help acf DIRECT COMPUTATION OF THE AUTOCORRELATION (AUTOCOVARIANCE) FUNCTION ESTIMATE (suitable for short sequences) CALLING: function [xro,xroE,mi,miE,xacf,stat] = acf(x,fig,alpha,K,P) INPUT: x ... time series data vector of length n fig ... figure handle for the acf plot (optional: default=[]=no plot) at most 50 acf values plotted. alpha ... a percentual confidence risk (optional: default=[]=5%) K ... length of leading part of xro to be used for the computation of Portmanteau statistics (see stat on output) (optional: default=[]=min(round(sqrt(n)),length(xro))) P ... x residuals => P=number of parameters in the associated model ... x is observed time series => P=0 (default=[]=0) OUTPUT: xro ... estimate of the autocorrelation function of length n/2 (column vector) (leading significant part only) xroE ... xroE(q+1)=asymptotic (1-alpha)% confidence halfwidth for xro(j+1), j>q (q=0,1,...n/2-1) under the hypothesis that x is an MA(q) process. (column vector) mi ... estimate of the mean miE ... = [miE1,miE2] = (1-alpha)% asymptotic confidence halfwidths of the mean miE2 is under assumption of normality of the generating noise miE20 (residuals) stat(2) = chi2inv(1-0.01*alpha,K-P) ... chi^2 quantile = NaN if K<=P (too many parameters in the model) stat(3) = K-P ... degrees of freedom Under the hypothesis xro(h)=0 for h>1 (x=white noise) stat(1) is approximately distributed as chi^2(K-p) and thus stat(1) > stat(2) => we reject with risk alpha % the hypothesis that x could be a white noise. datadir=[crpath,'/brockwel.dat'] datadir = q:/Matlab_Toolbox/cas_rady/brockwel.dat datafile=[datadir,'/airpass.dat'] datafile = q:/Matlab_Toolbox/cas_rady/brockwel.dat/airpass.dat format compact [x,xinfo] = getdata(datafile); xacf=acf(x); xacf(1:10) ans = 1.0000 0.9480 0.8756 0.8067 0.7526 0.7138 0.6817 0.6629 0.6556 0.6709 + ??? + | Error: Expression or statement is incomplete or incorrect. plotdata(xacf,'Autokorelacni funkce'); xacf=acf(x,figure); % Simulace n=1000; sigma=3; mi=10; wn=mi+sigma*randn(n,1); mean(wn) ans = 9.8708 std(wn) ans = 2.8305 help mean MEAN Average or mean value. For vectors, MEAN(X) is the mean value of the elements in X. For matrices, MEAN(X) is a row vector containing the mean value of each column. For N-D arrays, MEAN(X) is the mean value of the elements along the first non-singleton dimension of X. MEAN(X,DIM) takes the mean along the dimension DIM of X. Example: If X = [0 1 2 3 4 5] then mean(X,1) is [1.5 2.5 3.5] and mean(X,2) is [1 4] Class support for input X: float: double, single See also median, std, min, max, var, cov, mode. Overloaded methods: timeseries/mean fints/mean Reference page in Help browser doc mean help std STD Standard deviation. For vectors, Y = STD(X) returns the standard deviation. For matrices, Y is a row vector containing