Docstrings

AdditiveTimeSeriesModel(model1, model2)

Constructs an additive model from two timeseries models.

BartlettPeriodogram(timeseries, Δ, segmentlength)

Compute the Bartlett periodogram for the provided timeseries with sampling rate Δ.

Arguments

• timeseries: A Vector if univariate and an n by d Matrix if multivariate, where n is the number of observations and d is the dimension of the timeseries.
• Δ: A positive real number.
• segmentlength: the length of series used in each segment.

Computes an estimate of the spectral density function using Bartlett's method. Using the same normalisation as Periodogram.

External links

• Bartlett's method on Wikipedia

DebiasedWhittleData(model::Type{<:TimeSeriesModel{D}}, timeseries, Δ; lowerΩcutoff, upperΩcutoff, taper)

Create storage for timeseries data in a format which is useful for debiased Whittle methods.

Periodogram(timeseries, Δ)

Compute the periodogram for the provided timeseries with sampling rate Δ.

Arguments

• timeseries: A Vector if univariate and an n by d Matrix if multivariate, where n is the number of observations and d is the dimension of the timeseries.
• Δ: A positive real number.

The periodogram is defined as

\[\boldsymbol I(ω)&=\boldsymbol J(ω) \boldsymbol J(ω)^H \quad \text{where} \quad \boldsymbol J(ω) = \sqrt{\frac{Δ}{2π n}}\sum_{t=0}^{n-1} \boldsymbol{P}_{tΔ}e^{-itΔ ω}\]

Note the periodogram is in terms of angular frequency here, and uses the normalisation $Δ/2π$. The choice of normalisation is essentially arbitrary; however, this matches our definition for the spectral density function.

Time series model with a parameter for the dimension of the timeseries model.

TimeSeriesModelStorage

Preallocated storage for model manipulation including preplanned ffts.

Sub type to specify the need for additional routines approximating the acv from the sdf. Such routines require additional storage and have different default functions in some cases.

WhittleData(model::Type{<:TimeSeriesModel{D}}, timeseries, Δ; lowerΩcutoff, upperΩcutoff, taper)

Create storage for timeseries data in a format which is useful for Whittle methods.

M₁::Type{<:TimeSeriesModel{D,T}} + M₂::Type{<:TimeSeriesModel{D,T}} -> AdditiveTimeSeriesModel{M₁,M₂,D,T}

M₁::Type{<:TimeSeriesModel{D,T}} + M₂::Type{<:TimeSeriesModel{D,T}} -> Type{AdditiveTimeSeriesModel{M₁,M₂,D,T}}

ndims(::TimeSeriesModel) -> Integer
ndims(::Type{<:TimeSeriesModel}) -> Integer

Return the dimension of a timeseries model.

EI!(store, model::TimeSeriesModel)

Compute the expected periodogram and assign to appropriate place in memory (note this computes at twice the desired resolution).

EI(model::TimeSeriesModel, n, Δ)

Compute EI at Fourier frequencies fftshift(fftfreq(n,2π/Δ)).

Note internal computation provides values at twice the resolution, this function returns at the desired resolution.

acv!(out, model::TimeSeriesModel, τ)

Compute the acv for a single lag and when acv is known.

acv!(store::Acv2EIStorage, model::TimeSeriesModel, encodedtime::LagsEI)

Compute the acv at many lags and allocate to storage when acv is known.

acv!(store::AdditiveStorage, model::AdditiveTimeSeriesModel)

Compute the acv for an additive model.

Processes storage and model, and computes acv for each separately, then combines and stores in the leftmost storage. In other words, the sum of both autocovariances is stores in store 1. This is preferable as for some models the acv may be known.

acv!(store::Sdf2EIStorage, model::UnknownAcvTimeSeriesModel, encodedtime::FreqAcvEst)

Approximate the acv and allocate to storage when acv is unknown.

acv!(store::TimeSeriesModelStorageFunction, model::TimeSeriesModel)

Unwrap storage and pass to lower level acv! call.

acv!(store::Acv2EIStorageUni, model::TimeSeriesModel{1,T}, encodedtime::LagsEI)

Compute the acv at many lags and allocate to storage when acv is known in the univariate case.

acv!(store::Sdf2EIStorageUni, model::UnknownAcvTimeSeriesModel{1,T}, encodedtime::FreqAcvEst)

Approximate the acv and allocate to storage when acv is known in the univariate case.

acv(model::TimeSeriesModel, n, Δ)

Compute acv at lags -(n-1)*Δ:Δ:(n-1)*Δ.

acv(model::TimeSeriesModel, τ)

Compute acv at τ, provided acv is known.

add_asdf!(out, model::TimeSeriesModel, ω, Δ)

Add the aliased sdf of a model to a storage vector.

add_sdf!(out, model::TimeSeriesModel, ω)

