２次元ウェーブレット変換の例

Mathematica  パッケージや関数の指定

Mathematica  utilities

<<Utilities`MemoryConserve`
$MemoryIncrement=10000000;
Off[General::spell,General::spell1,
MemoryConserve::start,MemoryConserve::end];
<< LinearAlgebra`MatrixManipulation`;

<< Graphics`PlotField3D`;
<< Graphics`Colors`;
<< 
        Miscellaneous`WorldPlot`;

相関係数を求める関数

\(\(corRelation = 
        Compile[{{a, _Real, 1}, {b, _Real, 1}}, 
		Module[{av = 0. , 
              bv = 0. , aP = {0. }, bP = {0. }}, 
			av = 
              Apply[Plus, a]/Length[a]; 
			bv = 
              Apply[Plus, b]/Length[b]; 
			aP = a - av; 
			bP = 
              b - bv; 
			aP . 
                bP/\((Sqrt[aP . aP]*Sqrt[bP . bP])\)
			]];\)\)

離散値系ウェーブレット変換用の関数

高須賀義弘
2002/6/28

ウェーブレット基底関数

ドビッシーの基底関数

daub2={{1.,1.},{1.,-1.}}/Sqrt[2.];

daub4={{1.+Sqrt[3.], 3.+Sqrt[3.],3.-Sqrt[3.], 1.-Sqrt[3.]},
	{1.-Sqrt[3.],-3.+Sqrt[3.],3.+Sqrt[3.],-1.-Sqrt[3.]}}/
	(4 Sqrt[2.]);

daub6={{ .332670552950, .806891509311, .459877502118,
		-.135011020010, -.085441273882, .035226291882},
	   { .035226291882, .085441273882,-.135011020010,
	    -.459877502118,.806891509311,-.332670552950}};


daub8={{ 0.325803428051, 1.010945715092, 0.892200138246,
	    -0.039575026236,-0.264507167369, 0.043616300475,
	     0.046503601071,-0.014986989330},
	   {-0.014986989330,-0.046503601071, 0.043616300475,
		 0.264507167369,-0.039575026236,-0.892200138246,
		 1.010945715092,-0.325803428051}}/Sqrt[2.];

daub10={{ .160102397974, .603829269797, .724308528438,
		  .138428145901,-.242294887066,-.032244869585,
		  .077571493840,-.006241490213,-.012580751999,
		  .003335725285},
	    { .003335725285, .012580751999,-.006241490213,
	     -.077571493840,-.032244869585, .242294887066,
	      .138428145901,-.724308528438, .603829269797,
	     -.160102397974}};
	 
daub12={{ 0.111540743350, 0.494623890398, 0.751133908021,
		  0.315250351709,-0.226264693965,-0.129766867567,
		  0.097501605587, 0.027522865530,-0.031582039318,
		  0.000553842201, 0.004777257511,-0.001077301085},
		{-0.001077301085,-0.004777257511, 0.000553842201,
		  0.031582039318, 0.027522865530,-0.097501605587,
		 -0.129766867567, 0.226264693965, 0.315250351709,
		 -0.751133908021, 0.494623890398,-0.111540743350}};
		 
daub14={{ .077852054085, .396539319482, .729132090846, 
 		  .469782287405,-.143906003929,-.224036184994,
 		  .071309219267, .080612609151,-.038029936935, 
		 -.016574541631, .012550998556, .000429577973, 
		 -.001801640704, .000353713800},
		{ .000353713800, .001801640704, .000429577973, 
		 -.012550998556,-.016574541631, .038029936935,
		  .080612609151,-.071309219267,-.224036184994,
		  .143906003929, .469782287405,-.729132090846,
		  .396539319482,-.077852054085}};
		  
daub16={{ .054415842243, .312871590914, .675630736297, 
 		  .585354683654,-.015829105256,-.284015542962,
 		  .000472484574, .128747426620,-.017369301002, 
		 -.044088253931, .013981027917, .008746094047, 
		 -.004870352993,-.000391740373, .000675449406,
		 -.000117476784},
		{-.000117476784,-.000675449406,-.000391740373,
		  .004870352993, .008746094047,-.013981027917,
		 -.044088253931, .017369301002, .128747426620,
		 -.000472484574,-.284015542962, .015829105256,
		  .585354683654,-.675630736297, .312871590914,
		 -.054415842243}};

daub18={{ .038077947364, .243834674613, .604823123690, 
 		  .657288078051, .133197385825,-.293273783279, 
		 -.096840783223, .148540749338, .030725681479,
		 -.067632829061, .000250947115, .022361662124,
		 -.004723204758,-.004281503682, .001847646883,
          .000230385764,-.000251963189, .000039347320},
        { .000039347320, .000251963189, .000230385764,
         -.001847646883,-.004281503682, .004723204758,
          .022361662124,-.000250947115,-.067632829061,
         -.030725681479, .148540749338, .096840783223,
         -.293273783279,-.133197385825, .657288078051, 
         -.604823123690, .243834674613,-.038077947364}};
	
daub20={{ 0.026670057901, 0.188176800078, 0.527201188932,
		  0.688459039454, 0.281172343661,-0.249846424327,
		 -0.195946274377, 0.127369340336, 0.093057364604,
		 -0.071394147166,-0.029457536822, 0.033212674059,
		  0.003606553567,-0.010733175483, 0.001395351747,
		  0.001992405295,-0.000685856695,-0.000116466855,
		  0.000093588670,-0.000013264203},
		{-0.000013264203,-0.000093588670,-0.000116466855,
		  0.000685856695, 0.001992405295,-0.001395351747,
		 -0.010733175483,-0.003606553567, 0.033212674059,
		  0.029457536822,-0.071394147166,-0.093057364604,
		  0.127369340336, 0.195946274377,-0.249846424327,
		 -0.281172343661, 0.688459039454,-0.527201188932,
		  0.188176800078,-0.026670057901}};

コフマン(Coifman)の基底関数

coif6={{-.051429728471, .238929728471, .602859456942, 
 		 .272140543058,-.051429972847,-.011070271529},
 	   {-.011070271529, .051429972847, .272140543058,
		-.602859456942, .238929728471, .051429728471}}*
		Sqrt[2.];

coif12={{ .011587596739,-.029320137980,-.047639590310, 
 		  .273021046535, .574682393857, .294867193696,
		 -.054085607092,-.042026480461, .016744410163, 
 		  .003967883613,-.001289203356,-.000509505399},
 		{-.000509505399, .001289203356, .003967883613,
 		 -.016744410163,-.042026480461, .054085607092,
 		  .294867193696,-.574682393857, .273021046535,
		  .047639590310,-.029320137980,-.011587596739}}*
		 Sqrt[2.];
		  
coif18={{-.002682418671, .005503126709, .016583560479, 
		 -.046507764479,-.043220763560, .286503335274,
 		  .561285256870, .302983571773,-.050770140755, 
		 -.058196250762, .024434094321, .011229240962, 
		 -.006369601011,-.001820458916, .000790205101,
 		  .000329665174,-.000050192775,-.000024465734},
 		{-.000024465734, .000050192775, .000329665174,
 		 -.000790205101,-.001820458916, .006369601011,
 		  .011229240962,-.024434094321,-.058196250762,
 		  .050770140755, .302983571773,-.561285256870,
 		  .286503335274, .043220763560,-.046507764479,
 		 -.016583560479, .005503126709, .002682418671}}*
 		Sqrt[2.];
 		 
coif24={{ .000630961046,-.001152224852,-.005194524026, 
 		  .011362459244, .018867235378,-.057464234429, 
		 -.039652648517, .293667390895, .553126452562, 
 		  .307157326198,-.047112738865,-.068038127051,
 		  .027813640153, .017735837438,-.010756318517,
		 -.004001012886, .002652665946, .000895594529,
		 -.000416500571,-.000183829769, .000044080354, 
 		  .000022082857,-.000002304942,-.000001262175},
 		{-.000001262175, .000002304942, .000022082857,
 		 -.000044080354,-.000183829769, .000416500571,
 		  .000895594529,-.002652665946,-.004001012886,
 		  .010756318517, .017735837438,-.027813640153,
 		 -.068038127051, .047112738865, .307157326198,
 		 -.553126452562, .293667390895, .039652648517,
 		 -.057464234429,-.018867235378, .011362459244,
 		  .005194524026,-.001152224852,-.000630961046}}*
 		Sqrt[2.];
 		  
