[2] | 1 | ;+ |
---|
| 2 | ; NAME: |
---|
| 3 | ; C_TIMECORRELATE |
---|
| 4 | ; |
---|
| 5 | ; PURPOSE: |
---|
| 6 | ; This function computes the "time cross correlation" Pxy(L) or |
---|
| 7 | ; the "time cross covariance" between 2 arrays (this is some |
---|
| 8 | ; kind of c_correlate but for multidimenstionals arrays) as a |
---|
| 9 | ; function of the lag (L). |
---|
| 10 | ; |
---|
| 11 | ; CATEGORY: |
---|
| 12 | ; Statistics. |
---|
| 13 | ; |
---|
| 14 | ; CALLING SEQUENCE: |
---|
| 15 | ; Result = c_timecorrelate(X, Y, Lag) |
---|
| 16 | ; |
---|
| 17 | ; INPUTS: |
---|
| 18 | ; X: an Array which last dimension is the time dimension of |
---|
| 19 | ; size n, float or double. |
---|
| 20 | ; |
---|
| 21 | ; Y: an Array which last dimension is the time dimension of |
---|
| 22 | ; size n, float or double. |
---|
| 23 | ; |
---|
| 24 | ; LAG: A scalar or n-element vector, in the interval [-(n-2), (n-2)], |
---|
| 25 | ; of type integer that specifies the absolute distance(s) between |
---|
| 26 | ; indexed elements of X. |
---|
| 27 | ; |
---|
| 28 | ; KEYWORD PARAMETERS: |
---|
| 29 | ; COVARIANCE: If set to a non-zero value, the sample cross |
---|
| 30 | ; covariance is computed. |
---|
| 31 | ; |
---|
| 32 | ; DOUBLE: If set to a non-zero value, computations are done in |
---|
| 33 | ; double precision arithmetic. |
---|
| 34 | ; |
---|
| 35 | ; EXAMPLE |
---|
| 36 | ; Define two n-element sample populations. |
---|
| 37 | ; x = [3.73, 3.67, 3.77, 3.83, 4.67, 5.87, 6.70, 6.97, 6.40, 5.57] |
---|
| 38 | ; y = [2.31, 2.76, 3.02, 3.13, 3.72, 3.88, 3.97, 4.39, 4.34, 3.95] |
---|
| 39 | ; |
---|
| 40 | ; Compute the cross correlation of X and Y for LAG = -5, 0, 1, 5, 6, 7 |
---|
| 41 | ; lag = [-5, 0, 1, 5, 6, 7] |
---|
| 42 | ; result = c_timecorrelate(x, y, lag) |
---|
| 43 | ; |
---|
| 44 | ; The result should be: |
---|
| 45 | ; [-0.428246, 0.914755, 0.674547, -0.405140, -0.403100, -0.339685] |
---|
| 46 | ; |
---|
| 47 | ; PROCEDURE: |
---|
| 48 | ; |
---|
| 49 | ; |
---|
| 50 | ; FOR L>=0 |
---|
| 51 | ; |
---|
| 52 | ; n-L-1 |
---|
| 53 | ; sigma (X[k]-Xmean)(Y[k+L]-Ymean) |
---|
| 54 | ; k=0 |
---|
| 55 | ; correlation(X,Y,L)=------------------------------------------------------ |
---|
| 56 | ; n-1 n-1 |
---|
| 57 | ; sqrt( (sigma (X[k]-Xmean)^2)*(sigma (Y[k]-Ymean)^2)) |
---|
| 58 | ; k=0 k=0 |
---|
| 59 | ; |
---|
| 60 | ; |
---|
| 61 | ; |
---|
| 62 | ; n-L-1 |
---|
| 63 | ; sigma (X[k]-Xmean)(Y[k+L]-Ymean) |
---|
| 64 | ; k=0 |
---|
| 65 | ; covariance(X,Y,L)=------------------------------------------------------ |
---|
| 66 | ; n |
---|
| 67 | ; |
---|
| 68 | ; FOR L<0 |
