### Quick Start: Fit LocalLevel and Forecast

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Quick start workflow: install the package, create a LocalLevel model, fit it, print results, forecast 10 steps, and run Kalman filter and smoother to extract innovations and smoothed states.

```julia
import Pkg

Pkg.add("StateSpaceModels")

using StateSpaceModels

y = randn(100)

model = LocalLevel(y)

fit!(model)

print_results(model)

forec = forecast(model, 10)

kf = kalman_filter(model)

v = get_innovations(kf)

ks = kalman_smoother(model)

alpha = get_smoothed_state(ks)
```

--------------------------------

### LocalLevelExplanatory constructor and example

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructs a local level model with explanatory variables. The example creates a model from 100 random observations and a 100x1 matrix of explanatory variables.

```julia
LocalLevelExplanatory(y::Vector{Fl}, X::Matrix{Fl}) where Fl
```

```julia
julia> model = LocalLevelExplanatory(rand(100), rand(100, 1))
LocalLevelExplanatory
```

--------------------------------

### Install StateSpaceModels.jl with Pkg

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable

Install the registered StateSpaceModels.jl package using Julia's Pkg manager.

```julia
pkg> add StateSpaceModels
```

--------------------------------

### Example: set_initial_hyperparameters! on LocalLevel

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Example usage of set_initial_hyperparameters! on a LocalLevel model. It shows the hyperparameter names and the resulting constrained_values vector with NaN for unset values.

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

julia> get_names(model)
2-element Vector{String}:
 "sigma2_ε"
 "sigma2_η"

julia> set_initial_hyperparameters!(model, Dict("sigma2_η" => 100.0))
LocalLevel

julia> model.hyperparameters.constrained_values
2-element Vector{Float64}:
 NaN  
 100.0
```

--------------------------------

### SARIMA model construction example

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Shows how to create a SARIMA model instance using random data and specified orders. The output displays the model summary.

```julia
julia> model = SARIMA(rand(100); order=(1,1,1), seasonal_order=(1,2,3,12))
SARIMA(1, 1, 1)x(1, 2, 3, 12) with zero mean
```

--------------------------------

### LocalLevel constructor and example

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructs a local level model with the given vector of observations. The example creates a model from 100 random values.

```julia
LocalLevel(y::Vector{Fl}) where Fl
```

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

--------------------------------

### BasicStructural model construction example

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Shows how to create a BasicStructural model instance using random data and a seasonal period of 12. The output displays the model name.

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

--------------------------------

### MultivariateBasicStructural constructor and example

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructs a multivariate basic structural model with trend and seasonal components. The example creates a model from a 100x2 matrix of observations and a seasonal period of 12.

```julia
MultivariateBasicStructural(y::Matrix{Fl}, s::Int) where Fl
```

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

--------------------------------

### LocalLevelCycle constructor and example

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructs a local level model with a cycle component. The example creates a model from 100 random values.

```julia
LocalLevelCycle(y::Vector{Fl}) where Fl
```

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

--------------------------------

### LinearRegression model construction example

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Shows how to create a LinearRegression model instance using random data. The output displays the model name.

```julia
julia> model = LinearRegression(rand(100, 2), rand(100))
LinearRegression
```

--------------------------------

### Creating an Optimizer with BFGS

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Example of creating an Optimizer with BFGS and showing the trace. The semicolon suppresses the large log output.

```julia
julia> using Optim

# use a semicolon to avoid displaying the big log
julia> opt = Optimizer(Optim.BFGS(), Optim.Options(show_trace = true));
```

--------------------------------

### LocalLinearTrend constructor and example

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructs a local linear trend model. The example creates a model from 100 random values.

```julia
LocalLinearTrend(y::Vector{Fl}) where Fl
```

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

--------------------------------

### Example: fix_hyperparameters! on LocalLevel

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Example usage of fix_hyperparameters! on a LocalLevel model. It shows the hyperparameter names and the resulting fixed_constrained_values dictionary.

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

julia> get_names(model)
2-element Vector{String}:
 "sigma2_ε"
 "sigma2_η"

julia> fix_hyperparameters!(model, Dict("sigma2_ε" => 100.0))
LocalLevel

julia> model.hyperparameters.fixed_constrained_values
Dict{String, Float64} with 1 entry:
  "sigma2_ε" => 100.0
```

--------------------------------

### BasicStructuralExplanatory model construction example

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Shows how to create a BasicStructuralExplanatory model instance using random data, a seasonal period of 12, and a random matrix of explanatory variables. The output displays the model name.

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

--------------------------------

### Fitting LocalLevel and LocalLinearTrend models

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Example of fitting a LocalLevel model with default settings and a LocalLinearTrend model with a NelderMead optimizer. The output shows the model type after each call.