coif30={{-.000149963800, .000253561200, .001540245700, 
		 -.002941110800,-.007163781900, .016552066400, 
 		  .019917804300,-.064997262800,-.036800073600,
 		  .298092323500, .547505429400, .309706849000, 
		 -.043866050800,-.074652238900, .029195879500, 
 		  .023110777000,-.013973687900,-.006480090000, 
 		  .004783001400, .001720654700,-.001175822200,
		 -.000451227000, .000213729800, .000099377600,
		 -.000029232100,-.000015072000, .000002640800, 
 		  .000001459300,-.000000118400,-.000000067300},
 		{-.000000067300, .000000118400, .000001459300,
 		 -.000002640800,-.000015072000, .000029232100,
 		  .000099377600,-.000213729800,-.000451227000,
 		  .001175822200, .001720654700,-.004783001400,
 		 -.006480090000, .013973687900, .023110777000,
 		 -.029195879500,-.074652238900, .043866050800,
 		  .309706849000,-.547505429400, .298092323500,
 		  .036800073600,-.064997262800,-.019917804300,
 		  .016552066400, .007163781900,-.002941110800,
 		 -.001540245700, .000253561200, .000149963800}}*
 		Sqrt[2.];

ベイキン(Baylkin)の基底関数

bay6={{ .038580777748,-.126969125396,-.077161555496, 
 		.607491641386, .745687558934, .226584265197},
 	  { .226584265197,-.745687558934, .607491641386,
 	    .077161555496,-.126969125396,-.038580777748}};
 	    
bay12={{ .001194572696,-.012845579553, .024804330519, 
 		 .050023519962,-.155357222860,-.071638282295, 
 		 .570465001450, .750336305853, .280611651902,
		-.007410383519,-.014611552521,-.001358799059},
	   {-.001358799059, .014611552521,-.007410383519,
	    -.280611651902, .750336305853,-.570465001450,
	    -.071638282295, .155357222860, .050023519962,
	    -.024804330519,-.012845579553,-.001194572696}};
	    
bay18={{-.001691851019,-.003487876220, .019191160680, 
 		 .021671094636,-.098507213321,-.056997424478, 
 		 .456787122173, .789319409004, .380557130852,
		-.070438748795,-.056514193868, .036409962613,
 		 .008760130709,-.011194759274,-.001921335414,
 		 .002041380977, .000445830398,-.000216257277},
 	   {-.000216257277,-.000445830398, .002041380977,
 	     .001921335414,-.011194759274,-.008760130709,
 	     .036409962613, .056514193868,-.070438748795,
 	    -.380557130852, .789319409004,-.456787122173,
 	    -.056997424478, .098507213321, .021671094636,
 	    -.019191160680,-.003487876220, .001691851019}};

1次元ウェーブレット変換関数定義

\(\(waveletTransform[daTa_, 
          baseType_] := 
Module[
{i, 
            n = Length[daTa], nn = Length[baseType[\([1]\)]], tempData, 
            dataHead = {}, 
            dataLast = {}}, 
Do[
tempData = Take[\ RotateLeft[daTa, i], 
                nn]; \ 
dataHead = {dataHead\ , \((baseType[\([1]\)]\  . \ \ tempData)\)\ }; 
dataLast = {dataLast, \((baseType[\([2]\)]\  . \ \ \ tempData\ )\)}, 
	{i, 0, n - 1, 
              2}
\ ]; 
dataHead = 
            Flatten[dataHead]; 
If[Length[dataHead] >= nn, 
            Flatten[{waveletTransform[dataHead, baseType], 
                dataLast}], 
Flatten[{dataHead, 
                dataLast}]
]
];\)\)

1次元ウェーブレット逆変換関数定義

InverseWaveletTransform[inData_, 
          baseType_] := 
Module[
{i, j, 
            tempData, n = Length[inData], 
            nn = 2 ^ Ceiling[Log[2, Length[baseType[[1]]]]], d1, d2, 
            sumArray = {}}, 

daTa = 
            inData; 

For[j = nn, 
            j ≤ 
              n, 
tmpData = {}; 
Do[
              tmpData = Append[tmpData, daTa[[i]]]; 
              tmpData = Append[tmpData, daTa[[j/2 + i]]], {i, 
                j/2}]; 
d1 = 
              PadRight[baseType[[1]], j]; 
d2 = 
              PadRight[baseType[[2]], j]; 
sumArray = 
              Transpose[
                  Table[RotateRight[If[OddQ[i], d1, d2], 
                      i + If[OddQ[i], -1, -2]], {i, j}] ]  .  
                tmpData; 
daTa = 
              Join[sumArray, 
                Take[daTa, {j + 1, Length[daTa]}]]; 
j *= 
              2;
]; 
daTa
];

ｎ次元ウェーブレット変換関数関数定義

waveletTransformND[dataIn_, 
          baseTypeIn_] := 
Module[{nn, n, tl, data, i, k, 
            d1, d2, dataHead, dataLast, retVal, tempData, 
            dataSub}, 

nn = 
            Depth[dataIn] - 1; 
If[ 
            Depth[baseTypeIn] == 
              3, 
baseType = 
                Table[baseTypeIn, {nn}];, 
If[
                Depth[baseTypeIn] == 4   &&  
                  Length[baseTypeIn] == 
                    nn , 
baseType = 
                    baseTypeIn;, 
Abort[]];
]; 
tl = 
            Table[k, {k, nn}]  //  
              RotateRight; 

data = 
            dataIn; 
Do[
d1 = 
              baseType[[k, 1]]; 
d2 = 
              baseType[[k, 
                  2]]; 
retVal = {}; 
dataSub = data; 
While[
              True, 
dataHead = {}; 
              dataLast = {}; 
Do[
tempData = 
                  Take[RotateLeft[dataSub, i], 
                    Length[d1]]; 
dataHead = 
                  Join[dataHead, {d1  .  
                        tempData}]; 
dataLast = 
                  Join[dataLast, {d2  .  
                        tempData}];
, {i, 0, 
                  Length[dataSub] - 1, 
                  2}
]; 
retVal = 
                Join[dataLast, retVal]; 
If[
                Length[dataHead] >= Length[d1], dataSub = dataHead; 
                Continue[];]; 
data = 
                Transpose[Join[dataHead, retVal], 
                  tl]; 
Break[];
]
, {k, 
              nn}]; 
data
];

ｎ次元ウェーブレット逆変換関数関数定義

InverseWaveletTransformND[dataIn_, 
          baseTypeIn_] := 
Module[{nnn, nn, n, tl, data, i, 
            j, k, d1, d2, dataHead, tempData, dataSub, 
            sumArray}, 

nn = 
            Depth[dataIn] - 1; 
If[ 
            Depth[baseTypeIn] == 
              3, 
baseType = 
                Table[baseTypeIn, {nn}];, 
If[
                Depth[baseTypeIn] == 4   &&  
                  Length[baseTypeIn] == 
                    nn , 
baseType = 
                    baseTypeIn;, 
Abort[]];
]; 
tl = 
            Table[k, {k, nn}]  //  RotateRight; 
data = 
            dataIn; 
Do[
dataSub  = 
              data; 
n = 
              Length[dataSub]; 
nnn = 
              2 ^ Ceiling[
                  Log[2, Length[
                      baseType[[k, 1]]]]]; 
For[
              j = nnn, 
              j ≤ 
                n, 
tmpData = {}; 
Do[
                tmpData = Append[tmpData, dataSub[[i]]]; 
                tmpData = Append[tmpData, dataSub[[j/2 + i]]], {i, 
                  j/2}]; 
d1 = 
                PadRight[baseType[[k, 1]], j]; 
d2 = 
                PadRight[baseType[[k, 2]], 
                  j]; 
sumArray = 
                Transpose[
                    Table[RotateRight[If[OddQ[i], d1, d2], 
                        i + If[OddQ[i], -1, -2]], {i, j}] ]  .  
                  tmpData; 
dataSub = 
                Join[sumArray, 
                  Take[dataSub, {j + 1, 
                      Length[dataSub]}]]; 
j *= 
                2;
]; 
data = 
              Transpose[dataSub, tl]
, {k, 
              nn}]; 
data
];

