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

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

```{r setup, output=FALSE}
library(serosv)
```

## Local estimation by polynomial

Refer to `Chapter 7.1`

**Proposed model**

Within the local polynomial framework, the linear predictor $\eta(a)$ is approximated locally at one particular value $a_0$ for age by a line (local linear) or a parabola (local quadratic).

The estimator for the $k$-th derivative of $\eta(a_0)$, for $k = 0,1,…,p$ (degree of local polynomial) is as followed:

$$
\hat{\eta}^{(k)}(a_0) = k!\hat{\beta}_k(a_0) 
$$

The estimator for the prevalence at age $a_0$ is then given by

$$
\hat{\pi}(a_0) = g^{-1}\{ \hat{\beta}_0(a_0) \}
$$

-   Where $g$ is the link function

The estimator for the force of infection at age $a_0$ by assuming $p \ge 1$ is as followed

$$
\hat{\lambda}(a_0) = \hat{\beta}_1(a_0) \delta \{ \hat{\beta}_0 (a_0) \} 
$$

-   Where $\delta \{ \hat{\beta}_0(a_0) \} = \frac{dg^{-1} \{ \hat{\beta}_0(a_0) \} } {d\hat{\beta}_0(a_0)}$

**Fitting data**

```{r}
mump <- mumps_uk_1986_1987
age <- mump$age
pos <- mump$pos
tot <- mump$tot
y <- pos/tot
```

Use `plot_gcv()` to show GCV curves for the nearest neighbor method (left) and constant bandwidth (right).

```{r, fig.width=7, fig.height=3}
plot_gcv(
   age, pos, tot,
   nn_seq = seq(0.2, 0.8, by=0.1),
   h_seq = seq(5, 25, by=1)
 )
```

Use `lp_model()` to fit a local estimation by polynomials.

```{r}
lp <- lp_model(age, pos = pos, tot = tot, kern="tcub", nn=0.7, deg=2)
plot(lp)
```