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

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

The purpose of this vignette is to present the calculations for a peicewise quantile regression where for each time step there are multiple independent observations.

In the follow variables identified by Greek letters are considered unknown.

# Quantile regression

Data belongs to group $k$ whose time stamps are the set $t \in T_{k}$ which have common regression parameters $\theta_{k}$ and residual variance $\sigma_{k}$
At time step $t$ the vector of iid observations $\mathbf{y}_{t}=\left\{y_{t,1},\ldots,t_{t,n_{t}}\right\}$ is explained by the design matrix $\mathbf{X}_{t}$.

For a given quantile $\tau$ and using the check function $\rho\left(u,\tau\right) = u\left(\tau - I\left(u<0\right)\right)$ Koenker and Bassett (1978) show that an estimate of $\theta$ in QR model can be obtained by solving the convex optimization problem
\[
\min_{\theta} \left( \sum_{i=1}^{n_{t}} \rho\left(\mathbf{y}_{t,i}- \mathbf{X}_{t,i}\left(\mathbf{m}_{t}+\theta_{k}\right),\tau\right)
\right)
\]

Solving this gives the maximum likelihood estimator of the asymmetric Laplace (AL) distributions (Geraci and
Bottai, 2007 and Yu, Lu, and Stander, 2003) which has likelihood
\[
L\left(\mathbf{y}_{t} \left| \theta_k\right.\right) = \tau^{n_{t}}\left(1-\tau\right)^{n_{t}}\exp\left(-
\sum_{t=1}^{n_{t}} \rho\left(\mathbf{y}_{t,i}- \mathbf{X}_{t,i}\left(\mathbf{m}_{t}+\theta_{k}\right),\tau\right)
\right)
\]

With $\hat{\mathbf{y}}_{t} = \mathbf{y}_{t} - \mathbf{X}_{t} \mathbf{m}_{t}$ the log likelihood is given by
\[
l\left(\mathbf{y}_{t} \left| \theta_k,\sigma_k \right.\right) = 
n_{j}\log \left(\tau \left(1-\tau\right)\right) - 
\sum_{i=1}^{n_{t}} \rho\left(\hat{\mathbf{y}}_{t,i} - \mathbf{X}_{t,i}\theta_{k},\tau\right)
\]


The log-likelihood of $\mathbf{y}_{t \in T_{k}}$ is with $n_{k}=\sum\limits_{t\in T_{k}} n_{t}$
\[
l\left(\mathbf{y}_{t \in T_{k}} \left| \theta_k,\sigma_k,\mathbf{X}_{t}\right.\right) = 
n_{k}\log\left(\tau \left(1-\tau\right)\right) - 
\sum_{t \in T_{k}}\sum_{i=1}^{n_{t}} \rho\left(\hat{\mathbf{y}}_{t,i} - \mathbf{X}_{t,i}\theta_{k},\tau\right)
\]

with the cost being twice the negative log likelihood plus a penalty $\beta$ giving 

\[
C\left(\mathbf{y}_{t \in T_{k}} \left| \mu_t,m_k,\sigma_k,s_k\right.\right) = 
\sum_{t \in T_{k}}\sum_{i=1}^{n_{t}} \rho\left(\hat{\mathbf{y}}_{t,i} - \mathbf{X}_{t,i}\theta_{k},\tau\right)
- 2n_{k}\log\left(\tau \left(1-\tau\right)\right) 
+ \beta
\]

## Baseline: No Anomaly

Here $\theta_{k}=0$ and is no penalty so $\beta = 0$

## Collective anomaly

Estimate $\theta_{k}$ using ??? and then with penalty $\beta$
\[
C\left(\mathbf{y}_{t \in T_{k}} \left| \mu_t,m_k,\sigma_k,s_k\right.\right) = 
\sum_{t \in T_{k}}\sum_{i=1}^{n_{t}} \rho\left(\hat{\mathbf{y}}_{t,i} - \mathbf{X}_{t,i}\hat{\theta}_{k},\tau\right)
- 2n_{k}\log\left(\tau \left(1-\tau\right)\right) 
+ \beta
\]

## Point Anomaly

if $n_t > 0$ then could proceed like a collective anomaly. Otherwise select $\hat{\theta}$ such that
$\mathbf{y}}_{t,i} - \mathbf{X}_{t,i}\hat{\theta}_{k}= 0$