---|
| 69 | ; |
---|
| 70 | ; |
---|
| 71 | ; n-L-1 |
---|
| 72 | ; sigma (X[k+L]-Xmean)(Y[k]-Ymean) |
---|
| 73 | ; k=0 |
---|
| 74 | ; correlation(X,Y,L)=------------------------------------------------------ |
---|
| 75 | ; n-1 n-1 |
---|
| 76 | ; sqrt( (sigma (X[k]-Xmean)^2)*(sigma (Y[k]-Ymean)^2)) |
---|
| 77 | ; k=0 k=0 |
---|
| 78 | ; |
---|
| 79 | ; |
---|
| 80 | ; |
---|
| 81 | ; n-L-1 |
---|
| 82 | ; sigma (X[k+L]-Xmean)(Y[k]-Ymean) |
---|
| 83 | ; k=0 |
---|
| 84 | ; covariance(X,Y,L)=------------------------------------------------------ |
---|
| 85 | ; n |
---|
| 86 | ; |
---|
| 87 | ; Where Xmean and Ymean are the time means of the sample populations |
---|
| 88 | ; x=(x[t=0],x[t=1],...,x[t=n-1]) and y=(y[t=0],y[t=1],...,y[t=n-1]), |
---|
| 89 | ; respectively. |
---|
| 90 | ; |
---|
| 91 | ; |
---|
| 92 | ; |
---|
| 93 | ; REFERENCE: |
---|
| 94 | ; INTRODUCTION TO STATISTICAL TIME SERIES |
---|
| 95 | ; Wayne A. Fuller |
---|
| 96 | ; ISBN 0-471-28715-6 |
---|
| 97 | ; |
---|
| 98 | ; MODIFICATION HISTORY: |
---|
| 99 | ; 01/03/2000 Sebastien Masson (smasson@lodyc.jussieu.fr) |
---|
| 100 | ; Based on the C_CORRELATE procedure of IDL |
---|
| 101 | ;- |
---|
| 102 | |
---|
| 103 | FUNCTION TimeCross_Cov, X, Y, M, nT, Double = Double, ZERO2NAN = zero2nan |
---|
| 104 | ; |
---|
| 105 | if double then one = 1.0d ELSE one = 1.0 |
---|
| 106 | ; Sample cross covariance function. |
---|
| 107 | TimeDim = size(X, /n_dimensions) |
---|
| 108 | Xmean = TOTAL(X, TimeDim, Double = Double) / nT |
---|
| 109 | Xmean = Xmean[*]#replicate(one, nT - M) |
---|
| 110 | Ymean = TOTAL(Y, TimeDim, Double = Double) / nT |
---|
| 111 | Ymean = Ymean[*]#replicate(one, nT - M) |
---|
| 112 | ; |
---|
| 113 | case TimeDim of |
---|
| 114 | 1:res = TOTAL((X[0:nT - M - 1L] - Xmean) * (Y[M:nT - 1L] - Ymean) $ |
---|
| 115 | , Double = Double) |
---|
| 116 | 2:res = TOTAL((X[*, 0:nT - M - 1L] - Xmean) * (Y[*, M:nT - 1L] - Ymean) $ |
---|
| 117 | , TimeDim, Double = Double) |
---|
| 118 | 3:res = TOTAL((X[*, *, 0:nT - M - 1L] - Xmean) * (Y[*, *, M:nT - 1L] - Ymean) $ |
---|
| 119 | , TimeDim, Double = Double) |
---|
| 120 | 4:res = TOTAL((X[*, *, *, 0:nT - M - 1L] - Xmean) * (Y[*, *, *, M:nT - 1L] - Ymean) $ |
---|
| 121 | , TimeDim, Double = Double) |
---|
| 122 | ENDCASE |
---|
| 123 | if keyword_set(zero2nan) then begin |
---|
| 124 | zero = where(res EQ 0) |
---|
| 125 | if zero[0] NE -1 then res[zero] = !values.f_nan |
---|
| 126 | ENDIF |
---|
| 127 | ; |
---|
| 128 | RETURN, res |
---|
| 129 | |
---|
| 130 | END |
---|
| 131 | |
---|
| 132 | FUNCTION C_Timecorrelate, X, Y, Lag, Covariance = Covariance, Double = Double |
