https://github.com/cran/dtw
Tip revision: 532d2199f056d70ec695498b26d577e901bc7734 authored by Toni Giorgino on 01 September 2019, 05:50:02 UTC
version 1.21-3
version 1.21-3
Tip revision: 532d219
stepPattern.R
##
## Copyright (c) 2006-2019 of Toni Giorgino
##
## This file is part of the DTW package.
##
## DTW is free software: you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## DTW is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
## or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
## License for more details.
##
## You should have received a copy of the GNU General Public License
## along with DTW. If not, see <http://www.gnu.org/licenses/>.
##
## For pre-defined step patterns see below.
#############################
## Methods for accessing and creating step.patterns
## TODO: validate norm
#' Step patterns for DTW
#'
#' A `stepPattern` object lists the transitions allowed while searching
#' for the minimum-distance path. DTW variants are implemented by passing one
#' of the objects described in this page to the `stepPattern` argument of
#' the [dtw()] call.
#'
#' A step pattern characterizes the matching model and slope constraint
#' specific of a DTW variant. They also known as local- or slope-constraints,
#' transition types, production or recursion rules (GiorginoJSS).
#'
#' **Pre-defined step patterns**
#'
#' ```
#' ## Well-known step patterns
#' symmetric1
#' symmetric2
#' asymmetric
#'
#' ## Step patterns classified according to Rabiner-Juang (Rabiner1993)
#' rabinerJuangStepPattern(type,slope.weighting="d",smoothed=FALSE)
#'
#' ## Slope-constrained step patterns from Sakoe-Chiba (Sakoe1978)
#' symmetricP0; asymmetricP0
#' symmetricP05; asymmetricP05
#' symmetricP1; asymmetricP1
#' symmetricP2; asymmetricP2
#'
#' ## Step patterns classified according to Rabiner-Myers (Myers1980)
#' typeIa; typeIb; typeIc; typeId;
#' typeIas; typeIbs; typeIcs; typeIds; # smoothed
#' typeIIa; typeIIb; typeIIc; typeIId;
#' typeIIIc; typeIVc;
#'
#' ## Miscellaneous
#' mori2006;
#' rigid;
#' ```
#'
#'
#' A variety of classification schemes have been proposed for step patterns, including
#' Sakoe-Chiba (Sakoe1978); Rabiner-Juang (Rabiner1993); and Rabiner-Myers
#' (Myers1980). The `dtw` package implements all of the transition types
#' found in those papers, with the exception of Itakura's and
#' Velichko-Zagoruyko's steps, which require subtly different algorithms (this
#' may be rectified in the future). Itakura recursion is almost, but not quite,
#' equivalent to `typeIIIc`.
#'
#' For convenience, we shall review pre-defined step patterns grouped by
#' classification. Note that the same pattern may be listed under different
#' names. Refer to paper (GiorginoJSS) for full details.
#'
#' **1. Well-known step patterns**
#'
#' Common DTW implementations are based on one of the following transition
#' types.
#'
#' `symmetric2` is the normalizable, symmetric, with no local slope
#' constraints. Since one diagonal step costs as much as the two equivalent
#' steps along the sides, it can be normalized dividing by `N+M`
#' (query+reference lengths). It is widely used and the default.
#'
#' `asymmetric` is asymmetric, slope constrained between 0 and 2. Matches
#' each element of the query time series exactly once, so the warping path
#' `index2~index1` is guaranteed to be single-valued. Normalized by
#' `N` (length of query).
#'
#' `symmetric1` (or White-Neely) is quasi-symmetric, no local constraint,
#' non-normalizable. It is biased in favor of oblique steps.
#'
#' **2. The Rabiner-Juang set**
#'
#' A comprehensive table of step patterns is proposed in Rabiner-Juang's book
#' (Rabiner1993), tab. 4.5. All of them can be constructed through the
#' `rabinerJuangStepPattern(type,slope.weighting,smoothed)` function.
#'
#' The classification foresees seven families, labelled with Roman numerals
#' I-VII; here, they are selected through the integer argument `type`.
#' Each family has four slope weighting sub-types, named in sec. 4.7.2.5 as
#' "Type (a)" to "Type (d)"; they are selected passing a character argument
#' `slope.weighting`, as in the table below. Furthermore, each subtype can
#' be either plain or smoothed (figure 4.44); smoothing is enabled setting the
#' logical argument `smoothed`. (Not all combinations of arguments make
#' sense.)
