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\title{Global constraints and windowing functions for DTW}
  Various global constraints (windows) which can be applied to
  the \code{window.type} argument of \code{\link{dtw}},
  including the Sakoe-Chiba band, the Itakura parallelogram,
  and custom functions.

noWindow(iw, jw, ...);
sakoeChibaWindow(iw,jw,  window.size,...);
slantedBandWindow(iw,jw,query.size,reference.size,  window.size,...);
itakuraWindow(iw,jw,query.size,reference.size,  ...); 


%- maybe also 'usage' for other objects documented here.
  \item{iw}{ index in the query (row) -- automatically set }
  \item{jw}{ index in the reference (column) -- automatically set }
  \item{query.size}{ size of the query time series -- automatically set }
  \item{reference.size}{ size of the reference time series -- automatically set }
  \item{window.size}{ window size, used by some windowing functions -- must be set}
  \item{fun}{ a windowing function }
  \item{...}{additional arguments passed to windowing functions}


  Windowing functions can be passed to the
  \code{window.type} argument in \code{\link{dtw}} to put a global
  constraint to the warping paths allowed. They take two integer
  arguments (plus optional parameters) and must return a boolean value 
  \code{TRUE} if the coordinates fall within the allowed region 
  for warping paths, \code{FALSE} otherwise.

  User-defined functions can read variables \code{reference.size},
  \code{query.size} and \code{window.size}; these are pre-set upon
  invocation.  Some functions require additional parameters which must
  be set (e.g. \code{window.size}).  User-defined functions are free to
  implement any window shape, as long as at least one path is allowed
  between the initial and final alignment points, i.e., they are
  compatible with the DTW constraints.

  The \code{sakoeChibaWindow} function implements the Sakoe-Chiba band,
  i.e. \code{window.size} elements around the \code{main} diagonal. If
  the window size is too small, i.e. if
  \code{reference.size}-\code{query.size} > \code{window.size}, warping
  becomes impossible.

  An \code{itakuraWindow}  global constraint is still provided with this
  package.    See example below for a demonstration of
  the difference between a local the two.

  The \code{slantedBandWindow} (package-specific) is a band centered
  around the (jagged) line segment which joins element \code{[1,1]} to
  element \code{[query.size,reference.size]}, and will be
  \code{window.size} columns wide. In other words, the "diagonal" goes
  from one corner to the other of the possibly rectangular cost matrix,
  therefore having a slope of \code{M/N}, not 1.

    \code{dtwWindow.plot} visualizes a windowing function. By default
  it plots a 200 x 220 rectangular region, which can
  be changed via \code{reference.size} and \code{query.size} arguments.


  Although  \code{dtwWindow.plot} resembles object-oriented notation,
  there is not a such a dtwWindow class currently.

    A widely held misconception is that the "Itakura parallelogram" (as
  described in reference [2]) is a \emph{global} constraint, i.e. a
  window.  To the author's knowledge, it instead arises from the local
  slope restrictions imposed to the warping path, such as the one
  implemented by the \code{\link{typeIIIc}} step pattern.

  Windowing functions return \code{TRUE} if the coordinates passed as
  arguments fall within the chosen warping window, \code{FALSE}
  otherwise. User-defined functions should do the same.

[1] Sakoe, H.; Chiba, S., \emph{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: \url{http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1163055} \cr
[2] Itakura, F., \emph{Minimum prediction residual principle applied to speech
recognition,} Acoustics, Speech, and Signal Processing [see also IEEE
Transactions on Signal Processing], IEEE Transactions on , vol.23, no.1, pp.
67-72, Feb 1975. URL: \url{http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1162641} 


\author{Toni Giorgino}


## Display some windowing functions
dtwWindow.plot(itakuraWindow, main="So-called Itakura parallelogram window")
dtwWindow.plot(slantedBandWindow, window.size=2,
  reference=13, query=17, main="The slantedBandWindow at window.size=2")

## Asymmetric step with Sakoe-Chiba band


             window.size=30                  );

dtwPlot(asyband,type="density",main="Sine/cosine: asymmetric step, S-C window")


\concept{Dynamic Time Warp}
\concept{Global contraint}
\concept{Sakoe-Chiba Band}
\concept{Itakura Parallelogram}

\keyword{ ts }
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