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

julia> fit!(model)
LocalLevel

julia> model = LocalLinearTrend(LinRange(1, 100, 100) + rand(100))
LocalLinearTrend

julia> fit!(model; optimizer = Optimizer(StateSpaceModels.Optim.NelderMead()))
LocalLinearTrend
```

--------------------------------

### Get WHOLESALE_PRICE_INDEX dataset path

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Returns the absolute path to the WHOLESALE_PRICE_INDEX dataset stored inside StateSpaceModels.jl. The dataset is marked as TODO in the documentation.

```julia
WHOLESALE_PRICE_INDEX
```

--------------------------------

### SARIMA type definition

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Defines the SARIMA model constructor with keyword arguments for order, seasonal_order, and include_mean. The example shows creating a SARIMA model with order (1,1,1) and seasonal order (1,2,3,12).

```julia
SARIMA(
    y::Vector{Fl}; 
    order::Tuple{Int,Int,Int} = (1, 0, 0), 
    seasonal_order::Tuple{Int, Int, Int, Int} = (0, 0, 0, 0),
    include_mean::Bool = false,
) where Fl
```

--------------------------------

### BasicStructuralExplanatory type definition

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Defines the BasicStructuralExplanatory model constructor that takes a vector of observations, a seasonal period, and a matrix of explanatory variables. The example shows creating a model with 100 observations, seasonal period 12, and two explanatory variables.

```julia
BasicStructuralExplanatory(y::Vector{Fl}, s::Int, X::Matrix{Fl}) where Fl
```

--------------------------------

### Get hyperparameter names with get_names

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Returns the names of the hyperparameters registered on a StateSpaceModel.

```julia
get_names(model::StateSpaceModel)
```

--------------------------------

### BasicStructural type definition

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Defines the BasicStructural model constructor that takes a vector of observations and a seasonal period. The example shows creating a model with 100 observations and a seasonal period of 12.

```julia
BasicStructural(y::Vector{Fl}, s::Int) where Fl
```

--------------------------------

### Get INTERNET dataset path

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

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

```julia
INTERNET
```

--------------------------------

### Get smoothed state estimates from Kalman smoother

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/examples

Runs the Kalman smoother on the fitted model and retrieves the smoothed state estimates for the level of the annual flow. The output is a vector of Float64 values.

```julia
smoother_output = kalman_smoother(model)
get_smoothed_state(smoother_output)
```

--------------------------------

### Get filtered state estimates from Kalman filter

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/examples

Runs the Kalman filter on the fitted model and retrieves the filtered state estimates for the level of the annual flow. The output is a 100x1 matrix of Float64 values.

```julia
filter_output = kalman_filter(model)
get_filtered_state(filter_output)
```

--------------------------------

### Get SUNSPOTS_YEAR dataset path

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Returns the absolute path to the SUNSPOTS_YEAR dataset stored inside StateSpaceModels.jl. The dataset provides yearly numbers of sunspots from 1700 to 1988 (rounded to one digit).

```julia
SUNSPOTS_YEAR
```

--------------------------------

### Get RJ_TEMPERATURE dataset path

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Returns the absolute path to the RJ_TEMPERATURE dataset stored inside StateSpaceModels.jl. The dataset provides weekly mean temperature in Rio de Janeiro in Kelvin (K).

```julia
RJ_TEMPERATURE
```

--------------------------------

### Get FRONT_REAR_SEAT_KSI dataset path

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Returns the absolute path to the FRONT_REAR_SEAT_KSI dataset stored inside StateSpaceModels.jl. The dataset provides the log of British people killed or seriously injured in road accidents across the UK.

```julia
FRONT_REAR_SEAT_KSI
```

--------------------------------

### Get US_CHANGE dataset path

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Returns the absolute path to the US_CHANGE dataset stored inside StateSpaceModels.jl. The dataset provides percentage changes in quarterly personal consumption expenditure, personal disposable income, production, savings and the unemployment rate for the US, 1960 to 2016.

```julia
US_CHANGE
```

--------------------------------

### Set initial hyperparameters with set_initial_hyperparameters!

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

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

```julia
set_initial_hyperparameters!(model::StateSpaceModel,
                             initial_hyperparameters::Dict{String, <:Real})
```

--------------------------------

### BasicStructuralExplanatory

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructor for the BasicStructuralExplanatory model type.

```APIDOC
## BasicStructuralExplanatory

### Description
Constructs a basic structural state-space model with explanatory variables. The model includes a trend, seasonal component, and regression effects from the explanatory variables.

