---
title: "Regular periodic cubic splines"
author: "Jean Palate"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Regular periodic cubic splines}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

## Use of regular periodic cubic splines

```{r echo=FALSE}

suppressPackageStartupMessages(library(rjd3toolkit))
suppressPackageStartupMessages(library(rjd3sts))
library(knitr)
```

```{r data}
s<-log(ABS$X0.2.09.10.M)
```

### Usual BSM with Harrison-Stevens seasonal component

```{r bsm}
model<-model()

llt<-locallineartrend('l')
seas<-seasonal("s", 12, "HarrisonStevens")
n<-noise('n')
add(model, llt)
add(model, seas)
add(model, n)

rslt<-estimate(model, s)

sa1<-result(rslt, "ssf.smoothing.components")
```
###  BSM with full periodic splines seasonal component

The results of the first two models should be identical (up to numerical precision)


```{r bsm_splines1}
model<-model()
seas<-splines_regular("s", 12, nodes=c(0:11))
add(model, llt)
add(model, seas)
add(model, n)

rslt<-estimate(model, s)

sa2<-result(rslt, "ssf.smoothing.components")

summary(sa1[,2]-sa2[,2])
```

###  BSM with partial periodic splines seasonal component

The splines are computed on 8 points (instead of 12)
```{r bsm_splines2, fig.width=6, fig.height=5}
model<-model()
seas<-splines_regular("s", 12, nodes=c(1,2,6,7,8,9,10,11))
add(model, llt)
add(model, seas)
add(model, n)

rslt<-estimate(model, s)

sa3<-result(rslt, "ssf.smoothing.components")


matplot(cbind(sa1[301:336,2],sa2[301:336,2],sa3[301:336,2]), type='l', ylab="Seasonal")
```