% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/dtw.R
\name{dtw}
\alias{dtw}
\alias{is.dtw}
\alias{print.dtw}
\title{Dynamic Time Warp}
\usage{
dtw(x, y = NULL, dist.method = "Euclidean",
  step.pattern = symmetric2, window.type = "none",
  keep.internals = FALSE, distance.only = FALSE, open.end = FALSE,
  open.begin = FALSE, ...)

is.dtw(d)

\method{print}{dtw}(x, ...)
}
\arguments{
\item{x}{query vector \emph{or} local cost matrix}

\item{y}{reference vector, unused if \code{x} given as cost matrix}

\item{dist.method}{pointwise (local) distance function to use. See
\code{\link[proxy]{dist}} in package \pkg{proxy}}

\item{step.pattern}{a stepPattern object describing the local warping steps
allowed with their cost (see \code{\link{stepPattern}})}

\item{window.type}{windowing function. Character: "none", "itakura",
"sakoechiba", "slantedband", or a function (see details).}

\item{keep.internals}{preserve the cumulative cost matrix, inputs, and other
internal structures}

\item{distance.only}{only compute distance (no backtrack, faster)}

\item{open.begin, open.end}{perform open-ended alignments}

\item{...}{additional arguments, passed to \code{window.type}}

\item{d}{an arbitrary R object}
}
\value{
An object of class \code{dtw} with the following items:
\item{distance}{the minimum global distance computed, \emph{not}
normalized.} \item{normalizedDistance}{distance computed, \emph{normalized}
for path length, if normalization is known for chosen step pattern.}
\item{N,M}{query and reference length} \item{call}{the function call that
created the object} \item{index1}{matched elements: indices in \code{x}}
\item{index2}{corresponding mapped indices in \code{y}}
\item{stepPattern}{the \code{stepPattern} object used for the computation}
\item{jmin}{last element of reference matched, if \code{open.end=TRUE}}
\item{directionMatrix}{if \code{keep.internals=TRUE}, the directions of
steps that would be taken at each alignment pair (integers indexing
production rules in the chosen step pattern)} \item{stepsTaken}{the list of
steps taken from the beginning to the end of the alignment (integers
indexing chosen step pattern)} \item{index1s, index2s}{same as
\code{index1/2}, excluding intermediate steps for multi-step patterns like
\code{\link{asymmetricP05}} } \item{costMatrix}{if
\code{keep.internals=TRUE}, the cumulative cost matrix} \item{query,
reference}{if \code{keep.internals=TRUE} and passed as the \code{x} and
\code{y} arguments, the query and reference timeseries.}
}
\description{
Compute Dynamic Time Warp and find optimal alignment between two time
series.
}
\details{
The function performs Dynamic Time Warp (DTW) and computes the optimal
alignment between two time series \code{x} and \code{y}, given as numeric
vectors.  The ``optimal'' alignment minimizes the sum of distances between
aligned elements. Lengths of \code{x} and \code{y} may differ.

The local distance between elements of \code{x} (query) and \code{y}
(reference) can be computed in one of the following ways:

\enumerate{ \item if \code{dist.method} is a string, \code{x} and \code{y}
are passed to the \code{\link[proxy]{dist}} function in package \pkg{proxy}
with the method given; \item if \code{dist.method} is a function of two
arguments, it invoked repeatedly on all pairs \code{x[i],y[j]} to build the
local cost matrix; \item multivariate time series and arbitrary distance
metrics can be handled by supplying a local-distance matrix. Element
\code{[i,j]} of the local-distance matrix is understood as the distance
between element \code{x[i]} and \code{y[j]}. The distance matrix has
therefore \code{n=length(x)} rows and \code{m=length(y)} columns (see note
below).  }

Several common variants of the DTW recursion are supported via the
\code{step.pattern} argument, which defaults to \code{symmetric2}. Step
patterns are commonly used to \emph{locally} constrain the slope of the
alignment function. See \code{\link{stepPattern}} for details.

Windowing enforces a \emph{global} constraint on the envelope of the warping
path. It is selected by passing a string or function to the
\code{window.type} argument. Commonly used windows are (abbreviations
allowed):

\itemize{ \item\code{"none"}No windowing (default) \item\code{"sakoechiba"}A
band around main diagonal \item\code{"slantedband"}A band around slanted
diagonal \item\code{"itakura"}So-called Itakura parallelogram }

\code{window.type} can also be an user-defined windowing function.  See
\code{\link{dtwWindowingFunctions}} for all available windowing functions,
details on user-defined windowing, and a discussion of the (mis)naming of
the "Itakura" parallelogram as a global constraint.  Some windowing
functions may require parameters, such as the \code{window.size} argument.

Open-ended alignment, i.e. semi-unconstrained alignment, can be selected via
the \code{open.end} switch.  Open-end DTW computes the alignment which best
matches all of the query with a \emph{leading part} of the reference. This
is proposed e.g. by Mori (2006), Sakoe (1979) and others. Similarly,
open-begin is enabled via \code{open.begin}; it makes sense when
\code{open.end} is also enabled (subsequence finding). Subsequence
alignments are similar e.g. to UE2-1 algorithm by Rabiner (1978) and others.
Please find a review in Tormene et al. (2009).

If the warping function is not required, computation can be sped up enabling
the \code{distance.only=TRUE} switch, which skips the backtracking step. The
output object will then lack the \code{index{1,2,1s,2s}} and
\code{stepsTaken} fields.

