~ Function: Frechet distance ~

Description

Compute Frechet distance and Frechet path between two trajectories.

Usage

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pathFrechet(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 path is only a set of vertical segment. If timeScale is very small, then it is equal to the path find by Dynamic Time Warping.

FrechetSumOrMax

[character]: Like Frechet's distance, the Frechet's path 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 distFrechet(P,Q)=inf_{a,b} max_{t} d(P(a(t)),Q(b(t))).

The Frechet path is the couple of function (a(t),b(t)) that realize the previous equality : distFrechet(P,Q)=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 path 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 pathFrechet is C compiled whereas the function pathFrechetR is in R.

Value

A numeric value and the Frechet path in a matrix.

Author

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

References

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

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

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

See Also

distFrechet

Examples

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   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(pathFrechet(Px,Py,Qx,Qy))

   ### Same thing in R
   system.time(pathFrechet(Px,Py,Qx,Qy))

   ### Frechet using sum instead of max.
   pathFrechet(Px,Py,Qx,Qy,FrechetSumOrMax="sum")