the standard deviation of each column. For N-D arrays, STD operates along the first non-singleton dimension of X. STD normalizes Y by (N-1), where N is the sample size. This is the sqrt of an unbiased estimator of the variance of the population from which X is drawn, as long as X consists of independent, identically distributed samples. Y = STD(X,1) normalizes by N and produces the square root of the second moment of the sample about its mean. STD(X,0) is the same as STD(X). Y = STD(X,FLAG,DIM) takes the standard deviation along the dimension DIM of X. Pass in FLAG==0 to use the default normalization by N-1, or 1 to use N. Example: If X = [4 -2 1 9 5 7] then std(X,0,1) is [3.5355 4.9497 4.2426] and std(X,0,2) is [3.0 2.0] Class support for input X: float: double, single See also cov, mean, var, median, corrcoef. 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0.3029i 0 + 0.2978i 0 + 0.2982i 0 + 0.2973i 0 + 0.2974i 0 + 0.3038i 0 + 0.2902i 0 + 0.2890i 0 + 0.2965i 0 + 0.2996i 0 + 0.2947i 0 + 0.2979i 0 + 0.2955i 0 + 0.3046i 0 + 0.2871i 0 + 0.2942i sqrt((wn-mean(wn)).^2/(n-1)) ans = 0.0370 0.1540 0.0160 0.0314 0.1047 0.1171 0.1170 0.0005 0.0352 0.0207 0.0136 0.0730 0.0518 0.2113 0.0089 0.0149 0.1053 0.0097 0.0050 0.0749 0.0320 0.1227 0.0719 0.1582 0.0616 0.0855 0.1231 0.1472 0.1327 0.0583 0.0339 0.0696 0.0815 0.0717 0.1266 0.0675 0.1171 0.1100 0.0022 0.0108 0.1482 0.0285 0.0962 0.1384 0.0723 0.0543 0.0249 0.0834 0.2019 0.0015 0.0918 0.0624 0.0523 0.1647 0.0602 0.0570 0.0402 0.0917 0.0022 0.0005 0.0041 0.0261 0.1080 0.1738 0.0447 0.0891 0.0735 0.0589 0.0079 0.0684 0.0581 0.0202 0.0317 0.0240 0.1359 0.0181 0.0153 0.0340 0.1411 0.0292 0.0632 0.0799 0.0934 0.0901 0.0242 0.0267 0.0916 0.0663 0.1068 0.0084 0.0411 0.0124 0.0562 0.0490 0.0462 0.0861 0.0782 0.0581 0.0739 0.0211 0.1086 0.2049 0.0977 0.0451 0.0352 0.0263 0.0061 0.0912 0.0858 0.0315 0.1085 0.0961 0.1439 0.0094 0.1115 0.0002 0.1030 0.1240 0.0207 0.0946 0.0163 0.0664 0.1068 0.0396 0.0208 0.1111 0.1211 0.0925 0.0052 0.0571 0.0806 0.0261 0.0899 0.1312 0.0316 0.1445 0.1121 0.0608 0.1185 0.0028 0.0273 0.0760 0.0513 0.1454 0.0478 0.0763 0.0193 0.0670 0.0770 0.1099 0.0073 0.0021 0.0502 0.0524 0.0101 0.0372 0.0034 0.1498 0.0535 0.1238 0.0486 0.0817 0.0075 0.0555 0.0549 0.0566 0.0152 0.1909 0.0167 0.1553 0.1008 0.1459 0.0034 0.0606 0.0932 0.1131 0.0315 0.0367 0.0094 0.0308 0.0400 0.0393 0.0732 0.2046 0.1247 0.0930 0.1026 0.0329 0.1270 0.0340 0.1515 0.0713 0.1899 0.0520 0.1811 0.0282 0.1041 0.0159 0.1171 0.1019 0.0644 0.0530 0.0564 0.1003 0.0123 0.1862 0.0427 0.0479 0.0264 0.1215 0.0558 0.2166 0.1128 0.1043 0.0067 0.0401 0.0937 0.1972 0.0571 0.0628 0.0925 0.0132 0.1485 0.0004 0.1206 0.0620 0.0048 0.0702 0.0598 0.0198 0.0497 0.0675 0.0033 0.0885 0.2233 0.0539 0.0030 0.0908 0.0094 0.1010 0.0502 0.0036 0.0221 0.1252 