### Constructor
`BasicStructuralExplanatory(y::Vector{Fl}, s::Int, X::Matrix{Fl}) where Fl`

### Parameters
- **y** (Vector{Fl}) - Required - The time series data.
- **s** (Int) - Required - The seasonal period.
- **X** (Matrix{Fl}) - Required - The matrix of explanatory variables.

### Example
```julia
julia> model = BasicStructuralExplanatory(rand(100), 12, rand(100, 2))
BasicStructuralExplanatory
```
```

--------------------------------

### Julia: Run cross_validation on BasicStructural model

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/examples

Loads the air passengers dataset, applies a log transformation, fits a BasicStructural model with a 12-month seasonal period, and runs cross_validation with a 24-step horizon and 50 rolling windows.

```julia
using CSV, DataFrames
air_passengers = CSV.File(StateSpaceModels.AIR_PASSENGERS) |> DataFrame
log_air_passengers = log.(air_passengers.passengers)
model = BasicStructural(log_air_passengers, 12)
b = cross_validation(model, 24, 50)
```

--------------------------------

### LinearRegression type definition

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Defines the LinearRegression model constructor that takes a matrix of predictors and a vector of responses. The example shows creating a model with 100 observations and two predictors.

```julia
LinearRegression(X::Matrix{Fl}, y::Vector{Fl}) where Fl
```

--------------------------------

### ExperimentalSeasonalNaive model constructor

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructor for an experimental seasonal naive model that simulates S scenarios and uses the mean as the forecast. The default number of simulations is 10,000.

```julia
ExperimentalSeasonalNaive(y::Vector{<:Real}, seasonal::Int; S::Int = 10_000)
```

--------------------------------

### BasicStructural

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructor for the BasicStructural model type.

```APIDOC
## BasicStructural

### Description
Constructs a basic structural state-space model consisting of a trend (level + slope) and a seasonal component. The model is defined by the equations for the observation, level, slope, and seasonal components.

### Constructor
`BasicStructural(y::Vector{Fl}, s::Int) where Fl`

### Parameters
- **y** (Vector{Fl}) - Required - The time series data.
- **s** (Int) - Required - The seasonal period.

### Example
```julia
julia> model = BasicStructural(rand(100), 12)
BasicStructural
```
```

--------------------------------

### Get VEHICLE_FATALITIES dataset path

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

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

```julia
VEHICLE_FATALITIES
```

--------------------------------

### simulate_scenarios(model::StateSpaceModel, steps_ahead::Int, n_scenarios::Int; filter::KalmanFilter=default_filter(model))

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Samples future scenarios via Monte Carlo simulation for a given number of steps ahead.

```APIDOC
## simulate_scenarios(model::StateSpaceModel, steps_ahead::Int, n_scenarios::Int; filter::KalmanFilter=default_filter(model)) -> Array{<:AbstractFloat, 3}

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

### Method
Function call

### Endpoint
`simulate_scenarios(model::StateSpaceModel, steps_ahead::Int, n_scenarios::Int; filter::KalmanFilter=default_filter(model))`

### Parameters
- **model** (StateSpaceModel) - Required - The state space model to simulate scenarios from.
- **steps_ahead** (Int) - Required - Number of steps ahead to simulate.
- **n_scenarios** (Int) - Required - Number of scenarios to sample.
- **filter** (KalmanFilter) - Optional - The filter to use; defaults to `default_filter(model)`.

### Response
Returns an array of shape (steps_ahead, n_scenarios, ...) with simulated scenarios.
```

--------------------------------

### Get AIR_PASSENGERS dataset path

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

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

```julia
AIR_PASSENGERS
```

--------------------------------

### Naive model constructor

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructor for a naive model where the h-step ahead forecast is the last observed value. Use as a benchmark for forecasting.

```julia
Naive(y::Vector{<:Real})
```

--------------------------------

### set_initial_hyperparameters!

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Sets initial values for hyperparameters to be used in optimization.

```APIDOC
## set_initial_hyperparameters!(model::StateSpaceModel, initial_hyperparameters::Dict{String, <:Real})

### Description
Fill a model with user inputed initial points for hyperparameter optimization.

### Method
Function

### Endpoint
`StateSpaceModels.set_initial_hyperparameters!(model::StateSpaceModel, initial_hyperparameters::Dict{String, <:Real})`

### Parameters
- **model** (`StateSpaceModel`) - Required - The state space model.
- **initial_hyperparameters** (`Dict{String, <:Real}`) - Required - Dictionary mapping hyperparameter names to initial values.

