https://github.com/cran/dtw
Tip revision: 5f6fe253ca22e98effc9eda4c1cec74ff97e5e31 authored by Toni Giorgino on 11 December 2013, 00:00:00 UTC
version 1.17-1
version 1.17-1
Tip revision: 5f6fe25
stepPattern.R
###############################################################
# #
# Author: Toni Giorgino <toni.giorgino,gmail.com> #
# Istituto di Ingegneria Biomedica (ISIB-CNR) #
# Consiglio Nazionale delle Ricerche #
# www.isib.cnr.it #
# #
# $Id: stepPattern.R 343 2013-12-11 11:04:41Z tonig $
# #
###############################################################
## For pre-defined step patterns see below.
#############################
## Methods for accessing and creating step.patterns
## TODO: validate norm
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);
}
is.stepPattern <- function(x) {
return(inherits(x,"stepPattern"));
}
## Transpose - exchange role of query and reference
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
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*head(x[ss,2],-1) + (1-alpha)*x[ss,2][-1];
yh <- alpha*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
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);
}
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");
## 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 ));
## 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")