From 31c51549474f25f8a53b7a41c40d5b13876fb05a Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Thu, 3 Oct 2024 16:42:52 +0000 Subject: [PATCH] build based on d2ced26 --- previews/PR329/examples/index.html | 1284 ++++++++++++++-------------- previews/PR329/index.html | 2 +- previews/PR329/manual/index.html | 284 +++--- previews/PR329/search/index.html | 2 +- 4 files changed, 786 insertions(+), 786 deletions(-) diff --git a/previews/PR329/examples/index.html b/previews/PR329/examples/index.html index 003fa66..6a20b0b 100644 --- a/previews/PR329/examples/index.html +++ b/previews/PR329/examples/index.html @@ -4,81 +4,81 @@ plt = plot(nile.year, nile.flow, label = "Nile river annual flow") - + - - + - - + - - - - - - - - - - - - - - - - - - - - - - - +

Next, we fit a LocalLevel model:

model = LocalLevel(nile.flow)
 fit!(model)
LocalLevel

We can analyze the filtered estimates for the level of the annual flow:

filter_output = kalman_filter(model)
 plot!(plt, nile.year, get_filtered_state(filter_output), label = "Filtered level")
- + - - + - - + - - - - - - - - - - - - - - - - - - - - - - - - - +

We can do the same for the smoothed estimates for the level of the annual flow:

smoother_output = kalman_smoother(model)
 plot!(plt, nile.year, get_smoothed_state(smoother_output), label = "Smoothed level")
- + - - + - - + - - - - - - - - - - - - - - - - - - - - - - - - - - - +

StateSpaceModels.jl can also be used to obtain forecasts. Here we forecast 10 steps ahead:

steps_ahead = 10
 dates = collect(nile.year[end] + Year(1):Year(1):nile.year[end] + Year(10))
 forec = forecast(model, 10)
@@ -359,81 +359,81 @@
 plot!(plt, dates, expected_value, label = "Forecast")
- + - - + - - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - +

We can also simulate multiple scenarios for the forecasting horizon based on the estimated predictive distributions.

scenarios = simulate_scenarios(model, 10, 100)
 plot!(plt, dates, scenarios[:, 1, :], label = "", color = "grey", width = 0.2)
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The package also handles missing values automatically. To that end, the package considers that any NaN entries in the observations are missing values.

nile.flow[[collect(21:40); collect(61:80)]] .= NaN
 plt = plot(nile.year, nile.flow, label = "Annual nile river flow")
- + - - + - - + - - - - - - - - - - - - - - - - - - - - - - - - - +

Even though the series has several missing values, the same analysis is possible:

model = LocalLevel(nile.flow)
 fit!(model)
LocalLevel

And the exact same code can be used for filtering and smoothing:

filter_output = kalman_filter(model)
 smoother_output = kalman_smoother(model)
@@ -1145,96 +1145,96 @@
 plot!(plt, nile.year, get_smoothed_state(smoother_output), label = "Smoothed level")
- + - - + - - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - +

Airline passengers

We often write the model SARIMA model as an ARIMA $(p,d,q) \times (P,D,Q,s)$, where the lowercase letters indicate the specification for the non-seasonal component, and the uppercase letters indicate the specification for the seasonal component; $s$ is the periodicity of the seasons (e.g. it is often 4 for quarterly data or 12 for monthly data). The data process can be written generically as

\[\begin{equation} \phi_p (L) \tilde \phi_P (L^s) \Delta^d \Delta_s^D y_t = A(t) + \theta_q (L) \tilde \theta_Q (L^s) \epsilon_t \end{equation}\]

where

  • $\phi_p (L)$ is the non-seasonal autoregressive lag polynomial,
  • $\tilde \phi_P (L^s)$ is the seasonal autoregressive lag polynomial,
  • $\Delta^d \Delta_s^D y_t$ is the time series, differenced $d$ times, and seasonally differenced $D$ times.,
  • $A(t)$ is the trend polynomial (including the intercept),
  • $\theta_q (L)$ is the non-seasonal moving average lag polynomial,
  • $\tilde \theta_Q (L^s)$ is the seasonal moving average lag polynomial

sometimes we rewrite this as:

\[\begin{equation} @@ -1306,81 +1306,81 @@

- + - - + - - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - +

The text from this example is based on Python`s statsmodels library. The estimates for this example match up to the 3th decimal place the results of the paper State Space Methods in Ox/SsfPack from the journal of statistical software.

Finland road traffic fatalities

In this example, we will follow what is illustrated on Commandeur, Jacques J.F. & Koopman, Siem Jan, 2007. "An Introduction to State Space Time Series Analysis," OUP Catalogue, Oxford University Press (Chapter 3). We will study the LocalLinearTrend model with a series of the log of road traffic fatalities in Finalnd and analyse its slope to tell if the trend of fatalities was increasing or decrasing during different periods of time.

using StateSpaceModels, Plots, CSV, DataFrames
 df = CSV.File(StateSpaceModels.VEHICLE_FATALITIES) |> DataFrame
 log_ff = log.(df.ff)
 plt = plot(df.date, log_ff, label = "Log of Finland road traffic fatalities")
- + - - + - - + - - - - - - - - - - - - - - - - - - - - - - - - - +

We fit a LocalLinearTrend

model = LocalLinearTrend(log_ff)
 fit!(model)
LocalLinearTrend

By extracting the smoothed slope we conclude that according to our model the trend of fatalities in Finland was increasing in the years 1970, 1982, 1984 through to 1988, and in 1998

smoother_output = kalman_smoother(model)
 plot(df.date, get_smoothed_state(smoother_output)[:, 2], label = "slope")
- + - - + - - + - - - - - - - - - - - - - - - - - - - - - - - - - +

Vehicle tracking

This example illustrates how to perform vehicle tracking from noisy data.

using Random
 Random.seed!(1)
 
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diff --git a/previews/PR329/index.html b/previews/PR329/index.html
index 737ed69..db53e8b 100644
--- a/previews/PR329/index.html
+++ b/previews/PR329/index.html
@@ -10,4 +10,4 @@ 
Star us on GitHub!

 author={Raphael Saavedra and Guilherme Bodin and Mario Souto},
 journal={arXiv preprint arXiv:1908.01757},
 year={2019}
-}
+} diff --git a/previews/PR329/manual/index.html b/previews/PR329/manual/index.html index d476caf..7b223d8 100644 --- a/previews/PR329/manual/index.html +++ b/previews/PR329/manual/index.html @@ -67,12 +67,12 @@ c_{t+1} &= \rho_c \left(c_{t} \cos(\lambda_c) + c_{t}^{*} \sin(\lambda_c)\right) \quad & \tilde\omega_{t} \sim \mathcal{N}(0, \sigma^2_{\tilde\omega})\\ c_{t+1}^{*} &= \rho_c \left(-c_{t} \sin(\lambda_c) + c_{t}^{*} \sin(\lambda_c)\right) \quad &\tilde\omega^*_{t} \sim \mathcal{N}(0, \sigma^2_{\tilde\omega})\\ \end{aligned} -\end{gather*}\]

The cyclical component is intended to capture cyclical effects at time frames much longer than captured by the seasonal component. The parameter $\lambda_c$ is the frequency of the cycle and it is estimated via maximum likelihood. The inclusion of error terms allows the cycle effects to vary over time. The modelling options can be expressed in terms of "deterministic" or "stochastic" and the damping effect as a string, i.e., cycle = "stochastic", cycle = "deterministic" or cycle = "stochastic damped".