0.1810 0.0455 0.0139 0.0726 0.0687 0.2203 0.0981 0.0249 0.0289 0.1193 0.0220 0.0085 0.1165 0.1538 0.0627 0.0307 0.0473 0.1225 0.1059 0.0635 0.0030 0.0040 0.0196 0.0417 0.0210 0.1539 0.0936 0.0272 0.1152 0.0289 0.0853 0.1074 0.0928 0.0340 0.0206 0.0069 0.1051 0.0192 0.1399 0.0050 0.0109 0.0341 0.0515 0.1254 0.0479 0.0288 0.0028 0.0510 0.2069 0.0204 0.1297 0.1721 0.0350 0.1021 0.0630 0.1246 0.0810 0.0169 0.0091 0.1063 0.1164 0.0026 0.0550 0.0639 0.0581 0.0339 0.0142 0.1795 0.0220 0.2141 0.1473 0.1805 0.1554 0.0503 0.0135 0.0049 0.0835 0.0645 0.0644 0.0150 0.0021 0.0306 0.1045 0.0631 0.1621 0.0703 0.0811 0.0645 0.1284 0.0351 0.0598 0.0101 0.2284 0.0490 0.0152 0.0520 0.0581 0.0985 0.0004 0.0401 0.0273 0.0434 0.0007 0.0125 0.0868 0.1268 0.0459 0.1257 0.0432 0.1021 0.0807 0.0080 0.0677 0.0044 0.1866 0.1070 0.0890 0.0613 0.1312 0.0822 0.0351 0.0440 0.1578 0.0118 0.0985 0.1026 0.1688 0.1880 0.1593 0.1151 0.0162 0.0148 0.0333 0.0502 0.0887 0.0383 0.1068 0.2293 0.0259 0.0212 0.0707 0.0422 0.1809 0.1091 0.1124 0.0595 0.1314 0.0409 0.0414 0.1580 0.0257 0.0692 0.0564 0.0911 0.0135 0.0960 0.0027 0.0306 0.1344 0.0212 0.0474 0.1592 0.0824 0.0260 0.0678 0.0441 0.0854 0.0296 0.0634 0.0953 0.1498 0.0453 0.1779 0.0487 0.1250 0.0647 0.1351 0.1294 0.0822 0.2147 0.1739 0.0412 0.0060 0.0344 0.1416 0.0251 0.1264 0.0756 0.0157 0.0758 0.0398 0.1236 0.1447 0.0072 0.1816 0.1107 0.0702 0.0687 0.0061 0.0886 0.0874 0.2217 0.0755 0.0285 0.0134 0.0118 0.0070 0.0201 0.0435 0.0628 0.0523 0.0359 0.0258 0.0870 0.0098 0.0748 0.0804 0.0173 0.0183 0.0031 0.0299 0.1931 0.0095 0.1363 0.0660 0.0317 0.0587 0.0277 0.0323 0.0461 0.0093 0.1231 0.2351 0.0596 0.0916 0.0937 0.2260 0.0172 0.0096 0.0362 0.0152 0.1395 0.1028 0.0733 0.0389 0.0515 0.1500 0.0174 0.1727 0.0390 0.0578 0.0139 0.0169 0.0224 0.0655 0.0020 0.1330 0.0622 0.1215 0.0587 0.0098 0.0276 0.0032 0.1691 0.1580 0.0635 0.0128 0.0726 0.0397 0.1294 0.0315 0.0406 0.1703 0.0756 0.0103 0.0237 0.0120 0.0768 0.2164 0.0351 0.0860 0.0686 0.0567 0.0992 0.1236 0.0083 0.0257 0.0256 0.0987 0.1195 0.0474 0.0907 0.0122 0.0278 0.0555 0.0926 0.0500 0.1382 0.0007 0.0566 0.0120 0.1538 0.0273 0.0796 0.0704 0.1158 0.0674 0.1281 0.1194 0.0533 0.1372 0.0571 0.0222 0.1187 0.0802 0.0895 0.0027 0.2251 0.0618 0.1280 0.0354 0.0609 0.0181 0.0055 0.2460 0.0068 0.0791 0.0211 0.0270 0.1058 0.0591 0.0268 0.0292 0.0888 0.1539 0.1011 0.0016 0.1013 0.0754 0.0516 0.0450 0.0490 0.0674 0.0920 0.0195 0.0101 0.1154 0.0338 0.2594 0.0316 0.1310 0.0275 0.1322 0.0182 0.1566 0.0724 0.1125 0.1514 0.1354 0.0679 0.0461 0.0906 0.0979 0.0232 0.0765 0.1182 0.0864 0.0780 0.0035 0.0539 0.1336 0.0498 0.0706 0.0755 0.0118 0.0734 0.2028 0.0117 0.0849 0.0644 0.1637 0.0604 0.0791 0.0141 0.0110 0.0868 0.0144 0.0113 0.0459 0.0610 0.0214 0.1087 0.0277 0.0138 0.0002 0.2092 0.0444 0.0277 0.0391 0.0211 0.0006 