total disk. In particular, the radial velocity dispersion of the halo fits perfectly into the empirical Titius-Bode like law U = 6.6 ( 4 3 )3n+2, which was previously proposed for discrete kinematic components, where the values n = 0, 1, 2, 3 stands for early-type stars, thin disk, thick disk, and halo populations. Population statistics are used to segregate thin disk, thick disk, and halo, and to obtain a more accurate bayesian estimation of the population fractions