Milstein function #615
Milstein function #615
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| # %% | ||
| # Langevin dynamics is a mathematical model used to describe the behaviour of | ||
| # particles in a fluid, particularly in the context of statistical mechanic | ||
| # and molecular dynamics. It’s named after Paul Langevin a French physicist. |
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| # and molecular dynamics. It’s named after Paul Langevin a French physicist. | |
| # and molecular dynamics. It’s named after Paul Langevin, a French physicist. |
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Why is it named after him? I think the sentence is incomplete without that information.
| # a probability distribution: a 2-d Gaussian mixture. | ||
| # | ||
| # Langevin dynamics enable us to sample from probability distributions from | ||
| # which a non-normalised pdf is known. This is possible thanks to the use |
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| # which a non-normalised pdf is known. This is possible thanks to the use | |
| # which a non-normalised pdf is known. This is possible thanks to the use |
| # of :math:`N` Gaussians of mean :math:`\mu_n` and covarianze matrix | ||
| # :math:`\Sigma_n`. For the sake of simplicity, we will suppose the covarianze |
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| # of :math:`N` Gaussians of mean :math:`\mu_n` and covarianze matrix | |
| # :math:`\Sigma_n`. For the sake of simplicity, we will suppose the covarianze | |
| # of :math:`N` Gaussians of mean :math:`\mu_n` and covariance matrix | |
| # :math:`\Sigma_n`. For the sake of simplicity, we will suppose the covariance |
| # | ||
| # .. math:: | ||
| # | ||
| # \nabla_x \log p(\mathbf{x}). |
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Why it does suddenly take a vector as input?
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It should be always a vector. I will change it.
| # \nabla_\mathbf{x} \log p(\mathbf{x}) = \nabla_\mathbf{x} \log q(\mathbf{x}) | ||
| # - \nabla_\mathbf{x} \log Z = \nabla_\mathbf{x} \log q(\mathbf{x}), | ||
| # | ||
| # which is known. Once we know the score function, we can sample from the |
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I think it is better to split the paragraph here.
| # which is known. Once we know the score function, we can sample from the | |
| # which is known. | |
| # | |
| # Once we know the score function, we can sample from the |
| ... ) | ||
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| References: | ||
| Paromita Banerjee, Brynjarsdottir Jenny, Gurarie David, and |
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For references we are using the sphinxcontrib-bibtex functionality, in particular its footbibliography functionality.
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| References: | ||
| Paromita Banerjee, Brynjarsdottir Jenny, Gurarie David, and | ||
| Albert Jeffrey. Numerical Methods for Stochastic Differential Equations |
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It is better to reference an article (if there is any) than a thesis, both in term of correctness (an article has been peer-reviewed, while in a thesis there can be less scrutiny) and easiness of finding the relevant information (an article is much shorter and to-the-point).
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We haven't found any article about the Milstein L=0 method apart from this thesis. Alberto has checked the thesis and he has told me that it is probably correct as the institution has high quality standards. However, there is almost no literature about the Milstein method for multidimensional data. Most people use it for one-dimensional SDEs.
| with lower order of convergence. | ||
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| Examples: | ||
| Example of the use of milstein for a 1-d Geometric Brownian motion |
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| Example of the use of milstein for a 1-d Geometric Brownian motion | |
| Example of the use of Milstein's method for a 1-d Geometric Brownian motion |
| and Postintervention in Structural Equation Models. PhD thesis, USA, | ||
| 2020. AAI29605510 | ||
| """ | ||
| random_state = validate_random_state(random_state) |
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I do not have the bandwidth now to check for code correctness in depth. I will assume that Alberto has reviewed the code and tests with you thoroughly. If that is not true, please let me now to recheck it later before merging.
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| """Tests of milstein.""" | |||
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| """Tests of milstein.""" | |
| """Tests of Milstein's method.""" |
References to issues or other PRs
Describe the proposed changes
Added milstein function which computes the Milstein scheme for Stochastic Differential Equations (SDEs). Additionally, examples which show the functionality of the SDE solvers are added.
Additional information
Checklist before requesting a review