0.1481 0.0363 0.0084 0.0501 0.0609 0.0041 0.0350 0.0277 0.0265 0.0322 0.0864 0.0263 0.1213 0.0508 0.0435 0.0727 0.0078 0.1504 0.1574 0.0940 0.0684 0.2107 0.0451 0.0380 0.0069 0.0259 0.0969 0.0058 0.0817 0.0708 0.0179 0.0067 0.0162 0.0718 0.0186 0.0044 0.0930 0.0931 0.1033 0.0634 0.1069 0.1052 0.0605 0.1597 0.0813 0.1409 0.0679 0.0173 0.0775 0.0673 0.1208 0.1133 0.0536 0.0806 0.0247 0.0314 0.0749 0.0313 0.1686 0.1452 0.1954 0.0027 0.2536 0.0190 0.0205 0.0917 0.0129 0.0454 0.1400 0.0785 0.0807 0.0444 0.0747 0.0846 0.0747 0.0547 0.1021 0.0958 0.0385 0.0006 0.1124 0.0220 0.0111 0.0988 0.1814 0.0822 0.0036 0.1595 0.1240 0.0529 0.1918 0.0146 0.1453 0.0091 0.0195 0.0015 0.0206 0.0878 0.0214 0.0758 0.0523 0.0166 0.0307 0.0892 0.0855 0.0028 0.0377 0.0808 0.0812 0.0145 0.2634 0.0431 0.1200 0.0405 0.0515 0.0444 0.0264 0.0527 0.0061 0.0439 0.1171 0.0773 0.2469 0.0312 0.0825 0.0033 0.0856 0.0776 0.1280 0.1210 0.0951 0.1530 0.0899 0.0691 0.0884 0.0535 0.0693 0.0060 0.1051 0.1232 0.0496 0.1510 0.1328 0.0320 0.0092 0.1032 0.0237 0.0512 0.0810 0.0277 0.1373 0.0338 0.1881 0.0543 0.0367 0.0761 0.0616 0.0686 0.0977 0.0895 0.1974 0.0311 0.0655 0.0693 0.0185 0.0279 0.0962 0.0061 0.1003 0.0410 0.0106 0.0419 0.1100 0.0630 0.0232 0.1261 0.0610 0.0356 0.0906 0.0317 0.1092 0.0273 0.0197 0.0426 0.1195 0.1416 0.0932 0.0236 0.0600 0.0210 0.2409 0.0853 0.0767 0.0811 0.0706 0.0762 0.1585 0.1500 0.1487 0.1094 0.1012 0.0407 0.0957 0.0818 0.0076 0.0838 0.0065 0.0721 0.1539 0.0815 0.0599 0.0567 0.0353 0.0100 0.0475 0.0230 0.0285 0.2016 0.2121 0.0363 0.0316 0.0669 0.0510 0.0884 0.0204 0.0847 0.0956 0.1296 0.0020 0.1291 0.0257 0.1316 0.0101 0.0438 0.1600 0.0355 0.0543 0.0725 0.0363 0.0879 0.0311 0.0097 0.0044 0.0315 0.1147 0.0806 0.1246 0.0156 0.0170 0.0584 0.0244 0.1117 0.0129 0.1352 0.1365 0.0459 0.0578 0.0617 0.0832 0.2083 0.1083 0.0039 0.1492 0.1125 0.0238 0.0251 0.0914 0.0389 0.1369 0.0398 0.0072 0.0799 0.0892 0.0172 0.1496 0.1522 0.0448 0.0659 0.0577 0.1273 0.0478 0.0638 0.0401 0.0921 0.0289 0.0460 0.1468 0.0625 0.0982 0.0992 0.1682 0.0714 0.0669 0.0258 0.0171 0.0769 0.0369 0.0145 0.1035 0.0608 0.0223 0.0662 0.1144 0.0526 0.0416 0.0974 0.1713 0.1396 0.0906 0.0351 0.0107 0.1383 0.0356 0.0021 0.0258 0.0916 0.0590 0.0571 0.1087 0.0695 0.0298 0.1498 0.0958 0.0635 0.0716 0.0257 0.0531 0.1236 0.0856 0.1957 0.0302 0.0566 0.1436 0.0155 0.0363 0.0510 0.0786 0.0117 0.0454 0.1301 0.0324 0.0394 0.0236 0.0245 0.1467 0.1080 0.1298 0.0079 0.0659 0.0244 0.0339 0.0103 0.1628 0.1642 0.0352 sqrt(sum((wn-mean(wn)).