dtw_basic: Basic DTW distance

In dtwclust: Time Series Clustering Along with Optimizations for the Dynamic Time Warping Distance

Basic DTW distance

Description

This is a custom implementation of the DTW algorithm without all the functionality included in dtw::dtw(). Because of that, it should be faster, while still supporting the most common options.

Usage

dtw_basic(
  x,
  y,
  window.size = NULL,
  norm = "L1",
  step.pattern = dtw::symmetric2,
  backtrack = FALSE,
  normalize = FALSE,
  sqrt.dist = TRUE,
  ...,
  error.check = TRUE
)

Arguments

x, y	Time series. Multivariate series must have time spanning the rows and variables spanning the columns.
window.size	Size for slanted band window. NULL means no constraint.
norm	Norm for the LCM calculation, "L1" for Manhattan or "L2" for (squared) Euclidean. See notes.
step.pattern	Step pattern for DTW. Only symmetric1 or symmetric2 supported here. Note that these are not characters. See dtw::stepPattern.
backtrack	Also compute the warping path between series? See details.
normalize	Should the distance be normalized? Only supported for symmetric2.
sqrt.dist	Only relevant for norm = "L2", see notes.
...	Currently ignored.
error.check	Logical indicating whether the function should try to detect inconsistencies and give more informative errors messages. Also used internally to avoid repeating checks.

Details

If backtrack is TRUE, the mapping of indices between series is returned in a list.

The windowing constraint uses a centered window. The calculations expect a value in window.size that represents the distance between the point considered and one of the edges of the window. Therefore, if, for example, window.size = 10, the warping for an observation x_i considers the points between x_{i-10} and x_{i+10}, resulting in 10(2) + 1 = 21 observations falling within the window.

Value

The DTW distance. For backtrack = TRUE, a list with:

• distance: The DTW distance.

• index1: x indices for the matched elements in the warping path.

• index2: y indices for the matched elements in the warping path.

Proxy version

The version registered with proxy::dist() is custom (loop = FALSE in proxy::pr_DB). The custom function handles multi-threaded parallelization directly with RcppParallel. It uses all available threads by default (see RcppParallel::defaultNumThreads()), but this can be changed by the user with RcppParallel::setThreadOptions().

An exception to the above is when it is called within a foreach parallel loop made by dtwclust. If the parallel workers do not have the number of threads explicitly specified, this function will default to 1 thread per worker. See the parallelization vignette for more information - browseVignettes("dtwclust")

It also includes symmetric optimizations to calculate only half a distance matrix when appropriate—only one list of series should be provided in x. Starting with version 6.0.0, this optimization means that the function returns an array with the lower triangular values of the distance matrix, similar to what stats::dist() does; see DistmatLowerTriangular for a helper to access elements as it if were a normal matrix. If you want to avoid this optimization, call proxy::dist by giving the same list of series in both x and y.

In order for symmetry to apply here, the following must be true: no window constraint is used (window.size is NULL) or, if one is used, all series have the same length.

Note

The elements of the local cost matrix are calculated by using either Manhattan or squared Euclidean distance. This is determined by the norm parameter. When the squared Euclidean version is used, the square root of the resulting DTW distance is calculated at the end (as defined in Ratanamahatana and Keogh 2004; Lemire 2009; see vignette references). This can be avoided by passing FALSE in sqrt.dist.

The DTW algorithm (and the functions that depend on it) might return different values in 32 bit installations compared to 64 bit ones.

An infinite distance value indicates that the constraints could not be fulfilled, probably due to a too small window.size or a very large length difference between the series.