### Response
Returns the modified model.

### Example
```julia
julia> model = LocalLevel(rand(100))
LocalLevel

julia> get_names(model)
2-element Vector{String}:
 "sigma2_ε"
 "sigma2_η"

julia> set_initial_hyperparameters!(model, Dict("sigma2_η" => 100.0))
LocalLevel

julia> model.hyperparameters.constrained_values
2-element Vector{Float64}:
 NaN  
 100.0
```
```

--------------------------------

### 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))

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Performs rolling window cross-validation to benchmark forecasting skill.

```APIDOC
## 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))

### Description
Makes rolling window estimating and forecasting to benchmark the forecasting skill of the model for different time periods and different lead times. Returns a struct with the MAE and mean CRPS per lead time.

### Method
Function call

### Endpoint
`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))`

### Parameters
- **model** (StateSpaceModel) - Required - The state space model to cross-validate.
- **steps_ahead** (Int) - Required - Number of steps ahead to forecast.
- **start_idx** (Int) - Required - Starting index for the rolling window.
- **n_scenarios** (Int) - Optional - Number of scenarios for simulation; default is 10,000.
- **filter** (KalmanFilter) - Optional - The filter to use; defaults to `default_filter(model)`.
- **optimizer** (Optimizer) - Optional - The optimizer to use; defaults to `default_optimizer(model)`.

### Response
Returns a struct with the MAE and mean CRPS per lead time.
```

--------------------------------

### StateSpaceModels.ExperimentalSeasonalNaive

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructor for an experimental seasonal naive model that simulates multiple scenarios to produce a forecast.

```APIDOC
## ExperimentalSeasonalNaive(y::Vector{<:Real}, seasonal::Int; S::Int = 10_000)

### Description
A seasonal naive model where the h step ahead forecast is the mean of the simulation of S scenarios: yT+h|T = yT+h−m(k+1) + εt, where m is the seasonal period, k is the integer part of (h-1)/m and εt is a sampled error.

### Method
Constructor

### Endpoint
`StateSpaceModels.ExperimentalSeasonalNaive(y::Vector{<:Real}, seasonal::Int; S::Int = 10_000)`

### Parameters
- **y** (Vector{<:Real}) - Required - The time series data.
- **seasonal** (Int) - Required - The seasonal period (m).
- **S** (Int) - Optional - Number of simulation scenarios (default: 10,000).

### References
- None provided.
```

--------------------------------

### Get NILE dataset path

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Returns the absolute path to the NILE dataset stored inside StateSpaceModels.jl. The dataset provides measurements of the annual flow of the Nile river at Aswan from 1871 to 1970, in 108m3.

```julia
NILE
```

--------------------------------

### StateSpaceModels.Naive

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructor for a naive model that uses the last observed value as the forecast for all future horizons.

```APIDOC
## Naive(y::Vector{<:Real})

### Description
A naive model where the h step ahead forecast is yT+h|T = yT.

### Method
Constructor

### Endpoint
`StateSpaceModels.Naive(y::Vector{<:Real})`

### Parameters
- **y** (Vector{<:Real}) - Required - The time series data.

### References
- Hyndman, Rob J., Athanasopoulos, George. "Forecasting: Principles and Practice"
```

--------------------------------

### SARIMA

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructor for the SARIMA model type.

```APIDOC
## SARIMA

### Description
Constructs a SARIMA model (Seasonal AutoRegressive Integrated Moving Average) within the state-space framework. The model is specified by (p,d,q)×(P,D,Q,s) and can include a constant term via the `include_mean` keyword.

### Constructor
`SARIMA(y::Vector{Fl}; order::Tuple{Int,Int,Int} = (1, 0, 0), seasonal_order::Tuple{Int, Int, Int, Int} = (0, 0, 0, 0), include_mean::Bool = false) where Fl`

### Parameters
- **y** (Vector{Fl}) - Required - The time series data.
- **order** (Tuple{Int,Int,Int}) - Optional - The non-seasonal (p,d,q) order. Default: (1,0,0).
- **seasonal_order** (Tuple{Int,Int,Int,Int}) - Optional - The seasonal (P,D,Q,s) order. Default: (0,0,0,0).
- **include_mean** (Bool) - Optional - Whether to include a constant term. Default: false.

### Example
```julia
julia> model = SARIMA(rand(100); order=(1,1,1), seasonal_order=(1,2,3,12))
SARIMA(1, 1, 1)x(1, 2, 3, 12) with zero mean
```
```

--------------------------------

### kalman_filter

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Runs the Kalman filter on a state space model and returns the filter output.

```APIDOC
## kalman_filter

### Description
Filter the data using the desired `filter` and the estimated hyperparameters in `model`.

### Method
Function call

### Endpoint
`kalman_filter(model::StateSpaceModel; filter::KalmanFilter=default_filter(model)) -> FilterOutput`

### Parameters
- **model** (StateSpaceModel) - Required - The state space model with estimated hyperparameters.
- **filter** (KalmanFilter) - Optional - The filter to use. Defaults to `default_filter(model)`.