\code{is.dtw} tests whether the argument is of class \code{dtw}.
}
\note{
Cost matrices (both input and output) have query elements arranged
row-wise (first index), and reference elements column-wise (second index).
They print according to the usual convention, with indexes increasing down-
and rightwards.  Many DTW papers and tutorials show matrices according to
plot-like conventions, i.e.  reference index growing upwards. This may be
confusing.

A fast compiled version of the function is normally used.  Should it be
unavailable, the interpreted equivalent will be used as a fall-back with a
warning.
}
\examples{


## A noisy sine wave as query
idx<-seq(0,6.28,len=100);
query<-sin(idx)+runif(100)/10;

## A cosine is for reference; sin and cos are offset by 25 samples
reference<-cos(idx)
plot(reference); lines(query,col="blue");

## Find the best match
alignment<-dtw(query,reference);


## Display the mapping, AKA warping function - may be multiple-valued
## Equivalent to: plot(alignment,type="alignment")
plot(alignment$index1,alignment$index2,main="Warping function");

## Confirm: 25 samples off-diagonal alignment
lines(1:100-25,col="red")




#########
##
## Partial alignments are allowed.
##

alignmentOBE <-
  dtw(query[44:88],reference,
      keep=TRUE,step=asymmetric,
      open.end=TRUE,open.begin=TRUE);
plot(alignmentOBE,type="two",off=1);


#########
##
## Subsetting allows warping and unwarping of
## timeseries according to the warping curve. 
## See first example below.
##

## Most useful: plot the warped query along with reference 
plot(reference)
lines(query[alignment$index1]~alignment$index2,col="blue")

## Plot the (unwarped) query and the inverse-warped reference
plot(query,type="l",col="blue")
points(reference[alignment$index2]~alignment$index1)



#########
##
## Contour plots of the cumulative cost matrix
##    similar to: plot(alignment,type="density") or
##                dtwPlotDensity(alignment)
## See more plots in ?plot.dtw 
##

## keep = TRUE so we can look into the cost matrix

alignment<-dtw(query,reference,keep=TRUE);

contour(alignment$costMatrix,col=terrain.colors(100),x=1:100,y=1:100,
	xlab="Query (noisy sine)",ylab="Reference (cosine)");

lines(alignment$index1,alignment$index2,col="red",lwd=2);




#########
##
## An hand-checkable example
##

ldist<-matrix(1,nrow=6,ncol=6);  # Matrix of ones
ldist[2,]<-0; ldist[,5]<-0;      # Mark a clear path of zeroes
ldist[2,5]<-.01;		 # Forcely cut the corner

ds<-dtw(ldist);			 # DTW with user-supplied local
                                 #   cost matrix
da<-dtw(ldist,step=asymmetric);	 # Also compute the asymmetric 
plot(ds$index1,ds$index2,pch=3); # Symmetric: alignment follows
                                 #   the low-distance marked path
points(da$index1,da$index2,col="red");  # Asymmetric: visiting
                                        #   1 is required twice

ds$distance;
da$distance;





}
\references{
Toni Giorgino. \emph{Computing and Visualizing Dynamic Time
Warping Alignments in R: The dtw Package.} Journal of Statistical Software,
31(7), 1-24. \url{http://www.jstatsoft.org/v31/i07/} \cr \cr Tormene, P.;
Giorgino, T.; Quaglini, S. & Stefanelli, M. \emph{Matching incomplete time
series with dynamic time warping: an algorithm and an application to
post-stroke rehabilitation.} Artif Intell Med, 2009, 45, 11-34.
\url{http://dx.doi.org/10.1016/j.artmed.2008.11.007} \cr \cr Sakoe, H.;
Chiba, S., \emph{Dynamic programming algorithm optimization for spoken word
recognition,} Acoustics, Speech, and Signal Processing [see also IEEE
Transactions on Signal Processing], IEEE Transactions on , vol.26, no.1, pp.
43-49, Feb 1978.
\url{http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1163055} \cr \cr
Mori, A.; Uchida, S.; Kurazume, R.; Taniguchi, R.; Hasegawa, T. & Sakoe, H.
\emph{Early Recognition and Prediction of Gestures} Proc. 18th International
Conference on Pattern Recognition ICPR 2006, 2006, 3, 560-563 \cr \cr Sakoe,
H. \emph{Two-level DP-matching--A dynamic programming-based pattern matching
algorithm for connected word recognition} Acoustics, Speech, and Signal
Processing [see also IEEE Transactions on Signal Processing], IEEE
Transactions on, 1979, 27, 588-595 \cr \cr Rabiner L, Rosenberg A, Levinson
S (1978). \emph{Considerations in dynamic time warping algorithms for
discrete word recognition.} IEEE Trans. Acoust., Speech, Signal Process.,
26(6), 575-582. ISSN 0096-3518. \cr \cr Muller M. \emph{Dynamic Time
Warping} in \emph{Information Retrieval for Music and Motion}. Springer
Berlin Heidelberg; 2007. p. 69-84.
\url{http://link.springer.com/chapter/10.1007/978-3-540-74048-3_4}
}
\seealso{
\code{\link{dtwDist}}, for iterating dtw over a set of timeseries;
\code{\link{dtwWindowingFunctions}}, for windowing and global constraints;
\code{\link{stepPattern}}, step patterns and local constraints;
\code{\link{plot.dtw}}, plot methods for DTW objects.  To generate a local
distance matrix, the functions \code{\link[proxy]{dist}} in package
\pkg{proxy}, \code{\link[analogue]{distance}} in package \pkg{analogue},
\code{\link{outer}} may come handy.
}
\author{
Toni Giorgino
}
\keyword{ts}