#'
#' ```
#' Subtype | Rule | Norm | Unbiased
#' --------|------------|------|---------
#' a | min step | -- | NO
#' b | max step | -- | NO
#' c | Di step | N | YES
#' d | Di+Dj step | N+M | YES
#' ```
#'
#'
#' **3. The Sakoe-Chiba set**
#'
#' Sakoe-Chiba (Sakoe1978) discuss a family of slope-constrained patterns; they
#' are implemented as shown in page 47, table I. Here, they are called
#' `symmetricP<x>` and `asymmetricP<x>`, where `<x>` corresponds
#' to Sakoe's integer slope parameter *P*. Values available are
#' accordingly: `0` (no constraint), `1`, `05` (one half) and
#' `2`. See (Sakoe1978) for details.
#'
#' **4. The Rabiner-Myers set**
#'
#' The `type<XX><y>` step patterns follow the older Rabiner-Myers'
#' classification proposed in (Myers1980) and (MRR1980). Note that this is a
#' subset of the Rabiner-Juang set (Rabiner1993), and the latter should be
#' preferred in order to avoid confusion. `<XX>` is a Roman numeral
#' specifying the shape of the transitions; `<y>` is a letter in the range
#' `a-d` specifying the weighting used per step, as above; `typeIIx`
#' patterns also have a version ending in `s`, meaning the smoothing is
#' used (which does not permit skipping points). The `typeId, typeIId` and
#' `typeIIds` are unbiased and symmetric.
#'
#' **5. Others**
#'
#' The `rigid` pattern enforces a fixed unitary slope. It only makes sense
#' in combination with `open.begin=TRUE`, `open.end=TRUE` to find gapless
#' subsequences. It may be seen as the `P->inf` limiting case in Sakoe's classification.
#'
#' `mori2006` is Mori's asymmetric step-constrained pattern (Mori2006). It
#' is normalized by the matched reference length.
#'
#' [mvmStepPattern()] implements Latecki's Minimum Variance
#' Matching algorithm, and it is described in its own page.
#'
#'
#' **Methods**
#'
#' `print.stepPattern` prints an user-readable description of the
#' recurrence equation defined by the given pattern.
#'
#' `plot.stepPattern` graphically displays the step patterns productions
#' which can lead to element (0,0). Weights are shown along the step leading to
#' the corresponding element.
#'
#' `t.stepPattern` transposes the productions and normalization hint so
#' that roles of query and reference become reversed.
#'
#'
#' @aliases stepPattern is.stepPattern print.stepPattern t.stepPattern
#' plot.stepPattern symmetric1 symmetric2 asymmetric rabinerJuangStepPattern
#' symmetricP0 asymmetricP0 symmetricP05 asymmetricP05 symmetricP1 asymmetricP1
#' symmetricP2 asymmetricP2 typeIa typeIas typeIb typeIbs typeIc typeIcs typeId
#' typeIds typeIIa typeIIb typeIIc typeIId typeIIIc typeIVc mori2006 rigid
#' @export symmetric1 symmetric2 asymmetric rabinerJuangStepPattern symmetricP0 asymmetricP0 symmetricP05 asymmetricP05 symmetricP1 asymmetricP1 symmetricP2 asymmetricP2 typeIa typeIas typeIb typeIbs typeIc typeIcs typeId typeIds typeIIa typeIIb typeIIc typeIId typeIIIc typeIVc mori2006 rigid
#' @param x a step pattern object
#' @param type path specification, integer 1..7 (see (Rabiner1993), table 4.5)
#' @param slope.weighting slope weighting rule: character `"a"` to `"d"` (see (Rabiner1993), sec. 4.7.2.5)
#' @param smoothed logical, whether to use smoothing (see (Rabiner1993), fig. 4.44)
#' @param ... additional arguments to [print()].
#' @note Constructing `stepPattern` objects is tricky and thus undocumented. For a commented example please see source code for `symmetricP1`.
#' @author Toni Giorgino
#' @seealso [mvmStepPattern()], implementing Latecki's Minimal
#' Variance Matching algorithm.