The UnobservedComponents model has some dedicated Plot Recipes, see Visualization

References

  • Durbin, James, & Siem Jan Koopman. Time Series Analysis by State Space Methods: Second Edition. Oxford University Press, 2012
source
StateSpaceModels.ExponentialSmoothing — Type
ExponentialSmoothing(
+\end{gather*}\]
The cyclical component is intended to capture cyclical effects at time frames much longer  than captured by the seasonal component. The parameter $\lambda_c$ is the frequency of the cycle and it is estimated via maximum likelihood. The inclusion of error terms allows the cycle effects to vary over time. The modelling options can be expressed in terms of "deterministic" or "stochastic" and the damping effect as a string, i.e.,  cycle = "stochastic", cycle = "deterministic" or cycle = "stochastic damped".
The UnobservedComponents model has some dedicated Plot Recipes, see Visualization
References
  • Durbin, James, & Siem Jan Koopman. Time Series Analysis by State Space Methods: Second Edition.  Oxford University Press, 2012
source
StateSpaceModels.ExponentialSmoothing — Type
ExponentialSmoothing(
     y::Vector{Fl}; 
     trend::Bool = false,
     damped_trend::Bool = false,
     seasonal::Int = 0
-) where Fl

Linear exponential smoothing models. These models are also known as ETS in the literature. This model is estimated using the Kalman filter for linear Gaussian state space models, for this reason the possible models are the following ETS with additive errors:

  • ETS(A, N, N)
  • ETS(A, A, N)
  • ETS(A, Ad, N)
  • ETS(A, N, A)
  • ETS(A, A, A)
  • ETS(A, Ad, A)

Other softwares have use the augmented least squares approach and have all the possible ETS combinations. The Kalman filter approach might be slower than others but have the advantages of filtering the components.

References

  • Hyndman, Rob, Anne B. Koehler, J. Keith Ord, and Ralph D. Snyder. Forecasting with exponential smoothing: the state space approach. Springer Science & Business Media, 2008.
  • Hyndman, Robin John; Athanasopoulos, George. Forecasting: Principles and Practice. 2nd ed. OTexts, 2018.
source
StateSpaceModels.SARIMA — Type
SARIMA(
+) where Fl

Linear exponential smoothing models. These models are also known as ETS in the literature. This model is estimated using the Kalman filter for linear Gaussian state space models, for this reason the possible models are the following ETS with additive errors:

  • ETS(A, N, N)
  • ETS(A, A, N)
  • ETS(A, Ad, N)
  • ETS(A, N, A)
  • ETS(A, A, A)
  • ETS(A, Ad, A)

Other softwares have use the augmented least squares approach and have all the possible ETS combinations. The Kalman filter approach might be slower than others but have the advantages of filtering the components.

References

  • Hyndman, Rob, Anne B. Koehler, J. Keith Ord, and Ralph D. Snyder. Forecasting with exponential smoothing: the state space approach. Springer Science & Business Media, 2008.
  • Hyndman, Robin John; Athanasopoulos, George. Forecasting: Principles and Practice. 2nd ed. OTexts, 2018.
source
StateSpaceModels.SARIMA — Type
SARIMA(
     y::Vector{Fl}; 
     order::Tuple{Int,Int,Int} = (1, 0, 0), 
     seasonal_order::Tuple{Int, Int, Int, Int} = (0, 0, 0, 0),
@@ -88,13 +88,13 @@
     \phi_p (L) \tilde \phi_P (L^s) \Delta^d \Delta_s^D u_t & = A(t) + \theta_q (L) \tilde \theta_Q (L^s) \zeta_t
     \end{aligned}
 \end{gather*}\]
Example
julia> model = SARIMA(rand(100); order=(1,1,1), seasonal_order=(1,2,3,12))
-SARIMA(1, 1, 1)x(1, 2, 3, 12) model
See more on Airline passengers
References
  • Durbin, James, & Siem Jan Koopman. Time Series Analysis by State Space Methods: Second Edition.  Oxford University Press, 2012
source
StateSpaceModels.DAR — Type
DAR(y::Vector{Fl}, lags::Int) where Fl

A Dynamic Autorregressive model is defined by:

\[\begin{gather*} +SARIMA(1, 1, 1)x(1, 2, 3, 12) model

See more on Airline passengers

References

  • Durbin, James, & Siem Jan Koopman. Time Series Analysis by State Space Methods: Second Edition. Oxford University Press, 2012
source
StateSpaceModels.DAR — Type
DAR(y::Vector{Fl}, lags::Int) where Fl

A Dynamic Autorregressive model is defined by:

\[\begin{gather*} \begin{aligned} y_{t} &= \phi_0 + \sum_{i=1}^{lags} \phi_{i, t} y_{t - i} + \varepsilon_{t} \quad \varepsilon_{t} \sim \mathcal{N}(0, \sigma^2_{\varepsilon})\\ \phi_{i, t+1} &= \phi_{i, t} + \eta{i, t} \quad \eta{i, t} \sim \mathcal{N}(0, \sigma^2_{i, \eta})\\ \end{aligned} \end{gather*}\]

!!!! note System matrices When building the system matrices we get rid of the first lags observations in order to build all system matrices respecting the correspondent timestamps

!!! warning Forecasting The dynamic autorregressive model does not have the forecast method implemented yet. If you wish to perform forecasts with this model please open an issue.

!!! warning Missing values The dynamic autorregressive model currently does not support missing values (NaN observations.)