### Returns
- **FilterOutput** - Structure with the results of the Kalman filter.
```

--------------------------------

### Forecast output for 24 steps ahead

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/examples

Output of the forecast command, showing the forecasted values and standard errors for the next 24 time points.

```text
StateSpaceModels.Forecast{Float64}([[6.1100783269760806], [6.053643535089317], [6.171326207071635], [6.199284566888648], [6.232663436588682], [6.368794427137307], [6.50742299366151], [6.503004703603158], [6.324632038853844], [6.209032320028253]  …  [6.26744855378555], [6.295406913602562], [6.328785783302597], [6.464916773851222], [6.603545340375425], [6.5991270503170725], [6.420754385567759], [6.305154666742168], [6.159527964824452], [6.263985795085841]], [[0.0013606492643246676;;], [0.0018485279562723811;;], [0.0023364066482185275;;], [0.002824285340164674;;], [0.0033121640321108203;;], [0.0038000427240569667;;], [0.004287921530288826;;], [0.004775800322234974;;], [0.0052636791141811206;;], [0.005751557906128267;;]  …  [0.010255711840981315;;], [0.0112756523907784;;], [0.012295592969832864;;], [0.013315533581722543;;], [0.01433547439154342;;], [0.015355415366095434;;], [0.01637535614434456;;], [0.017395296922618417;;], [0.01842137626486757;;], [0.019434891531997466;;]])
```

--------------------------------

### Forecast 24 steps ahead with SARIMA model

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/examples

Creates a forecast of 24 steps ahead from the fitted SARIMA model. The output shows the forecasted values and their standard errors.

```julia
forec = forecast(model, 24)
```

--------------------------------

### SeasonalNaive model constructor

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructor for a seasonal naive model where the forecast is the value from the same season in the previous cycle. The seasonal period m is provided as the second argument.

```julia
SeasonalNaive(y::Vector{<:Real}, seasoanl::Int)
```

--------------------------------

### Unconstrain box with unconstrain_box!

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Map an unconstrained hyperparameter ψ∗∈R to a constrained hyperparameter ψ∈[lb,ub]. The mapping is ψ∗=−lnub−lbψ−lb−1

```julia
unconstrain_box!(model::StateSpaceModel, str::String, lb::Fl, ub::Fl) where Fl
```

--------------------------------

### forecast(model::StateSpaceModel, steps_ahead::Int; kwargs...)

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

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

```APIDOC
## forecast(model::StateSpaceModel, steps_ahead::Int; kwargs...)

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

### Method
Function call

### Endpoint
`forecast(model::StateSpaceModel, steps_ahead::Int; kwargs...)`

### Parameters
- **model** (StateSpaceModel) - Required - The state space model to forecast from.
- **steps_ahead** (Int) - Required - Number of steps ahead to forecast.
- **kwargs** - Optional - Additional keyword arguments (not specified).

### Response
Returns the forecast mean and covariance for future observations.
```

--------------------------------

### StateSpaceModels.print_results

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

The `print_results` function prints the optimization results stored in a fitted state-space model.

```APIDOC
## `StateSpaceModels.print_results` — Function

### Description
Query the results of the optimization called by `fit!`.

### Signature
```julia
print_results(model::StateSpaceModel)
```

### Parameters
- **model** (`StateSpaceModel`) - Required - The fitted state-space model whose optimization results are to be printed.
```

--------------------------------

### Constrain box with constrain_box!

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Map a constrained hyperparameter ψ∈[lb,ub] to an unconstrained hyperparameter ψ∗∈R. The mapping is ψ=lb+ub−lb1+exp(−ψ∗)

```julia
constrain_box!(model::StateSpaceModel, str::String, lb::Fl, ub::Fl) where Fl
```

--------------------------------

### SARIMA model estimation results

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/examples

Output of fitting the SARIMA(0,1,1)x(0,1,1,12) model, showing parameter estimates, standard errors, z-statistics, and p-values. The estimates match the paper 'State Space Methods in Ox/SsfPack' up to the 3rd decimal place.

```text
                             Results
===============================================================
Model:                        SARIMA(0, 1, 1)x(0, 1, 1, 12) with zero mean
Number of observations:       144
Number of unknown parameters: 3
Log-likelihood:               244.6323
AIC:                          -483.2645
AICc:                         -483.0931
BIC:                          -474.3551
---------------------------------------------------------------
Parameter      Estimate      Std.Error      z stat      p-value
ma_L1           -0.4011         0.1291     -3.1079       0.0019
s_ma_L12        -0.5561         0.1462     -3.8029       0.0001
sigma2_η         0.0014         0.0024      0.5646       0.5724
```

--------------------------------

### Output of fitting LocalLevel model

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/examples

Displays the fitted model object, which is a LocalLevel model.