#' @references
#' * (GiorginoJSS) Toni Giorgino. *Computing and Visualizing
#' Dynamic Time Warping Alignments in R: The dtw Package.* Journal of
#' Statistical Software, 31(7), 1-24. <http://www.jstatsoft.org/v31/i07/>
#' * (Itakura1975) Itakura, F., *Minimum prediction residual
#' principle applied to speech recognition,* Acoustics, Speech, and Signal
#' Processing, IEEE
#' Transactions on , vol.23, no.1, pp. 67-72, Feb 1975. URL:
#' <http://dx.doi.org/10.1109/TASSP.1975.1162641>
#' * (MRR1980) Myers,
#' C.; Rabiner, L. & Rosenberg, A. *Performance tradeoffs in dynamic time
#' warping algorithms for isolated word recognition*, IEEE Trans. Acoust.,
#' Speech, Signal Process., 1980, 28, 623-635. URL:
#' <http://dx.doi.org/10.1109/TASSP.1980.1163491>
#' * (Mori2006) 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. URL:
#' <http://dx.doi.org/10.1109/ICPR.2006.467>
#' * (Myers1980) Myers,
#' Cory S. *A Comparative Study Of Several Dynamic Time Warping
#' Algorithms For Speech Recognition*, MS and BS thesis, Dept. of Electrical
#' Engineering and Computer Science, Massachusetts Institute of Technology,
#' archived Jun 20 1980, <http://hdl.handle.net/1721.1/27909>
#' * (Rabiner1993) Rabiner, L. R., & Juang, B.-H. (1993). *Fundamentals of
#' speech recognition.* Englewood Cliffs, NJ: Prentice Hall.
#' * (Sakoe1978) 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 URL:
#' <http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1163055>
#' @keywords ts
#' @examples
#'
#'
#' #########
#' ##
#' ## The usual (normalizable) symmetric step pattern
#' ## Step pattern recursion, defined as:
#' ## g[i,j] = min(
#' ## g[i,j-1] + d[i,j] ,
#' ## g[i-1,j-1] + 2 * d[i,j] ,
#' ## g[i-1,j] + d[i,j] ,
#' ## )
#'
#' print(symmetric2) # or just "symmetric2"
#'
#'
#'
#' #########
#' ##
#' ## The well-known plotting style for step patterns
#'
#' plot(symmetricP2,main="Sakoe's Symmetric P=2 recursion")
#'
#'
#'
#' #########
#' ##
#' ## Same example seen in ?dtw , now with asymmetric step pattern
#'
#' idx<-seq(0,6.28,len=100);
#' query<-sin(idx)+runif(100)/10;
#' reference<-cos(idx);
#'
#' ## Do the computation
#' asy<-dtw(query,reference,keep=TRUE,step=asymmetric);
#'
#' dtwPlot(asy,type="density",main="Sine and cosine, asymmetric step")
#'
#'
#' #########
#' ##
#' ## Hand-checkable example given in [Myers1980] p 61
#' ##
#'
#' `tm` <-
#' structure(c(1, 3, 4, 4, 5, 2, 2, 3, 3, 4, 3, 1, 1, 1, 3, 4, 2,
#' 3, 3, 2, 5, 3, 4, 4, 1), .Dim = c(5L, 5L))
#'
#' @name stepPattern
NULL
stepPattern <- function(v,norm=NA) {
obj <- NULL;
if(is.vector(v)) {
obj <- matrix(v,ncol=4,byrow=TRUE);
} else if(is.matrix(v)) {
obj <- v;
} else {
stop("stepPattern constructor only supports vector or matrix");
}
class(obj)<-"stepPattern";
attr(obj,"npat") <- max(obj[,1]);
attr(obj,"norm") <- norm;
return(obj);
}
#' @export
is.stepPattern <- function(x) {
return(inherits(x,"stepPattern"));
}
## Transpose - exchange role of query and reference
#' @rdname stepPattern
#' @export
t.stepPattern <- function(x) {
# exchange dx <-> dy
tsp <- x[,c(1,3,2,4)];
tsp <- stepPattern(tsp);
# fix normalization, if available
on <- attr(x,"norm");
if(! is.na(on) ) {
if(on == "N") {
attr(tsp,"norm") <- "M";
} else if(on == "M") {
attr(tsp,"norm") <- "N";
}
}
return(tsp);
}
## plot the step pattern
#' @rdname stepPattern
#' @export
plot.stepPattern <- function(x,...) {
pats <- unique(x[,1]); #list of patterns
xr <- max(x[,2]);
yr <- max(x[,3]);
#for weight labels
fudge <- c(-.5,1.2);
alpha <- .5; # 1 start, 0 end
## dummy plot to fix the plot limits
plot(-x[,2],-x[,3],type="n",
xlab="Query index",ylab="Reference index",
asp=1,lab=c(xr+1,yr+1,1),
ax=FALSE,
...);
for(i in pats) {
ss <- x[,1]==i;
lines(-x[ss,2],-x[ss,3],type="o", ...);
if(sum(ss)==1) {
next;
}
xh <- alpha*utils::head(x[ss,2],-1) + (1-alpha)*x[ss,2][-1];
yh <- alpha*utils::head(x[ss,3],-1) + (1-alpha)*x[ss,3][-1];
text(-xh,-yh,
labels=round(x[ss,4][-1],2),
adj=fudge,
...);
}
axis(1,at=c(-xr:0), ...)