Example

julia> model = DAR(randn(100), 2)
-DAR
source
StateSpaceModels.BasicStructural — Type
BasicStructural(y::Vector{Fl}, s::Int) where Fl

The basic structural state-space model consists of a trend (level + slope) and a seasonal component. It is defined by:

\[\begin{gather*} +DAR

source
StateSpaceModels.BasicStructural — Type
BasicStructural(y::Vector{Fl}, s::Int) where Fl

The basic structural state-space model consists of a trend (level + slope) and a seasonal component. It is defined by:

\[\begin{gather*} \begin{aligned} y_{t} &= \mu_{t} + \gamma_{t} + \varepsilon_{t} \quad &\varepsilon_{t} \sim \mathcal{N}(0, \sigma^2_{\varepsilon})\\ \mu_{t+1} &= \mu_{t} + \nu_{t} + \xi_{t} \quad &\xi_{t} \sim \mathcal{N}(0, \sigma^2_{\xi})\\ @@ -102,7 +102,7 @@ \gamma_{t+1} &= -\sum_{j=1}^{s-1} \gamma_{t+1-j} + \omega_{t} \quad & \omega_{t} \sim \mathcal{N}(0, \sigma^2_{\omega})\\ \end{aligned} \end{gather*}\]

Example

julia> model = BasicStructural(rand(100), 12)
-BasicStructural

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press.
source
StateSpaceModels.BasicStructuralExplanatory — Type
BasicStructuralExplanatory(y::Vector{Fl}, s::Int, X::Matrix{Fl}) where Fl

It is defined by:

\[\begin{gather*} +BasicStructural

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press.
source
StateSpaceModels.BasicStructuralExplanatory — Type
BasicStructuralExplanatory(y::Vector{Fl}, s::Int, X::Matrix{Fl}) where Fl

It is defined by:

\[\begin{gather*} \begin{aligned} y_{t} &= \mu_{t} + \gamma_{t} + \beta_{t, i}X_{t, i} \varepsilon_{t} \quad &\varepsilon_{t} \sim \mathcal{N}(0, \sigma^2_{\varepsilon})\\ \mu_{t+1} &= \mu_{t} + \nu_{t} + \xi_{t} \quad &\xi_{t} \sim \mathcal{N}(0, \sigma^2_{\xi})\\ @@ -111,7 +111,7 @@ \beta_{t+1} &= \beta_{t} \end{aligned} \end{gather*}\]

Example

julia> model = BasicStructuralExplanatory(rand(100), 12, rand(100, 2))
-BasicStructuralExplanatory

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press.
source
StateSpaceModels.LinearRegression — Type
LinearRegression(X::Matrix{Fl}, y::Vector{Fl}) where Fl

The linear regression state-space model is defined by:

\[\begin{gather*} +BasicStructuralExplanatory

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press.
source
StateSpaceModels.LinearRegression — Type
LinearRegression(X::Matrix{Fl}, y::Vector{Fl}) where Fl

The linear regression state-space model is defined by:

\[\begin{gather*} \begin{aligned} y_{t} &= X_{1,t} \cdot \beta_{1,t} + \dots + X_{n,t} \cdot \beta_{n,t} + \varepsilon_{t} \quad &\varepsilon_{t} \sim \mathcal{N}(0, \sigma^2_{\varepsilon})\\ \beta_{1,t+1} &= \beta_{1,t}\\ @@ -119,13 +119,13 @@ \beta_{n,t+1} &= \beta_{n,t}\\ \end{aligned} \end{gather*}\]

Example

julia> model = LinearRegression(rand(100, 2), rand(100))
-LinearRegression
source
StateSpaceModels.LocalLevel — Type
LocalLevel(y::Vector{Fl}) where Fl

The local level model is defined by:

\[\begin{gather*} \begin{aligned} y_{t} &= \mu_{t} + \varepsilon_{t} \quad \varepsilon_{t} \sim \mathcal{N}(0, \sigma^2_{\varepsilon})\\ \mu_{t+1} &= \mu_{t} + \eta_{t} \quad \eta_{t} \sim \mathcal{N}(0, \sigma^2_{\eta})\\ \end{aligned} \end{gather*}\]

Example

julia> model = LocalLevel(rand(100))
-LocalLevel

See more on Nile river annual flow

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press. pp. 9
source
StateSpaceModels.LocalLevelCycle — Type
LocalLevelCycle(y::Vector{Fl}) where Fl

The local level model with a cycle component is defined by:

\[\begin{gather*} +LocalLevel

See more on Nile river annual flow

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press. pp. 9
source
StateSpaceModels.LocalLevelCycle — Type
LocalLevelCycle(y::Vector{Fl}) where Fl

The local level model with a cycle component is defined by:

\[\begin{gather*} \begin{aligned} y_{t} &= \mu_{t} + c_{t} + \varepsilon_{t} \quad &\varepsilon_{t} \sim \mathcal{N}(0, \sigma^2_{\varepsilon})\\ \mu_{t+1} &= \mu_{t} + \eta_{t} \quad &\eta_{t} \sim \mathcal{N}(0, \sigma^2_{\eta})\\ @@ -133,7 +133,7 @@ c_{t+1}^{*} &= -c_{t} \sin(\lambda_c) + c_{t}^{*} \sin(\lambda_c) \quad &\tilde\omega^*_{t} \sim \mathcal{N}(0, \sigma^2_{\tilde\omega})\\ \end{aligned} \end{gather*}\]

Example

julia> model = LocalLevelCycle(rand(100))
-LocalLevelCycle

See more on TODO RJ_TEMPERATURE

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press. pp. 48
source
StateSpaceModels.LocalLevelExplanatory — Type
LocalLevelExplanatory(y::Vector{Fl}, X::Matrix{Fl}) where Fl

A local level model with explanatory variables is defined by:

\[\begin{gather*} +LocalLevelCycle

See more on TODO RJ_TEMPERATURE

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press. pp. 48
source
StateSpaceModels.LocalLevelExplanatory — Type
LocalLevelExplanatory(y::Vector{Fl}, X::Matrix{Fl}) where Fl

A local level model with explanatory variables is defined by:

\[\begin{gather*} \begin{aligned} y_{t} &= \mu_{t} + X_{1,t} \cdot \beta_{1,t} + \dots + X_{n,t} \cdot \beta_{n,t} + \varepsilon_{t} \quad &\varepsilon_{t} \sim \mathcal{N}(0, \sigma^2_{\varepsilon})\\ \mu_{t+1} &= \mu_{t} + \xi_{t} &\xi_{t} \sim \mathcal{N}(0, \sigma^2_{\xi})\\ @@ -142,14 +142,14 @@ \beta_{n,t+1} &= \beta_{n,t} &\tau_{n, t} \sim \mathcal{N}(0, \sigma^2_{\tau_{n}})\\\\ \end{aligned} \end{gather*}\]

Example

julia> model = LocalLevelExplanatory(rand(100), rand(100, 1))
-LocalLevelExplanatory
source
StateSpaceModels.LocalLinearTrend — Type
LocalLinearTrend(y::Vector{Fl}) where Fl

The linear trend model is defined by:

\[\begin{gather*} \begin{aligned} y_{t} &= \mu_{t} + \gamma_{t} + \varepsilon_{t} \quad &\varepsilon_{t} \sim \mathcal{N}(0, \sigma^2_{\varepsilon})\\ \mu_{t+1} &= \mu_{t} + \nu_{t} + \xi_{t} \quad &\xi_{t} \sim \mathcal{N}(0, \sigma^2_{\xi})\\ \nu_{t+1} &= \nu_{t} + \zeta_{t} \quad &\zeta_{t} \sim \mathcal{N}(0, \sigma^2_{\zeta})\\ \end{aligned} \end{gather*}\]

