Stochastic Processes (Spectral Representation)

Stochastic process generation based on the Spectral Representation Method which utilizes Power Spectral Density Functions. Correpsonding theory and literature can be found here Stochastic-Process-Generation.

Index

• UncertaintyQuantification.SpectralRepresentation
• Base.names
• UncertaintyQuantification.dimensions
• UncertaintyQuantification.evaluate
• UncertaintyQuantification.sample
• UncertaintyQuantification.to_physical_space!
• UncertaintyQuantification.to_standard_normal_space!

Types and Spectral Representation functions

UncertaintyQuantification.SpectralRepresentation Method

SpectralRepresentation(psd::AbstractPowerSpectralDensity, time::AbstractVector{<:Real}, name::Symbol) -> SpectralRepresentation

Constructs a SpectralRepresentation instance representing a stochastic process generated using the spectral representation method.

Arguments

• psd::AbstractPowerSpectralDensity: An instance of a power spectral density model.

• time::AbstractVector{<:Real}: A vector of time points.

• name::Symbol: A symbol representing the name of the process.

Returns

A SpectralRepresentation instance with the given arguments (parameters).

Example

w = 0:0.1:10
S_0 = 1.0
ω_f = 2.0
ζ_f = 0.05
ω_g = 3.0
ζ_g = 0.1
psd = CloughPenzien(w, S_0, ω_f, ζ_f, ω_g, ζ_g)
t = 0:0.1:10
name = :process1
sr = SpectralRepresentation(psd, t, name)

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UncertaintyQuantification.sample Function

sample(sr::SpectralRepresentation, n::Integer=1) -> DataFrame

Generates samples of random phase angles for a given SpectralRepresentation instance.

Arguments

• sr::SpectralRepresentation: An instance of the SpectralRepresentation struct.

• n::Integer=1: The number of samples to generate (default is 1).

Returns

A DataFrame containing the generated samples of random phase angles.

Example

w = 0:0.1:10
psd = CloughPenzien(w, 1.0, 2.0, 0.05, 3.0, 0.1)
t = 0:0.1:10
name = :process1
sr = SpectralRepresentation(psd, t, name)
samples = sample(sr, 5)

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UncertaintyQuantification.evaluate Method

evaluate(sr::SpectralRepresentation, ϕ::AbstractVector{<:Real}) -> AbstractVector{<:Real}

Evaluates the stochastic process for a given SpectralRepresentation instance and a vector of random phase angles.

Arguments

• sr::SpectralRepresentation: An instance of the SpectralRepresentation struct.

• ϕ::AbstractVector{<:Real}: A vector of random phase angles.

Returns

A vector of real numbers representing the evaluated stochastic process.

Example

w = 0:0.1:10
psd = CloughPenzien(w, 1.0, 2.0, 0.05, 3.0, 0.1)
t = 0:0.1:10
name = :process1
sr = SpectralRepresentation(psd, t, name)
ϕ = rand(Uniform(0, 2π), length(psd.ω))
process_values = evaluate(sr, ϕ)

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UncertaintyQuantification.to_standard_normal_space! Method

to_standard_normal_space!(sr::SpectralRepresentation, df::DataFrame) -> Nothing

Transforms the random phase angles in the given DataFrame from a uniform distribution to a standard normal distribution.

Arguments

• sr::SpectralRepresentation: An instance of the SpectralRepresentation struct.

• df::DataFrame: A DataFrame containing the random phase angles to be transformed.

Returns

Nothing. The DataFrame is modified in place.

Example

w = 0:0.1:10
psd = CloughPenzien(w, 1.0, 2.0, 0.05, 3.0, 0.1)
t = 0:0.1:10
name = :process1
sr = SpectralRepresentation(psd, t, name)
samples = sample(sr, 5)
to_standard_normal_space!(sr, samples)

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UncertaintyQuantification.to_physical_space! Method

to_physical_space!(sr::SpectralRepresentation, df::DataFrame) -> Nothing

Transforms the random phase angles in the given DataFrame from a standard normal distribution to a uniform distribution.

Arguments

• sr::SpectralRepresentation: An instance of the SpectralRepresentation struct.

• df::DataFrame: A DataFrame containing the random phase angles to be transformed.

Returns

Nothing. The DataFrame is modified in place.

Example

w = 0:0.1:10
psd = CloughPenzien(w, 1.0, 2.0, 0.05, 3.0, 0.1)
t = 0:0.1:10
name = :process1
sr = SpectralRepresentation(psd, t, name)
samples = sample(sr, 5)
to_standard_normal_space!(sr, samples)  # Transform to standard normal space
to_physical_space!(sr, samples)         # Transform back to physical space

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UncertaintyQuantification.dimensions Method

dimensions(sr::SpectralRepresentation) -> Int

Returns the number of dimensions (frequencies) in the given SpectralRepresentation instance.

Arguments

• sr::SpectralRepresentation: An instance of the SpectralRepresentation struct.

Returns

An integer representing the number of dimensions (frequencies).

Example

w = 0:0.1:10
psd = CloughPenzien(w, 1.0, 2.0, 0.05, 3.0, 0.1)
t = 0:0.1:10
name = :process1
sr = SpectralRepresentation(psd, t, name)
num_dimensions = dimensions(sr)

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Base.names Method

names(sr::SpectralRepresentation) -> Vector{Symbol}

Returns the names of the random phase angles for a given SpectralRepresentation instance.

Arguments

• sr::SpectralRepresentation: An instance of the SpectralRepresentation struct.

Returns

A vector of symbols representing the names of the random phase angles.

Example

w = 0:0.1:10
psd = CloughPenzien(w, 1.0, 2.0, 0.05, 3.0, 0.1)
t = 0:0.1:10
name = :process1
sr = SpectralRepresentation(psd, t, name)
phase_angle_names = names(sr)

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