axis(2,at=c(-yr:0), ...)
endpts <- x[,4]==-1;
points(-x[endpts,2],-x[endpts,3],pch=16, ...);
}
## pretty-print the matrix meaning,
## so it will not be as write-only as now
#' @rdname stepPattern
#' @export
print.stepPattern <-function(x,...) {
step.pattern<-x; # for clarity
np<-max(step.pattern[,1]); #no. of patterns
head<-"g[i,j] = min(\n";
body<-"";
## cycle over available step patterns
for(p in 1:np) {
steps<-.extractpattern(step.pattern,p);
ns<-dim(steps)[1];
## restore row order
steps<-matrix(steps[ns:1,],ncol=3); # enforce a matrix
## cycle over steps s in the current pattern p
for(s in 1:ns) {
di<-steps[s,1]; # delta in query
dj<-steps[s,2]; # delta in templ
cc<-steps[s,3]; # step cost multiplier
## make pretty-printable negative increments
dis<-ifelse(di==0,"",-di); # 4 -> -4; 0 -> .
djs<-ifelse(dj==0,"",-dj); # 0 maps to empty string
## cell origin, as coordinate pair
dijs<-sprintf("i%2s,j%2s",dis,djs);
if(cc==-1) { # g
gs<-sprintf(" g[%s]",dijs);
body<-paste(body,gs);
} else {
## prettyprint step cost multiplier in ccs: 1 -> .; 2 -> 2 *
ccs<-ifelse(cc==1," ",sprintf("%2.2g *",cc));
ds<-sprintf("+%s d[%s]",ccs,dijs);
body<-paste(body,ds);
}
}
body<-paste(body,",\n",s="");
}
tail<-")\n\n";
norm <- attr(x,"norm");
ntxt <- sprintf("Normalization hint: %s\n",norm);
rv<-paste(head,body,tail,ntxt);
cat("Step pattern recursion:\n");
cat(rv);
}
## TODO: sanity check on the step pattern definition
.checkpattern <- function(sp) {
## must have 4 x n elements
## all integers
## first column in ascending order from 1, no missing steps
## 2nd, 3rd row non-negative
## 4th: first for each step is -1
}
# Auxiliary function to easily map pattern -> delta
.mkDirDeltas <- function(dir) {
m1 <- dir[ dir[,4]==-1, ,drop=FALSE ];
m1 <- m1[,-4];
m1 <- m1[,-1];
return(m1);
}
## Extract rows belonging to pattern no. sn
## with first element stripped
## in reverse order
.extractpattern <- function(sp,sn) {
sbs<-sp[,1]==sn; # pick only rows beginning by sn
spl<-sp[sbs,-1,drop=FALSE];
# of those: take only column Di, Dj, cost
# (drop first - pattern no. column)
nr<-nrow(spl); # how many are left
spl<-spl[nr:1,,drop=FALSE]; # invert row order
return(spl);
}
##################################################
##################################################
## Utility inner functions to manipulate
## step patterns. Could be implemented as
## a grammar, a'la ggplot2
.Pnew <- function(p,subt,smoo) {
sp <- list();
sp$i <- 0;
sp$j <- 0;
sp$p <- p;
sp$subt <- subt;
sp$smoo <- smoo;
return(sp);
}
.Pstep <- function(sp,di,dj) {
sp$i <- c(sp$i,di);
sp$j <- c(sp$j,dj);
return(sp);
}
.Pend <- function(sp,subt,smoo) {
sp$si <- cumsum(sp$i);
sp$sj <- cumsum(sp$j);
sp$ni <- max(sp$si)-sp$si;
sp$nj <- max(sp$sj)-sp$sj;
w <- NULL;
# smallest of i,j jumps
if(sp$subt=="a") {
w <- pmin(sp$i,sp$j);
} else if(sp$subt=="b") {
# largest of Di, Dj
w <- pmax(sp$i,sp$j);
} else if(sp$subt=="c") {
# Di exactly
w <- sp$i;
} else if(sp$subt=="d") {
