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Concept# Probability density function

Summary

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would be equal to that sample. Probability density is the probability per unit length, in other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample.
In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values,

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We study the regularity of the probability density function of the supremum of the solution to the linear stochastic heat equation. Using a general criterion for the smoothness of densities for locally nondegenerate random variables, we establish the smoothness of the joint density of the random vector whose components are the solution and the supremum of an increment in time of the solution over an interval (at a fixed spatial position), and the smoothness of the density of the supremum of the solution over a space-time rectangle that touches thet=0 axis. Applying the properties of the divergence operator, we establish a Gaussian-type upper bound on these two densities respectively, which presents a close connection with the Holder-continuity properties of the solution.

In this thesis, we study systems of linear and/or non-linear stochastic heat equations and fractional heat equations in spatial dimension $1$ driven by space-time white noise. The main topic is the study of hitting probabilities for the solutions to these systems.
We first study the properties of the probability density functions of the solution to non-linear systems of stochastic fractional heat equations driven by multiplicative space-time white noise. Using the techniques of Malliavin calculus, we prove that the one-point probability density function of the solution is infinitely differentiable, uniformly bounded and positive everywhere. Moreover, a Gaussian-type upper bound on the two-point probability density function is obtained by a detailed analysis of the small eigenvalues of the Malliavin matrix. We establish an optimal lower bound on hitting probabilities for the (non-Gaussian) solution, which is as sharp as that for the Gaussian solution to a system of linear equations.
We develop a new method to study the upper bound on hitting probabilities, from the perspective of probability density functions. For the solution to the linear stochastic heat equation, we prove that the random vector, which consists of the solution and the supremum of a linear increment of the solution over a time segment, has an infinitely differentiable probability density function. We derive a formula for this density and establish a Gaussian-type upper bound. The smoothness property and Gaussian-type upper bound for the density of the supremum of the solution over a space-time rectangle touching the $t = 0$ axis are also studied. Furthermore, we extend these results to the solutions of systems of linear stochastic fractional heat equations.
For a system of linear stochastic heat equations with Dirichlet boundary conditions, we present a sufficient condition for certain sets to be hit with probability one.

We introduce an online outlier detection algorithm to detect outliers in a sequentially observed data stream. For this purpose, we use a two-stage filtering and hedging approach. In the first stage, we construct a multimodal probability density function to model the normal samples. In the second stage, given a new observation, we label it as an anomaly if the value of aforementioned density function is below a specified threshold at the newly observed point. In order to construct our multimodal density function, we use an incremental decision tree to construct a set of subspaces of the observation space. We train a single component density function of the exponential family using the observations, which fall inside each subspace represented on the tree. These single component density functions are then adaptively combined to produce our multimodal density function, which is shown to achieve the performance of the best convex combination of the density functions defined on the subspaces. As we observe more samples, our tree grows and produces more subspaces. As a result, our modeling power increases in time, while mitigating overfitting issues. In order to choose our threshold level to label the observations, we use an adaptive thresholding scheme. We show that our adaptive threshold level achieves the performance of the optimal prefixed threshold level, which knows the observation labels in hindsight. Our algorithm provides significant performance improvements over the state of the art in our wide set of experiments involving both synthetic as well as real data.

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