「ウェーブレット変換 ｎ次の関数」の使用法
                                             ３次元ウェーブレット変換の例

ウェーブレット変換 ｎ次の関数  waveletTransformND は ｎ次元データのウェーブレット変換をウェーブレット変換行列 wMat を用いて行う関数です。
     たとえば、32×16×64 の大きさの配列データを基底関数  daub2(ドビッシーの2次基底関数), daub4 ,daub8を用いてウェーブレット変換し、ウェーブレットスペクトラムを求める手順は次のとおりです。
1. ウェーブレット変換行列  wMat を作成します。
                  wMat={daub2,daub4,daub8};
2. ウェーブレット変換し、ウェーブレットスペクトラム wSpect を求めます。
                  wSpect=waveletTransformND[data,wMat];

      ウェーブレット逆変換の ｎ次の関数  InverseWaveletTransformND は ｎ次元データのウェーブレット逆変換をウェーブレット変換行列 wMat を用いて行う関数です。	ウェーブレットスペクトラム wSpect を逆変換してデータを復元する手順は次のとおりです
1. ウェーブレット変換行列 wMat を作成します。
                  wMat={daub2,daub4,daub8};
2. 逆変します。
                 recover=InverseWaveletTransformND[wSpect,wMat];
 このとき wMat=daub4 のように 基底関数を１つだけ記述すると、
          wMat={daub4,daub4,daub4}; 
 と同じになります。

多重解像度計算の関数

多重解像度計算の関数（１次元）

expanMatrix[v_,wMat_]:=
 Block[{mdata=Length[v],is=Length[wMat[[1]]],
	ndata=Ceiling[Log[2.,mdata]],mother,emat,exPan},
	mother=Ceiling[Log[2.,is]]-1;	
    emat=InverseWaveletTransform[Join[Take[v,{1,2^mother}],Table[0,{mdata-
    		2^mother}]],wMat];	
   	partInverse[i_]:=(exPan=Table[0,{mdata}];
   		Do[exPan[[j]]=v[[j]],{j,1+2^(i-1),2^i}];
   		InverseWaveletTransform[exPan,wMat]);
    exPan=Table[partInverse[i],{i,mother+1,ndata}];
    emat=AppendColumns[{emat},exPan]		
  ];

多重解像度計算の関数（２次元）

expanMatrix2D[spect2D_,baseTypeIn_]:=
	Block[{dim=Dimensions[spect2D],wMattable,isL,isR,
	       motherL,motherR,emat2D,exPan2D},
	
    If[ Depth[baseTypeIn]==3,
                    wMattable={baseTypeIn,baseTypeIn},
                      wMattable=baseTypeIn];
    isL=Length[wMattable[[1]][[1]]]; isR=Length[wMattable[[2]][[1]]];
    motherL=Ceiling[Log[2.,isL]]-1; motherR=Ceiling[Log[2.,isR]]-1;
	nLdataL=Log[2,dim[[1]]]; nLdataR=Log[2,dim[[2]]];
	 sizeL=dim[[1]]; sizeR=dim[[2]];
	 nL=nLdataL-motherL; nR=nLdataR-motherR;
	  Which[
		nL>=nR, nLdata=nL;sizeR=2^(motherR+nL),
		nR>nL, nLdata=nR;sizeL=2^(motherL+nR)
		];

	 inputMatrix=ZeroMatrix[sizeL,sizeR];
			Do[inputMatrix[[i,j]]=spect2D[[i,j]],
				{i,dim[[1]]},{j,dim[[2]]}];
		
	  exPan2D=ZeroMatrix[dim[[1]],dim[[2]]];
			Do[Do[exPan2D[[i,j]]=spect2D[[i,j]],{i,2^motherL}],
				{j,2^motherR}];
			emat2D=InverseWaveletTransformND[exPan2D,wMattable];
		
	  partInverse2D[i_]:=
			(	iL=motherL+i; iR=motherR+i;
			    dummyMat=ZeroMatrix[sizeL,sizeR];
				Do[dummyMat[[k,m]]=inputMatrix[[k,m]],
					{k,2^iL},{m,1+2^(iR-1),2^iR}];
   				Do[dummyMat[[k,m]]=inputMatrix[[k,m]],
   					{k,1+2^(iL-1),2^iL},{m,2^(iR-1)}];
   				InverseWaveletTransformND[Table[dummyMat[[k,m]],
   					{k,dim[[1]]},{m,dim[[2]]}],wMattable]
   			);
    	
      exPan2D=Table[partInverse2D[i],{i,nLdata}];
    		Flatten[{{emat2D},exPan2D},1]
    ];

多重解像度計算の関数（３次元）

expanMatrix3D[spectrum_,baseTypeIn_]:=
	Block[{dim=Dimensions[spectrum],
	   n=Depth[spectrum]-1,wMattable,is,
	   mother,nwdata,ndata,size,emat3D,exPan3D},
	 
   If[ Depth[baseTypeIn]==3,
                    wMattable=Table[baseTypeIn,{n}],
                      wMattable=baseTypeIn];
   is=Table[Length[wMattable[[i]][[1]]],
	                        {i,Length[wMattable]}];
   mother=Table[Ceiling[Log[2.,is[[i]]]]-1,{i,n}];
   nwdata=Table[Log[2,dim[[i]]]-mother[[i]],{i,n}];
   ndata=Max[nwdata]; size=2^(mother+ndata);
   
   inputMatrix=Table[0,{size[[1]]},{size[[2]]},{size[[3]]}];
   Do[inputMatrix[[k,j,i]]=spectrum[[k,j,i]],
				{k,dim[[1]]},{j,dim[[2]]},{i,dim[[3]]}];
		
   exPanM=Table[0,{dim[[1]]},{dim[[2]]},{dim[[3]]}];;
   Do[exPanM[[k,j,i]]=spectrum[[k,j,i]],{k,2^mother[[1]]},
				{j,2^mother[[2]]},{i,2^mother[[3]]}];
	emat=InverseWaveletTransformND[exPanM,wMattable];
		
   partInverse[i_]:=
	(s=mother+i;
	 dummyMat=Table[0,{size[[1]]},{size[[2]]},{size[[3]]}];
	 Do[dummyMat[[k,l,m]]=inputMatrix[[k,l,m]],
	  {k,1+2^(s[[1]]-1),2^s[[1]]},{l,2^s[[2]]},{m,2^s[[3]]}];
	 Do[dummyMat[[k,l,m]]=inputMatrix[[k,l,m]],
   	  {k,2^(s[[1]]-1)},{l,2^s[[2]]},{m,1+2^(s[[3]]-1),2^s[[3]]}];
   	 Do[dummyMat[[k,l,m]]=inputMatrix[[k,l,m]],
      {k,2^(s[[1]]-1)},{l,1+2^(s[[2]]-1),2^s[[2]]},{m,2^(s[[3]]-1)}];
     InverseWaveletTransformND[Table[dummyMat[[k,l,m]],
   		{k,dim[[1]]},{l,dim[[2]]},{m,dim[[3]]}],wMattable]
   	 );
    	
      exPanM=Table[partInverse[i],{i,ndata}];
      Join[{emat},exPanM]
    ];