# Di+Dj
w <- sp$i+sp$j;
} else {
stop("Unsupported subtype");
}
# drop first element in w
w <- w[-1];
if(sp$smoo)
w <- rep(mean(w),length(w));
# prepend -1
w <- c(-1,w);
sp$w <- w;
return(sp);
}
.PtoMx <- function(sp) {
nr <- length(sp$i);
mx <- matrix(nrow=nr,ncol=4)
mx[,1] <- sp$p;
mx[,2] <- sp$ni;
mx[,3] <- sp$nj;
mx[,4] <- sp$w;
return(mx);
}
#' @export
#' @rdname stepPattern
rabinerJuangStepPattern <- function(type,slope.weighting="d",smoothed=FALSE) {
sw <- slope.weighting;
sm <- smoothed;
## Actually build the step
r <- switch(type,
.RJtypeI(sw,sm),
.RJtypeII(sw,sm),
.RJtypeIII(sw,sm),
.RJtypeIV(sw,sm),
.RJtypeV(sw,sm),
.RJtypeVI(sw,sm),
.RJtypeVII(sw,sm)
);
norm <- NA;
if(sw=="c") {
norm <- "N";
} else if(sw=="d") {
norm <- "N+M";
}
# brain-damaged legacy
rv <- as.vector(t(r));
rs <- stepPattern(rv);
attr(rs,"norm") <- norm;
attr(rs,"call") <- match.call();
return(rs);
}
.RJtypeI <- function(s,m) {
t <- .Pnew(1,s,m)
t <- .Pstep(t,1,0)
t <- .Pend(t);
m1 <- .PtoMx(t);
t <- .Pnew(2,s,m)
t <- .Pstep(t,1,1)
t <- .Pend(t);
m2 <- .PtoMx(t);
t <- .Pnew(3,s,m)
t <- .Pstep(t,0,1)
t <- .Pend(t);
m3 <- .PtoMx(t)
return(rbind(m1,m2,m3));
}
.RJtypeII <- function(s,m) {
t <- .Pnew(1,s,m)
t <- .Pstep(t,1,1)
t <- .Pstep(t,1,0)
t <- .Pend(t);
m1 <- .PtoMx(t);
t <- .Pnew(2,s,m)
t <- .Pstep(t,1,1)
t <- .Pend(t);
m2 <- .PtoMx(t);
t <- .Pnew(3,s,m)
t <- .Pstep(t,1,1)
t <- .Pstep(t,0,1)
t <- .Pend(t);
m3 <- .PtoMx(t)
return(rbind(m1,m2,m3));
}
.RJtypeIII <- function(s,m) {
t <- .Pnew(1,s,m)
t <- .Pstep(t,2,1)
t <- .Pend(t);
m1 <- .PtoMx(t);
t <- .Pnew(2,s,m)
t <- .Pstep(t,1,1)
t <- .Pend(t);
m2 <- .PtoMx(t);
t <- .Pnew(3,s,m)
t <- .Pstep(t,1,2)
t <- .Pend(t);
m3 <- .PtoMx(t)
return(rbind(m1,m2,m3));
}
.RJtypeIV <- function(s,m) {
t <- .Pnew(1,s,m)
t <- .Pstep(t,1,1)
t <- .Pstep(t,1,0)
t <- .Pend(t);
m1 <- .PtoMx(t);
t <- .Pnew(2,s,m)
t <- .Pstep(t,1,2)
t <- .Pstep(t,1,0)
t <- .Pend(t);
m2 <- .PtoMx(t);
t <- .Pnew(3,s,m)
t <- .Pstep(t,1,1)
t <- .Pend(t);
m3 <- .PtoMx(t)
t <- .Pnew(4,s,m)
t <- .Pstep(t,1,2)
t <- .Pend(t);
m4 <- .PtoMx(t)
return(rbind(m1,m2,m3,m4));
}
.RJtypeV <- function(s,m) {
t <- .Pnew(1,s,m)
t <- .Pstep(t,1,1)
t <- .Pstep(t,1,0)
t <- .Pstep(t,1,0)
t <- .Pend(t);
m1 <- .PtoMx(t);
t <- .Pnew(2,s,m)
t <- .Pstep(t,1,1)
t <- .Pstep(t,1,0)
t <- .Pend(t);
m2 <- .PtoMx(t);
t <- .Pnew(3,s,m)
t <- .Pstep(t,1,1)
t <- .Pend(t);
m3 <- .PtoMx(t)
t <- .Pnew(4,s,m)
t <- .Pstep(t,1,1)
t <- .Pstep(t,0,1)
t <- .Pend(t);
m4 <- .PtoMx(t)
t <- .Pnew(5,s,m)
t <- .Pstep(t,1,1)
t <- .Pstep(t,0,1)
t <- .Pstep(t,0,1)
t <- .Pend(t);
m5 <- .PtoMx(t)
return(rbind(m1,m2,m3,m4,m5));
}
.RJtypeVI <- function(s,m) {
t <- .Pnew(1,s,m)
t <- .Pstep(t,1,1)
t <- .Pstep(t,1,1)
t <- .Pstep(t,1,0)
t <- .Pend(t);
m1 <- .PtoMx(t);
t <- .Pnew(2,s,m)
t <- .Pstep(t,1,1)
t <- .Pend(t);
m2 <- .PtoMx(t);
t <- .Pnew(3,s,m)
t <- .Pstep(t,1,1)
t <- .Pstep(t,1,1)
t <- .Pstep(t,0,1)
t <- .Pend(t);
m3 <- .PtoMx(t)
return(rbind(m1,m2,m3));
