dtw {dtw} | R Documentation |
Compute Dynamic Time Warp and find optimal alignment between two time series.
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)
x |
query vector or local cost matrix |
y |
reference vector, unused if x given as cost matrix |
dist.method |
pointwise (local) distance function to use. See
dist in package proxy |
step.pattern |
a stepPattern object describing the
local warping steps allowed with their cost (see stepPattern ) |
window.type |
windowing function. Character: "none", "itakura", "sakoechiba", "slantedband", or a function (see details). |
open.begin, open.end |
perform open-ended alignments |
keep.internals |
preserve the cumulative cost matrix, inputs, and other internal structures |
distance.only |
only compute distance (no backtrack, faster) |
d |
an arbitrary R object |
... |
additional arguments, passed to window.type |
The function performs Dynamic Time Warp (DTW) and computes the optimal
alignment between two time series x
and y
, given as
numeric vectors. The ``optimal'' alignment minimizes the sum of
distances between aligned elements. Lengths of x
and y
may differ.
The local distance between elements of x
(query) and y
(reference) can be computed in one of the following ways:
dist.method
is a string, x
and
y
are passed to the dist
function in
package proxy with the method given;
dist.method
is a function of two arguments, it invoked
repeatedly on all pairs x[i],y[j]
to build the local cost matrix;
[i,j]
of the
local-distance matrix is understood as the distance between element
x[i]
and y[j]
. The distance matrix has therefore
n=length(x)
rows and m=length(y)
columns (see note
below).
Several common variants of DTW are supported via the
step.pattern
argument, which defaults to
symmetric2
. Most common step patterns are pre-defined: see
stepPattern
for details.
Open-ended alignment, i.e. semi-unconstrained alignment, can be
selected via the open.end
argument. Open-end DTW computes the
alignment which best matches all of the query with a leading
part of the reference. This is proposed e.g. by Mori (2006), Sakoe
(1979) and others. Similarly, open-begin is enabled via
open.begin
. Open-begin alignments usually only make sense when
open.end
is enabled, as well (subsequence alignment); otherwise,
it is just as easy to reverse the query sequence. Subsequence
alignments are similar e.g. to UE2-1 algorithm by Rabiner (1978) and
others.
Windowing (enforcing a global constraint) is supported by passing a
string or function window.type
argument. Commonly used windows
are (abbreviations allowed):
"none"
"sakoechiba"
"slantedband"
"itakura"
window.type
can also be an user-defined windowing function.
See 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
window.size
argument.
A native (fast, compiled) version of the function is normally available. If it is not, an interpreted equivalent will be used as a fall-back, with a warning.
is.dtw
tests whether the argument is of class dtw
.
An object of class dtw
with the following items:
distance |
the minimum global distance computed, not normalized. |
normalizedDistance |
distance computed, normalized for path length, if normalization is known for chosen step pattern. |
N,M |
query and reference length |
call |
the function call that created the object |
index1 |
matched elements: indices in x |
index2 |
corresponding mapped indices in y |
stepPattern |
the stepPattern object used for the computation |
jmin |
last element of reference matched, if open.end=TRUE |
directionMatrix |
if keep.internals=TRUE , the
directions of steps that would be taken at each alignment pair
(integers indexing step patterns) |
costMatrix |
if keep.internals=TRUE , the cumulative
cost matrix |
query, reference |
if keep.internals=TRUE and passed as the x
and y arguments, the query and reference timeseries. |
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.
The partial
argument has been deprecated and renamed
open.end
as of version 1.12.
Toni Giorgino
Sakoe, H.; Chiba, S., 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
Mori, A.; Uchida, S.; Kurazume, R.; Taniguchi, R.; Hasegawa, T. &
Sakoe, H. Early Recognition and Prediction of Gestures
Proc. 18th International Conference on Pattern Recognition ICPR 2006,
2006, 3, 560-563
Sakoe, H. 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
Rabiner L, Rosenberg A, Levinson S (1978). Considerations in
dynamic time warping algorithms for discrete word recognition.
IEEE Trans. Acoust., Speech, Signal Process.,
26(6), 575-582. ISSN 0096-3518.
dtwDist
, for iterating dtw over a set of timeseries;
dtwWindowingFunctions
, for windowing and global constraints;
stepPattern
, step patterns and local constraints;
plot.dtw
, plot methods for DTW objects.
To generate a local distance matrix, the functions
dist
in package proxy,
distance
in package analogue,
outer
may come handy.
## 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;