風データ作図用の関数

plot vector function

vector[{x1_,y1_},{x2_,y2_},th_,ya_]:=
 Module[{a,b,z,aa,bb,sen,c1,c2,c11,c12,d1,d2,d11,d12,e1,e2,
         p,ten},
  a=x2; b=y2; aa=x1+x2; bb=y1+y2; z=N[Sqrt[a^2+b^2]];
  If[z!=0.0,
  (sen={Thickness[th],Line[{{x1,y1},{aa,bb}}]};
   c1=-a Cos[15 Degree] + b Sin[15 Degree];
   c2=-a Sin[15 Degree] - b Cos[15 Degree];
   c11=aa+ya c1/Sqrt[c1^2+c2^2];c12=bb+ya c2/Sqrt[c1^2+c2^2];
   d1=-a Cos[-15 Degree] + b Sin[-15 Degree];
   d2=-a Sin[-15 Degree] - b Cos[-15 Degree];
   d11=aa+ya d1/Sqrt[d1^2+d2^2];d12=bb+ya d2/Sqrt[d1^2+d2^2];
   e1=aa-0.6 ya a/z; e2=bb-0.6 ya b/z;
   p=Polygon[{{aa,bb},{c11,c12},{e1,e2},{d11,d12},{aa,bb}}];
   Graphics[{sen,p}]),
  (ten={PointSize[th],Point[{x1,y1}]};Graphics[{ten}])
  ]
  ];
vectors[vectorT_,th_,ya_]:=
 Module[{n,v1},
  n=Dimensions[vectorT];
  v1=Table[{N[i],0.0},{i,n[[1]]}];
  Table[vector[v1[[i]],vectorT[[i]],th,ya],{i,n[[1]]}]
 ];
vectors2[vectorT_,th_,ya_]:=
Module[{n,v1,v2,v3},
  n=Dimensions[vectorT];
  v1=Flatten[vectorT,1];
  v2=Table[{N[j],N[i]},{i,n[[1]]},{j,n[[2]]}];
  v3=Flatten[v2,1];
  Table[vector[v3[[i]],v1[[i]],th,ya],{i,n[[1]] n[[2]]}]
  ];

plot vector function2

vectorz[{x1_,y1_},{x2_,y2_},th_,ya_]:=
 Module[{a,b,z,aa,bb,sen,c1,c2,c11,c12,d1,d2,d11,d12,e1,e2,
         p,ten},
  a=x2; b=y2; aa=x1+x2; bb=y1+y2; z=N[Sqrt[a^2+b^2]];
  If[z!=0.0,
  (sen={Thickness[th],Line[{{x1,y1},{aa,bb}}]};
   c1=-a Cos[15 Degree] + b Sin[15 Degree];
   c2=-a Sin[15 Degree] - b Cos[15 Degree];
   c11=aa+ya c1/Sqrt[c1^2+c2^2];c12=bb+ya c2/Sqrt[c1^2+c2^2];
   d1=-a Cos[-15 Degree] + b Sin[-15 Degree];
   d2=-a Sin[-15 Degree] - b Cos[-15 Degree];
   d11=aa+ya d1/Sqrt[d1^2+d2^2];d12=bb+ya d2/Sqrt[d1^2+d2^2];
   e1=aa-0.6 ya a/z; e2=bb-0.6 ya b/z;
   p=Polygon[{{aa,bb},{c11,c12},{e1,e2},{d11,d12},{aa,bb}}];
   Graphics[{sen,p}]),
  (Graphics[{}])
  ]
  ];
vectorsz[vectorT_,th_,ya_]:=
 Module[{n,v1},
  n=Dimensions[vectorT];
  v1=Table[{N[i],0.0},{i,n[[1]]}];
  Table[vectorz[v1[[i]],vectorT[[i]],th,ya],{i,n[[1]]}]
 ];
vectorsz2[vectorT_,th_,ya_]:=
 Module[{n,v1,v2,v3},
  n=Dimensions[vectorT];
  v1=Flatten[vectorT,1];
  v2=Table[{N[j],N[i]},{i,n[[1]]},{j,n[[2]]}];
  v3=Flatten[v2,1];
  Table[vectorz[v3[[i]],v1[[i]],th,ya],{i,n[[1]] n[[2]]}]
  ];
vectorszo[vectorT_,th_,ya_,o_,fac_]:=
 Module[{n,v1},
  n=Dimensions[vectorT];
  v1=Table[{N[i]*fac,0.0},{i,1-o,n[[1]]-o}];
  Table[vectorz[v1[[i]],vectorT[[i]]fac,th,ya*fac],{i,n[[1]]}]
 ];
vectorszo2[vectorT_,th_,ya_,xo_,yo_,fac_]:=
 Module[{n,v1,v2,v3},
  n=Dimensions[vectorT];
  v1=Flatten[vectorT,1]*fac;
  v2=Table[{N[j]*fac,N[i]*fac},
          {i,1-yo,n[[1]]-yo},{j,1-xo,n[[2]]-xo}];
  v3=Flatten[v2,1];
  Table[vectorz[v3[[i]],v1[[i]],th,ya*fac],{i,n[[1]] n[[2]]}]
  ];

WorldPlot1

Banana

wpaOrange=WorldPlot[{ World, Table[Banana,{174}] },
	WorldRotation->{0, 0, -180},
	WorldRange->{{-75,82.5},{0,-42.5}},
	WorldGrid->None,
	WorldFrame->None,
	WorldToGraphics->True];

緯度、経度用の目盛

fontc12={"Courier",12.};
fontc13={"Courier",13.};
faca=60*2.5; xa=72; ya=32;
xyFramea= {{{(0-xa)*faca,FontForm[0,fontc12]},
           {(24-xa)*faca,FontForm[60,fontc12]},
           {(48-xa)*faca,FontForm[120,fontc12]},
           {(72-xa)*faca,FontForm[180,fontc12]},
           {(96-xa)*faca,FontForm[240,fontc12]},
           {(120-xa)*faca,FontForm["    degree",fontc12]},
           {(144-xa)*faca,""}},
          {{(8-ya)*faca,FontForm[-60,fontc12]},
           {(20-ya)*faca,FontForm[-30,fontc12]},
           {(32-ya)*faca,FontForm[0,fontc12]},
           {(44-ya)*faca,FontForm[30,fontc12]},
           {(56-ya)*faca,FontForm[60,fontc12]},
           {(68-ya)*faca,""}},None,None};

