The Multi-angle Imaging SpectroRadiometer, one of five science instruments aboard NASA's Terra spacecraft, was launched into earth orbit in December 1999. Acquisition of earth imagery began in February 2000, and the quality of the data is excellent. Overall, MISR has been performing superbly, though the instrument exhibits several idiosyncrasies, some of which were anticipated prior to launch. Details regarding the in-flight performance of the instrument system are presented.
The design, characteristics, and first test flight results are described of the Portable Remote Imaging Spectrometer, an airborne sensor specifically designed to address the challenges of coastal ocean remote sensing. The sensor incorporates several technologies that are demonstrated for the first time, to the best of our knowledge, in a working system in order to achieve a high performance level in terms of uniformity, signal-to-noise ratio, low polarization sensitivity, low stray light, and high spatial resolution. The instrument covers the 350-1050 nm spectral range with a 2.83 nm sampling per pixel, and a 0.88 mrad instantaneous field of view, with 608 cross-track pixels in a pushbroom configuration. Two additional infrared channels (1240 and 1610 nm) are measured by a spot radiometer housed in the same head. The spectrometer design is based on an optically fast (F/1.8) Dyson design form coupled to a wide angle two-mirror telescope in a configuration that minimizes polarization sensitivity without the use of a depolarizer. A grating with minimum polarization sensitivity and broadband efficiency was fabricated as well as a slit assembly with black (etched) silicon surface to minimize backscatter. First flight results over calibration sites as well as Monterey Bay in California have demonstrated good agreement between in situ and remotely sensed data, confirming the potential value of the sensor to the coastal ocean science community.
We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all 2-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two singular points of type A 1 and provide explicit expressions that allow to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using A-discriminants and intersection theory. Finally, we show that toric Fano varieties associated to 3-valent phylogenetic trees have ML degree one and provide a formula for the maximum likelihood estimate. We prove it as a corollary to a more general result about the multiplicativity of ML degrees of codimension zero toric fiber products, and it also follows from a connection to a recent result about staged trees.The authors would like to thank ML degree of special complete intersections. Moreover, a geometric characterisation of the ML degree of a smooth variety in the case when the divisor corresponding to the rational function is a normal crossings divisor is given in [6]. In the same paper an explicit combinatorial formula for the ML degree of a toric variety is derived by relaxing the restrictive smoothness assumption and allowing some mild singularities. For an introduction to the geometry behind the MLE for algebraic statistical models for discrete data the interested reader is refered to [24], which includes most of the current results on the MLE problem from the perspective of algebraic geometry as well as statistical motivation.This article is concerned with the problem of MLE on toric Fano varieties. Toric varieties correpond to log-linear models in statistics. Since the seminal papers by L.A. Goodman in the 70s [16,17], log-linear models have been widely used in statistics and areas like natural language processing when analyzing crossclassified data in multidimensional contingency tables [4]. The ML degree of a toric variety is bounded above by its degree. Toric Fano varieties provide several interesting classes of toric varieties for investigating the ML degree drop. We focus on studying the maximum likelihood estimation for 2-dimensional Gorenstein toric Fano varieties, Veronese(2, 2) with different scalings and toric Fano varieties associated to 3-valent phylogenetic trees.Two-dimensional Gorenstein toric Fano varieties correspond to reflexive polygons and by the classification results there are exactly 16 isomorphism classes of such polygons, see for example [27]. Out of these 16 isomorphism classes five correspond to smooth del Pezzo surfaces and 11 correspond to del Pezzo surfaces with singularities. Our first main result Theorem 3.1 states that the ML degree is equal to the degree of the surface in all cases except for the quintic del Pezzo surface with two singular points of type A 1 . Furthermore, in Table 2, we provide explicit expressions that all...
In a recent paper, Hauenstein, Sturmfels, and the second author discovered a conjectural bijection between critical points of the likelihood function on the complex variety of matrices of rank r and critical points on the complex variety of matrices of co-rank r − 1. In this paper, we prove that conjecture for rectangular matrices and for symmetric matrices, as well as a variant for skew-symmetric matrices.
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