dtw: Dynamic Time Warp In dtw: Dynamic Time Warping Algorithms

 dtw R Documentation

Dynamic Time Warp

Description

Compute Dynamic Time Warp and find optimal alignment between two time series.

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)

## S3 method for class 'dtw'
print(x, ...)
```

Arguments

 `x` query vector or local cost matrix `y` reference vector, or NULL if `x` given as a local cost matrix `dist.method` pointwise (local) distance function to use. See `proxy::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). `keep.internals` preserve the cumulative cost matrix, inputs, and other internal structures `distance.only` only compute distance (no backtrack, faster) `open.begin, open.end` perform open-ended alignments `...` additional arguments, passed to `window.type` `d` an arbitrary R object

Details

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:

1. if `dist.method` is a string, `x` and `y` are passed to the `proxy::dist()` function in package proxy with the method given;

2. if `dist.method` is a function of two arguments, it invoked repeatedly on all pairs `x[i],y[j]` to build the local cost matrix;

3. multivariate time series and arbitrary distance metrics can be handled by supplying a precomputed local cost matrix. Element `[i,j]` of the local cost 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 the DTW recursion are supported via the `step.pattern` argument, which defaults to `symmetric2`. Step patterns are commonly used to locally constrain the slope of the alignment function. See `stepPattern()` for details.

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

• `"none"` No windowing (default)

• `"sakoechiba"` A band around main diagonal

• `"slantedband"` A band around slanted diagonal

• `"itakura"` So-called Itakura parallelogram

`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.

Open-ended alignment, i.e. semi-unconstrained alignment, can be selected via the `open.end` switch. 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`; it makes sense when `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 `distance.only=TRUE` switch, which skips the backtracking step. The output object will then lack the `index{1,2,1s,2s}` and `stepsTaken` fields.

`is.dtw` tests whether the argument is of class `dtw`.

Value

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 production rules in the chosen step pattern)

• `stepsTaken` the list of steps taken from the beginning to the end of the alignment (integers indexing chosen step pattern)

• `index1s, index2s` same as `index1/2`, excluding intermediate steps for multi-step patterns like `asymmetricP05()`

• `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.

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.

Toni Giorgino

References

1. Toni Giorgino. Computing and Visualizing Dynamic Time Warping Alignments in R: The dtw Package. Journal of Statistical Software, 31(7), 1-24. doi: 10.18637/jss.v031.i07

2. Tormene, P.; Giorgino, T.; Quaglini, S. & Stefanelli, M. Matching incomplete time series with dynamic time warping: an algorithm and an application to post-stroke rehabilitation. Artif Intell Med, 2009, 45, 11-34. doi: 10.1016/j.artmed.2008.11.007

3. Sakoe, H.; Chiba, S., Dynamic programming algorithm optimization for spoken word recognition, Acoustics, Speech, and Signal Processing, IEEE Transactions on , vol.26, no.1, pp. 43-49, Feb 1978. doi: 10.1109/TASSP.1978.1163055

4. 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 doi: 10.1109/ICPR.2006.467

5. Sakoe, H. Two-level DP-matchingâ€“A dynamic programming-based pattern matching algorithm for connected word recognition Acoustics, Speech, and Signal Processing, IEEE Transactions on, 1979, 27, 588-595 doi: 10.1109/TASSP.1979.1163310

6. 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. doi: 10.1109/TASSP.1978.1163164

7. Muller M. Dynamic Time Warping in Information Retrieval for Music and Motion. Springer Berlin Heidelberg; 2007. p. 69-84. doi: 10.1007/978-3-540-74048-3_4

`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 cost matrix, the functions `proxy::dist()`, `analogue::distance()`, `vegan::vegdist()`, or `outer()` may come handy.

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;

```

dtw documentation built on Sept. 20, 2022, 1:06 a.m.