離散値系ウェーブレット変換用（２次元の例）

Data    1989年7月 風データ（250hPa）

\(month = "July 1989"; 
    v = {{{1.7`, 10.7`}, {1.`, 10.5`}, {0.4`, 10.`}, {\(-0.2`\), 
            9.4`}, {\(-0.6`\), 8.6`}, {\(-1.`\), 7.7`}, {\(-1.3`\), 
            6.7`}, {\(-1.5`\), 5.7`}, {\(-1.5`\), 4.7`}, {\(-1.5`\), 
            3.7`}, {\(-1.4`\), 2.8`}, {\(-1.3`\), 1.9`}, {\(-1.1`\), 
            1.1`}, {\(-0.9`\), 
            0.4`}, {\(-0.7`\), \(-0.3`\)}, {\(-0.5`\), \(-0.9`\)}, {\(-0.2`\), \(-1.4`\)}, {0.`, \(-1.9`\)}, {0.3`, \(-2.3`\)}, {0.5`, \(-2.7`\)}, {0.8`, \(-2.9`\)}, {1.1`, \(-3.2`\)}, {1.4`, \(-3.3`\)}, {1.8`, \(-3.4`\)}, {2.1`, \(-3.4`\)}, {2.5`, \(-3.3`\)}, {2.8`, \(-3.2`\)}, {3.2`, \(-3.`\)}, {3.6`, \(-2.7`\)}, {3.9`, \(-2.3`\)}, {4.3`, \(-1.9`\)}, {4.6`, \(-1.5`\)}, {4.8`, \(-1.`\)}, {5.1`, \(-0.6`\)}, {5.3`, \(-0.2`\)}, {5.4`, 
            0.1`}, {5.6`, 0.3`}, {5.7`, 0.5`}, {5.8`, 0.5`}, {6.`, 
            0.6`}, {6.1`, 0.5`}, {6.3`, 0.4`}, {6.4`, 0.3`}, {6.6`, 
            0.2`}, {6.8`, 0.2`}, {6.9`, 0.1`}, {7.`, 0.1`}, {7.1`, 
            0.2`}, {7.2`, 0.2`}, {7.2`, 0.4`}, {7.1`, 0.4`}, {7.`, 
            0.6`}, {6.8`, 0.6`}, {6.7`, 0.7`}, {6.5`, 0.7`}, {6.3`, 
            0.6`}, {6.1`, 0.5`}, {5.9`, 0.4`}, {5.7`, 0.3`}, {5.6`, 
            0.2`}, {5.6`, 0.1`}, {5.6`, 0.1`}, {5.6`, 0.1`}, {5.7`, 
            0.2`}, {5.8`, 0.4`}, {5.9`, 0.7`}, {6.1`, 1.1`}, {6.2`, 
            1.5`}, {6.2`, 2.`}, {6.2`, 2.5`}, {6.2`, 2.9`}, {6.1`, 
            3.4`}, {5.9`, 3.7`}, {5.7`, 3.9`}, {5.5`, 4.1`}, {5.2`, 
            4.1`}, {4.9`, 3.9`}, {4.7`, 3.6`}, {4.4`, 3.2`}, {4.2`, 
            2.7`}, {4.`, 2.1`}, {3.8`, 1.5`}, {3.7`, 0.8`}, {3.7`, 
            0.1`}, {3.6`, \(-0.6`\)}, {3.7`, \(-1.2`\)}, {3.7`, \(-1.8`\)}, {3.7`, \(-2.4`\)}, {3.8`, \(-2.9`\)}, {3.9`, \(-3.3`\)}, {4.`, \(-3.7`\)}, {4.1`, \(-4.`\)}, {4.2`, \(-4.3`\)}, {4.3`, \(-4.6`\)}, {4.5`, \(-4.8`\)}, {4.6`, \(-5.`\)}, {4.7`, \(-5.2`\)}, {4.9`, \(-5.4`\)}, {5.1`, \(-5.6`\)}, {5.2`, \(-5.7`\)}, {5.4`, \(-5.9`\)}, {5.6`, \(-6.2`\)}, {5.8`, \(-6.4`\)}, {6.1`, \(-6.7`\)}, {6.4`, \(-7.`\)}, {6.7`, \(-7.2`\)}, {7.1`, \(-7.5`\)}, {7.5`, \(-7.8`\)}, {8.`, \(-8.`\)}, {8.5`, \(-8.2`\)}, {9.`, \(-8.3`\)}, {9.5`, \(-8.3`\)}, {10.`, \(-8.3`\)}, {10.5`, \(-8.1`\)}, {11.`, \(-7.8`\)}, {11.4`, \(-7.5`\)}, {11.8`, \(-7.1`\)}, {12.`, \(-6.5`\)}, {12.2`, \(-6.`\)}, {12.3`, \(-5.3`\)}, {12.2`, \(-4.7`\)}, {12.2`, \(-4.1`\)}, {12.`, \(-3.4`\)}, {11.8`, \(-2.8`\)}, {11.5`, \(-2.1`\)}, {11.1`, \(-1.5`\)}, {10.8`, \(-0.9`\)}, {10.4`, \(-0.2`\)}}, {{2.4`, 10.7`}, {1.9`, 
            10.2`}, {1.5`, 9.6`}, {1.2`, 8.7`}, {1.1`, 7.6`}, {1.`, 
            6.5`}, {1.`, 5.3`}, {1.1`, 4.2`}, {1.3`, 3.1`}, {1.6`, 
            2.1`}, {1.9`, 1.2`}, {2.2`, 
            0.5`}, {2.4`, \(-0.2`\)}, {2.7`, \(-0.8`\)}, {3.`, \(-1.3`\)}, {3.2`, \(-1.8`\)}, {3.5`, \(-2.2`\)}, {3.7`, \(-2.7`\)}, {3.9`, \(-3.`\)}, {4.2`, \(-3.3`\)}, {4.4`, \(-3.6`\)}, {4.6`, \(-3.9`\)}, {4.8`, \(-4.1`\)}, {5.1`, \(-4.2`\)}, {5.3`, \(-4.2`\)}, {5.5`, \(-4.2`\)}, {5.8`, \(-4.`\)}, {6.`, \(-3.8`\)}, {6.3`, \(-3.5`\)}, {6.6`, \(-3.1`\)}, {6.8`, \(-2.6`\)}, {7.1`, \(-2.1`\)}, {7.3`, \(-1.6`\)}, {7.5`, \(-1.1`\)}, {7.7`, \(-0.7`\)}, {7.8`, \(-0.4`\)}, {8.`, \(-0.2`\)}, {8.1`, \(-0.1`\)}, {8.3`, \(-0.1`\)}, {8.4`, \(-0.2`\)}, {8.7`, \(-0.4`\)}, {8.9`, \(-0.6`\)}, {9.2`, \(-0.8`\)}, {9.6`, \(-1.`\)}, {9.9`, \(-1.1`\)}, {10.3`, \(-1.1`\)}, {10.6`, \(-1.1`\)}, {10.9`, \(-0.9`\)}, {11.2`, \(-0.7`\)}, {11.4`, \(-0.4`\)}, {11.5`, \(-0.2`\)}, {11.6`, 0.1`}, {11.6`, 0.4`}, {11.6`, 0.6`}, {11.4`, 
            0.7`}, {11.3`, 0.7`}, {11.2`, 0.6`}, {11.`, 0.5`}, {10.9`, 
            0.3`}, {10.8`, 
            0.1`}, {10.7`, \(-0.2`\)}, {10.7`, \(-0.3`\)}, {10.8`, \(-0.5`\)}, {10.9`, \(-0.5`\)}, {11.`, \(-0.4`\)}, {11.2`, \(-0.2`\)}, {11.4`, 0.2`}, {11.5`, 0.6`}, {11.7`, 1.2`}, {11.8`, 1.7`}, {11.8`, 
            2.3`}, {11.8`, 2.9`}, {11.8`, 3.4`}, {11.6`, 3.8`}, {11.4`, 
            4.`}, {11.2`, 4.1`}, {10.9`, 4.1`}, {10.7`, 3.8`}, {10.4`, 
            3.5`}, {10.1`, 3.1`}, {9.8`, 2.5`}, {9.5`, 1.9`}, {9.2`, 
            1.3`}, {9.`, 0.6`}, {8.8`, 
            0.`}, {8.6`, \(-0.6`\)}, {8.4`, \(-1.2`\)}, {8.2`, \(-1.7`\)}, {8.`, \(-2.2`\)}, {7.9`, \(-2.6`\)}, {7.8`, \(-3.`\)}, {7.8`, \(-3.3`\)}, {7.7`, \(-3.6`\)}, {7.7`, \(-3.9`\)}, {7.8`, \(-4.2`\)}, {7.8`, \(-4.4`\)}, {7.9`, \(-4.6`\)}, {8.`, \(-4.8`\)}, {8.1`, \(-5.`\)}, {8.2`, \(-5.2`\)}, {8.4`, \(-5.5`\)}, {8.5`, \(-5.8`\)}, {8.7`, \(-6.1`\)}, {8.9`, \(-6.4`\)}, {9.2`, \(-6.8`\)}, {9.5`, \(-7.3`\)}, {9.9`, \(-7.7`\)}, {10.4`, \(-8.2`\)}, {10.9`, \(-8.6`\)}, {11.5`, \(-8.9`\)}, {12.2`, \(-9.1`\)}, {13.`, \(-9.1`\)}, {13.7`, \(-9.1`\)}, {14.4`, \(-8.8`\)}, {15.1`, \(-8.3`\)}, {15.6`, \(-7.7`\)}, {16.1`, \(-7.`\)}, {16.5`, \(-6.2`\)}, {16.6`, \(-5.2`\)}, {16.6`, \(-4.3`\)}, {16.5`, \(-3.3`\)}, {16.2`, \(-2.4`\)}, {15.9`, \(-1.5`\)}, {15.4`, \(-0.7`\)}, {14.8`, 0.`}, {14.1`, 0.7`}, {13.4`, 