---|
| 133 | |
---|
| 134 | ;Compute the sample cross correlation or cross covariance of |
---|
| 135 | ;(Xt, Xt+l) and (Yt, Yt+l) as a function of the lag (l). |
---|
| 136 | |
---|
| 137 | ON_ERROR, 2 |
---|
| 138 | |
---|
| 139 | xsize = SIZE(X) |
---|
| 140 | ysize = SIZE(Y) |
---|
| 141 | nt = float(xsize[xsize[0]]) |
---|
| 142 | NDim = xsize[0] |
---|
| 143 | |
---|
| 144 | if total(xsize[0:xsize[0]] NE ysize[0:ysize[0]]) NE 0 then $ |
---|
| 145 | MESSAGE, "X and Y arrays must have the same size and the same dimensions" |
---|
| 146 | |
---|
| 147 | ;Check length. |
---|
| 148 | if nt lt 2 then $ |
---|
| 149 | MESSAGE, "Time dimension of X and Y arrays must contain 2 or more elements." |
---|
| 150 | |
---|
| 151 | ;If the DOUBLE keyword is not set then the internal precision and |
---|
| 152 | ;result are identical to the type of input. |
---|
| 153 | if N_ELEMENTS(Double) eq 0 then $ |
---|
| 154 | Double = (Xsize[Xsize[0]+1] eq 5 or ysize[ysize[0]+1] eq 5) |
---|
| 155 | |
---|
| 156 | if n_elements(lag) EQ 0 then lag = 0 |
---|
| 157 | nLag = N_ELEMENTS(Lag) |
---|
| 158 | |
---|
| 159 | if nLag eq 1 then Lag = [Lag] ;Create a 1-element vector. |
---|
| 160 | |
---|
| 161 | case NDim of |
---|
| 162 | 1:if Double eq 0 then Correl = FLTARR(nLag) else Correl = DBLARR(nLag) |
---|
| 163 | 2:if Double eq 0 then Correl = FLTARR(Xsize[1], nLag) else Correl = DBLARR(Xsize[1], nLag) |
---|
| 164 | 3:if Double eq 0 then Correl = FLTARR(Xsize[1], Xsize[2], nLag) $ |
---|
| 165 | else Correl = DBLARR(Xsize[1], Xsize[2], nLag) |
---|
| 166 | 4:if Double eq 0 then Correl = FLTARR(Xsize[1], Xsize[2], Xsize[3], nLag) $ |
---|
| 167 | else Correl = DBLARR(Xsize[1], Xsize[2], Xsize[3], nLag) |
---|
| 168 | endcase |
---|
| 169 | |
---|
| 170 | if KEYWORD_SET(Covariance) eq 0 then begin ;Compute Cross Correlation. |
---|
| 171 | for k = 0, nLag-1 do begin |
---|
| 172 | if Lag[k] ge 0 then BEGIN |
---|
| 173 | case NDim of |
---|
| 174 | 1:Correl[k] = TimeCross_Cov(X, Y, Lag[k], nT, Double = Double) / $ |
---|
| 175 | sqrt(TimeCross_Cov(X,X, 0L, nT, Double = Double, /zero2nan) * $ |
---|
| 176 | TimeCross_Cov(Y,Y, 0L, nT, Double = Double, /zero2nan)) |
---|
| 177 | 2:Correl[*, k] = TimeCross_Cov(X, Y, Lag[k], nT, Double = Double) / $ |
---|
| 178 | sqrt(TimeCross_Cov(X,X, 0L, nT, Double = Double, /zero2nan) * $ |
---|
| 179 | TimeCross_Cov(Y,Y, 0L, nT, Double = Double, /zero2nan)) |
---|
| 180 | 3:Correl[*, *, k] = TimeCross_Cov(X, Y, Lag[k], nT, Double = Double) / $ |
---|
| 181 | sqrt(TimeCross_Cov(X,X, 0L, nT, Double = Double, /zero2nan) * $ |
---|
