BayesianNonparametricStatistics.jl is a Julia package to sample from nonparametric posteriors with data from a diffusion process (SDE). You can simulate SDEs, simulate Gaussian processes, calculate the posterior associated with diffusion process and a Gaussian process prior.

This package works with Julia 0.6, 0.7 and any version 1.X.

When you use any version of Julia 1 press ] and copy-paste

`add BayesianNonparametricStatistics`

alternatively you can use

```
using Pkg
Pkg.add("BayesianNonparametricStatistics")
```

When using Julia 0.6, execute the following code:

`Pkg.clone("https://github.com/Jan-van-Waaij/BayesianNonparametricStatistics.jl.git", "BayesianNonparametricStatistics")`

when using Julia 0.7, execute

```
using Pkg
Pkg.clone("https://github.com/Jan-van-Waaij/BayesianNonparametricStatistics.jl.git", "BayesianNonparametricStatistics")
```

After installation, type the following in your Julia script, or in a Julia REPL.

` using BayesianNonparametricStatistics`

to use the package.

Sample from an SDE dX_t=sin(2\pi X_t)dt+dW_t:

```
using BayesianNonparametricStatistics, Plots
# implement SDE dX_t=sin(2\pi X_t)dt+dW_t,
# starting at zero till time 1000.0, discretised
# with precision 0.01.
model = SDEModel(1.0,0.0,1000.0,0.01)
sde = SDE(x->sinpi(2*x),model)
# Sample from sde.
X = rand(sde)
# Plot X.
plot(X.timeinterval, X.samplevalues)
```

To recover the drift function, using Gaussian process posterior: (the code is for Julia 0.7 or 1.X. When using 0.6 leave the line "using LinearAlgebra, SparseArrays" out and replace "Diagonal" in the second line by "diagm")

```
using BayesianNonparametricStatistics, LinearAlgebra, SparseArrays, Plots
distribution = GaussianVector(sparse(Diagonal([k^(-1.0) for k in 1.0:50.0])))
Π = GaussianProcess([fourier(k) for k in 1:50], distribution)
postΠ = calculateposterior(Π, X, model)
# sample 10 times from posterior
plot()
x = 0.0:0.01:1.0
for k in 1:10
f = rand(postΠ)
y = f.(x)
plot!(x,y,show=true)
end
```

Julia 1.X now also works with the Distributions package:

```
using BayesianNonparametricStatistics, LinearAlgebra, SparseArrays, Plots, Distributions
distribution = MvNormal([k^(-1.0) for k in 1.0:50.0])
Π = GaussianProcess([fourier(k) for k in 1:50], distribution)
postΠ = calculateposterior(Π, X, model)
# sample 10 times from posterior
plot()
x = 0.0:0.01:1.0
for k in 1:10
f = rand(postΠ)
y = f.(x)
plot!(x,y,show=true)
end
```

Go to the Wiki.

https://github.com/Jan-van-Waaij/BayesianNonparametricStatistics.jl.git

The BayesianNonparametricStatistics.jl package is licensed under the MIT "Expat" License:

Copyright (c) 2017-2021: Jan van Waaij.

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.