}
.RJtypeVII <- function(s,m) {
t <- .Pnew(1,s,m)
t <- .Pstep(t,1,1)
t <- .Pstep(t,1,0)
t <- .Pstep(t,1,0)
t <- .Pend(t);
m1 <- .PtoMx(t);
t <- .Pnew(2,s,m)
t <- .Pstep(t,1,2)
t <- .Pstep(t,1,0)
t <- .Pstep(t,1,0)
t <- .Pend(t);
m2 <- .PtoMx(t);
t <- .Pnew(3,s,m)
t <- .Pstep(t,1,3)
t <- .Pstep(t,1,0)
t <- .Pstep(t,1,0)
t <- .Pend(t);
m3 <- .PtoMx(t)
t <- .Pnew(4,s,m)
t <- .Pstep(t,1,1)
t <- .Pstep(t,1,0)
t <- .Pend(t);
m4 <- .PtoMx(t)
t <- .Pnew(5,s,m)
t <- .Pstep(t,1,2)
t <- .Pstep(t,1,0)
t <- .Pend(t);
m5 <- .PtoMx(t)
t <- .Pnew(6,s,m)
t <- .Pstep(t,1,3)
t <- .Pstep(t,1,0)
t <- .Pend(t);
m6 <- .PtoMx(t);
t <- .Pnew(7,s,m)
t <- .Pstep(t,1,1)
t <- .Pend(t);
m7 <- .PtoMx(t)
t <- .Pnew(8,s,m)
t <- .Pstep(t,1,2)
t <- .Pend(t);
m8 <- .PtoMx(t)
t <- .Pnew(9,s,m)
t <- .Pstep(t,1,3)
t <- .Pend(t);
m9 <- .PtoMx(t)
return(rbind(m1,m2,m3,m4,m5,m6,m7,m8,m9));
}
##################################################
##################################################
##
## Various step patterns, defined as internal variables
##
## First column: enumerates step patterns.
## Second step in query index
## Third step in reference index
## Fourth weight if positive, or -1 if starting point
##
## For \cite{} see dtw.bib in the package
##
## Widely-known variants
## White-Neely symmetric (default)
## aka Quasi-symmetric \cite{White1976}
## normalization: no (N+M?)
symmetric1 <- stepPattern(c(
1,1,1,-1,
1,0,0,1,
2,0,1,-1,
2,0,0,1,
3,1,0,-1,
3,0,0,1
));
## Normal symmetric
## normalization: N+M
symmetric2 <- stepPattern(c(
1,1,1,-1,
1,0,0,2,
2,0,1,-1,
2,0,0,1,
3,1,0,-1,
3,0,0,1
),"N+M");
## classic asymmetric pattern: max slope 2, min slope 0
## normalization: N
asymmetric <- stepPattern(c(
1,1,0,-1,
1,0,0,1,
2,1,1,-1,
2,0,0,1,
3,1,2,-1,
3,0,0,1
),"N");
# % \item{\code{symmetricVelichkoZagoruyko}}{symmetric, reproduced from %
# [Sakoe1978]. Use distance matrix \code{1-d}}
#
## normalization: max[N,M]
## note: local distance matrix is 1-d
## \cite{Velichko}
.symmetricVelichkoZagoruyko <- stepPattern(c(
1, 0, 1, -1,
2, 1, 1, -1,
2, 0, 0, -1.001,
3, 1, 0, -1 ));
# % \item{\code{asymmetricItakura}}{asymmetric, slope contrained 0.5 -- 2
# from reference [Itakura1975]. This is the recursive definition % that
# generates the Itakura parallelogram; }
#
## Itakura slope-limited asymmetric \cite{Itakura1975}
## Max slope: 2; min slope: 1/2
## normalization: N
.asymmetricItakura <- stepPattern(c(
1, 1, 2, -1,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 1,
3, 2, 1, -1,
3, 1, 0, 1,
3, 0, 0, 1,
4, 2, 2, -1,
4, 1, 0, 1,
4, 0, 0, 1
));
#############################
## Slope-limited versions
##
## Taken from Table I, page 47 of "Dynamic programming algorithm
## optimization for spoken word recognition," Acoustics, Speech, and
## Signal Processing, vol.26, no.1, pp. 43-49, Feb 1978 URL:
## http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1163055
##
## Mostly unchecked
## Row P=0
symmetricP0 <- symmetric2;
