\HeaderA{PsiJ}{Compute discrete autocorrelation wavelets.}{PsiJ}
\keyword{manip}{PsiJ}
\begin{Description}\relax
This function computes discrete autocorrelation wavelets. 

The inner products of the discrete autocorrelation wavelets are computed by the routine \code{\LinkA{ipndacw}{ipndacw}}.
\end{Description}
\begin{Usage}
\begin{verbatim}
PsiJ(J, filter.number = 10, family = "DaubLeAsymm", tol = 1e-100, OPLENGTH=2000)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{J}] Discrete autocorrelation wavelets will be computed for scales -1 up to scale J. This number should be a negative integer.
\item[\code{filter.number}] The index of the wavelet used to compute the discrete autocorrelation wavelets.
\item[\code{family}] The family of wavelet used to compute the discrete autocorrelation wavelets.
\item[\code{tol}] In the brute force computation for Daubechies compactly supported wavelets many inner product computations are performed. This tolerance discounts any results which are smaller than \code{tol} which effectively defines how long the inner product/autocorrelation products are.
\item[\code{OPLENGTH}] This integer variable defines some workspace of length OPLENGTH. The code uses this workspace. If the workspace is not long enough then the routine will stop and probably tell you what OPLENGTH should be set to.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
This function computes the discrete autocorrelation wavelets. It does not have any direct use for time-scale analysis (e.g. \code{\LinkA{ewspec}{ewspec}}). However, it is useful to be able to numerically compute the discrete autocorrelation wavelets for arbitrary wavelets and scales as there are still unanswered theoretical questions concerning the wavelets. The method is a brute force -- a more elegant solution would probably be based on interpolatory schemes. 

\bold{Horizontal scale}. This routine returns only the values of the discrete autocorrelation wavelets and not their horiztonal positions. Each discrete autocorrelation wavelet is compactly supported with the support determined from the compactly supported wavelet that generates it. See the paper by Nason, von Sachs and Kroisandt which defines the horiztonal scale (but basically the finer scale discrete autocorrelation wavelets are interpolated versions of the coarser ones. When one goes from scale j to j-1 (negative j remember) an extra point is inserted between all of the old points and the discrete autocorrelation wavelet value is computed there. Thus as J tends to negative infinity the numerical approximation tends towards the continuous autocorrelation wavelet. 

This function stores any discrete autocorrelation wavelet sets that it computes. The storage mechanism is not as advanced as that for \code{\LinkA{ipndacw}{ipndacw}} and its subsidiary routines \code{\LinkA{rmget}{rmget}} and \code{\LinkA{firstdot}{firstdot}} but helps a little bit. The \code{\LinkA{Psiname}{Psiname}} function defines the naming convention for objects returned by this function. 