Example

julia> model = LocalLinearTrend(rand(100))
-LocalLinearTrend

See more on Finland road traffic fatalities

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press. pp. 44
source
StateSpaceModels.MultivariateBasicStructural — Type
MultivariateBasicStructural(y::Matrix{Fl}, s::Int) where Fl

An implementation of a non-homogeneous seemingly unrelated time series equations for basic structural state-space model consists of trend (local linear trend) and seasonal components. It is defined by:

\[\begin{gather*} +LocalLinearTrend

See more on Finland road traffic fatalities

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press. pp. 44
source
StateSpaceModels.MultivariateBasicStructural — Type
MultivariateBasicStructural(y::Matrix{Fl}, s::Int) where Fl

An implementation of a non-homogeneous seemingly unrelated time series equations for basic structural state-space model consists of trend (local linear trend) and seasonal components. It is defined by:

\[\begin{gather*} \begin{aligned} y_{t} &= \mu_{t} + \gamma_{t} + \varepsilon_{t} \quad &\varepsilon_{t} \sim \mathcal{N}(0, \Sigma_{\varepsilon})\\ \mu_{t+1} &= \mu_{t} + \nu_{t} + \xi_{t} \quad &\xi_{t} \sim \mathcal{N}(0, \Sigma_{\xi})\\ @@ -157,7 +157,7 @@ \gamma_{t+1} &= -\sum_{j=1}^{s-1} \gamma_{t+1-j} + \omega_{t} \quad & \omega_{t} \sim \mathcal{N}(0, \Sigma_{\omega})\\ \end{aligned} \end{gather*}\]

Example

julia> model = MultivariateBasicStructural(rand(100, 2), 12)
-MultivariateBasicStructural

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press.
source
StateSpaceModels.VehicleTracking — Type
VehicleTracking(y::Matrix{Fl}, ρ::Fl, H::Matrix{Fl}, Q::Matrix{Fl}) where Fl

The vehicle tracking example illustrates a model where there are no hyperparameters, the user defines the parameters $\rho$, $H$ and $Q$ and the model gives the predicted and filtered speed and position. In this case, $y_t$ is a $2 \times 1$ observation vector representing the corrupted measurements of the vehicle's position on the two-dimensional plane in instant $t$.

The position and speed in each dimension compose the state of the vehicle. Let us refer to $x_t^{(d)}$ as the position on the axis $d$ and to $\dot{x}^{(d)}_t$ as the speed on the axis $d$ in instant $t$. Additionally, let $\eta^{(d)}_t$ be the input drive force on the axis $d$, which acts as state noise. For a single dimension, we can describe the vehicle dynamics as

\[\begin{equation} +MultivariateBasicStructural

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press.
source
StateSpaceModels.VehicleTracking — Type
VehicleTracking(y::Matrix{Fl}, ρ::Fl, H::Matrix{Fl}, Q::Matrix{Fl}) where Fl

The vehicle tracking example illustrates a model where there are no hyperparameters, the user defines the parameters $\rho$, $H$ and $Q$ and the model gives the predicted and filtered speed and position. In this case, $y_t$ is a $2 \times 1$ observation vector representing the corrupted measurements of the vehicle's position on the two-dimensional plane in instant $t$.

The position and speed in each dimension compose the state of the vehicle. Let us refer to $x_t^{(d)}$ as the position on the axis $d$ and to $\dot{x}^{(d)}_t$ as the speed on the axis $d$ in instant $t$. Additionally, let $\eta^{(d)}_t$ be the input drive force on the axis $d$, which acts as state noise. For a single dimension, we can describe the vehicle dynamics as

\[\begin{equation} \begin{aligned} & x_{t+1}^{(d)} = x_t^{(d)} + \Big( 1 - \frac{\rho \Delta_t}{2} \Big) \Delta_t \dot{x}^{(d)}_t + \frac{\Delta^2_t}{2} \eta_t^{(d)}, \\ & \dot{x}^{(d)}_{t+1} = (1 - \rho) \dot{x}^{(d)}_{t} + \Delta_t \eta^{(d)}_t, @@ -165,7 +165,7 @@ \end{equation}\]

We can cast the dynamical system as a state-space model in the following manner:

\[\begin{align*} y_t &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \alpha_{t+1} + \varepsilon_t, \\ \alpha_{t+1} &= \begin{bmatrix} 1 & (1 - \tfrac{\rho \Delta_t}{2}) \Delta_t & 0 & 0 \\ 0 & (1 - \rho) & 0 & 0 \\ 0 & 0 & 1 & (1 - \tfrac{\rho \Delta_t}{2}) \\ 0 & 0 & 0 & (1 - \rho) \end{bmatrix} \alpha_{t} + \begin{bmatrix} \tfrac{\Delta^2_t}{2} & 0 \\ \Delta_t & 0 \\ 0 & \tfrac{\Delta^2_t}{2} \\ 0 & \Delta_t \end{bmatrix} \eta_{t}, -\end{align*}\]

See more on Vehicle tracking

source

Naive models

Naive models are not state space models but are good benchmarks for forecasting, for this reason we implemented them here.

StateSpaceModels.Naive — Type
Naive(y::Vector{<:Real})

A naive model where the h step ahead forecast is

\[y_{T+h|T} = y_T\]

References

  • Hyndman, Rob J., Athanasopoulos, George. "Forecasting: Principles and Practice"
source
StateSpaceModels.SeasonalNaive — Type
SeasonalNaive(y::Vector{<:Real}, seasoanl::Int)

A seasonal naive model where the h step ahead forecast is

\[y_{T+h|T} = y_{T + h - m(k+1)}\]

where m is the seasonal period and k is the integer part of (h-1)/m.

References

  • Hyndman, Rob J., Athanasopoulos, George. "Forecasting: Principles and Practice"
source
StateSpaceModels.ExperimentalSeasonalNaive — Type
ExperimentalSeasonalNaive(y::Vector{<:Real}, seasonal::Int; S::Int = 10_000)

A seasonal naive model where the h step ahead forecast is the mean of the simulation of S scenarios

\[y_{T+h|T} = y_{T + h - m(k+1)} + \varepsilon_t\]

where m is the seasonal period, k is the integer part of (h-1)/m and $\varepsilon_t$ is a sampled error.

We call it experimental because so far we could not find a good reference and implementation. If you know something please post it as an issue.

source

Automatic forecasting

Some models have various parameters and modelling options. The package provides simple functions that search through different parameters to obtain the best fit for your data without a deeper understanding. The search procedures can be published in scientific papers or purely heuristic designed by the developers. In any case if the documentation explains the procedures and indicates if there are any references.

