D2ACWmat  R Documentation 
This function computes twodimensional discrete autocorrelation wavelets, but results in a matrix form, rather than a list.
D2ACWmat(J, filter.number = 10, family = "DaubLeAsymm", switch = "direction", OPLENGTH = 1e+05)
J 
Discrete autocorrelation wavelets will be computed for scales 1 up to scale J within each decomposition direction (horiznotal, vertical and diagonal). 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. 
switch 
Allows the user to define how they wish their inner product matrix to be formed. There are two available options:

 structures the matrix by direction within each scale. Thus the ordering is as follows (1,V), (1, H), (1, D), (2, V), (2, H), ...
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. 
This function computes the 2D discrete autocorrelation wavelets. It does not have any direct use for spacescale analysis (e.g. cddews). 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 (see Eckley and Nason (2005) for example).
This routine returns only the values of the discrete autocorrelation wavelets and not their spatial positions. Each discrete autocorrelation wavelet is compactly supported with the support determined from the compactly supported wavelet that generates it. See the paper by Eckley, Nason, and Treloar which defines the spatial scale (but basically the finer scale discrete autocorrelation wavelets are interpolated versions of the coarser ones. When one goes from scale j to j1 (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 2D discrete autocorrelation wavelet sets that it computes. The Psiname2D function defines the naming convention for objects returned by this function. The storage mechanism is not as advanced as that for ipndacw and its subsidiary routines rmname and firstdot but helps a little bit.
Sometimes it is useful to have the discrete autocorrelation wavelets stored in matrix form. The D2ACWmat does this.
A matrix containing 3J submatrices, each of dimension 2L_J1 x 2L_J1, where L_J denotes the support of the coarsest discret autocorrelation wavelet. Each submatrix, contains the values of the discrete autocorrelation wavelet for a different scaledirection pair.
The middle position of each submatrix is the value of the discrete autocorrelation wavelet at zero — by definition, this is always 1. The discrete autocorrelation wavelet is symmetric about this point.
If switch="direction"
, the first J submatrices contain the vertical autocorrelation wavelet coefficients, the second set of J components
contains
the horizontal autocorrelation wavelet coefficients (scales 1, ...J) and the last J components constitute the diagonal autocorrelation wavelet
coefficients.
However, if switch="level"
, then the first 3 rows contain the finest scale autocorrelation wavelet coefficients in the vertical, horizontal and
diagonal decomposition directions respectively. The second set of 3 contains the vertical, horizontal and diagonal coefficients at scale 2 etc, etc.
Idris Eckley
Eckley IA, Nason GP (2005). Efficient computation of the innerproduct matrix of discrete autocorrelation wavelets. Statistics and Computing, 15, 8392.
Eckley, I.A., Nason, G.P. and Treloar, R.L. (2010) Locally stationary wavelet fields with application to the modelling and analysis of image texture. Journal of the Royal Statistical Society (Series C), 59, 595  616.
Eckley, I.A. and Nason, G.P. (2011). LS2W: Implementing the Locally Stationary 2D Wavelet Process Approach in R, Journal of Statistical Software, 43(3), 123. URL http://www.jstatsoft.org/v43/i03/.
