# distFrechet: ~ Function: Frechet distance ~ In kmlShape: K-Means for Longitudinal Data using Shape-Respecting Distance

## Description

Compute Frechet distance between two trajectories.

## Usage

 ```1 2 3``` ```distFrechet(Px,Py,Qx, Qy, timeScale=0.1, FrechetSumOrMax = "sum") distFrechetR(Px,Py,Qx, Qy, timeScale=0.1, FrechetSumOrMax = "sum") distFrechetRec(Px,Py,Qx, Qy, timeScale=0.1, FrechetSumOrMax = "sum") ```

## Arguments

 `Px` [vector(numeric)] Times (abscisse) of the first trajectories. `Py` [vector(numeric)] Values of the first trajectories. `Qx` [vector(numeric)] Times of the second trajectories. `Qy` [vector(numeric)] Values of the second trajectories. `timeScale` [`numeric`]: allow to modify the time scale, increasing or decreasing the cost of the horizontal shift. If timeScale is very big, then the Frechet's distance is equal to the euclidienne distance. If timeScale is very small, then it is equal to the Dynamic Time Warping. `FrechetSumOrMax` [`character`]: The Frechet's distance can be define using the 'sum' function or the 'max' function. This option let the user to chose one or the other.

## Details

Given two curve P and Q, Frechet distance between P and Q is define as `inf_{a,b} max_{t} d(P(a(t)),Q(b(t)))`. It's computation is a NP-complex problem. When P and Q are trajectories (discrete curve), the problem is polynomial.

The Frechet distance can also be define using a sum instead of a max: `inf_{a,b} sum_{t} d(P(a(t)),Q(b(t)))`

The function `distFrechet` is C compiled, the function `distFrechetR` is in R, the function `distFrechetRec` is in recursive (the slowest) in R.

A numeric value.

## Author

Christophe Genolini
1. UMR U1027, INSERM, Universit<e9> Paul Sabatier / Toulouse III / France
2. CeRSM, EA 2931, UFR STAPS, Universit<e9> de Paris Ouest-Nanterre-La D<e9>fense / Nanterre / France

## References

[1] Thomas Eiter & Heikki Mannila:
"Computing Discrete Fr<b4>echet Distance"

[2] C. Genolini and B. Falissard
"KmL: k-means for longitudinal data"
Computational Statistics, vol 25(2), pp 317-328, 2010

[3] C. Genolini and B. Falissard
"KmL: A package to cluster longitudinal data"
Computer Methods and Programs in Biomedicine, 104, pp e112-121, 2011

distTraj

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ``` Px <- 1:20 Py <- dnorm(1:20,12,2) Qx <- 1:20 Qy <- dnorm(1:20,8,2) ### Function from Eiter and Mannila compiled in C system.time(cat("\n",distFrechet(Px,Py,Qx,Qy))) ### Same thing in R system.time(cat("\n",distFrechetR(Px,Py,Qx,Qy))) ### Frechet using sum instead of max. distFrechet(Px,Py,Qx,Qy,FrechetSumOrMax="sum") ```

### Example output

```Loading required package: class

Attaching package: 'kmlShape'

The following object is masked from 'package:longitudinalData':

distFrechet

Warning messages:
1: In rgl.init(initValue, onlyNULL) : RGL: unable to open X11 display
2: 'rgl_init' failed, running with rgl.useNULL = TRUE
call: fun(...)