            1.4`}, {12.7`, 2.`}}, {{4.8`, 10.`}, {4.4`, 9.4`}, {4.2`, 
            8.6`}, {4.`, 7.5`}, {3.8`, 6.3`}, {3.8`, 5.`}, {3.9`, 
            3.8`}, {4.`, 2.7`}, {4.2`, 1.7`}, {4.3`, 0.9`}, {4.5`, 
            0.2`}, {4.6`, \(-0.4`\)}, {4.7`, \(-0.9`\)}, {4.8`, \(-1.3`\)}, {5.`, \(-1.8`\)}, {5.2`, \(-2.2`\)}, {5.4`, \(-2.6`\)}, {5.6`, \(-3.`\)}, {5.9`, \(-3.3`\)}, {6.2`, \(-3.7`\)}, {6.4`, \(-3.9`\)}, {6.7`, \(-4.1`\)}, {6.9`, \(-4.2`\)}, {7.`, \(-4.3`\)}, {7.1`, \(-4.3`\)}, {7.2`, \(-4.2`\)}, {7.3`, \(-4.1`\)}, {7.4`, \(-3.9`\)}, {7.5`, \(-3.6`\)}, {7.7`, \(-3.2`\)}, {7.9`, \(-2.8`\)}, {8.`, \(-2.3`\)}, {8.2`, \(-1.8`\)}, {8.4`, \(-1.3`\)}, {8.5`, \(-0.8`\)}, {8.6`, \(-0.5`\)}, {8.8`, \(-0.3`\)}, {8.9`, \(-0.2`\)}, {9.`, \(-0.4`\)}, {9.1`, \(-0.6`\)}, {9.4`, \(-1.`\)}, {9.6`, \(-1.4`\)}, {10.`, \(-1.8`\)}, {10.4`, \(-2.2`\)}, {10.9`, \(-2.4`\)}, {11.4`, \(-2.6`\)}, {11.9`, \(-2.6`\)}, {12.5`, \(-2.5`\)}, {13.`, \(-2.3`\)}, {13.6`, \(-2.`\)}, {14.1`, \(-1.7`\)}, {14.6`, \(-1.4`\)}, {15.1`, \(-1.1`\)}, {15.6`, \(-0.9`\)}, {15.9`, \(-0.7`\)}, {16.3`, \(-0.6`\)}, {16.6`, \(-0.5`\)}, {16.8`, \(-0.6`\)}, {17.`, \(-0.6`\)}, {17.1`, \(-0.8`\)}, {17.3`, \(-0.9`\)}, {17.4`, \(-1.1`\)}, {17.5`, \(-1.2`\)}, {17.7`, \(-1.2`\)}, {17.9`, \(-1.1`\)}, {18.1`, \(-0.9`\)}, {18.2`, \(-0.5`\)}, {18.4`, 0.`}, {18.6`, 0.6`}, {18.7`, 1.2`}, {18.7`, 
            1.8`}, {18.7`, 2.5`}, {18.7`, 3.`}, {18.6`, 3.5`}, {18.4`, 
            3.8`}, {18.2`, 4.1`}, {17.9`, 4.2`}, {17.6`, 4.2`}, {17.2`, 
            4.`}, {16.8`, 3.7`}, {16.3`, 3.4`}, {15.9`, 3.`}, {15.3`, 
            2.5`}, {14.8`, 1.9`}, {14.3`, 1.4`}, {13.8`, 0.8`}, {13.3`, 
            0.2`}, {12.8`, \(-0.3`\)}, {12.4`, \(-0.8`\)}, {12.1`, \(-1.4`\)}, {11.8`, \(-1.9`\)}, {11.6`, \(-2.3`\)}, {11.5`, \(-2.8`\)}, {11.4`, \(-3.2`\)}, {11.4`, \(-3.6`\)}, {11.5`, \(-3.9`\)}, {11.5`, \(-4.1`\)}, {11.6`, \(-4.4`\)}, {11.7`, \(-4.6`\)}, {11.8`, \(-4.8`\)}, {11.8`, \(-5.`\)}, {11.8`, \(-5.3`\)}, {11.9`, \(-5.6`\)}, {11.9`, \(-6.1`\)}, {12.1`, \(-6.6`\)}, {12.2`, \(-7.2`\)}, {12.6`, \(-7.8`\)}, {13.`, \(-8.5`\)}, {13.5`, \(-9.1`\)}, {14.2`, \(-9.5`\)}, {14.9`, \(-9.9`\)}, {15.8`, \(-10.`\)}, {16.6`, \(-9.9`\)}, {17.5`, \(-9.5`\)}, {18.3`, \(-8.9`\)}, {19.`, \(-8.1`\)}, {19.6`, \(-7.`\)}, {20.1`, \(-5.9`\)}, {20.4`, \(-4.6`\)}, {20.5`, \(-3.3`\)}, {20.4`, \(-2.1`\)}, {20.1`, \(-0.9`\)}, {19.8`, 0.2`}, {19.3`, 1.1`}, {18.7`, 2.`}, {18.`, 2.8`}, {17.3`, 
            3.5`}, {16.5`, 4.2`}}, {{8.3`, 9.6`}, {7.8`, 9.`}, {7.4`, 
            8.`}, {7.`, 6.9`}, {6.7`, 5.7`}, {6.5`, 4.4`}, {6.3`, 
            3.2`}, {6.1`, 2.2`}, {5.9`, 1.4`}, {5.8`, 0.8`}, {5.6`, 
            0.2`}, {5.4`, \(-0.2`\)}, {5.3`, \(-0.6`\)}, {5.2`, \(-1.1`\)}, {5.2`, \(-1.6`\)}, {5.3`, \(-2.1`\)}, {5.5`, \(-2.7`\)}, {5.8`, \(-3.2`\)}, {6.2`, \(-3.7`\)}, {6.7`, \(-4.`\)}, {7.`, \(-4.2`\)}, {7.4`, \(-4.3`\)}, {7.6`, \(-4.2`\)}, {7.8`, \(-4.1`\)}, {7.8`, \(-3.9`\)}, {7.8`, \(-3.8`\)}, {7.8`, \(-3.6`\)}, {7.8`, \(-3.4`\)}, {7.8`, \(-3.2`\)}, {7.8`, \(-3.`\)}, {7.9`, \(-2.7`\)}, {8.1`, \(-2.3`\)}, {8.2`, \(-1.9`\)}, {8.4`, \(-1.4`\)}, {8.6`, \(-1.`\)}, {8.6`, \(-0.6`\)}, {8.7`, \(-0.4`\)}, {8.8`, \(-0.4`\)}, {8.8`, \(-0.6`\)}, {8.9`, \(-0.9`\)}, {9.1`, \(-1.4`\)}, {9.3`, \(-2.`\)}, {9.6`, \(-2.5`\)}, {9.9`, \(-3.1`\)}, {10.4`, \(-3.5`\)}, {10.9`, \(-3.8`\)}, {11.5`, \(-4.`\)}, {12.2`, \(-4.1`\)}, {13.`, \(-4.1`\)}, {13.8`, \(-4.1`\)}, {14.7`, \(-4.`\)}, {15.8`, \(-3.9`\)}, {16.8`, \(-3.8`\)}, {17.9`, \(-3.7`\)}, {19.`, \(-3.5`\)}, {20.`, \(-3.3`\)}, {20.9`, \(-3.2`\)}, {21.8`, \(-3.`\)}, {22.5`, \(-2.8`\)}, {23.1`, \(-2.6`\)}, {23.7`, \(-2.5`\)}, {24.1`, \(-2.3`\)}, {24.5`, \(-2.1`\)}, {24.8`, \(-1.9`\)}, {25.1`, \(-1.6`\)}, {25.3`, \(-1.2`\)}, {25.5`, \(-0.7`\)}, {25.6`, \(-0.1`\)}, {25.6`, 0.5`}, {25.6`, 1.1`}, {25.5`, 1.8`}, {25.5`, 2.4`}, {25.3`, 
            2.9`}, {25.1`, 3.4`}, {24.9`, 3.8`}, {24.7`, 4.1`}, {24.4`, 
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            3.3`}}, {{6.3`, 1.4`}, {6.6`, 1.8`}, {6.9`, 2.2`}, {7.3`, 
            2.5`}, {7.7`, 2.8`}, {8.1`, 3.`}, {8.6`, 3.2`}, {9.`, 
            3.3`}, {9.4`, 3.4`}, {9.8`, 3.5`}, {10.2`, 3.6`}, {10.6`, 
            3.6`}, {10.9`, 3.6`}, {11.3`, 3.6`}, {11.7`, 3.6`}, {12.`, 