```text
LocalLevel
```

--------------------------------

### LinearUnivariateTimeInvariant constructor

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Definition of the system matrices y, Z, d, T, c, R, H, Q for linear univariate time invariant state space models. The model equations are yt = Zαt + d + εt with εt ~ N(0, H) and αt+1 = Tαt + c + Rηt with ηt ~ N(0, Q).

```julia
LinearUnivariateTimeInvariant{Fl}(
    y::Vector{Fl},
    Z::Vector{Fl},
    T::Matrix{Fl},
    R::Matrix{Fl},
    d::Fl,
    c::Vector{Fl},
    H::Fl,
    Q::Matrix{Fl},
) where Fl <: AbstractFloat
```

--------------------------------

### LinearUnivariateTimeVariant constructor

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Definition of the system matrices y, Z, d, T, c, R, H, Q for linear univariate time variant state space models. The model equations are yt = Ztαt + dt + εt with εt ~ N(0, Ht) and αt+1 = Ttαt + ct + Rtηt with ηt ~ N(0, Qt).

```julia
LinearUnivariateTimeVariant{Fl}(
    y::Vector{Fl},
    Z::Vector{Vector{Fl}},
    T::Vector{Matrix{Fl}},
    R::Vector{Matrix{Fl}},
    d::Vector{Fl},
    c::Vector{Vector{Fl}},
    H::Vector{Fl},
    Q::Vector{Matrix{Fl}},
) where Fl <: AbstractFloat
```

--------------------------------

### StateSpaceModels.fit!

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

The `fit!` function estimates state-space model parameters via maximum likelihood, allowing customization of the filter and optimizer.

```APIDOC
## `StateSpaceModels.fit!` — Function

### Description
Estimate the state-space model parameters via maximum likelihood. The resulting optimal hyperparameters and the corresponding log-likelihood are stored within the model. You can choose the desired filter method (`UnivariateKalmanFilter`, `ScalarKalmanFilter`, etc.) and the Optim.jl optimization algorithm.

### Signature
```julia
fit!(
    model::StateSpaceModel;
    filter::KalmanFilter=default_filter(model),
    optimizer::Optimizer=default_optimizer(model),
    save_hyperparameter_distribution::Bool=true
)
```

### Parameters
- **model** (`StateSpaceModel`) - Required - The state-space model to be fitted.
- **filter** (`KalmanFilter`) - Optional - The filter method to use (e.g., `UnivariateKalmanFilter`, `ScalarKalmanFilter`). Defaults to `default_filter(model)`.
- **optimizer** (`Optimizer`) - Optional - The Optim.jl optimization algorithm to use. Defaults to `default_optimizer(model)`.
- **save_hyperparameter_distribution** (`Bool`) - Optional - Whether to save the hyperparameter distribution. Defaults to `true`.

### Example
```julia
julia> model = LocalLevel(rand(100))
LocalLevel

julia> fit!(model)
LocalLevel

julia> model = LocalLinearTrend(LinRange(1, 100, 100) + rand(100))
LocalLinearTrend

julia> fit!(model; optimizer = Optimizer(StateSpaceModels.Optim.NelderMead()))
LocalLinearTrend
```
```

--------------------------------

### Cite StateSpaceModels.jl with BibTeX

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable

BibTeX entry for citing StateSpaceModels.jl in academic work.

```bibtex
@article{SaavedraBodinSouto2019,
title={StateSpaceModels.jl: a Julia Package for Time-Series Analysis in a State-Space Framework},
author={Raphael Saavedra and Guilherme Bodin and Mario Souto},
journal={arXiv preprint arXiv:1908.01757},
year={2019}
}
```

--------------------------------

### Unconstrain variance with unconstrain_variance!

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Map an unconstrained hyperparameter ψ∗∈R to a constrained hyperparameter ψ∈R+. The mapping is ψ∗=√ψ.

```julia
unconstrain_variance!(model::StateSpaceModel, str::String)
```

--------------------------------

### get_innovations

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Retrieves the innovations from the Kalman filter output.

```APIDOC
## get_innovations

### Description
Returns the innovations `v` obtained with the Kalman filter.

### Method
Function call

### Endpoint
`get_innovations`

### Returns
- **v** - Innovations from the Kalman filter.
```

--------------------------------

### StateSpaceModels.auto_ets

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Automatically selects the best ExponentialSmoothing model based on AIC.