Sometimes it is useful to have the discrete autocorrelation wavelets stored in matrix form. The \code{\LinkA{PsiJmat}{PsiJmat}} does this.
\end{Details}
\begin{Value}
A list containing -J components, numbered from 1 to -J. The [[j]]th component contains the discrete autocorrelation wavelet at scale j.
\end{Value}
\begin{Section}{RELEASE}
Version 3.9 Copyright Guy Nason 1998
\end{Section}
\begin{References}\relax
Nason, G.P., von Sachs, R. and Kroisandt, G. (1998). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. \code{echnical Report}, Department of Mathematics University of Bristol/ Fachbereich Mathematik, Kaiserslautern.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{ewspec}{ewspec}}, \code{\LinkA{ipndacw}{ipndacw}}, \code{\LinkA{PsiJmat}{PsiJmat}}, \code{\LinkA{Psiname}{Psiname}}.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
#
# Let us create the discrete autocorrelation wavelets for the Haar wavelet.
# We shall create up to scale 4.
#
#PsiJ(-4, filter.number=1, family="DaubExPhase")
#Computing PsiJ
#Returning precomputed version
#Took  0.00999999  seconds
#[[1]]:
#[1] -0.5  1.0 -0.5
#
#[[2]]:
#[1] -0.25 -0.50  0.25  1.00  0.25 -0.50 -0.25
#
#[[3]]:
# [1] -0.125 -0.250 -0.375 -0.500 -0.125  0.250  0.625  1.000  0.625  0.250
#[11] -0.125 -0.500 -0.375 -0.250 -0.125
#
#[[4]]:
# [1] -0.0625 -0.1250 -0.1875 -0.2500 -0.3125 -0.3750 -0.4375 -0.5000 -0.3125
#[10] -0.1250  0.0625  0.2500  0.4375  0.6250  0.8125  1.0000  0.8125  0.6250
#[19]  0.4375  0.2500  0.0625 -0.1250 -0.3125 -0.5000 -0.4375 -0.3750 -0.3125
#[28] -0.2500 -0.1875 -0.1250 -0.0625
#
# You can plot the fourth component to get an idea of what the
# autocorrelation wavelet looks like.
#
# Note that the previous call stores the autocorrelation wavelet
# in Psi.4.1.DaubExPhase. This is mainly so that it doesn't have to
# be recomputed.  
#
# Note that the x-coordinates in the following are approximate.
#
#plot(seq(from=-1, to=1, length=length(Psi.4.1.DaubExPhase[[4]])),
#       Psi.4.1.DaubExPhase[[4]], type="l",
#       xlab = "t", ylab = "Haar Autocorrelation Wavelet")
#
#
# Now let us repeat the above for the Daubechies Least-Asymmetric wavelet
# with 10 vanishing moments.
# We shall create up to scale 6, a higher resolution version than last
# time.
#
#PsiJ(-6, filter.number=10, family="DaubLeAsymm")
#[[1]]:
# [1]  3.537571e-07  5.699601e-16 -7.512135e-06 -7.705013e-15  7.662378e-05
# [6]  5.637163e-14 -5.010016e-04 -2.419432e-13  2.368371e-03  9.976593e-13
#[11] -8.684028e-03 -1.945435e-12  2.605208e-02  6.245832e-12 -6.773542e-02
#[16]  4.704777e-12  1.693386e-01  2.011086e-10 -6.209080e-01  1.000000e+00
#[21] -6.209080e-01  2.011086e-10  1.693386e-01  4.704777e-12 -6.773542e-02
#[26]  6.245832e-12  2.605208e-02 -1.945435e-12 -8.684028e-03  9.976593e-13
#[31]  2.368371e-03 -2.419432e-13 -5.010016e-04  5.637163e-14  7.662378e-05
#[36] -7.705013e-15 -7.512135e-06  5.699601e-16  3.537571e-07
#
#[[2]]
#       scale 2 etc. etc.
#
#[[3]]   scale 3 etc. etc.
#
#scales [[4]] and [[5]]...
#
#[[6]]
#...
#   remaining scale 6 elements...
#...
#[2371] -1.472225e-31 -1.176478e-31 -4.069848e-32 -2.932736e-41  6.855259e-33
#[2376]  5.540202e-33  2.286296e-33  1.164962e-42 -3.134088e-35  3.427783e-44
#[2381] -1.442993e-34 -2.480298e-44  5.325726e-35  9.346398e-45 -2.699644e-36
#[2386] -4.878634e-46 -4.489527e-36 -4.339365e-46  1.891864e-36  2.452556e-46
#[2391] -3.828924e-37 -4.268733e-47  4.161874e-38  3.157694e-48 -1.959885e-39
##
# Let's now plot the 6th component (6th scale, this is the finest
# resolution, all the other scales will be coarser representations)
#
# Note that the previous call stores the autocorrelation wavelet
# in Psi.6.10.DaubLeAsymm.
#
# Note that the x-coordinates in the following are non-existant!
#
#ts.plot(Psi.6.10.DaubExPhase[[6]], xlab = "t",
# ylab = "Daubechies N=10 least-asymmetric Autocorrelation Wavelet")
\end{ExampleCode}
\end{Examples}