StateSpaceModels.auto_ets — Function
auto_ets(y::Vector{Fl}; seasonal::Int = 0) where Fl

Automatically fits the best ExponentialSmoothing model according to the best AIC between the models:

  • ETS(A, N, N)
  • ETS(A, A, N)
  • ETS(A, Ad, N)

If the user provides the time series seasonality it will search between the models

  • ETS(A, N, A)
  • ETS(A, A, A)
  • ETS(A, Ad, A)

References

  • Hyndman, Robin John; Athanasopoulos, George.

Forecasting: Principles and Practice. 2nd ed. OTexts, 2018.

source

Naive models

Naive models are not state space models but are good benchmarks for forecasting, for this reason we implemented them here.

StateSpaceModels.Naive — Type
Naive(y::Vector{<:Real})

A naive model where the h step ahead forecast is

\[y_{T+h|T} = y_T\]

References

  • Hyndman, Rob J., Athanasopoulos, George. "Forecasting: Principles and Practice"
source
StateSpaceModels.SeasonalNaive — Type
SeasonalNaive(y::Vector{<:Real}, seasoanl::Int)

A seasonal naive model where the h step ahead forecast is

\[y_{T+h|T} = y_{T + h - m(k+1)}\]

where m is the seasonal period and k is the integer part of (h-1)/m.

References

  • Hyndman, Rob J., Athanasopoulos, George. "Forecasting: Principles and Practice"
source
StateSpaceModels.ExperimentalSeasonalNaive — Type
ExperimentalSeasonalNaive(y::Vector{<:Real}, seasonal::Int; S::Int = 10_000)

A seasonal naive model where the h step ahead forecast is the mean of the simulation of S scenarios

\[y_{T+h|T} = y_{T + h - m(k+1)} + \varepsilon_t\]

where m is the seasonal period, k is the integer part of (h-1)/m and $\varepsilon_t$ is a sampled error.

We call it experimental because so far we could not find a good reference and implementation. If you know something please post it as an issue.

source

Automatic forecasting

Some models have various parameters and modelling options. The package provides simple functions that search through different parameters to obtain the best fit for your data without a deeper understanding. The search procedures can be published in scientific papers or purely heuristic designed by the developers. In any case if the documentation explains the procedures and indicates if there are any references.

StateSpaceModels.auto_ets — Function
auto_ets(y::Vector{Fl}; seasonal::Int = 0) where Fl

Automatically fits the best ExponentialSmoothing model according to the best AIC between the models:

  • ETS(A, N, N)
  • ETS(A, A, N)
  • ETS(A, Ad, N)

If the user provides the time series seasonality it will search between the models

  • ETS(A, N, A)
  • ETS(A, A, A)
  • ETS(A, Ad, A)

References

  • Hyndman, Robin John; Athanasopoulos, George.

Forecasting: Principles and Practice. 2nd ed. OTexts, 2018.

source
StateSpaceModels.auto_arima — Function
auto_arima(y::Vector{Fl};
            seasonal::Int = 0,
            max_p::Int = 5,
            max_q::Int = 5,
@@ -179,7 +179,7 @@
            show_trace::Bool = false,
            integration_test::String = "kpss",
            seasonal_integration_test::String = "seas"
-           ) where Fl

Automatically fits the best SARIMA model according to the best information criteria

TODO write example

References

  • Hyndman, RJ and Khandakar.

Automatic time series forecasting: The forecast package for R. Journal of Statistical Software, 26(3), 2008.

source

Implementing a custom model

Users are able to implement any custom user-defined model.

Systems

The StateSpaceModel matrices are represented as a StateSpaceSystem.

StateSpaceModels.LinearUnivariateTimeInvariant — Type
LinearUnivariateTimeInvariant{Fl}(
+           ) where Fl

Automatically fits the best SARIMA model according to the best information criteria

TODO write example

References

  • Hyndman, RJ and Khandakar.

Automatic time series forecasting: The forecast package for R. Journal of Statistical Software, 26(3), 2008.

source

Implementing a custom model

Users are able to implement any custom user-defined model.

Systems

The StateSpaceModel matrices are represented as a StateSpaceSystem.

StateSpaceModels.LinearUnivariateTimeInvariant — Type
LinearUnivariateTimeInvariant{Fl}(
     y::Vector{Fl},
     Z::Vector{Fl},
     T::Matrix{Fl},
@@ -193,7 +193,7 @@
         y_{t} &=  Z\alpha_{t} + d + \varepsilon_{t} \quad &\varepsilon_{t} \sim \mathcal{N}(0, H)\\
         \alpha_{t+1} &= T\alpha_{t} + c + R\eta_{t} \quad &\eta_{t} \sim \mathcal{N}(0, Q)\\
     \end{aligned}
-\end{gather*}\]
where:
  • $y_{t}$ is a scalar
  • $Z$ is a $m \times 1$ vector
  • $d$ is a scalar
  • $T$ is a $m \times m$ matrix
  • $c$ is a $m \times 1$ vector
  • $R$ is a $m \times r$ matrix
  • $H$ is a scalar
  • $Q$ is a $r \times r$ matrix
source
StateSpaceModels.LinearUnivariateTimeVariant — Type
LinearUnivariateTimeVariant{Fl}(
+\end{gather*}\]
where:
  • $y_{t}$ is a scalar
  • $Z$ is a $m \times 1$ vector
  • $d$ is a scalar
  • $T$ is a $m \times m$ matrix
  • $c$ is a $m \times 1$ vector
  • $R$ is a $m \times r$ matrix
  • $H$ is a scalar
  • $Q$ is a $r \times r$ matrix
source
StateSpaceModels.LinearUnivariateTimeVariant — Type
LinearUnivariateTimeVariant{Fl}(
     y::Vector{Fl},
     Z::Vector{Vector{Fl}},
     T::Vector{Matrix{Fl}},
@@ -207,7 +207,7 @@
         y_{t} &=  Z_{t}\alpha_{t} + d_{t} + \varepsilon_{t} \quad &\varepsilon_{t} \sim \mathcal{N}(0, H_{t})\\
         \alpha_{t+1} &= T_{t}\alpha_{t} + c_{t} + R_{t}\eta_{t} \quad &\eta_{t} \sim \mathcal{N}(0, Q_{t})\\
     \end{aligned}
-\end{gather*}\]
where:
  • $y_{t}$ is a scalar
  • $Z_{t}$ is a $m \times 1$ vector
  • $d_{t}$ is a scalar
  • $T_{t}$ is a $m \times m$ matrix
  • $c_{t}$ is a $m \times 1$ vector
  • $R_{t}$ is a $m \times r$ matrix
  • $H_{t}$ is a scalar
  • $Q_{t}$ is a $r \times r$ matrix
source

Hyperparameters

The model hyperparameters are constant (non-time-varying) parameters that are optimized when fit! is called. The package provides some useful functions to accelerate experimentation and custom model development.