```APIDOC
## auto_ets(y::Vector{Fl}; seasonal::Int = 0) where Fl

### Description
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).

### Method
Function

### Endpoint
`StateSpaceModels.auto_ets(y::Vector{Fl}; seasonal::Int = 0) where Fl`

### Parameters
- **y** (Vector{Fl}) - Required - The time series data.
- **seasonal** (Int) - Optional - The seasonal period (default: 0).

### References
- Rob J. Hyndman, and George Athanasopoulos. Forecasting: Principles and Practice. 2nd ed. OTexts, 2018.
```

--------------------------------

### Load and prepare Finland road traffic fatalities data

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/examples

Loads the vehicle fatalities dataset and computes the natural logarithm of the fatalities column. The result is stored in `log_ff` for subsequent modeling.

```julia
using StateSpaceModels, CSV, DataFrames
df = CSV.File(StateSpaceModels.VEHICLE_FATALITIES) |> DataFrame
log_ff = log.(df.ff)
```

--------------------------------

### Load Nile river flow data with CSV and DataFrames

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/examples

Loads the built-in Nile river annual flow dataset using CSV.File and converts it to a DataFrame. The data contains 100 rows with columns for year (Date) and flow (Float64).

```julia
using CSV, DataFrames
nile = CSV.File(StateSpaceModels.NILE) |> DataFrame
```

--------------------------------

### StateSpaceModels.has_fit_methods

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

The `has_fit_methods` function checks whether a given state-space model type supports the `fit!` operation.

```APIDOC
## `StateSpaceModels.has_fit_methods` — Function

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

### Signature
```julia
has_fit_methods(model_type::Type{<:StateSpaceModel}) -> Bool
```

### Parameters
- **model_type** (`Type{<:StateSpaceModel}`) - Required - The type of the state-space model to check.

### Returns
- **Bool** - `true` if the model type has the necessary methods for `fit!`, `false` otherwise.
```

--------------------------------

### Simulate scenarios function signature

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Samples n_scenarios future scenarios via Monte Carlo simulation for steps_ahead using the desired filter. Returns a 3-dimensional array of floats.

```julia
simulate_scenarios(
    model::StateSpaceModel, steps_ahead::Int, n_scenarios::Int;
    filter::KalmanFilter=default_filter(model)
) -> Array{<:AbstractFloat, 3}
```

--------------------------------

### Fit LocalLinearTrend model to log fatalities

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/examples

Fits a LocalLinearTrend model to the log-transformed fatalities data. The model is stored in `model` and then fitted using `fit!`.

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

--------------------------------

### StateSpaceModels.Optimizer

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

The `Optimizer` type wraps Optim.jl optimizers for use in StateSpaceModels.jl, providing a straightforward way to select optimization algorithms.

```APIDOC
## `StateSpaceModels.Optimizer` — Type

### Description
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
julia> using Optim

# use a semicolon to avoid displaying the big log
julia> opt = Optimizer(Optim.BFGS(), Optim.Options(show_trace = true));
```
```

--------------------------------

### Filter and smooth Nile flow with missing values

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/examples

Runs the Kalman filter and smoother on the fitted model, then extracts the smoothed state. The exact same code works as when there are no missing values.

```julia
filter_output = kalman_filter(model)
smoother_output = kalman_smoother(model)
get_smoothed_state(smoother_output)
```

--------------------------------

### unconstrain_box!

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Applies inverse box constraint mapping to a hyperparameter.

```APIDOC
## unconstrain_box!(model::StateSpaceModel, str::String, lb::Fl, ub::Fl) where Fl

### Description
Map an unconstrained hyperparameter ψ∗∈R to a constrained hyperparameter ψ∈[lb,ub]. The mapping is ψ∗=−lnub−lbψ−lb−1.

### Method
Function

### Endpoint
`StateSpaceModels.unconstrain_box!(model::StateSpaceModel, str::String, lb::Fl, ub::Fl) where Fl`

### Parameters
- **model** (`StateSpaceModel`) - Required - The state space model.
- **str** (`String`) - Required - The name of the hyperparameter to unconstrain.
- **lb** (`Fl`) - Required - Lower bound of the box constraint.
- **ub** (`Fl`) - Required - Upper bound of the box constraint.

### Response
Returns the modified model.
```

--------------------------------

### Extract smoothed slope from LocalLinearTrend model

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/examples

Runs the Kalman smoother on the fitted model and extracts the smoothed state for the slope component (second column). The resulting vector indicates the trend direction over time.

```julia
smoother_output = kalman_smoother(model)
get_smoothed_state(smoother_output)[:, 2]
```

