343,257 research outputs found
The Fundamental Nature of the Log Loss Function
The standard loss functions used in the literature on probabilistic
prediction are the log loss function, the Brier loss function, and the
spherical loss function; however, any computable proper loss function can be
used for comparison of prediction algorithms. This note shows that the log loss
function is most selective in that any prediction algorithm that is optimal for
a given data sequence (in the sense of the algorithmic theory of randomness)
under the log loss function will be optimal under any computable proper mixable
loss function; on the other hand, there is a data sequence and a prediction
algorithm that is optimal for that sequence under either of the two other
standard loss functions but not under the log loss function.Comment: 12 page
Critical Market Crashes
This review is a partial synthesis of the book ``Why stock market crash''
(Princeton University Press, January 2003), which presents a general theory of
financial crashes and of stock market instabilities that his co-workers and the
author have developed over the past seven years. The study of the frequency
distribution of drawdowns, or runs of successive losses shows that large
financial crashes are ``outliers'': they form a class of their own as can be
seen from their statistical signatures. If large financial crashes are
``outliers'', they are special and thus require a special explanation, a
specific model, a theory of their own. In addition, their special properties
may perhaps be used for their prediction. The main mechanisms leading to
positive feedbacks, i.e., self-reinforcement, such as imitative behavior and
herding between investors are reviewed with many references provided to the
relevant literature outside the confine of Physics. Positive feedbacks provide
the fuel for the development of speculative bubbles, preparing the instability
for a major crash. We demonstrate several detailed mathematical models of
speculative bubbles and crashes. The most important message is the discovery of
robust and universal signatures of the approach to crashes. These precursory
patterns have been documented for essentially all crashes on developed as well
as emergent stock markets, on currency markets, on company stocks, and so on.
The concept of an ``anti-bubble'' is also summarized, with two forward
predictions on the Japanese stock market starting in 1999 and on the USA stock
market still running. We conclude by presenting our view of the organization of
financial markets.Comment: Latex 89 pages and 38 figures, in press in Physics Report
The Big World of Nanothermodynamics
Nanothermodynamics extends standard thermodynamics to facilitate finite-size
effects on the scale of nanometers. A key ingredient is Hill's subdivision
potential that accommodates the non-extensive energy of independent small
systems, similar to how Gibbs' chemical potential accommodates distinct
particles. Nanothermodynamics is essential for characterizing the thermal
equilibrium distribution of independently relaxing regions inside bulk samples,
as is found for the primary response of most materials using various
experimental techniques. The subdivision potential ensures strict adherence to
the laws of thermodynamics: total energy is conserved by including an
instantaneous contribution from the entropy of local configurations, and total
entropy remains maximized by coupling to a thermal bath. A unique feature of
nanothermodynamics is the completely-open nanocanonical ensemble. Another
feature is that particles within each region become statistically
indistinguishable, which avoids non-extensive entropy, and mimics
quantum-mechanical behavior. Applied to mean-field theory, nanothermodynamics
gives a heterogeneous distribution of regions that yields stretched-exponential
relaxation and super-Arrhenius activation. Applied to Monte Carlo simulations,
there is a nonlinear correction to Boltzmann's factor that improves agreement
between the Ising model and measured non-classical critical scaling in magnetic
materials. Nanothermodynamics also provides a fundamental mechanism for the 1/f
noise found in many materials.Comment: 22 pages, 14 figures, revie
An extensive grid of DARWIN models for M-type AGB stars I. Mass-loss rates and other properties of dust-driven winds
The purpose of this work is to present an extensive grid of dynamical
atmosphere and wind models for M-type AGB stars, covering a wide range of
relevant stellar parameters. We used the DARWIN code, which includes
frequency-dependent radiation-hydrodynamics and a time-dependent description of
dust condensation and evaporation, to simulate the dynamical atmosphere. The
wind-driving mechanism is photon scattering on submicron-sized MgSiO
grains. The grid consists of models, with luminosities from
to and
effective temperatures from 2200K to 3400K. For the first time different
current stellar masses are explored with M-type DARWIN models, ranging from
0.75M to 3M. The modelling results are radial atmospheric
structures, dynamical properties such as mass-loss rates and wind velocities,
and dust properties (e.g. grain sizes, dust-to-gas ratios, and degree of
condensed Si). We find that the mass-loss rates of the models correlate
strongly with luminosity. They also correlate with the ratio :
increasing by an order of magnitude increases the mass-loss rates by
about three orders of magnitude, which may naturally create a superwind regime
in evolution models. There is, however, no discernible trend of mass-loss rate
with effective temperature, in contrast to what is found for C-type AGB stars.