^2)/(n-1)) ans = 2.8305 std(wn) ans = 2.8305 norm(wn-mean(wn))/sqrt(n-1) ans = 2.8305 sum(wn)/n ans = 9.8708 mean(wn) ans = 9.8708 plotdata(wn,'Bily sum') acf(wn,figure) WARNING: The process behaves like non-stationary => the estimate miE may not be correct ans = 1.0000 0.0111 -0.0230 0.0194 0.0068 -0.0371 0.0241 0.0101 -0.0011 0.0577 0.0526 -0.0141 0.0325 0.0397 -0.0457 0.0148 0.0369 -0.0287 0.0346 -0.0011 0.0209 -0.0252 -0.0082 -0.0622 0.0076 -0.0424 -0.0258 -0.0358 0.0188 -0.0401 0.0019 -0.0471 -0.0641 -0.0092 -0.0435 -0.0403 -0.0172 0.0170 -0.0150 -0.0235 -0.0031 -0.0351 -0.0432 -0.0255 -0.0394 0.0037 0.0191 -0.0299 0.0253 0.0189 -0.0372 -0.0191 0.0081 0.0180 -0.0050 0.0423 -0.0093 0.0255 0.0378 -0.0001 -0.0228 0.0338 -0.0337 -0.0598 0.0108 0.0335 0.0084 0.0421 -0.0414 -0.0267 0.0665 0.0188 -0.0499 -0.0000 -0.0011 -0.0446 0.0352 0.0020 -0.0608 0.0137 -0.0192 -0.0118 -0.0186 0.0341 -0.0302 -0.0110 -0.0146 0.0202 0.0262 0.0345 -0.0493 0.0204 0.0105 -0.0284 -0.0268 0.0148 0.0110 0.0489 0.0085 -0.0135 -0.0423 0.0273 -0.0173 0.0041 0.0808 -0.0455 -0.0164 0.0431 -0.0458 0.0071 -0.0103 0.0177 -0.0304 0.0164 -0.0356 -0.0541 0.0435 0.0302 0.0359 0.0183 0.0050 -0.0322 0.0392 -0.0100 0.0233 0.0284 -0.0218 -0.0030 0.0344 0.0157 -0.0338 0.0113 -0.0371 -0.0309 -0.0129 -0.0103 -0.0166 -0.0074 0.0162 -0.0747 0.0720 0.0467 -0.0242 -0.0606 0.0317 -0.0205 -0.0236 -0.0174 -0.0265 -0.0262 0.0013 0.0026 0.0166 0.0141 -0.0273 -0.0121 0.0023 0.0562 0.0026 -0.0156 -0.0239 -0.0270 0.0264 0.0386 0.0070 -0.0494 0.0029 -0.0238 0.0544 -0.0014 -0.0038 -0.0179 -0.0027 0.0340 0.0134 -0.0244 0.0096 0.0249 0.0275 -0.0231 0.0149 0.0067 -0.0185 0.0351 -0.0006 -0.0638 -0.0375 -0.0073 -0.0080 -0.0047 0.0384 -0.0112 -0.0161 -0.0108 -0.0049 -0.0289 0.0216 -0.0406 0.0374 0.0416 0.0160 -0.0340 -0.0084 -0.0479 -0.0465 -0.0006 -0.0465 -0.0321 0.0500 -0.0229 0.0190 0.0533 0.0125 -0.0437 -0.0239 0.0210 0.0295 0.0258 0.0406 -0.0047 0.0250 0.0014 -0.0082 0.0096 0.0174 0.0039 -0.0019 0.0141 0.0575 0.0233 0.0123 -0.0065 -0.0288 0.0285 0.0093 0.0440 0.0342 0.0040 0.0203 0.0110 0.0042 -0.0297 0.0024 0.0200 0.0142 0.0134 0.0377 -0.0046 0.0093 -0.0109 -0.0817 -0.0221 -0.0123 -0.0114 -0.0025 0.0245 -0.0295 0.0088 -0.0464 -0.0384 0.0099 -0.0256 -0.0627 -0.0560 -0.0371 -0.0038 -0.0389 -0.0161 -0.0272 -0.0053 -0.0072 -0.0492 -0.0127 -0.0032 -0.0163 0.0277 -0.0417 -0.0152 -0.0147 -0.0077 -0.0014 0.0316 0.0445 -0.0485 0.0182 0.0332 0.0153 -0.0062 0.0106 -0.0119 0.0159 0.0567 0.0173 0.0196 0.0195 0.0240 -0.0053 0.0234 -0.0332 0.0026 0.0612 0.0274 -0.0083 0.0123 -0.0010 0.0148 0.0328 -0.0138 -0.0298 0.0150 -0.0212 -0.0432 0.0283 -0.0243 -0.0103 0.0061 0.0162 -0.0275 -0.0071 0.0084 -0.0459 0.0152 -0.0081 -0.0223 0.0093 0.0528 -0.0323 -0.0013 -0.0098 -0.0115 0.0008 -0.0115 -0.0212 -0.0410 0.0358 0.0184 0.0359 0.0256 -0.0155 0.0586 0.0372 0.0302 -0.0261 -0.0109 -0.0066 -0.0073 -0.0067 0.0156 0.0077 0.0316 -0.0125 0.0267 0.0221 -0.0088 -0.0072 -0.0363 0.0412 -0.0192 0.0450 -0.0464 -0.0064 0.0373 -0.0184 -0.0004 0.0083 0.0153 -0.0271 0.0403 -0.0079 -0.0026 0.0016 0.0166 -0.0407 -0.0024 0.0133 -0.0007 -0.0042 -0.0140 