---
title: "sdrt-vignette"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{sdrt-vignette}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

## Overview

The **sdrt** R package provides powerful capabilities for estimating a basis for the Time Series Central Mean Subspace (TS-CMS). This `vignette` serves as a comprehensive guide to the functionalities offered by the **sdrt**  package, utilizing the `lynx` dataset as an illustrative example.

## Chapter 1: Installation of **itdr** package

The **itdr** R package can be installed in two ways:  

* From the Comprehensive R Archive Network (CRAN):

The following code chunk demonstrate the installation of the package using the `install.packages()` function in R. Then, the library can be import into the current R session using the  `library()` function. 

```{r eval=FALSE, include=TRUE}
install.packages("sdrt")
library(sdrt)
```

* From GitHub:

The development version of **sdrt** R package can be installed form GitHub.  

```{r eval=FALSE, include=TRUE}
library(devtools)
install_github("TharinduPDeAlwis/sdrt")
library(sdrt)
```


## Chapter 2: Fourier Transformation Method to Estimate the TS-CMS.

In this section, we demonstrate the functions within the **sdrt** package that utilize the Fourier method to estimate the sufficient dimension reduction (SDR) subspaces in time series. 

In Section 2.1 brings the estimation of model parameters. The tuning the hyper parameter in Section 2.2 and the estimation of a basis of the TS-CMS explains in Section 2.3.  

### 2.1: Estimating Model Parameters

The first step is to estimate the model parameters, specifically `p` and `d`. The block bootstrap procedure is employed to estimate the unknown lags number (`p`) and the unknown dimension of the TS-CMS  (`d`). For more details refer to Samadi and De Alwis (2023). The following R code chunk demonstrate the use of the  `pd.boots()` function to estimate `p` and `d`. 

```{r eval=FALSE, include=TRUE}
data("lynx")
y <- log10(lynx)
p_list=seq(2,5,by=1)
fit.model=pd.boots(y,p_list,w1=0.1,B=10)
fit.model$dis_pd
fit.model$p_hat
fit.model$d_hat
```

In this example, the estimated lag number and the dimension of the TS-CMS are denoted as `p_hat=3` and `d_hat=1` for the `lynx` dataset. 

### 2.2: Tuning the Model Parameter
Within the Fourier method for estimating the TS-CMS, there exists a hyperparameter \(\sigma_u^2\). While a recommended value of \(\sigma_u^2=0.1\) exists, the **sdrt** package  offers users the flexibility to fine-tune this parameter for each dataset. The following R code chunk outlines the procedure using the `sigma_u()` function to tune this parameter for the `lynx` dataset. 

```{r eval=FALSE, include=TRUE}
set.seed(1)
data("lynx")
y <- log10(lynx)
p <- 3
d <- 1
w1_list=seq(0.1,0.5,by=0.1)
Tunning.model=sigma_u(y, p, d, w1_list=w1_list, std=FALSE, B=10)
Tunning.model$sigma_u_hat
```

In this example, the estimated tuning parameter for the `lynx` dataset is \(\sigma_u^2=0.3\). Users can adapt this process to optimize the hyperparameter for their specific dataset. 

### 2.3: Estimating the TS-CMS

We have described the estimation procedure of model parameters, `p` and `d` in Section 2.1 and the tuning parameter \(\sigma_u^2\) in Section 2.2. Now, we are ready to estimate the Time Series Central Mean Subspace (TS-CMS). The TS-CMS can be estimated using the `sdrt()` function by passing the `method` argument as `"FM"`. The following R code chunk illustrate the use of the `sdrt()` function for this purpose.   

```{r eval=FALSE, include=TRUE}
library(sdrt)
data("lynx")
y <- log10(lynx)
p <- 3
d <- 1
fit.model <- sdrt(y, p, d=1,method="FM",density = "kernel")
fit.model$eta_hat
```

In this example, the estimated TS-CMS is obtained using the Fourier method ("FM"). Users can customize the parameters according to their dataset and analysis needs.

## Chapter 3: Nadaraya-Watson Method to Estimate the TS-CMS.

In this section, we demonstrate the application of the `sdrt()` function within the **sdrt** R package to estimate the TS-CMS using the Nadaraya-Watson (NW) method, as proposed by Park et al. (2009). The following R code chunk illustrates the utilization of the `sdrt()` on the `lynx` dataset. Here, we specify the `method` argument as `"NW"` to indicate the Nadaraya-Watson estimation method. 


```{r eval=FALSE, include=TRUE}
library(pracma)
data("lynx")
y <- log10(lynx)
p <- 4
d <- 1
fit.model <- sdrt(y, p, d,method="NW")
fit.model$eta_hat
```
In this example, the Nadaraya-Watson method is employed to estimate the TS-CMS, and users can adjust parameters based on their specific dataset requirements.