The getters are:

The setters are:

StateSpaceModels.fix_hyperparameters! — Function
fix_hyperparameters!(model::StateSpaceModel, fixed_hyperparameters::Dict)

Fixes the desired hyperparameters so that they are not considered as decision variables in the model estimation.

Example

julia> model = LocalLevel(rand(100))
+\end{gather*}\]
where:
  • $y_{t}$ is a scalar
  • $Z_{t}$ is a $m \times 1$ vector
  • $d_{t}$ is a scalar
  • $T_{t}$ is a $m \times m$ matrix
  • $c_{t}$ is a $m \times 1$ vector
  • $R_{t}$ is a $m \times r$ matrix
  • $H_{t}$ is a scalar
  • $Q_{t}$ is a $r \times r$ matrix
source

Hyperparameters

The model hyperparameters are constant (non-time-varying) parameters that are optimized when fit! is called. The package provides some useful functions to accelerate experimentation and custom model development.

The getters are:

The setters are:

StateSpaceModels.fix_hyperparameters! — Function
fix_hyperparameters!(model::StateSpaceModel, fixed_hyperparameters::Dict)

Fixes the desired hyperparameters so that they are not considered as decision variables in the model estimation.

Example

julia> model = LocalLevel(rand(100))
 LocalLevel model
 
 julia> get_names(model)
@@ -220,7 +220,7 @@
 
 julia> model.hyperparameters.fixed_constrained_values
 Dict{String,Float64} with 1 entry:
-  "sigma2_ε" => 100.0
source
StateSpaceModels.set_initial_hyperparameters! — Function
set_initial_hyperparameters!(model::StateSpaceModel,
                              initial_hyperparameters::Dict{String, <:Real})

Fill a model with user inputed initial points for hyperparameter optimzation.

Example

julia> model = LocalLevel(rand(100))
 LocalLevel model
 
@@ -235,7 +235,7 @@
 julia> model.hyperparameters.constrained_values
 2-element Array{Float64,1}:
  NaN  
- 100.0
source

Mappings:

StateSpaceModels.constrain_variance! — Function
constrain_variance!(model::StateSpaceModel, str::String)

Map a constrained hyperparameter $\psi \in \mathbb{R}^+$ to an unconstrained hyperparameter $\psi_* \in \mathbb{R}$.

The mapping is $\psi = \psi_*^2$

source
StateSpaceModels.unconstrain_variance! — Function
unconstrain_variance!(model::StateSpaceModel, str::String)

Map an unconstrained hyperparameter $\psi_{*} \in \mathbb{R}$ to a constrained hyperparameter $\psi \in \mathbb{R}^+$.

The mapping is $\psi_{*} = \sqrt{\psi}$.

source
StateSpaceModels.constrain_box! — Function
constrain_box!(model::StateSpaceModel, str::String, lb::Fl, ub::Fl) where Fl

Map a constrained hyperparameter $\psi \in [lb, ub]$ to an unconstrained hyperparameter $\psi_* \in \mathbb{R}$.

The mapping is $\psi = lb + \frac{ub - lb}{1 + \exp(-\psi_{*})}$

source
StateSpaceModels.unconstrain_box! — Function
unconstrain_box!(model::StateSpaceModel, str::String, lb::Fl, ub::Fl) where Fl

Map an unconstrained hyperparameter $\psi_* \in \mathbb{R}$ to a constrained hyperparameter $\psi \in [lb, ub]$.

The mapping is $\psi_* = -\ln \frac{ub - lb}{\psi - lb} - 1$

source
StateSpaceModels.constrain_identity! — Function
constrain_identity!(model::StateSpaceModel, str::String)

Map an constrained hyperparameter $\psi \in \mathbb{R}$ to an unconstrained hyperparameter $\psi_* \in \mathbb{R}$. This function is necessary to copy values from a location to another inside HyperParameters

The mapping is $\psi = \psi_*$

source
StateSpaceModels.unconstrain_identity! — Function
unconstrain_identity!(model::StateSpaceModel, str::String)

Map an unconstrained hyperparameter $\psi_* \in \mathbb{R}$ to a constrained hyperparameter $\psi \in \mathbb{R}$. This function is necessary to copy values from a location to another inside HyperParameters

The mapping is $\psi_* = \psi$

source

Filters and smoothers

StateSpaceModels.jl lets users define tailor-made filters in an easy manner. TODO docs here

StateSpaceModels.UnivariateKalmanFilter — Type
UnivariateKalmanFilter{Fl <: AbstractFloat}

A Kalman filter that is tailored to univariate systems, exploiting the fact that the dimension of the observations at any time period is 1.

TODO equations and descriptions of a1 and P1

source
StateSpaceModels.SparseUnivariateKalmanFilter — Type
SparseUnivariateKalmanFilter{Fl <: AbstractFloat}

A Kalman filter that is tailored to sparse univariate systems, exploiting the fact that the dimension of the observations at any time period is 1 and that Z, T and R are sparse.

TODO equations and descriptions of a1 and P1

source
StateSpaceModels.FilterOutput — Type
FilterOutput{Fl<:Real}

Structure with the results of the Kalman filter:

  • v: innovations
  • F: variance of innovations
  • a: predictive state
  • att: filtered state
  • P: variance of predictive state
  • Ptt: variance of filtered state
  • Pinf: diffuse part of the covariance
source

Fitting and Optimizers

StateSpaceModels.jl has an interface for Optim.jl algorithms. The models can be estimated using different algorithms and tunned to the user needs

Mappings:

StateSpaceModels.constrain_variance! — Function
constrain_variance!(model::StateSpaceModel, str::String)

Map a constrained hyperparameter $\psi \in \mathbb{R}^+$ to an unconstrained hyperparameter $\psi_* \in \mathbb{R}$.

The mapping is $\psi = \psi_*^2$

source
StateSpaceModels.unconstrain_variance! — Function
unconstrain_variance!(model::StateSpaceModel, str::String)

Map an unconstrained hyperparameter $\psi_{*} \in \mathbb{R}$ to a constrained hyperparameter $\psi \in \mathbb{R}^+$.

The mapping is $\psi_{*} = \sqrt{\psi}$.

source
StateSpaceModels.constrain_box! — Function
constrain_box!(model::StateSpaceModel, str::String, lb::Fl, ub::Fl) where Fl

Map a constrained hyperparameter $\psi \in [lb, ub]$ to an unconstrained hyperparameter $\psi_* \in \mathbb{R}$.

The mapping is $\psi = lb + \frac{ub - lb}{1 + \exp(-\psi_{*})}$

source
StateSpaceModels.unconstrain_box! — Function
unconstrain_box!(model::StateSpaceModel, str::String, lb::Fl, ub::Fl) where Fl

Map an unconstrained hyperparameter $\psi_* \in \mathbb{R}$ to a constrained hyperparameter $\psi \in [lb, ub]$.