D2ACW
, D2Amat
, D2autoplot
# # Let us create the discrete autocorrelation wavelets for the Haar wavelet. # We shall create up to scale 4. # D2ACWmat(J=2, filter.number=1, family="DaubExPhase") #Computing The twodimensional (discrete) autocorrelation coefficients: # #The output will be structured as follows .... # # # #Levels 1 to 2 contain the vertical autocorrelation wavelet coefficients. # #Levels 3 to 4 contain the horizontal autocorrelation wavelet coefficients. # #Levels 5 to 6 contain the horizontal autocorrelation wavelet coefficients. # # # #Returning precomputed version #Returning precomputed version #Returning precomputed version #Took NA seconds # [,1] [,2] [,3] [,4] [,5] [,6] [,7] # [1,] 0.0000 0.000 0.0000 0.00 0.0000 0.000 0.0000 # [2,] 0.0000 0.000 0.0000 0.00 0.0000 0.000 0.0000 # [3,] 0.0000 0.000 0.2500 0.50 0.2500 0.000 0.0000 # [4,] 0.0000 0.000 0.5000 1.00 0.5000 0.000 0.0000 # [5,] 0.0000 0.000 0.2500 0.50 0.2500 0.000 0.0000 # [6,] 0.0000 0.000 0.0000 0.00 0.0000 0.000 0.0000 # [7,] 0.0000 0.000 0.0000 0.00 0.0000 0.000 0.0000 # [8,] 0.0625 0.125 0.1875 0.25 0.1875 0.125 0.0625 # [9,] 0.1250 0.250 0.3750 0.50 0.3750 0.250 0.1250 #[10,] 0.0625 0.125 0.1875 0.25 0.1875 0.125 0.0625 #[11,] 0.2500 0.500 0.7500 1.00 0.7500 0.500 0.2500 #[12,] 0.0625 0.125 0.1875 0.25 0.1875 0.125 0.0625 #[13,] 0.1250 0.250 0.3750 0.50 0.3750 0.250 0.1250 #[14,] 0.0625 0.125 0.1875 0.25 0.1875 0.125 0.0625 #[15,] 0.0000 0.000 0.0000 0.00 0.0000 0.000 0.0000 #[16,] 0.0000 0.000 0.0000 0.00 0.0000 0.000 0.0000 #[17,] 0.0000 0.000 0.2500 0.50 0.2500 0.000 0.0000 #[18,] 0.0000 0.000 0.5000 1.00 0.5000 0.000 0.0000 #[19,] 0.0000 0.000 0.2500 0.50 0.2500 0.000 0.0000 #[20,] 0.0000 0.000 0.0000 0.00 0.0000 0.000 0.0000 #[21,] 0.0000 0.000 0.0000 0.00 0.0000 0.000 0.0000 #[22,] 0.0625 0.125 0.0625 0.25 0.0625 0.125 0.0625 #[23,] 0.1250 0.250 0.1250 0.50 0.1250 0.250 0.1250 #[24,] 0.1875 0.375 0.1875 0.75 0.1875 0.375 0.1875 #[25,] 0.2500 0.500 0.2500 1.00 0.2500 0.500 0.2500 #[26,] 0.1875 0.375 0.1875 0.75 0.1875 0.375 0.1875 #[27,] 0.1250 0.250 0.1250 0.50 0.1250 0.250 0.1250 #[28,] 0.0625 0.125 0.0625 0.25 0.0625 0.125 0.0625 #[29,] 0.0000 0.000 0.0000 0.00 0.0000 0.000 0.0000 #[30,] 0.0000 0.000 0.0000 0.00 0.0000 0.000 0.0000 #[31,] 0.0000 0.000 0.2500 0.50 0.2500 0.000 0.0000 #[32,] 0.0000 0.000 0.5000 1.00 0.5000 0.000 0.0000 #[33,] 0.0000 0.000 0.2500 0.50 0.2500 0.000 0.0000 #[34,] 0.0000 0.000 0.0000 0.00 0.0000 0.000 0.0000 #[35,] 0.0000 0.000 0.0000 0.00 0.0000 0.000 0.0000 #[36,] 0.0625 0.125 0.0625 0.25 0.0625 0.125 0.0625 #[37,] 0.1250 0.250 0.1250 0.50 0.1250 0.250 0.1250 #[38,] 0.0625 0.125 0.0625 0.25 0.0625 0.125 0.0625 #[39,] 0.2500 0.500 0.2500 1.00 0.2500 0.500 0.2500 #[40,] 0.0625 0.125 0.0625 0.25 0.0625 0.125 0.0625 #[41,] 0.1250 0.250 0.1250 0.50 0.1250 0.250 0.1250 #[42,] 0.0625 0.125 0.0625 0.25 0.0625 0.125 0.0625
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