| 182 | TimeCross_Cov(Y,Y, 0L, nT, Double = Double, /zero2nan)) |
---|
| 183 | 4:Correl[*, *, *, k] = TimeCross_Cov(X, Y, Lag[k], nT, Double = Double) / $ |
---|
| 184 | sqrt(TimeCross_Cov(X,X, 0L, nT, Double = Double, /zero2nan) * $ |
---|
| 185 | TimeCross_Cov(Y,Y, 0L, nT, Double = Double, /zero2nan)) |
---|
| 186 | endcase |
---|
| 187 | ENDIF else BEGIN |
---|
| 188 | case NDim of |
---|
| 189 | 1:Correl[k] = TimeCross_Cov(y, x, ABS(Lag[k]), nT, Double = Double) / $ |
---|
| 190 | sqrt(TimeCross_Cov(X,X, 0L, nT, Double = Double, /zero2nan) * $ |
---|
| 191 | TimeCross_Cov(Y,Y, 0L, nT, Double = Double, /zero2nan)) |
---|
| 192 | 2:Correl[*, k] = TimeCross_Cov(y, x, ABS(Lag[k]), nT, Double = Double) / $ |
---|
| 193 | sqrt(TimeCross_Cov(X,X, 0L, nT, Double = Double, /zero2nan) * $ |
---|
| 194 | TimeCross_Cov(Y,Y, 0L, nT, Double = Double, /zero2nan)) |
---|
| 195 | 3:Correl[*, *, k] = TimeCross_Cov(y, x, ABS(Lag[k]), nT, Double = Double) / $ |
---|
| 196 | sqrt(TimeCross_Cov(X,X, 0L, nT, Double = Double, /zero2nan) * $ |
---|
| 197 | TimeCross_Cov(Y,Y, 0L, nT, Double = Double, /zero2nan)) |
---|
| 198 | 4:Correl[*, *, *, k] = TimeCross_Cov(y, x, ABS(Lag[k]), nT, Double = Double) / $ |
---|
| 199 | sqrt(TimeCross_Cov(X,X, 0L, nT, Double = Double, /zero2nan) * $ |
---|
| 200 | TimeCross_Cov(Y,Y, 0L, nT, Double = Double, /zero2nan)) |
---|
| 201 | endcase |
---|
| 202 | ENDELSE |
---|
| 203 | endfor |
---|
| 204 | endif else begin ;Compute Cross Covariance. |
---|
| 205 | for k = 0, nLag-1 do begin |
---|
| 206 | if Lag[k] ge 0 then BEGIN |
---|
| 207 | case NDim of |
---|
| 208 | 1:Correl[k] = TimeCross_Cov(X, Y, Lag[k], nT, Double = Double) / nT |
---|
| 209 | 2:Correl[*, k] = TimeCross_Cov(X, Y, Lag[k], nT, Double = Double) / nT |
---|
| 210 | 3:Correl[*, *, k] = TimeCross_Cov(X, Y, Lag[k], nT, Double = Double) / nT |
---|
| 211 | 4:Correl[*, *, *, k] = TimeCross_Cov(X, Y, Lag[k], nT, Double = Double) / nT |
---|
| 212 | ENDCASE |
---|
| 213 | ENDIF else BEGIN |
---|
| 214 | case NDim of |
---|
| 215 | 1:Correl[k] = TimeCross_Cov(y, x, ABS(Lag[k]), nT, Double = Double) / nT |
---|
| 216 | 2:Correl[*, k] = TimeCross_Cov(y, x, ABS(Lag[k]), nT, Double = Double) / nT |
---|
| 217 | 3:Correl[*, *, k] = TimeCross_Cov(y, x, ABS(Lag[k]), nT, Double = Double) / nT |
---|
| 218 | 4:Correl[*, *, *, k] = TimeCross_Cov(y, x, ABS(Lag[k]), nT, Double = Double) / nT |
---|
| 219 | ENDCASE |
---|
| 220 | ENDELSE |
---|
| 221 | endfor |
---|
| 222 | endelse |
---|
| 223 | |
---|
| 224 | if Double eq 0 then RETURN, FLOAT(Correl) else $ |
---|
| 225 | RETURN, Correl |
---|
| 226 | |
---|
| 227 | END |
---|
| 228 | |
---|