We also find that the mass-loss rates level off at luminosities higher than
, and consequently at pulsation periods longer
than days. The final grain radii range from 0.25 micron to 0.6
micron. The amount of condensed Si is typically between 10% and 40%, with
gas-to-dust mass ratios between 500 and 4000.Comment: Accepted to A&A, 17 pages, 15 figure
From Galaxy Clusters to Ultra-Faint Dwarf Spheroidals: A Fundamental Curve Connecting Dispersion-supported Galaxies to Their Dark Matter Halos
We examine scaling relations of dispersion-supported galaxies over more than
eight orders of magnitude in luminosity by transforming standard fundamental
plane parameters into a space of mass (M1/2), radius (r1/2), and luminosity
(L1/2). We find that from ultra-faint dwarf spheroidals to giant cluster
spheroids, dispersion-supported galaxies scatter about a one-dimensional
"fundamental curve" through this MRL space. The weakness of the M1/2-L1/2 slope
on the faint end may imply that potential well depth limits galaxy formation in
small galaxies, while the stronger dependence on L1/2 on the bright end
suggests that baryonic physics limits galaxy formation in massive galaxies. The
mass-radius projection of this curve can be compared to median dark matter halo
mass profiles of LCDM halos in order to construct a virial mass-luminosity
relationship (Mvir-L) for galaxies that spans seven orders of magnitude in
Mvir. Independent of any global abundance or clustering information, we find
that (spheroidal) galaxy formation needs to be most efficient in halos of Mvir
~ 10^12 Msun and to become inefficient above and below this scale. Moreover,
this profile matching technique is most accurate at the high and low luminosity
extremes (where dark matter fractions are highest) and is therefore quite
complementary to statistical approaches that rely on having a well-sampled
luminosity function. We also consider the significance and utility of the
scatter about this relation, and find that in the dSph regime observational
errors are almost at the point where we can explore the intrinsic scatter in
the luminosity-virial mass relation. Finally, we note that purely stellar
systems like Globular Clusters and Ultra Compact Dwarfs do not follow the
fundamental curve relation. This allows them to be easily distinguished from
dark-matter dominated dSph galaxies in MRL space. (abridged)Comment: 27 pages, 18 figures, ApJ accepted. High-res movies of 3D figures are
available at http://www.physics.uci.edu/~bullock/fcurve/movies.htm
Fundamental rate-loss tradeoff for optical quantum key distribution
Since 1984, various optical quantum key distribution (QKD) protocols have
been proposed and examined. In all of them, the rate of secret key generation
decays exponentially with distance. A natural and fundamental question is then
whether there are yet-to-be discovered optical QKD protocols (without quantum
repeaters) that could circumvent this rate-distance tradeoff. This paper
provides a major step towards answering this question. We show that the
secret-key-agreement capacity of a lossy and noisy optical channel assisted by
unlimited two-way public classical communication is limited by an upper bound
that is solely a function of the channel loss, regardless of how much optical
power the protocol may use. Our result has major implications for understanding
the secret-key-agreement capacity of optical channels---a long-standing open
problem in optical quantum information theory---and strongly suggests a real
need for quantum repeaters to perform QKD at high rates over long distances.Comment: 9+4 pages, 3 figures. arXiv admin note: text overlap with
arXiv:1310.012
On the Universality of the Logistic Loss Function
A loss function measures the discrepancy between the true values
(observations) and their estimated fits, for a given instance of data. A loss
function is said to be proper (unbiased, Fisher consistent) if the fits are
defined over a unit simplex, and the minimizer of the expected loss is the true
underlying probability of the data. Typical examples are the zero-one loss, the
quadratic loss and the Bernoulli log-likelihood loss (log-loss). In this work
we show that for binary classification problems, the divergence associated with
smooth, proper and convex loss functions is bounded from above by the
Kullback-Leibler (KL) divergence, up to a multiplicative normalization
constant. It implies that by minimizing the log-loss (associated with the KL
divergence), we minimize an upper bound to any choice of loss functions from
this set. This property justifies the broad use of log-loss in regression,
decision trees, deep neural networks and many other applications. In addition,
we show that the KL divergence bounds from above any separable Bregman
divergence that is convex in its second argument (up to a multiplicative
normalization constant). This result introduces a new set of divergence
inequalities, similar to the well-known Pinsker inequality
- …