Spatio-temporal evolution of global surface temperature distributions

Climate is known for being characterised by strong non-linearity and chaotic behaviour. Nevertheless, few studies in climate science adopt statistical methods specifically designed for non-stationary or non-linear systems. Here we show how the use of statistical methods from Information Theory can describe the non-stationary behaviour of climate fields, unveiling spatial and temporal patterns that may otherwise be difficult to recognize. We study the maximum temperature at two meters above ground using the NCEP CDAS1 daily reanalysis data, with a spatial resolution of 2.5 by 2.5 degree and covering the time period from 1 January 1948 to 30 November 2018. The spatial and temporal evolution of the temperature time series are retrieved using the Fisher Information Measure, which quantifies the information in a signal, and the Shannon Entropy Power, which is a measure of its uncertainty -- or unpredictability. The results describe the temporal behaviour of the analysed variable. Our findings suggest that tropical and temperate zones are now characterized by higher levels of entropy. Finally, Fisher-Shannon Complexity is introduced and applied to study the evolution of the daily maximum surface temperature distributions.Comment: 7 pages, 4 figure

Efficiently Learning Structured Distributions from Untrusted Batches

We study the problem, introduced by Qiao and Valiant, of learning from untrusted batches. Here, we assume $m$ users, all of whom have samples from some underlying distribution $p$ over $1, \ldots, n$. Each user sends a batch of $k$ i.i.d. samples from this distribution; however an $\epsilon$-fraction of users are untrustworthy and can send adversarially chosen responses. The goal is then to learn $p$ in total variation distance. When $k = 1$ this is the standard robust univariate density estimation setting and it is well-understood that $\Omega (\epsilon)$ error is unavoidable. Suprisingly, Qiao and Valiant gave an estimator which improves upon this rate when $k$ is large. Unfortunately, their algorithms run in time exponential in either $n$ or $k$. We first give a sequence of polynomial time algorithms whose estimation error approaches the information-theoretically optimal bound for this problem. Our approach is based on recent algorithms derived from the sum-of-squares hierarchy, in the context of high-dimensional robust estimation. We show that algorithms for learning from untrusted batches can also be cast in this framework, but by working with a more complicated set of test functions. It turns out this abstraction is quite powerful and can be generalized to incorporate additional problem specific constraints. Our second and main result is to show that this technology can be leveraged to build in prior knowledge about the shape of the distribution. Crucially, this allows us to reduce the sample complexity of learning from untrusted batches to polylogarithmic in $n$ for most natural classes of distributions, which is important in many applications. To do so, we demonstrate that these sum-of-squares algorithms for robust mean estimation can be made to handle complex combinatorial constraints (e.g. those arising from VC theory), which may be of independent technical interest.Comment: 46 page

Cramer-Rao type integral inequalities for general loss functions

Bayes risk, Cramer-Rao type integral inequality, Hajek-LeCam lower bound, locally asymptotic minimax error, lower bound, 62G05,

Nonparametric Density Estimation for Stochastic Processes from Sampled Data

International audienceLet X m CXt,t>0] be a stationary stochastic process and suppose XQ has a probability density f. Suppose the process X is observed at jump times Cr^.i^l) of a point process [Nt,t>03. Nonpararaetric density estimation of f based on the sampled data £X(r^),l<i<n3 is studied. Asymptotic properties of estimators of delta-family type for f based on [X(Ti),l<i<n) are investigated

ESTIMATION OF THE INTEGRATED SQUARED DENSITY DERIVATIVES BY WAVELETS

The problem of estimation of the integral of the squared derivative of a probability density $f$ is considered using wavelet orthonormal bases. For $f$ such that $f^{(d)}$, the $d$-th derivative belongs to the Sobolev space $H_2^s , s > 0$, we obtain the precise asymptotic expression for the mean integrated squared error of the wavelet estimator

NONPARAMETRIC ESTIMATION OF LINEAR MULTIPLIER FOR STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL LÉVY PROCESS WITH SMALL NOISE

We discuss nonparametric estimation of the linear multiplier in a trend coefficient in models governed by a stochastic differential equation driven by a fractional Lévy process with small noise

Nonparametric Density Estimation for Stochastic Processes from Sampled Data

International audienceLet X m CXt,t>0] be a stationary stochastic process and suppose XQ has a probability density f. Suppose the process X is observed at jump times Cr^.i^l) of a point process [Nt,t>03. Nonpararaetric density estimation of f based on the sampled data £X(r^),l<i<n3 is studied. Asymptotic properties of estimators of delta-family type for f based on [X(Ti),l<i<n) are investigated

ESTIMATION OF THE INTEGRATED SQUARED DENSITY DERIVATIVES BY WAVELETS

The problem of estimation of the integral of the squared derivative of a probability density $f$ is considered using wavelet orthonormal bases. For $f$ such that $f^{(d)}$, the $d$-th derivative belongs to the Sobolev space $H_2^s , s > 0$, we obtain the precise asymptotic expression for the mean integrated squared error of the wavelet estimator