The mapping is $\psi_* = -\ln \frac{ub - lb}{\psi - lb} - 1$

source
StateSpaceModels.constrain_identity! — Function
constrain_identity!(model::StateSpaceModel, str::String)

Map an constrained hyperparameter $\psi \in \mathbb{R}$ to an unconstrained hyperparameter $\psi_* \in \mathbb{R}$. This function is necessary to copy values from a location to another inside HyperParameters

The mapping is $\psi = \psi_*$

source
StateSpaceModels.unconstrain_identity! — Function
unconstrain_identity!(model::StateSpaceModel, str::String)

Map an unconstrained hyperparameter $\psi_* \in \mathbb{R}$ to a constrained hyperparameter $\psi \in \mathbb{R}$. This function is necessary to copy values from a location to another inside HyperParameters

The mapping is $\psi_* = \psi$

source

Filters and smoothers

StateSpaceModels.jl lets users define tailor-made filters in an easy manner. TODO docs here

StateSpaceModels.UnivariateKalmanFilter — Type
UnivariateKalmanFilter{Fl <: AbstractFloat}

A Kalman filter that is tailored to univariate systems, exploiting the fact that the dimension of the observations at any time period is 1.

TODO equations and descriptions of a1 and P1

source
StateSpaceModels.SparseUnivariateKalmanFilter — Type
SparseUnivariateKalmanFilter{Fl <: AbstractFloat}

A Kalman filter that is tailored to sparse univariate systems, exploiting the fact that the dimension of the observations at any time period is 1 and that Z, T and R are sparse.

TODO equations and descriptions of a1 and P1

source
StateSpaceModels.FilterOutput — Type
FilterOutput{Fl<:Real}

Structure with the results of the Kalman filter:

  • v: innovations
  • F: variance of innovations
  • a: predictive state
  • att: filtered state
  • P: variance of predictive state
  • Ptt: variance of filtered state
  • Pinf: diffuse part of the covariance
source

Fitting and Optimizers

StateSpaceModels.jl has an interface for Optim.jl algorithms. The models can be estimated using different algorithms and tunned to the user needs

StateSpaceModels.fit! — Function
fit!(
     model::StateSpaceModel;
     filter::KalmanFilter=default_filter(model),
     optimizer::Optimizer=Optimizer(Optim.LBFGS()),
@@ -250,17 +250,17 @@
 LocalLinearTrend model
 
 julia> fit!(model; optimizer = Optimizer(StateSpaceModels.Optim.NelderMead()))
-LocalLinearTrend model
source
StateSpaceModels.Optimizer — Type
Optimizer

An Optim.jl wrapper to make the choice of the optimizer straightforward in StateSpaceModels.jl Users can choose among all suitable Optimizers in Optim.jl using very similar syntax.

Example

julia> using Optim
+LocalLinearTrend model
source
StateSpaceModels.Optimizer — Type
Optimizer

An Optim.jl wrapper to make the choice of the optimizer straightforward in StateSpaceModels.jl Users can choose among all suitable Optimizers in Optim.jl using very similar syntax.

Example

julia> using Optim
 
 # use a semicolon to avoid displaying the big log
-julia> opt = Optimizer(Optim.LBFGS(), Optim.Options(show_trace = true));
source
Missing docstring.

Missing docstring for results. Check Documenter's build log for details.

StateSpaceModels.has_fit_methods — Function
has_fit_methods(model_type::Type{<:StateSpaceModel}) -> Bool

Verify if a certain StateSpaceModel has the necessary methods to perform fit!`.

source
StateSpaceModels.isfitted — Function
isfitted(model::StateSpaceModel) -> Bool

Verify if model is fitted, i.e., returns false if there is at least one NaN entry in the hyperparameters.

source

Forecasting and simulating

StateSpaceModels.jl has functions to make forecasts of the predictive densities multiple steps ahead and to simulate scenarios based on those forecasts. The package also has a functions to benchmark the model forecasts using cross_validation techniques.

StateSpaceModels.forecast — Function
forecast(model::StateSpaceModel, steps_ahead::Int; kwargs...)
-forecast(model::StateSpaceModel, exogenous::Matrix{Fl}; kwargs...) where Fl

Forecast the mean and covariance for future observations from a StateSpaceModel.

source
Missing docstring.

Missing docstring for results. Check Documenter's build log for details.

StateSpaceModels.has_fit_methods — Function
has_fit_methods(model_type::Type{<:StateSpaceModel}) -> Bool

Verify if a certain StateSpaceModel has the necessary methods to perform fit!`.

source
StateSpaceModels.isfitted — Function
isfitted(model::StateSpaceModel) -> Bool

Verify if model is fitted, i.e., returns false if there is at least one NaN entry in the hyperparameters.

source

Forecasting and simulating

StateSpaceModels.jl has functions to make forecasts of the predictive densities multiple steps ahead and to simulate scenarios based on those forecasts. The package also has a functions to benchmark the model forecasts using cross_validation techniques.

StateSpaceModels.forecast — Function
forecast(model::StateSpaceModel, steps_ahead::Int; kwargs...)
+forecast(model::StateSpaceModel, exogenous::Matrix{Fl}; kwargs...) where Fl

Forecast the mean and covariance for future observations from a StateSpaceModel.

source
StateSpaceModels.simulate_scenarios — Function
simulate_scenarios(
     model::StateSpaceModel, steps_ahead::Int, n_scenarios::Int;
     filter::KalmanFilter=default_filter(model)
-) -> Array{<:AbstractFloat, 3}

Samples n_scenarios future scenarios via Monte Carlo simulation for steps_ahead using the desired filter.

source
StateSpaceModels.cross_validation — Function
cross_validation(model::StateSpaceModel, steps_ahead::Int, start_idx::Int;
+) -> Array{<:AbstractFloat, 3}

Samples n_scenarios future scenarios via Monte Carlo simulation for steps_ahead using the desired filter.

source
StateSpaceModels.cross_validation — Function
cross_validation(model::StateSpaceModel, steps_ahead::Int, start_idx::Int;
          n_scenarios::Int = 10_000,
          filter::KalmanFilter=default_filter(model),
-         optimizer::Optimizer=default_optimizer(model)) where Fl

Makes rolling window estimating and forecasting to benchmark the forecasting skill of the model in for different time periods and different lead times. The function returns a struct with the MAE and mean CRPS per lead time. See more on CrossValidation the forecasts of a model

References

  • DTU course "31761 - Renewables in electricity markets" available on youtube https://www.youtube.com/watch?v=Ffo8XilZAZw&t=556s
source

Visualization

Some user friendly plot recipes are defined using RecipesBase.jl. If you have any suggestions do not hesitate to post it as an issue.

using StateSpaceModels, CSV, DataFrames, Plots
+         optimizer::Optimizer=default_optimizer(model)) where Fl

