---
title: "Precipitation forecasts"
author: "Patrick Schmidt"
date: "`r Sys.Date()`"
output:
  rmarkdown::html_vignette:
    fig_caption: yes

vignette: >
  %\VignetteIndexEntry{Precipitation}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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


# Data

```{r, fig.show='hold', fig.width = 6, fig.cap = 'Time series of one-day ahead HRES Precipitation forecasts (crosses) and respective observations (solid line) over London.'}
  
library(PointFore)
library(ggplot2)
library(lubridate)

precipitation$Date <- as.Date(row.names(precipitation),format = "%d-%m-%Y")


ggplot(subset(precipitation, month(Date)< 7 & year(Date)==2013))+
  geom_line(aes(x=Date,y=Y))+
  geom_point(aes(x=Date,y=X), color = 'red', size = 2, shape=4)
  
```

For more information on the data see `?precipitation`.


# Analysis

Now, let us analyse the forecasts. We begin with the constant expectile model.

```{r, fig.show='hold', fig.cap = 'Constant expectile analysis.'}

instruments <- c("lag(lag(Y))","X")

res <- estimate.functional(iden.fct = expectiles, model = constant,
                           instruments = instruments,
                           Y = precipitation$Y, X=precipitation$X)
summary(res)
plot(res,hline = TRUE)
```

Optimality is rejected with a p-value of `r round(summary(res$gmm)$stest$test[2],digits=2)`. On average the forecast tends to overestimation compared to an optimal mean forecast with an expectile level of `r round(summary(res$gmm)$coefficients[1],digits=2)`.


Next, we consider state-dependent forecasting behavior. Instead of using the conventional state-dependence models we rely on the linear probit specification model but enforce an expectile level of $0$ for the forecast $0$. This is a logical consequence of precipitation being a positive random variable.

```{r, fig.show='hold', fig.cap = 'linear probit model.'}
probit0 <- function(stateVariable,theta) probit_linear(stateVariable, theta)*(stateVariable>0)

res <- estimate.functional(iden.fct =   expectiles ,
                           model = probit0,
                           theta0 = c(0,0),
                           instruments = instruments,
                           state = precipitation$X,
                           Y = precipitation$Y, X=precipitation$X)
summary(res)
```

To replicate the result plot in the paper , we need to adjust the standard plot function of the PointFore package to the probit0 specification model.


```{r, fig.show='hold', fig.cap = 'Plot linear probit model.'}
plot(res,limits = c(0.001,15),hline = TRUE)+
  geom_point(data=data.frame(x=c(0,0),y=c(0,.395),shape=c(1,2)),
             aes(x=x,y=y,shape=as.factor(shape)),
             ,size=3,show.legend = FALSE)+
  scale_shape_manual(values=c(16,1))
```