## normalization: N ?
asymmetricP0 <- stepPattern(c(
1,0,1,-1,
1,0,0,0,
2,1,1,-1,
2,0,0,1,
3,1,0,-1,
3,0,0,1
),"N");
## alternative implementation
.asymmetricP0b <- stepPattern(c(
1,0,1,-1,
2,1,1,-1,
2,0,0,1,
3,1,0,-1,
3,0,0,1
),"N");
## Row P=1/2
symmetricP05 <- stepPattern(c(
1 , 1, 3 , -1,
1 , 0, 2 , 2,
1 , 0, 1 , 1,
1 , 0, 0 , 1,
2 , 1, 2 , -1,
2 , 0, 1 , 2,
2 , 0, 0 , 1,
3 , 1, 1 , -1,
3 , 0, 0 , 2,
4 , 2, 1 , -1,
4 , 1, 0 , 2,
4 , 0, 0 , 1,
5 , 3, 1 , -1,
5 , 2, 0 , 2,
5 , 1, 0 , 1,
5 , 0, 0 , 1
),"N+M");
asymmetricP05 <- stepPattern(c(
1 , 1 , 3 , -1,
1 , 0 , 2 ,1/3,
1 , 0 , 1 ,1/3,
1 , 0 , 0 ,1/3,
2 , 1 , 2 , -1,
2 , 0 , 1 , .5,
2 , 0 , 0 , .5,
3 , 1 , 1 , -1,
3 , 0 , 0 , 1 ,
4 , 2 , 1 , -1,
4 , 1 , 0 , 1 ,
4 , 0 , 0 , 1 ,
5 , 3 , 1 , -1,
5 , 2 , 0 , 1 ,
5 , 1 , 0 , 1 ,
5 , 0 , 0 , 1
),"N");
## Row P=1
## Implementation of Sakoe's P=1, Symmetric algorithm
symmetricP1 <- stepPattern(c(
1,1,2,-1, # First branch: g(i-1,j-2)+
1,0,1,2, # + 2d(i ,j-1)
1,0,0,1, # + d(i ,j)
2,1,1,-1, # Second branch: g(i-1,j-1)+
2,0,0,2, # +2d(i, j)
3,2,1,-1, # Third branch: g(i-2,j-1)+
3,1,0,2, # + 2d(i-1,j)
3,0,0,1 # + d( i,j)
),"N+M");
asymmetricP1 <- stepPattern(c(
1, 1 , 2 , -1 ,
1, 0 , 1 , .5 ,
1, 0 , 0 , .5 ,
2, 1 , 1 , -1 ,
2, 0 , 0 , 1 ,
3, 2 , 1 , -1 ,
3, 1 , 0 , 1 ,
3, 0 , 0 , 1
),"N");
## Row P=2
symmetricP2 <- stepPattern(c(
1, 2, 3, -1,
1, 1, 2, 2,
1, 0, 1, 2,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 2,
3, 3, 2, -1,
3, 2, 1, 2,
3, 1, 0, 2,
3, 0, 0, 1
),"N+M");
asymmetricP2 <- stepPattern(c(
1, 2 , 3 , -1,
1, 1 , 2 ,2/3,
1, 0 , 1 ,2/3,
1, 0 , 0 ,2/3,
2, 1 , 1 ,-1 ,
2, 0 , 0 ,1 ,
3, 3 , 2 ,-1 ,
3, 2 , 1 ,1 ,
3, 1 , 0 ,1 ,
3, 0 , 0 ,1
),"N");
