# riem.dtw: Dynamic Time Warping Distance In Riemann: Learning with Data on Riemannian Manifolds

## Description

Given two time series - a query X = (X_1,X_2,…,X_N) and a reference Y = (Y_1,Y_2,…,Y_M), `riem.dtw` computes the most basic version of Dynamic Time Warping (DTW) distance between two series using a symmetric step pattern, meaning no window constraints and others at all. Although the scope of DTW in Euclidean space-valued objects is rich, it is scarce for manifold-valued curves. If you are interested in the topic, we refer to dtw package.

## Usage

 `1` ```riem.dtw(riemobj1, riemobj2, geometry = c("intrinsic", "extrinsic")) ```

## Arguments

 `riemobj1` a S3 `"riemdata"` class for M manifold-valued data along the curve. `riemobj2` a S3 `"riemdata"` class for N manifold-valued data along the curve. `geometry` (case-insensitive) name of geometry; either geodesic (`"intrinsic"`) or embedded (`"extrinsic"`) geometry.

## Value

the distance value.

`dtw`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41``` ```#------------------------------------------------------------------- # Curves on Sphere # # curve1 : y = 0.5*cos(x) on the tangent space at (0,0,1) # curve2 : y = 0.5*sin(x) on the tangent space at (0,0,1) # # we will generate two sets for curves of different sizes. #------------------------------------------------------------------- ## GENERATION clist = list() for (i in 1:10){ # curve type 1 vecx = seq(from=-0.9, to=0.9, length.out=sample(10:50, 1)) vecy = 0.5*cos(vecx) + rnorm(length(vecx), sd=0.1) mats = cbind(vecx, vecy, 1) clist[[i]] = wrap.sphere(mats/sqrt(rowSums(mats^2))) } for (i in 1:10){ # curve type 2 vecx = seq(from=-0.9, to=0.9, length.out=sample(10:50, 1)) vecy = 0.5*sin(vecx) + rnorm(length(vecx), sd=0.1) mats = cbind(vecx, vecy, 1) clist[[i+10]] = wrap.sphere(mats/sqrt(rowSums(mats^2))) } ## COMPUTE DISTANCES outint = array(0,c(20,20)) outext = array(0,c(20,20)) for (i in 1:19){ for (j in 2:20){ outint[i,j] <- outint[j,i] <- riem.dtw(clist[[i]], clist[[j]], geometry="intrinsic") outext[i,j] <- outext[j,i] <- riem.dtw(clist[[i]], clist[[j]], geometry="extrinsic") } } ## VISUALIZE opar <- par(no.readonly=TRUE) par(mfrow=c(1,2), pty="s") image(outint[,20:1], axes=FALSE, main="intrinsic DTW Distance") image(outext[,20:1], axes=FALSE, main="extrinsic DTW Distance") par(opar) ```