intame_metric.Rmd

---
title: Intame Losses
author: 
  - name: Bodo Burger
    #affiliation: LMU Munich
    #address: >
    #  First line
    #  Second line
    #email: \email{bb@example.com}
    #url: http://example.com
  #- name: Second Author
    #affiliation: Affiliation
date: 2019-03-08
output:
  html_document: 
    toc: yes
  github_document:
    toc: yes
  pdf_document:
    toc: yes
urlcolor: blue
#bibliography: "literature.bib"
#link-citations: true
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, message = FALSE, warning = TRUE,
                      cache = FALSE, cache.path = "cache/intame_metrics")
library("ggplot2")
library("intame")
library("SimilarityMeasures")
theme_set(theme_light())
set.seed(4218)
```

## Overview

Metric Name    | Inputs         | Range of values | Default theshold | Optimum
-------------- | -------------- | --------------- | ---------------- | --------
R2int          | $f$, residuals | $(0, 1)$        | $0.95$           |  max
L2             | residuals      | $(0, +\infty($  | sd(FE)/5         |  min
L1             | residuals      | $(0, +\infty($  | sd(FE)/5         |  min
Frechet        | $f, \hat{f}$   | $(0, +\infty($  | *?*              |  min           

## Formulae

- **R2int**
    - **interval**: $1 - \frac{\sum \text{residual}_i^2}{\sum f - \overline{f}}$
    - **aggregate**: $\frac{\sum n_k part_k}{\sum n_k}$ (mean weighted by number of points per interval)
- **L2**: 
    - $\frac{\sum \text{residual}_i^2}{n}$

$\text{residual}_i = f(x_i) - \hat{f}(x_i)$

## Suggest threshold

```{r}
lm.mod = lm(mpg ~ ., data = mtcars)
features_of_interest = c("disp", "hp", "drat", "wt")

suggest_threshold(lm.mod, mtcars, features_of_interest, "R2int", fe_method = "ALE")
st = suggest_threshold(lm.mod, mtcars, features_of_interest, "L1", fe_method = "PD")
st
```


## Frechet Distance

```{r,collapse=TRUE}
x = c(0, 1, 2, 3)
y1 = c(3, 2, 1, 2)
y2 = c(6, 5, 6, 7)
y3 = c(7, 6, 6, 7)
y4 = c(2, 2, 2)

path1 = matrix(c(x, y1-mean(y1)), 4)
path2 = matrix(c(x, y2-mean(y2)+.5), 4)
path3 = matrix(c(x, y3-mean(y3)), 4)
path4 = matrix(c(0, 0, 2, y4-mean(y4)), 3)

plot(path1[,1], path1[,2], type = "l", col = "blue", ylim = c(-3, 3))
lines(path2[,1], path2[,2]+.03, col = "green")
lines(path3[,1], path3[,2], col = "red")
lines(path4[,1], path4[,2], col = "orange")

Frechet(path1, path2)
Frechet(path1, path3)
Frechet(path1, path4)
Frechet(path2, path3)
Frechet(path2, path4)
Frechet(path3, path4)
```