            3.5`}, {12.4`, 3.3`}, {12.8`, 3.1`}, {13.1`, 2.8`}, {13.4`, 
            2.4`}, {13.8`, 2.`}, {14.1`, 1.6`}, {14.3`, 1.`}, {14.5`, 
            0.5`}, {14.7`, 
            0.`}, {14.8`, \(-0.6`\)}, {14.9`, \(-1.1`\)}, {14.9`, \(-1.5`\)}, {14.9`, \(-2.`\)}, {14.9`, \(-2.3`\)}, {14.8`, \(-2.6`\)}, {14.7`, \(-2.8`\)}, {14.6`, \(-2.9`\)}, {14.4`, \(-3.`\)}, {14.3`, \(-3.`\)}, {14.2`, \(-3.`\)}, {14.1`, \(-3.1`\)}, {14.`, \(-3.1`\)}, {13.9`, \(-3.1`\)}, {13.8`, \(-3.2`\)}, {13.6`, \(-3.3`\)}, {13.5`, \(-3.5`\)}, {13.4`, \(-3.6`\)}, {13.2`, \(-3.8`\)}, {13.1`, \(-4.`\)}, {12.9`, \(-4.1`\)}, {12.8`, \(-4.3`\)}, {12.6`, \(-4.4`\)}, {12.4`, \(-4.5`\)}, {12.3`, \(-4.6`\)}, {12.1`, \(-4.5`\)}, {12.`, \(-4.5`\)}, {11.8`, \(-4.4`\)}, {11.7`, \(-4.2`\)}, {11.5`, \(-4.`\)}, {11.4`, \(-3.8`\)}, {11.3`, \(-3.5`\)}, {11.1`, \(-3.3`\)}, {10.9`, \(-3.`\)}, {10.8`, \(-2.7`\)}, {10.6`, \(-2.4`\)}, {10.5`, \(-2.`\)}, {10.4`, \(-1.7`\)}, {10.2`, \(-1.3`\)}, {10.1`, \(-0.9`\)}, {10.1`, \(-0.5`\)}, {10.1`, \(-0.1`\)}, {10.1`, 
            0.4`}, {10.2`, 0.8`}, {10.3`, 1.3`}, {10.5`, 1.7`}, {10.8`, 
            2.2`}, {11.`, 2.6`}, {11.4`, 2.9`}, {11.7`, 3.2`}, {12.1`, 
            3.5`}, {12.5`, 3.7`}, {12.9`, 3.8`}, {13.4`, 3.9`}, {13.8`, 
            3.8`}, {14.2`, 3.7`}, {14.7`, 3.5`}, {15.1`, 3.2`}, {15.4`, 
            2.8`}, {15.7`, 2.3`}, {15.9`, 1.9`}, {16.1`, 1.4`}, {16.3`, 
            0.8`}, {16.4`, 
            0.3`}, {16.5`, \(-0.1`\)}, {16.5`, \(-0.6`\)}, {16.5`, \(-1.`\)}, {16.5`, \(-1.4`\)}, {16.5`, \(-1.8`\)}, {16.4`, \(-2.1`\)}, {16.3`, \(-2.5`\)}, {16.1`, \(-2.8`\)}, {15.9`, \(-3.2`\)}, {15.5`, \(-3.5`\)}, {15.1`, \(-3.8`\)}, {14.7`, \(-4.`\)}, {14.1`, \(-4.2`\)}, {13.5`, \(-4.3`\)}, {12.8`, \(-4.3`\)}, {12.1`, \(-4.3`\)}, {11.3`, \(-4.1`\)}, {10.6`, \(-3.8`\)}, {9.8`, \(-3.5`\)}, {9.1`, \(-3.`\)}, {8.4`, \(-2.5`\)}, {7.8`, \(-1.9`\)}, {7.3`, \(-1.3`\)}, {6.8`, \(-0.7`\)}, {6.5`, 0.`}, {6.2`, 
            0.6`}, {5.9`, 1.2`}, {5.8`, 1.8`}, {5.7`, 2.3`}, {5.7`, 
            2.8`}, {5.8`, 3.2`}, {5.9`, 3.5`}, {6.2`, 3.8`}, {6.4`, 
            3.9`}, {6.7`, 3.9`}, {7.1`, 3.8`}, {7.4`, 3.5`}, {7.7`, 
            3.1`}, {8.`, 2.7`}}, {{7.6`, 0.9`}, {7.7`, 1.1`}, {7.9`, 
            1.3`}, {8.1`, 1.5`}, {8.3`, 1.6`}, {8.6`, 1.8`}, {8.9`, 
            1.9`}, {9.2`, 2.`}, {9.5`, 2.1`}, {9.8`, 2.1`}, {10.1`, 
            2.2`}, {10.4`, 2.2`}, {10.8`, 2.2`}, {11.1`, 2.2`}, {11.4`, 
            2.1`}, {11.8`, 2.`}, {12.1`, 1.9`}, {12.5`, 1.8`}, {12.8`, 
            1.5`}, {13.1`, 1.3`}, {13.4`, 1.`}, {13.7`, 0.6`}, {14.`, 
            0.3`}, {14.2`, \(-0.1`\)}, {14.3`, \(-0.5`\)}, {14.4`, \(-0.9`\)}, {14.5`, \(-1.3`\)}, {14.5`, \(-1.7`\)}, {14.4`, \(-2.`\)}, {14.4`, \(-2.3`\)}, {14.3`, \(-2.5`\)}, {14.2`, \(-2.7`\)}, {14.1`, \(-2.8`\)}, {13.9`, \(-3.`\)}, {13.8`, \(-3.`\)}, {13.7`, \(-3.1`\)}, {13.6`, \(-3.1`\)}, {13.4`, \(-3.2`\)}, {13.3`, \(-3.2`\)}, {13.2`, \(-3.3`\)}, {13.1`, \(-3.4`\)}, {12.9`, \(-3.5`\)}, {12.8`, \(-3.5`\)}, {12.6`, \(-3.7`\)}, {12.4`, \(-3.8`\)}, {12.2`, \(-3.9`\)}, {11.9`, \(-4.`\)}, {11.7`, \(-4.1`\)}, {11.4`, \(-4.1`\)}, {11.1`, \(-4.1`\)}, {10.8`, \(-4.1`\)}, {10.5`, \(-4.`\)}, {10.3`, \(-3.9`\)}, {10.`, \(-3.8`\)}, {9.8`, \(-3.6`\)}, {9.5`, \(-3.4`\)}, {9.3`, \(-3.2`\)}, {9.1`, \(-2.9`\)}, {9.`, \(-2.6`\)}, {8.9`, \(-2.4`\)}, {8.8`, \(-2.1`\)}, {8.7`, \(-1.8`\)}, {8.6`, \(-1.5`\)}, {8.6`, \(-1.2`\)}, {8.7`, \(-0.9`\)}, {8.7`, \(-0.6`\)}, {8.8`, \(-0.2`\)}, {8.9`, 0.1`}, {9.1`, 0.4`}, {9.3`, 0.7`}, {9.5`, 1.1`}, {9.8`, 
            1.4`}, {10.1`, 1.7`}, {10.4`, 1.9`}, {10.8`, 2.2`}, {11.2`, 
            2.4`}, {11.6`, 2.5`}, {12.`, 2.6`}, {12.4`, 2.7`}, {12.8`, 
            2.7`}, {13.1`, 2.6`}, {13.5`, 2.5`}, {13.8`, 2.3`}, {14.1`, 
            2.1`}, {14.4`, 1.8`}, {14.6`, 1.5`}, {14.8`, 1.2`}, {14.8`, 
            0.8`}, {14.9`, 0.5`}, {14.9`, 
            0.2`}, {14.9`, \(-0.2`\)}, {14.8`, \(-0.5`\)}, {14.6`, \(-0.8`\)}, {14.5`, \(-1.1`\)}, {14.2`, \(-1.3`\)}, {14.`, \(-1.5`\)}, {13.7`, \(-1.7`\)}, {13.4`, \(-1.9`\)}, {13.`, \(-2.`\)}, {12.6`, \(-2.1`\)}, {12.2`, \(-2.1`\)}, {11.7`, \(-2.1`\)}, {11.3`, \(-2.`\)}, {10.8`, \(-1.9`\)}, {10.3`, \(-1.7`\)}, {9.8`, \(-1.5`\)}, {9.4`, \(-1.2`\)}, {9.`, \(-0.9`\)}, {8.6`, \(-0.5`\)}, {8.2`, \(-0.1`\)}, {8.`, 0.3`}, {7.7`, 
            0.7`}, {7.6`, 1.2`}, {7.5`, 1.6`}, {7.4`, 2.`}, {7.5`, 
            2.3`}, {7.5`, 2.7`}, {7.7`, 3.`}, {7.8`, 3.2`}, {8.`, 
            3.4`}, {8.3`, 3.5`}, {8.5`, 3.5`}, {8.8`, 3.5`}, {9.1`, 
            3.4`}, {9.3`, 3.2`}, {9.5`, 3.`}, {9.7`, 2.7`}, {9.8`, 
            2.4`}}};\)