################################
## Taken from Table III, page 49.
## Four varieties of DP-algorithm compared
## 1st row: asymmetric
## 2nd row: symmetricVelichkoZagoruyko
## 3rd row: symmetric1
## 4th row: asymmetricItakura
#############################
## Classified according to Rabiner
##
## Taken from chapter 2, Myers' thesis [4]. Letter is
## the weighting function:
##
## rule norm unbiased
## a min step ~N NO
## b max step ~N NO
## c x step N YES
## d x+y step N+M YES
##
## Mostly unchecked
# R-Myers R-Juang
# type I type II
# type II type III
# type III type IV
# type IV type VII
typeIa <- stepPattern(c(
1, 2, 1, -1,
1, 1, 0, 1,
1, 0, 0, 0,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 2, -1,
3, 0, 1, 1,
3, 0, 0, 0
));
typeIb <- stepPattern(c(
1, 2, 1, -1,
1, 1, 0, 1,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 2, -1,
3, 0, 1, 1,
3, 0, 0, 1
));
typeIc <- stepPattern(c(
1, 2, 1, -1,
1, 1, 0, 1,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 2, -1,
3, 0, 1, 1,
3, 0, 0, 0
),"N");
typeId <- stepPattern(c(
1, 2, 1, -1,
1, 1, 0, 2,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 2,
3, 1, 2, -1,
3, 0, 1, 2,
3, 0, 0, 1
),"N+M");
## ----------
## smoothed variants of above
typeIas <- stepPattern(c(
1, 2, 1, -1,
1, 1, 0, .5,
1, 0, 0, .5,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 2, -1,
3, 0, 1, .5,
3, 0, 0, .5
));
typeIbs <- stepPattern(c(
1, 2, 1, -1,
1, 1, 0, 1,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 2, -1,
3, 0, 1, 1,
3, 0, 0, 1
));
typeIcs <- stepPattern(c(
1, 2, 1, -1,
1, 1, 0, 1,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 1,
3, 1, 2, -1,
3, 0, 1, .5,
3, 0, 0, .5
),"N");
typeIds <- stepPattern(c(
1, 2, 1, -1,
1, 1, 0, 1.5,
1, 0, 0, 1.5,
2, 1, 1, -1,
2, 0, 0, 2,
3, 1, 2, -1,
3, 0, 1, 1.5,
3, 0, 0, 1.5
),"N+M");
## ----------
typeIIa <- stepPattern(c(
1, 1, 1, -1,
1, 0, 0, 1,
2, 1, 2, -1,
2, 0, 0, 1,
3, 2, 1, -1,
3, 0, 0, 1
));
typeIIb <- stepPattern(c(
1, 1, 1, -1,
1, 0, 0, 1,
2, 1, 2, -1,
2, 0, 0, 2,
3, 2, 1, -1,
3, 0, 0, 2
));
typeIIc <- stepPattern(c(
1, 1, 1, -1,
1, 0, 0, 1,
2, 1, 2, -1,
2, 0, 0, 1,
3, 2, 1, -1,
3, 0, 0, 2
),"N");
typeIId <- stepPattern(c(
1, 1, 1, -1,
1, 0, 0, 2,
2, 1, 2, -1,
2, 0, 0, 3,
3, 2, 1, -1,
3, 0, 0, 3
),"N+M");
## ----------
## Rabiner [3] discusses why this is not equivalent to Itakura's
typeIIIc <- stepPattern(c(
1, 1, 2, -1,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 1,
3, 2, 1, -1,
3, 1, 0, 1,
3, 0, 0, 1,
4, 2, 2, -1,
4, 1, 0, 1,
4, 0, 0, 1
),"N");
## ----------
## numbers follow as production rules in fig 2.16
typeIVc <- stepPattern(c(
1, 1, 1, -1,
1, 0, 0, 1,
2, 1, 2, -1,
2, 0, 0, 1,
3, 1, 3, -1,
3, 0, 0, 1,
4, 2, 1, -1,
4, 1, 0, 1,
4, 0, 0, 1,
5, 2, 2, -1,
5, 1, 0, 1,
5, 0, 0, 1,
6, 2, 3, -1,
6, 1, 0, 1,
6, 0, 0, 1,
7, 3, 1, -1,
7, 2, 0, 1,
7, 1, 0, 1,
7, 0, 0, 1,
8, 3, 2, -1,
8, 2, 0, 1,
8, 1, 0, 1,
8, 0, 0, 1,
9, 3, 3, -1,
9, 2, 0, 1,
9, 1, 0, 1,
9, 0, 0, 1
),"N");
#############################
##
## Mori's asymmetric step-constrained pattern. Normalized in the
## reference length.
##
## 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
##
mori2006 <- stepPattern(c(
1, 2, 1, -1,
1, 1, 0, 2,
1, 0, 0, 1,
2, 1, 1, -1,
2, 0, 0, 3,
3, 1, 2, -1,
3, 0, 1, 3,
3, 0, 0, 3
),"M");
## Completely unflexible: fixed slope 1. Only makes sense with
## open.begin and open.end
rigid <- stepPattern(c(1,1,1,-1,
1,0,0,1 ),"N")