0.0212 0.0127 0.0078 -0.0041 -0.0381 0.0119 -0.0099 -0.0011 -0.0048 -0.0101 0.0334 0.0205 -0.0454 0.0182 -0.0114 0.0293 0.0077 0.0142 -0.0303 -0.0230 -0.0075 0.0144 -0.0427 -0.0054 0.0037 -0.0233 0.0101 -0.0597 0.0152 0.0048 0.0019 0.0136 -0.0047 -0.0074 -0.0006 0.0232 -0.0199 0.0007 -0.0386 -0.0581 0.0218 -0.0206 0.0015 -0.0368 0.0321 0.0192 -0.0079 -0.0335 0.0305 0.0009 0.0591 -0.0138 -0.0086 0.0126 0.0201 -0.0372 0.0251 0.0015 0.0070 -0.0267 0.0047 -0.0343 -0.0275 -0.0026 -0.0019 0.0103 0.0284 0.0440 0.0080 -0.0083 0.0255 -0.0459 -0.0117 0.0073 0.0174 -0.0206 0.0211 0.0148 0.0176 0.0080 0.0139 0.0118 -0.0006 -0.0189 0.0112 0.0250 0.0327 0.0166 -0.0211 0.0167 0.0217 -0.0046 -0.0283 -0.0155 -0.0012 -0.0122 0.0188 0.0080 0.0145 -0.0168 -0.0017 0.0047 0.0199 -0.0122 -0.0130 0.0154 0.0176 0.0287 -0.0233 -0.0065 -0.0193 -0.0056 -0.0158 0.0007 -0.0043 0.0189 -0.0216 -0.0477 -0.0022 -0.0082 0.0079 -0.0133 %Typicke prubehy acf: t=(1:n).'; k=0; x=k*t+randn(n,1); acf(x,figure); WARNING: The process behaves like non-stationary => the estimate miE may not be correct k=0.001; x=k*t+randn(n,1); acf(x,figure); k=0.0015; x=k*t+randn(n,1); acf(x,figure); k=0.002; x=k*t+randn(n,1); acf(x,figure); % Period. sl.: x=sin(100*pi*t./n)+1*randn(n,1); acf(x,figure); x=sin(100*pi*t./n)+2*randn(n,1); acf(x,figure); WARNING: The process behaves like non-stationary => the estimate miE may not be correct k=0.0015; x=k*t+sin(100*pi*t./n)+2*randn(n,1); acf(x,figure); k=0.003; x=k*t+sin(100*pi*t./n)+2*randn(n,1); acf(x,figure); close all % Simulace AR(1) procesu z prikl.CR1/2.23 a 2.33: % Simulace MA(1) procesu z prikl.CR1/2.23(2) a 2.33: n=1000; sigma=2; z=sigma*randn(n+1,1); r=0.5, theta1=(1+sqrt(1-4*r^2))/(2*r), theta2=(1+sqrt(1-4*r^2))/(2*r) r = 0.5000 theta1 = 1 theta2 = 1 r=0.5, theta1=(1+sqrt(1-4*r^2))/(2*r), theta2=(1-sqrt(1-4*r^2))/(2*r) r = 0.5000 theta1 = 1 theta2 = 1 x1=z(2:n+1)+theta1*z(1:n); x2=z(2:n+1)+theta2*z(1:n); acf(x1,figure) ans = 1.0000 0.5171 0.0577 0.1170 0.0904 0.0085 0.0153 0.0157 -0.0238 -0.0235 -0.0068 -0.0215 -0.0248 0.0081 0.0041 -0.0419 -0.0262 0.0288 0.0203 0.0090 0.0306 0.0002 -0.0331 -0.0309 -0.0577 -0.0614 -0.0075 0.0376 0.0202 -0.0384 -0.0412 0.0163 0.0331 0.0090 0.0262 0.0617 0.0698 0.0535 0.0297 0.0080 -0.0005 0.0020 -0.0194 -0.0415 -0.0678 -0.0717 -0.0193 0.0066 -0.0133 -0.0392 -0.0602 -0.0430 0.0159 0.0282 -0.0181 -0.0374 -0.0134 0.0108 0.0109 -0.0083 -0.0303 -0.0294 -0.0236 -0.0256 -0.0123 0.0074 0.0311 0.0321 0.0133 0.0337 0.0548 0.0320 0.0077 0.0111 0.0126 -0.0051 -0.0342 -0.0195 0.0232 0.0209 0.0049 -0.0324 -0.0630 -0.0195 0.0094 -0.0081 -0.0162 -0.0396 -0.0521 -0.0382 -0.0311 -0.0230 0.0004 0.0145 0.0089 -0.0029 -0.0233 -0.0058 0.0519 0.0640 0.0415 0.0461 0.0769 0.1058 0.1091 0.1033 0.0648 0.0213 0.0251 0.0088 -0.0222 0.0095 0.0452 0.0149 -0.0073 