Add the sdf of a model to a storage vector.

asdf!(out, model::TimeSeriesModel, ω, Δ)

Computes the asdf and overwrites a storage vector.

asdf!(store::TimeSeriesModelStorage, model::TimeSeriesModel, freq::FreqAcvEst)

Compute the asdf for all frequencies and allocates to appropriate location in storage.

asdf(model::TimeSeriesModel, ω, Δ)

Compute asdf with sampling rate Δ at ω.

checkparameterlength(x,model::Type{<:TimeSeriesModel})

checks if the parameter vector is the correct length for the given model.

checkposreal(x)

Checks if x is a positive real number.

coherance(model::TimeSeriesModel, ω)

Compute the coherance of a given model at frequency ω.

coherancy(model::TimeSeriesModel, ω)

Compute the coherancy of a given model at frequency ω.

debiasedwhittle!(store::TimeSeriesModelStorage, model::TimeSeriesModel, data::GenWhittleData)

Function to compute the debiased Whittle likelihood using a preallocated store.

debiasedwhittle_Ehess!(EH, store, data::GenWhittleData)

Compute the expected hessian of the de-biased Whittle likelihood.

debiasedwhittle_FG!(F, G, store, model::TimeSeriesModel, data::GenWhittleData)

Compute the debiased Whittle likelihood and its gradient using a preallocated store.

debiasedwhittle_FGH!(F, G, H, store, model::TimeSeriesModel, data::GenWhittleData)

Compute the debiased Whittle likelihood and its gradient and hessian using a preallocated store.

debiasedwhittle_Fisher!(F, G, H, store, model::TimeSeriesModel, data::GenWhittleData)

Compute the debiased Whittle likelihood, its gradient and fisher information.

extract_EI(store::Sdf2EIStorage)
extract_EI(store::Acv2EIStorage)
extract_EI(store::Sdf2EIStorageUni)
extract_EI(store::Acv2EIStorageUni)

Extract the EI from general storage.

extract_acv(store::Sdf2EIStorage)
extract_acv(store::Acv2EIStorage)
extract_acv(store::Sdf2EIStorageUni)
extract_acv(store::Acv2EIStorageUni)

Extract the acv from general storage.

extract_array(store::Sdf2EIStorage)
extract_array(store::Acv2EIStorage)
extract_array(store::Sdf2EIStorageUni)
extract_array(store::Acv2EIStorageUni)

Extract array from storage.

fit(ts,Δ;model::Type{<:TimeSeriesModel},x₀,lowerΩcutoff,upperΩcutoff,x_lowerbounds,x_upperbounds,method,taper)
fit(timeseries::TimeSeries;model::Type{<:TimeSeriesModel},x₀,lowerΩcutoff,upperΩcutoff,x_lowerbounds,x_upperbounds,method,taper)

Fit a time series model using the IPNewton method from Optim.jl.

Arguments

• ts: n by D matrix containing the timeseries (or vector if D=1), where n is the number of observations and D is the number of series.
• Δ: The sampling rate, which should be a positive real number.
• timeseries: Can be provided in place of ts and Δ.
• model: The model which will be fitted. Should be a type (not a realisation of the model) e.g. JONSWAP{k} not JONSWAP{K}(x).
• x₀: The initial parameter guess.
• lowerΩcutoff: The lower cutoff for the frequency range to be used in fitting. Default is 0.
• upperΩcutoff: The upper cutoff for the frequency range to be used in fitting. Default is Inf.
• x_lowerbounds: The lower bounds on the parameter space. If nothing is provided (the default) then these are set to default values based on the model.
• x_upperbounds: The upper bounds on the parameter space. If nothing is provided (the default) then these are set to default values based on the model.
• method: Either :Whittle or :debiasedWhittle.
• taper: The choice of tapering to be used. This should be nothing (in which case no taper is used) or dpss_nw where nw time-bandwith product (see DSP.dpss for more details).
• options: Options for optimisation of type Optim.Options.

getallderiv(store::AdditiveStorage)
getallderiv(store::TimeSeriesModelStorageGradient)

Extract all of the derivatives from a generic store.

grad_EI!(store::TimeSeriesModelStorageGradient, model::TimeSeriesModel)

Compute the gradient of the expected periodogram and assign to appropriate place in memory (note this computes at twice the desired resolution).

grad_acv!