データのグラフ

mVorg = v;  
    vx = Transpose[mVorg, {2, 3, 1}][[1]];
vy = Transpose[
            mVorg, {2, 3, 1}][[2]];
vDim = Dimensions[vx];  ny = vDim[[1]];  
    nx = vDim[[2]];

ww=Table[{0.0,0.0},{ny},{nx}];
mV=mVorg;
  Do[ ww[[j,i]]=mV[[j,i]], {j,1,ny,3},{i,1,nx,3}];
  gmV=vectorszo2[ww 0.3,0.001,1.5,xa,ya,faca];
  Show[wpaOrange,gmV,
   Frame->True, AspectRatio->0.5,
   PlotLabel->FontForm
     ["250hPa surface           "<>month,fontc12], 
	   PlotRange -> {{-11700,10300},{-5250,5580}},
	   ImageSize-> {300,180},  
	   FrameLabel->{FontForm["Longitude",fontc13],
				   FontForm["Latitude" ,fontc13]},
      FrameTicks->xyFramea,
      Epilog->Text[FontForm["degree",fontc12],
				             {-15550,4800},{-1,0}]];

データを２のべき乗個にそろえる

nvDim = 
        Table[   order = 2; 
While[
            vDim[[i]] > order, order = order*2];  
          order, {i, Length[vDim]}];
my = nvDim[[1]];  
    mx = nvDim[[2]]; 
If[(ny == my  &&  nx == mx ), h = 0, h = 1];

\(\(If[
        h\  ≠ \ 
          0, 
\ \((If\ \ [my > ny, 
            wtb = Table[0. , {my - ny}, {nx}]]; 
\ \ \ \ \ \ \ \ If\ \ [
            mx > nx, wtc = Table[0. , {mx - nx}, {my}]]\ \ )\)];\)\)

\(\(If[
        h\  ≠ \ 
          0, \((\ \ \ If\ \ [my > ny, w1 = Join[vx, wtb], w1 = vx]; 
          w2 = Transpose[
              w1\ ]\ ; 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ If\ \ [mx > nx, w1 = Join[w2, wtc], w1 = w2]; 
          nvx = Transpose[
              w1\ ]\ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ )\), \ \ \ nvx = vx\ \ ]\ ;\)\ \ \)
\(\(If[
        h\  ≠ \ 
          0, \((\ \ If\ \ [my > ny, w1 = Join[vy, wtb], w1 = vy]; 
          w2 = Transpose[
              w1\ ]\ ; 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ If\ \ [mx > nx, w1 = Join[w2, wtc], w1 = w2]; 
          nvy = Transpose[
              w1]\ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ )\), \ \ nvy = vy\ \ ];\)\)
nmVorg = Transpose[{nvx, nvy}, {3, 1, 2} ] ;

ウェーブレット変換    ２次元

ウェーブレット変換 ｎ次の関数  waveletTransformND と InverseWaveletTransformNDは それぞれ
ｎ次元データのウェーブレット変換と逆変換をウェーブレット変換行列 wMat を用いて行う関数です。
     ここではmy×mx の大きさの配列データを基底関数  daub20(ドビッシーの2次基底関数)とdaub12を用いてウェーブレット変換します。

wMattable = {daub20, daub12};

spectrumX=waveletTransformND[nvx,  wMattable];
spectrumY=waveletTransformND[nvy,  wMattable];

逆変換

recoverX=InverseWaveletTransformND[spectrumX, wMattable];
recoverY=InverseWaveletTransformND[spectrumY, wMattable];
recoverV = 
        Transpose[{recoverX, recoverY}, {3, 1, 2} ] ;

相関係数を求める

Print["  Cor.Coe.=",
	   corRelation[Flatten[nmVorg],Flatten[recoverV]]];

多重解像度解析

多重解像度解析の計算

xCompmul=expanMatrix2D[spectrumX,wMattable];//Timing
yCompmul=expanMatrix2D[spectrumY,wMattable];//Timing

各レベルに分解されたデータの表示（元のデータサイズ）
         時間、緯度、経度についてデータを間引きして表示）

dim=Dimensions[xCompmul];
muL=dim[[1]]; jL=dim[[2]]; iL=dim[[3]];
mN={"0","1","2","3","4","5","6","7","8","9","10"};

ww=Table[{0.0,0.0},{ny},{nx}];
Do[{xComp=xCompmul[[m]]; yComp=yCompmul[[m]]; 
  mV=Transpose[{xComp,yComp},{3,1,2} ] ; 
  Do[ ww[[j,i]]=mV[[j,i]], {j,1,ny,3},{i,1,nx,3}];
  gmV=vectorszo2[ww 0.3,0.001,1.5,xa,ya,faca];
  Show[wpaOrange,gmV,
   Frame->True, AspectRatio->0.5,
   PlotLabel->FontForm
     ["250hPa surface    "<>month<>" level "<>mN[[m]],fontc12], 
	   PlotRange -> {{-11700,10300},{-5250,5580}},
	   ImageSize-> {300,180},  
	   FrameLabel->{FontForm["Longitude",fontc13],
				   FontForm["Latitude" ,fontc13]},
      FrameTicks->xyFramea,
      Epilog->Text[FontForm["degree",fontc12],
				             {-15550,4800},{-1,0}]   ]}
	,{m,muL}];

すべてのレベルの和のデータの表示（元のデータサイズ）
         時間、緯度、経度についてデータを間引きして表示）

ww=Table[{0.0,0.0},{ny},{nx}];
xSum=Apply[Plus,xCompmul];ySum=Apply[Plus,yCompmul];
  mV=Transpose[{xSum,ySum},{3,1,2} ] ; 
  Do[ ww[[j,i]]=mV[[j,i]], {j,1,ny,3},{i,1,nx,3}];
  gmV=vectorszo2[ww 0.3,0.001,1.5,xa,ya,faca];
  Show[wpaOrange,gmV,
   Frame->True, AspectRatio->0.5,
   PlotLabel->FontForm
     ["250hPa surface    "<>month,fontc12], 
	   PlotRange -> {{-11700,10300},{-5250,5580}},
	   ImageSize-> {300,180},  
	   FrameLabel->{FontForm["Longitude",fontc13],
				   FontForm["Latitude" ,fontc13]},
      FrameTicks->xyFramea,
      Epilog->Text[FontForm["degree",fontc12],
				             {-15550,4800},{-1,0}]   ];

相関係数を求める

Print["  Cor.Coe.=",
	   corRelation[Flatten[nmVorg],Flatten[mV]]];

0レベルから各レベルまでの和のデータの表示（元のデータサイズ）
         時間、緯度、経度についてデータを間引きして表示）

ww=Table[{0.0,0.0},{ny},{nx}];
Do[{xComp1=Table[xCompmul[[m]],{m,md}];
    yComp1=Table[yCompmul[[m]],{m,md}];
    xSum=Apply[Plus,xComp1]; ySum=Apply[Plus,yComp1];
  mV=Transpose[{xSum,ySum},{3,1,2} ] ; 
  Do[ ww[[j,i]]=mV[[j,i]], {j,1,ny,3},{i,1,nx,3}];
  gmV=vectorszo2[ww 0.3,0.001,1.5,xa,ya,faca];
  Show[wpaOrange,gmV,
   Frame->True, AspectRatio->0.5,
   PlotLabel->FontForm
     ["250hPa surface    "<>month<>" level 0-"<>mN[[md]],fontc12], 
	   PlotRange -> {{-11700,10300},{-5250,5580}},
	   ImageSize-> {300,180},  
	   FrameLabel->{FontForm["Longitude",fontc13],
				   FontForm["Latitude" ,fontc13]},
      FrameTicks->xyFramea,
      Epilog->Text[FontForm["degree",fontc12],
				             {-15550,4800},{-1,0}]   ]}
	,{md,muL}];