--------------------------------

### LinearRegression

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructor for the LinearRegression model type.

```APIDOC
## LinearRegression

### Description
Constructs a linear regression state-space model where the observation is a linear combination of explanatory variables with time-varying coefficients that follow a random walk.

### Constructor
`LinearRegression(X::Matrix{Fl}, y::Vector{Fl}) where Fl`

### Parameters
- **X** (Matrix{Fl}) - Required - The matrix of explanatory variables.
- **y** (Vector{Fl}) - Required - The response variable.

### Example
```julia
julia> model = LinearRegression(rand(100, 2), rand(100))
LinearRegression
```
```

--------------------------------

### StateSpaceModels.isfitted

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

The `isfitted` function checks whether a state-space model has been successfully fitted by verifying that no hyperparameters are `NaN`.

```APIDOC
## `StateSpaceModels.isfitted` — Function

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

### Signature
```julia
isfitted(model::StateSpaceModel) -> Bool
```

### Parameters
- **model** (`StateSpaceModel`) - Required - The state-space model to check.

### Returns
- **Bool** - `true` if the model is fitted (no `NaN` hyperparameters), `false` otherwise.
```

--------------------------------

### Unconstrain identity with unconstrain_identity!

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Map an unconstrained hyperparameter ψ∗∈R to a constrained hyperparameter ψ∈R. This function is necessary to copy values from a location to another inside `HyperParameters`. The mapping is ψ∗=ψ

```julia
unconstrain_identity!(model::StateSpaceModel, str::String)
```

--------------------------------

### ExponentialSmoothing Constructor Signature

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Constructor signature for the ExponentialSmoothing (ETS) type. Accepts a vector y and optional keyword arguments to enable a trend, damped trend, and seasonal component with a given periodicity.

```julia
ExponentialSmoothing(
    y::Vector{Fl};
    trend::Bool = false,
    damped_trend::Bool = false,
    seasonal::Int = 0
) where Fl
```

--------------------------------

### Constrain variance with constrain_variance!

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Map a constrained hyperparameter ψ∈R+ to an unconstrained hyperparameter ψ∗∈R. The mapping is ψ=ψ2∗

```julia
constrain_variance!(model::StateSpaceModel, str::String)
```

--------------------------------

### StateSpaceModels.auto_arima

Source: https://lampspuc.github.io/StateSpaceModels.jl/stable/manual

Automatically selects the best SARIMA model based on information criteria.

```APIDOC
## auto_arima(y::Vector{Fl}; seasonal::Int = 0, max_p::Int = 5, max_q::Int = 5, max_P::Int = 2, max_Q::Int = 2, max_d::Int = 2, max_D::Int = 1, max_order::Int = 5, information_criteria::String = "aicc", allow_mean::Bool = true, show_trace::Bool = false, integration_test::String = "kpss", seasonal_integration_test::String = "seas") where Fl

### Description
Automatically fits the best `SARIMA` model according to the best information criteria.

### Method
Function

### Endpoint
`StateSpaceModels.auto_arima(y::Vector{Fl}; seasonal::Int = 0, max_p::Int = 5, max_q::Int = 5, max_P::Int = 2, max_Q::Int = 2, max_d::Int = 2, max_D::Int = 1, max_order::Int = 5, information_criteria::String = "aicc", allow_mean::Bool = true, show_trace::Bool = false, integration_test::String = "kpss", seasonal_integration_test::String = "seas") where Fl`

### Parameters
- **y** (Vector{Fl}) - Required - The time series data.
- **seasonal** (Int) - Optional - The seasonal period (default: 0).
- **max_p** (Int) - Optional - Maximum autoregressive order (default: 5).
- **max_q** (Int) - Optional - Maximum moving average order (default: 5).
- **max_P** (Int) - Optional - Maximum seasonal autoregressive order (default: 2).
- **max_Q** (Int) - Optional - Maximum seasonal moving average order (default: 2).
- **max_d** (Int) - Optional - Maximum number of non-seasonal differences (default: 2).
- **max_D** (Int) - Optional - Maximum number of seasonal differences (default: 1).
- **max_order** (Int) - Optional - Maximum order of the model (default: 5).
- **information_criteria** (String) - Optional - Information criterion to use (default: "aicc").
- **allow_mean** (Bool) - Optional - Whether to allow a mean term (default: true).
- **show_trace** (Bool) - Optional - Whether to show trace of the search (default: false).
- **integration_test** (String) - Optional - Test for non-seasonal integration (default: "kpss").
- **seasonal_integration_test** (String) - Optional - Test for seasonal integration (default: "seas").

### References
- Hyndman, RJ and Khandakar. Automatic time series forecasting: The forecast package for R. Journal of Statistical Software, 26(3), 2008.
```