Alternative way to derive the distribution of the multivariate Ornstein–Uhlenbeck process

Vatiwutipong, Pat; Phewchean, Nattakorn

doi:10.1186/s13662-019-2214-1

Cited by 29 publications

(19 citation statements)

References 7 publications

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“…We consider three stocks, A, B, and C, whose price follows a multidimensional Ornstein-Uhlenbeck process, [22]. Using the notation from [22], we can sample the prices by applying the Euler's discretization, X t+Δt μ + e −θΔt (X t − μ) + G. We assume that σ is a unitary matrix, therefore the random vector G follows a normal distribution with mean 0 and covariance (θ + θ T ) − 1 [I − e −(θ+θ T )Δt ]. For this example, we set the parameters to,…”

Section: Trading Examplementioning

confidence: 99%

An Explainable Bayesian Decision Tree Algorithm

Nuti¹,

Rugama²,

Cross³

2021

Front. Appl. Math. Stat.

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Bayesian Decision Trees provide a probabilistic framework that reduces the instability of Decision Trees while maintaining their explainability. While Markov Chain Monte Carlo methods are typically used to construct Bayesian Decision Trees, here we provide a deterministic Bayesian Decision Tree algorithm that eliminates the sampling and does not require a pruning step. This algorithm generates the greedy-modal tree (GMT) which is applicable to both regression and classification problems. We tested the algorithm on various benchmark classification data sets and obtained similar accuracies to other known techniques. Furthermore, we show that we can statistically analyze how was the GMT derived from the data and demonstrate this analysis with a financial example. Notably, the GMT allows for a technique that provides explainable simpler models which is often a prerequisite for applications in finance or the medical industry.

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Section: Trading Examplementioning

confidence: 99%

An Explainable Bayesian Decision Tree Algorithm

Nuti¹,

Rugama²,

Cross³

2021

Front. Appl. Math. Stat.

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“…Firstly we consider an equation with a linear convection term—Ornstein-Uhlenbeck process (OUP) ( Vatiwutipong and Phewchean, 2019 ) in one, three and five dimensions. For the one-dimensional case, which is presented for convention, we only solve equation using the dense format (not TT-format), hence the corresponding results are used to verify the general correctness and convergence properties of the proposed algorithm, but not its efficiency.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…6 in the d -dimensional case with

where

is invertible real matrix,

is the long-term mean,

and

(

). This equation is a well known multivariate OUP with the following properties [see for example ( Singh et al, 2018 ; Vatiwutipong and Phewchean, 2019 )]: • mean vector is

• covariance matrix is

and, in our case as noted above

; • transitional PDF is

• stationary solution is

where matrix

can be found from the following equation

• the (multivariate) OUP at any time is a (multivariate) normal random variable; • the OUP is mean-reverting (the solution tends to its long-term mean

as time t tends to infinity) if all eigenvalues of A are positive (if

in the one-dimensional case). …”

Section: Numerical Examplesmentioning

confidence: 99%

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Solution of the Fokker–Planck Equation by Cross Approximation Method in the Tensor Train Format

Chertkov

Oseledets

2021

Front. Artif. Intell.

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We propose the novel numerical scheme for solution of the multidimensional Fokker–Planck equation, which is based on the Chebyshev interpolation and the spectral differentiation techniques as well as low rank tensor approximations, namely, the tensor train decomposition and the multidimensional cross approximation method, which in combination makes it possible to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases. We demonstrate the effectiveness of the proposed approach on a number of multidimensional problems, including Ornstein-Uhlenbeck process and the dumbbell model. The developed computationally efficient solver can be used in a wide range of practically significant problems, including density estimation in machine learning applications.

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“…with A symmetric and definite positive, we obtain an Ornstein-Uhlenbeck process (Le Gall, 2016)[Chapter 8]. If we choose µ 0 as a Gaussian N (m 0 , Σ 0 ), then we know the Wasserstein gradient flow µ t in closed form (Wibisono, 2018;Vatiwutipong and Phewchean, 2019), for all t > 0, µ t = N (m t , Σ t ) with…”

Section: Relation Between Swgfs and Wgfsmentioning

confidence: 99%

Sliced-Wasserstein Gradient Flows

Bonet¹,

Courty²,

Septier³

et al. 2021

Preprint

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Minimizing functionals in the space of probability distributions can be done with Wasserstein gradient flows. To solve them numerically, a possible approach is to rely on the Jordan-Kinderlehrer-Otto (JKO) scheme which is analogous to the proximal scheme in Euclidean spaces. However, this bilevel optimization problem is known for its computational challenges, especially in high dimension. To alleviate it, very recent works propose to approximate the JKO scheme leveraging Brenier's theorem, and using gradients of Input Convex Neural Networks to parameterize the density (JKO-ICNN). However, this method comes with a high computational cost and stability issues. Instead, this work proposes to use gradient flows in the space of probability measures endowed with the sliced-Wasserstein (SW) distance. We argue that this method is more flexible than JKO-ICNN, since SW enjoys a closedform differentiable approximation. Thus, the density at each step can be parameterized by any generative model which alleviates the computational burden and makes it tractable in higher dimensions. Interestingly, we also show empirically that these gradient flows are strongly related to the usual Wasserstein gradient flows, and that they can be used to minimize efficiently diverse machine learning functionals.

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Alternative way to derive the distribution of the multivariate Ornstein–Uhlenbeck process

Cited by 29 publications

References 7 publications

An Explainable Bayesian Decision Tree Algorithm

An Explainable Bayesian Decision Tree Algorithm

Solution of the Fokker–Planck Equation by Cross Approximation Method in the Tensor Train Format

Sliced-Wasserstein Gradient Flows

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