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GAS.jl is a Julia package for modeling, forecasting, and simulating time series with generalized autoregressive score models (GAS), also known as score-driven models. Implementations are based on the paper [Generalized Autoregressive Models with Applications](http://dx.doi.org/10.1002/jae.1279) by D. Creal, S. J. Koopman, and A. Lucas.
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ScoreDrivenModels.jl is a Julia package for modeling, forecasting, and simulating time series with generalized autoregressive score models (ScoreDrivenModels), also known as score-driven models. Implementations are based on the paper [Generalized Autoregressive Models with Applications](http://dx.doi.org/10.1002/jae.1279) by D. Creal, S. J. Koopman, and A. Lucas.
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## Water inflow
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Let's model some monthly water inflow data from the Northeast of Brazil using a Lognormal GAS model. Since inflow is a highly seasonal phenomenon, we will utilize lags 1 and 12. The former aims to characterize the short-term evolution of the series, while the latter characterizes the seasonality. The full code is in the examples folder.
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Let's model some monthly water inflow data from the Northeast of Brazil using a Lognormal ScoreDrivenModels model. Since inflow is a highly seasonal phenomenon, we will utilize lags 1 and 12. The former aims to characterize the short-term evolution of the series, while the latter characterizes the seasonality. The full code is in the examples folder.
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```julia
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# Convert data to vector
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y =Vector{Float64}(vec(inflow'))
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# Specify GAS model: here we use lag 1 for trend characterization and
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# Specify ScoreDrivenModels model: here we use lag 1 for trend characterization and
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# lag 12 for seasonality characterization
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gas =GAS.Model([1, 12], [1, 12], LogNormal, 0.0)
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gas =ScoreDrivenModels.Model([1, 12], [1, 12], LogNormal, 0.0)
GAS.jl is a Julia package for modeling, forecasting and simulating time series with or generalized autoregressive score models (GAS) models, also known as dynamic conditional score models (DCS). Implementations are based on the paper [Generalized Autoregressive Models with Applications](http://dx.doi.org/10.1002/jae.1279) by D. Creal, S. J. Koopman and A. Lucas.
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ScoreDrivenModels.jl is a Julia package for modeling, forecasting and simulating time series with or generalized autoregressive score models (ScoreDrivenModels) models, also known as dynamic conditional score models (DCS). Implementations are based on the paper [Generalized Autoregressive Models with Applications](http://dx.doi.org/10.1002/jae.1279) by D. Creal, S. J. Koopman and A. Lucas.
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## Installation
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This package is not yet registered so you can `Pkg.add` it as follows:
Contributions to this package are more than welcome, if you find a bug or have any suggestions for the documentation please post it on the [github issue tracker](https://github.com/LAMPSPUC/GAS.jl/issues).
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Contributions to this package are more than welcome, if you find a bug or have any suggestions for the documentation please post it on the [github issue tracker](https://github.com/LAMPSPUC/ScoreDrivenModels.jl/issues).
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When contributing please note that the package follows the [JuMP style guide](https://www.juliaopt.org/JuMP.jl/stable/style/).
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## Links
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Links are reparametrizations utilized to ensure certain parameter is within its original domain, i.e. in a distribution one would like to ensure that the time varying parameter ``f \in \mathbb{R}^+``. The way to do this is to model ``\tilde{f} = \ln{f}``. More generally one can stablish that ``\tilde{f} = h(f)``. We refer to this procedure as **linking**. When the parameter is linked the GAS recursion happens in the domain of ``\tilde{f}`` and then one can recover the orginal parameter by ``f = \left(h\right)^-1(\tilde f)``. We refer to this procedure as **unlinking**. The new GAS recursion becomes.
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Links are reparametrizations utilized to ensure certain parameter is within its original domain, i.e. in a distribution one would like to ensure that the time varying parameter ``f \in \mathbb{R}^+``. The way to do this is to model ``\tilde{f} = \ln{f}``. More generally one can stablish that ``\tilde{f} = h(f)``. We refer to this procedure as **linking**. When the parameter is linked the ScoreDrivenModels recursion happens in the domain of ``\tilde{f}`` and then one can recover the orginal parameter by ``f = \left(h\right)^-1(\tilde f)``. We refer to this procedure as **unlinking**. The new ScoreDrivenModels recursion becomes.
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```math
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\begin{equation*}\left\{\begin{array}{ccl}
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\end{equation*}
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```
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Notice that the change in parametrization changes the dynamics of the model. The GAS(1,1) for a Normal distribution with inverse scaling ``d = 1`` is equivalent to the GARCH(1, 1) model, but only on the original parameter, if you work with a different parametrization the model is no longer equivalent.
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Notice that the change in parametrization changes the dynamics of the model. The ScoreDrivenModels(1,1) for a Normal distribution with inverse scaling ``d = 1`` is equivalent to the GARCH(1, 1) model, but only on the original parameter, if you work with a different parametrization the model is no longer equivalent.
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### Types of links
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The abstract type `Link` subsumes any type of link that can be expressed.
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