Noise Models

We are currently considering three different possible noise models. To understand their difference, consider again how we may have an observation $(x_i, y_i)$, and each $y_i$ may be multi-dimensional. The first question is: Is noise correlated /across/ multiple observations $y_i$ and $y_j$, $i \neq j$?

If there is no correlation between observations, we can use the noise models

UncorrGaussianNoiseModel for (possibly multivariate) Gaussian noise, and
UncorrProductNoiseModel for univariate noise with arbitrary distributions.

If there is correlation between observations, we can use

CorrGaussianNoiseModel and provide a single multivariate normal distribution.

Noise Models

ProbabilisticParameterEstimators.NoiseModel — Type

NoiseModel

Abstract base type for noise models.

Fields:

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ProbabilisticParameterEstimators.UncorrGaussianNoiseModel — Type

UncorrGaussianNoiseModel{DT}

Model Gaussian noise within observations, but without correlation between observations.

Fields:

noisedistributions::AbstractVector{DT} where DT<:(Union{var"#s1", var"#s5"} where {var"#s1"<:Distributions.Normal, var"#s5"<:Distributions.MvNormal}): A vector of (possibly multivariate) Gaussian noise distributions with type DT, one for each observation.

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ProbabilisticParameterEstimators.UncorrProductNoiseModel — Type

UncorrProductNoiseModel{DT}

Model arbitrary noise for univariate observations, without correlation between observations.

Fields:

noisedistributions::AbstractVector{DT} where DT<:Distributions.Distribution{Distributions.Univariate, Distributions.Continuous}: A vector of univariate noise distributions of any kind with type DT. Can not model correlations within a single observation.

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ProbabilisticParameterEstimators.CorrGaussianNoiseModel — Type

CorrGaussianNoiseModel{DT}

Model noise possibly correlated between observations with a single large MvNormal.

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Utility Functions

ProbabilisticParameterEstimators.mvnoisedistribution — Function

mvnoisedistribution(noisemodel)

Construct a samplable noise distribution (e.g. Product or MvNormal).

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ProbabilisticParameterEstimators.covmatrix — Function

covmatrix(noisemodel)

Construct a single N by N covariance matrix for flat vector of observations.

May be a special matrix type such as Diagonal or BlockDiagonal, depending on the noise model.

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Noise Model Examples

## UncorrGaussianNoiseModel
xs = rand(5)
# one (uni- or multivariate) normal distribution per observation
noises = [MvNormal(zeros(2), I) for _ in eachindex(xs)]
noisemodel = UncorrGaussianNoiseModel(noises)
θtrue = [1.0, 2.0]
ysmeas = f.(xs, [θtrue]) .+ rand.(noises)

## UncorrProductNoiseModel
xs = eachcol(rand(2,30))
# one univariate distribution per observation
productnoise = [truncated(0.1*Normal(), 0, Inf) for _ in 1:length(xs)]
noisemodel = UncorrProductNoiseModel(productnoise)
θtrue = [5., 10]
ysmeas = f.(xs, [θtrue]) .+ rand.(productnoise)

## CorrGaussianNoiseModel
xs = 5 * eachcol(rand(2,30))
n = length(xs)
# one large Gaussian relating all observations
corrnoise = MvNormal(zeros(n), I(n) + 1/5*hermitianpart(rand(n, n)))
noisemodel = CorrGaussianNoiseModel(corrnoise)
θtrue = [5., 10]
ysmeas = f.(xs, [θtrue]) .+ rand(corrnoise)