PsiJ               package:wavethresh               R Documentation

_C_o_m_p_u_t_e _d_i_s_c_r_e_t_e _a_u_t_o_c_o_r_r_e_l_a_t_i_o_n _w_a_v_e_l_e_t_s.

_D_e_s_c_r_i_p_t_i_o_n:

     This function computes discrete autocorrelation wavelets. 

     The inner products of the discrete autocorrelation wavelets are
     computed by the routine 'ipndacw'.

_U_s_a_g_e:

     PsiJ(J, filter.number = 10, family = "DaubLeAsymm", tol = 1e-100, OPLENGTH=2000)

_A_r_g_u_m_e_n_t_s:

       J: Discrete autocorrelation wavelets will be computed for scales
          -1 up to scale J. This number should be a negative integer.

filter.number: The index of the wavelet used to compute the discrete
          autocorrelation wavelets.

  family: The family of wavelet used to compute the discrete
          autocorrelation wavelets.

     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 'tol' which effectively defines how long the
          inner product/autocorrelation products are.

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.

_D_e_t_a_i_l_s:

     This function computes the discrete autocorrelation wavelets. It
     does not have any direct use for time-scale analysis (e.g.
     '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. 

     *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 'ipndacw' and its subsidiary routines 'rmget' and 'firstdot'
     but helps a little bit. The '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 'PsiJmat' does this.

_V_a_l_u_e:

     A list containing -J components, numbered from 1 to -J. The
     [[j]]th component contains the discrete autocorrelation wavelet at
     scale j.

_R_E_L_E_A_S_E:

     Version 3.9 Copyright Guy Nason 1998

_R_e_f_e_r_e_n_c_e_s:

     Nason, G.P., von Sachs, R. and Kroisandt, G. (1998). Wavelet
     processes and adaptive estimation of the evolutionary wavelet
     spectrum. 'echnical Report', Department of Mathematics University
     of Bristol/ Fachbereich Mathematik, Kaiserslautern.

_S_e_e _A_l_s_o:

     'ewspec', 'ipndacw', 'PsiJmat', 'Psiname'.

_E_x_a_m_p_l_e_s:

     #
     # 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")