Makes rolling window estimating and forecasting to benchmark the forecasting skill of the model in for different time periods and different lead times. The function returns a struct with the MAE and mean CRPS per lead time. See more on CrossValidation the forecasts of a model

References

  • DTU course "31761 - Renewables in electricity markets" available on youtube https://www.youtube.com/watch?v=Ffo8XilZAZw&t=556s
source

Visualization

Some user friendly plot recipes are defined using RecipesBase.jl. If you have any suggestions do not hesitate to post it as an issue.

using StateSpaceModels, CSV, DataFrames, Plots
 
 air_passengers = CSV.File(StateSpaceModels.AIR_PASSENGERS) |> DataFrame
 log_air_passengers = log.(air_passengers.passengers)
@@ -272,81 +272,81 @@
 plot(model, forec; legend = :topleft)
- + - - + - - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + 
using StateSpaceModels, CSV, DataFrames, Plots
 
 finland_fatalities = CSV.File(StateSpaceModels.VEHICLE_FATALITIES) |> DataFrame
@@ -416,271 +416,271 @@
 plot(model, ks)
- + - - + - - + - - - - - - - - - - - - - - - - - - - - - - - - - - - + - - - - - - - - - - - - - - - - - - - - - - - - - - - + - - - - - - - - - - - - - - - - - - - - - - - - -Datasets

The package provides some datasets to illustrate the funtionalities and models. These datasets are stored as csv files and the path to these files can be obtained through their names as seen below. In the examples we illustrate the datasets using DataFrames.jl and CSV.jl

StateSpaceModels.AIR_PASSENGERS — Constant
AIR_PASSENGERS

The absolute path for the AIR_PASSENGERS dataset stored inside StateSpaceModels.jl. This dataset provides monthly totals of a US airline passengers from 1949 to 1960.

See more on Airline passengers

References

  • https://www.stata-press.com/data/r12/ts.html
source
StateSpaceModels.FRONT_REAR_SEAT_KSI — Constant
FRONT_REAR_SEAT_KSI

The absolute path for the FRONT_REAR_SEAT_KSI dataset stored inside StateSpaceModels.jl. This dataset provides the log of british people killed or serious injuried in road accidents accross UK.

References

  • Commandeur, Jacques J.F. & Koopman, Siem Jan, 2007. "An Introduction to State Space Time Series Analysis," OUP Catalogue, Oxford University Press (Chapter 3)
  • http://staff.feweb.vu.nl/koopman/projects/ckbook/OxCodeAll.zip
source
StateSpaceModels.INTERNET — Constant
INTERNET

The absolute path for the INTERNET dataset stored inside StateSpaceModels.jl. This dataset provides the number of users logged on to an Internet server each minute over 100 minutes.

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press. (Chapter 9)
source
StateSpaceModels.NILE — Constant
NILE

The absolute path for the NILE dataset stored inside StateSpaceModels.jl. This dataset provides measurements of the annual flow of the Nile river at Aswan from 1871 to 1970, in $10^8 m^3$.

See more on Nile river annual flow

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press. (Chapter 2)
source
StateSpaceModels.VEHICLE_FATALITIES — Constant
VEHICLE_FATALITIES

The absolute path for the VEHICLE_FATALITIES dataset stored inside StateSpaceModels.jl. This dataset provides the number of annual road traffic fatalities in Norway and Finland.

See more on Finland road traffic fatalities

References

  • Commandeur, Jacques J.F. & Koopman, Siem Jan, 2007. "An Introduction to State Space Time Series Analysis," OUP Catalogue, Oxford University Press (Chapter 3)
  • http://staff.feweb.vu.nl/koopman/projects/ckbook/OxCodeAll.zip
source
StateSpaceModels.US_CHANGE — Constant
US_CHANGE

Percentage changes in quarterly personal consumption expenditure, personal disposable income, production, savings and the unemployment rate for the US, 1960 to 2016.

Federal Reserve Bank of St Louis.

References

  • Hyndman, Rob J., Athanasopoulos, George. "Forecasting: Principles and Practice"
source
+plotdiagnostics(kf)

Datasets

The package provides some datasets to illustrate the funtionalities and models. These datasets are stored as csv files and the path to these files can be obtained through their names as seen below. In the examples we illustrate the datasets using DataFrames.jl and CSV.jl

StateSpaceModels.AIR_PASSENGERS — Constant
AIR_PASSENGERS

The absolute path for the AIR_PASSENGERS dataset stored inside StateSpaceModels.jl. This dataset provides monthly totals of a US airline passengers from 1949 to 1960.

See more on Airline passengers

References

  • https://www.stata-press.com/data/r12/ts.html
source
StateSpaceModels.FRONT_REAR_SEAT_KSI — Constant
FRONT_REAR_SEAT_KSI

The absolute path for the FRONT_REAR_SEAT_KSI dataset stored inside StateSpaceModels.jl. This dataset provides the log of british people killed or serious injuried in road accidents accross UK.

References

  • Commandeur, Jacques J.F. & Koopman, Siem Jan, 2007. "An Introduction to State Space Time Series Analysis," OUP Catalogue, Oxford University Press (Chapter 3)
  • http://staff.feweb.vu.nl/koopman/projects/ckbook/OxCodeAll.zip
source
StateSpaceModels.INTERNET — Constant
INTERNET

The absolute path for the INTERNET dataset stored inside StateSpaceModels.jl. This dataset provides the number of users logged on to an Internet server each minute over 100 minutes.

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press. (Chapter 9)
source
StateSpaceModels.NILE — Constant
NILE

The absolute path for the NILE dataset stored inside StateSpaceModels.jl. This dataset provides measurements of the annual flow of the Nile river at Aswan from 1871 to 1970, in $10^8 m^3$.

See more on Nile river annual flow

References

  • Durbin, James, & Siem Jan Koopman. (2012). "Time Series Analysis by State Space Methods: Second Edition." Oxford University Press. (Chapter 2)
source
StateSpaceModels.VEHICLE_FATALITIES — Constant
VEHICLE_FATALITIES

The absolute path for the VEHICLE_FATALITIES dataset stored inside StateSpaceModels.jl. This dataset provides the number of annual road traffic fatalities in Norway and Finland.

See more on Finland road traffic fatalities

References

  • Commandeur, Jacques J.F. & Koopman, Siem Jan, 2007. "An Introduction to State Space Time Series Analysis," OUP Catalogue, Oxford University Press (Chapter 3)
  • http://staff.feweb.vu.nl/koopman/projects/ckbook/OxCodeAll.zip
source
StateSpaceModels.US_CHANGE — Constant
US_CHANGE

Percentage changes in quarterly personal consumption expenditure, personal disposable income, production, savings and the unemployment rate for the US, 1960 to 2016.

Federal Reserve Bank of St Louis.

References

  • Hyndman, Rob J., Athanasopoulos, George. "Forecasting: Principles and Practice"
source
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