Examples

## Not run: 
# ====================================================================================
# Understanding multivariate DTW
# ====================================================================================

# The variables for each multivariate time series are:
# tip force, x velocity, and y velocity
A1 <- CharTrajMV[[1L]] # A character
B1 <- CharTrajMV[[6L]] # B character

# Let's extract univariate time series
A1_TipForce <- A1[,1L] # first variable (column)
A1_VelX <- A1[,2L] # second variable (column)
A1_VelY <- A1[,3L] # third variable (column)
B1_TipForce <- B1[,1L] # first variable (column)
B1_VelX <- B1[,2L] # second variable (column)
B1_VelY <- B1[,3L] # third variable (column)

# Looking at each variable independently:

# Just force
dtw_basic(A1_TipForce, B1_TipForce, norm = "L1", step.pattern = symmetric1)
# Corresponding LCM
proxy::dist(A1_TipForce, B1_TipForce, method = "L1")

# Just x velocity
dtw_basic(A1_VelX, B1_VelX, norm = "L1", step.pattern = symmetric1)
# Corresponding LCM
proxy::dist(A1_VelX, B1_VelX, method = "L1")

# Just y velocity
dtw_basic(A1_VelY, B1_VelY, norm = "L1", step.pattern = symmetric1)
# Corresponding LCM
proxy::dist(A1_VelY, B1_VelY, method = "L1")

# NOTES:
# In the previous examples there was one LCM for each *pair* of series.
# Additionally, each LCM has dimensions length(A1_*) x length(B1_*)

# proxy::dist won't return the LCM for multivariate series,
# but we can do it manually:
mv_lcm <- function(mvts1, mvts2) {
    # Notice how the number of variables (columns) doesn't come into play here
    num_obs1 <- nrow(mvts1)
    num_obs2 <- nrow(mvts2)

    lcm <- matrix(0, nrow = num_obs1, ncol = num_obs2)

    for (i in 1L:num_obs1) {
        for (j in 1L:num_obs2) {
            # L1 norm for ALL variables (columns).
            # Consideration: mvts1 and mvts2 MUST have the same number of variables
            lcm[i, j] <- sum(abs(mvts1[i,] - mvts2[j,]))
        }
    }

    # return
    lcm
}

# Let's say we start with only x velocity and y velocity for each character
mvts1 <- cbind(A1_VelX, A1_VelY)
mvts2 <- cbind(B1_VelX, B1_VelY)

# DTW distance
dtw_d <- dtw_basic(mvts1, mvts2, norm = "L1", step.pattern = symmetric1)
# Corresponding LCM
lcm <- mv_lcm(mvts1, mvts2) # still 178 x 174
# Sanity check
all.equal(
    dtw_d,
    dtw::dtw(lcm, step.pattern = symmetric1)$distance # supports LCM as input
)

# Now let's consider all variables for each character
mvts1 <- cbind(mvts1, A1_TipForce)
mvts2 <- cbind(mvts2, B1_TipForce)

# Notice how the next code is exactly the same as before,
# even though we have one extra variable now

# DTW distance
dtw_d <- dtw_basic(mvts1, mvts2, norm = "L1", step.pattern = symmetric1)
# Corresponding LCM
lcm <- mv_lcm(mvts1, mvts2) # still 178 x 174
# Sanity check
all.equal(
    dtw_d,
    dtw::dtw(lcm, step.pattern = symmetric1)$distance # supports LCM as input
)

# By putting things in a list,
# proxy::dist returns the *cross-distance matrix*, not the LCM
series_list <- list(mvts1, mvts2)
distmat <- proxy::dist(series_list, method = "dtw_basic",
                       norm = "L1", step.pattern = symmetric1)
# So this should be TRUE
all.equal(distmat[1L, 2L], dtw_d)

# NOTE: distmat is a 2 x 2 matrix, because there are 2 multivariate series.
# Each *cell* in distmat has a corresponding LCM (not returned by the function).
# Proof:
manual_distmat <- matrix(0, nrow = 2L, ncol = 2L)
for (i in 1L:nrow(manual_distmat)) {
    for (j in 1L:ncol(manual_distmat)) {
        lcm_cell <- mv_lcm(series_list[[i]], series_list[[j]]) # LCM for this pair
        manual_distmat[i, j] <- dtw::dtw(lcm_cell, step.pattern = symmetric1)$distance
    }
}
# TRUE
all.equal(
    as.matrix(distmat),
    manual_distmat
)

## End(Not run)

dtwclust documentation built on Sept. 11, 2024, 9:07 p.m.