Compute the gradient of the acv and allocate to storage as appropriate.

grad_acv(model::TimeSeriesModel, τ)

Compute the gradient of the acv at τ.

grad_add_asdf!(out, model::TimeSeriesModel, ω, Δ)

Add the gradient of the aliased sdf of a model to an input storage vector.

grad_add_sdf!(out, model::TimeSeriesModel, ω)

Add the gradient of the sdf of a model to an input storage vector.

grad_asdf!(out, model::TimeSeriesModel, ω, Δ)

Compute the gradient of the aliased sdf of a model and store in storage vector.

grad_asdf!(store::TimeSeriesModelStorage, model::TimeSeriesModel, freq::FreqAcvEst)
grad_asdf!(store::TimeSeriesModelStorageGradient, model::TimeSeriesModel)

Compute the gradient of the asdf for all frequencies and allocate to appropriate location in storage.

grad_sdf(model::TimeSeriesModel, ω)

Compute the gradient of the sdf at ω.

groupdelay(model::TimeSeriesModel, ω)

Compute the groupdelay of a given model at frequency ω.

hess_EI!(store::TimeSeriesModelStorageHessian, model::TimeSeriesModel)

Compute the Hessian of the expected periodogram and assign to appropriate place in memory (note this computes at twice the desired resolution).

hess_acv!

Compute the Hessian of the acv and allocate to storage as appropriate.

hess_add_asdf!(out, model::Type{<:TimeSeriesModel}, ω, Δ)

Add the Hessian of the aliased sdf of a model to an input storage vector.

hess_add_sdf!(out, model::TimeSeriesModel, ω)

Add the Hessian of the sdf of a model to an input storage vector.

hess_asdf!(out, model::TimeSeriesModel, ω, Δ)

Compute the Hessian of the aliased sdf of a model and store in storage vector.

hess_asdf!(store::TimeSeriesModelStorage, model::TimeSeriesModel, freq::FreqAcvEst)
hess_asdf!(store::TimeSeriesModelStorageHessian, model::TimeSeriesModel)

Compute the Hessian of the asdf for all frequencies and allocate to appropriate location in storage.

hess_sdf(model::TimeSeriesModel, ω)

Compute the Hessian of the sdf at ω.

indexLT(i,j,d)

Return the lower triangle index for an elements of a symmetric matrix assuming the lower triangle is encoded down the columns.

minbins(model::TimeSeriesModel)

Returns the minimum number of bins required for a good approximation of the sdf from the acv for a given model.

nalias(model::TimeSeriesModel)

Returns the number of times a sdf should be alised for a given model.

nlowertriangle_dimension(::TimeSeriesModel) -> Integer
nlowertriangle_dimension(::Type{<:TimeSeriesModel}) -> Integer

Return the number of elements in the lower triangle the spectral density matrix function of a timeseries model.

nlowertriangle_parameter(::TimeSeriesModel) -> Integer
nlowertriangle_parameter(::Type{<:TimeSeriesModel}) -> Integer

Return the number of elements in the lower triangle of the hessian matrix for a model with given number of parameters.

parameternames(::TimeSeriesModel)
parameternames(::Type{<:TimeSeriesModel})

Return the parameter names for a given timeseries model.

plotacv(model)
plotacv!(model)
plotacv(model, τ)
plotacv!(model, τ)
plotacv(model, n, Δ)
plotacv!(model, n, Δ)

Plot the aliased spectral density function of a model at lags τ. If the model does not have known acv, then the number of lags n and sampling rate Δ should be provided. In this case, the lags are -(n-1)*Δ:Δ:(n-1)*Δ. If unprovided, n = 30 and Δ = 1.

plotasdf(model)
plotasdf!(model)
plotasdf(model, Ω, Δ)
plotasdf!(model, Ω, Δ)

Plot the aliased spectral density function of a model at the frequencies Ω. By default, Ω = range(-π,π,length=200) and Δ = 1.

plotcoh(model)
plotcoh!(model)
plotcoh(model, Ω)
plotcoh!(model, Ω)

Plot the coherance and group delay of a model at the frequencies Ω. If multivariate, above the diagonal is the coherance, and below is the group delay and on the diagonal is the spectral density function. By default, Ω = range(-π,π,length=200) and Δ = 1.

plotei(model)
plotei!(model)
plotei(model, n, Δ)
plotei!(model, n, Δ)

Plot the aliased spectral density function of a model at the angular Fourier frequencies fftfreq(n,2π/Δ). By default, n = 1024 and Δ = 1.

plotsdf(model)
plotsdf!(model)
plotsdf(model, Ω)
plotsdf!(model, Ω)

Plot the spectral density function of a model at the frequencies Ω. By default, Ω = range(-π,π,length=200) and Δ = 1.

sdf(model::TimeSeriesModel, ω)

Compute sdf at ω.

whittle!(store, model::TimeSeriesModel, data::GenWhittleData)

Compute the Whittle likelihood using a preallocated store.

whittle_FG!(F, G, store, model::TimeSeriesModel, data::GenWhittleData)

Compute the Whittle likelihood and its gradient using a preallocated store.

whittle_FGH!(F, G, H, store, model::TimeSeriesModel, data::GenWhittleData)

Compute the debiased Whittle likelihood and its gradient and hessian using a preallocated store.