0.0282 0.0513 0.0164 -0.0268 -0.0392 -0.0182 -0.0020 -0.0168 -0.0313 -0.0308 0.0129 0.0437 -0.0013 -0.0293 -0.0247 -0.0569 -0.0571 0.0064 0.0089 -0.0192 0.0262 0.0569 0.0207 0.0225 0.0493 0.0247 0.0125 0.0332 0.0409 0.0495 0.0184 -0.0325 -0.0048 0.0477 0.0421 -0.0000 -0.0356 -0.0282 0.0084 0.0163 0.0181 0.0038 -0.0213 -0.0179 -0.0336 -0.0559 -0.0449 -0.0261 -0.0074 -0.0089 -0.0381 -0.0219 0.0203 0.0099 -0.0244 -0.0476 -0.0524 -0.0031 0.0482 0.0411 0.0494 0.0575 0.0121 -0.0090 0.0096 0.0157 0.0054 0.0044 0.0281 0.0410 0.0314 0.0223 0.0044 0.0010 0.0179 0.0182 -0.0089 -0.0593 -0.0991 -0.0798 -0.0143 -0.0063 -0.0671 -0.0776 -0.0446 -0.0409 -0.0152 0.0330 0.0425 0.0374 0.0299 0.0338 0.0447 0.0277 0.0334 0.0479 0.0339 0.0061 -0.0272 -0.0116 0.0372 0.0371 -0.0235 -0.0428 -0.0128 -0.0205 -0.0314 -0.0168 -0.0156 -0.0168 0.0042 0.0270 0.0308 0.0208 -0.0122 -0.0625 -0.0702 -0.0363 -0.0068 -0.0055 -0.0265 -0.0178 0.0031 -0.0006 0.0093 0.0211 0.0088 0.0217 0.0466 0.0279 0.0152 0.0278 0.0177 -0.0057 -0.0165 -0.0093 0.0115 0.0258 0.0370 0.0278 -0.0113 -0.0105 0.0169 0.0060 0.0055 0.0215 0.0129 0.0126 0.0039 -0.0153 -0.0087 -0.0001 -0.0238 -0.0406 -0.0200 -0.0126 -0.0272 -0.0423 -0.0293 -0.0040 -0.0206 -0.0434 -0.0375 -0.0083 0.0322 0.0659 0.0729 0.0333 0.0192 0.0426 0.0255 -0.0069 -0.0090 -0.0099 -0.0085 0.0042 0.0108 0.0385 0.0508 0.0058 -0.0527 -0.0606 -0.0160 -0.0065 -0.0362 -0.0261 0.0116 0.0068 -0.0296 -0.0323 0.0012 0.0301 0.0308 0.0239 0.0186 0.0006 0.0080 0.0216 -0.0106 -0.0187 0.0082 0.0104 0.0142 0.0166 0.0036 -0.0041 0.0032 0.0125 0.0130 0.0182 0.0138 -0.0163 -0.0154 0.0262 0.0178 -0.0187 -0.0142 -0.0079 -0.0224 -0.0254 -0.0159 -0.0493 -0.0798 -0.0460 -0.0348 -0.0616 -0.0680 -0.0475 -0.0269 -0.0162 -0.0116 -0.0168 -0.0046 0.0223 0.0249 0.0258 0.0417 0.0289 0.0015 0.0005 -0.0187 -0.0397 -0.0185 -0.0008 0.0025 -0.0076 -0.0305 -0.0135 0.0378 0.0663 0.0534 0.0252 0.0208 0.0251 0.0294 0.0334 0.0366 0.0321 0.0070 -0.0078 0.0009 0.0124 0.0064 0.0057 0.0325 0.0315 -0.0093 0.0016 0.0403 0.0308 -0.0027 -0.0261 -0.0240 -0.0343 -0.0364 0.0082 0.0278 0.0055 -0.0196 -0.0298 -0.0076 -0.0068 -0.0310 -0.0101 0.0076 -0.0100 -0.0074 -0.0033 -0.0313 -0.0355 0.0006 0.0212 0.0091 -0.0339 -0.0455 -0.0066 -0.0049 -0.0195 -0.0148 -0.0305 -0.0302 -0.0205 -0.0384 -0.0358 -0.0103 -0.0116 -0.0372 -0.0335 -0.0215 -0.0377 -0.0434 -0.0046 0.0389 0.0357 0.0196 0.0251 0.0284 0.0298 0.0339 0.0104 -0.0168 -0.0086 0.0163 0.0156 -0.0019 0.0188 0.0377 0.0178 -0.0127 -0.0296 -0.0226 -0.0104 -0.0058 0.0051 0.0275 0.0325 0.0198 0.0139 -0.0005 -0.0337 -0.0456 -0.0213 -0.0010 -0.0173 -0.0259 0.0036 0.0055 -0.0264 -0.0175 0.0191 0.0348 0.0587 0.0895 0.0779 0.0528 0.0596 0.0553 0.0356 0.0395 0.0331 -0.0045 -0.0031 0.0177 -0.0022 -0.0091 0.0127 0.0191 0.0095 0.0086 0.0031 0.0065 0.0232 0.0190 0.0162 0.0090 -0.0158 -0.0356 -0.0336 -0.0124 -0.0211 -0.0249 -0.0008 -0.0188 -0.0426 -0.0195 sigma=2; z=sigma*randn(n+1,1); x1=z(2:n+1)+theta1*z(1:n); x2=z(2:n+1)+theta2*z(1:n); acf(x1,figure) ans = 1.0000 0.5011 0.0345 0.0323 -0.0249 -0.0589 -0.0613 -0.0197 0.0097 -0.0148 -0.0597 -0.0665 -0.0213 -0.0206 -0.0457 -0.0137 0.0303 0.0456 0.0262 -0.0216 -0.0169 0.0437 0.0476 0.0130 -0.0138 -0.0191 0.0034 0.0164 0.0227 0.0067 -0.0112 -0.0177 -0.0175 -0.0103 0.0008 0.0197 0.0039 -0.0094 -0.0016 -0.0192 -0.0411 -0.0173 0.0111 0.0276 0.0588 0.0305 -0.0164 -0.0150 -0.0185 -0.0148 -0.0186 -0.0356 -0.0090 0.0210 0.0346 0.0686 0.0960 0.0345 -0.0324 0.0033 -0.0009 -0.0813 -0.0741 -0.0242 -0.0778 -0.1058 -0.0457 0.0036 0.0178 -0.0019 -0.0092 -0.0150 -0.0433 -0.0710 -0.0559 -0.0222 -0.0324 -0.0183 0.0133 0.0132 -0.0142 -0.0433 -0.0033 0.0427 0.0380 0.0406 0.0277 0.0276 0.0357 -0.0317 -0.0855 -0.0724 -0.0532 -0.0554 -0.0581 -0.0522 -0.0394 -0.0369 -0.0279 0.0123 0.0428 0.0499 0.0362 0.0415 0.0511 0.0004 -0.0255 0.0084 -0.0026 -0.0254 -0.0048 -0.0193 -0.0584 -0.0506 -0.0400 -0.0410 -0.0318 -0.0378 -0.0476 -0.0228 -0.0093 -0.0080 0.0360 0.0472 0.0128 -0.0030 0.0095 0.0223 0.0107 0.0250 0.0648 0.0710 0.0391 -0.0171 -0.0399 0.0022 0.0083 -0.0116 0.0121 0.0315 0.0370 0.0302 0.0104 -0.0075 -0.0252 -0.0399 -0.0498 -0.0692 -0.0766 -0.0439 0.0000 0.0427 0.0542 0.0276 0.0114 0.0173 0.0216 0.0160 0.0189 0.0186 0.0027 -0.0153 -0.0293 -0.0322 -0.0238 -0.0184 -0.0247 -0.0084 0.0191 0.0203 0.0281 0.0318 0.0133 0.0003 -0.0151 -0.0277 0.0008 0.0479 0.0341 -0.0235 -0.0254 0.0191 0.0230 0.0127 0.0302 0.0359 0.0251 0.0147 0.0296 0.0304 -0.0044 -0.0335 -0.0485 -0.0442 -0.0499 -0.0434 -0.0034 0.0318 0.0359 -0.0176 -0.0516 -0.0426 -0.0570 -0.0683 -0.0276 0.0540 0.0772 0.0560 0.0374 0.0126 0.0046 -0.0120 -0.0100 0.0124 0.0005 -0.0251 -0.0599 -0.0565 -0.0212 -0.0299 -0.0193 -0.0029 -0.0231 0.0057 0.0513 0.0352 0.0206 0.0188 0.0005 0.0111 0.0243 0.0013 -0.0249 -0.0147 0.0069 0.0023 0.0237 0.0667 0.0708 0.0412 0.0231 0.0286 0.0174 -0.0088 -0.0264 -0.0296 0.0127 0.0270 -0.0194 -0.0321 -0.0097 0.0034 -0.0003 -0.0097 -0.0300 -0.0334 -0.0157 -0.0385 -0.0696 -0.0298 0.0078 0.0019 0.0154 0.0463 0.0639 0.0647 0.0226 -0.0161 0.0027 -0.0138 -0.0594 -0.0651 -0.0280 -0.0003 0.0060 0.0246 0.0252 0.0118 -0.0406 -0.0822 -0.0063 0.0596 0.0376 0.0204 0.0011 -0.0240 -0.0185 -0.0189 -0.0105 0.0307 0.0376 0.0180 0.0321 0.0495 0.0395 0.0233 0.0151 0.0083 0.0015 -0.0093 -0.0191 -0.0142 -0.0263 -0.0466 -0.0399 -0.0278 -0.0314 -0.0170 0.0080 0.0201 0.0065 -0.0128 -0.0086 -0.0281 -0.0314 -0.0082 -0.0118 -0.0099 0.0182 0.0483 0.0457 0.0228 0.0105 -0.0091 -0.0149 -0.0140 -0.0200 -0.0172 -0.0022 -0.0063 -0.0290 -0.0120 -0.0047 -0.0154 -0.0253 -0.0169 0.0142 0.0041 -0.0177 -0.0059 0.0136 -0.0103 -0.0319 -0.0160 -0.0075 -0.0016 -0.0155 -0.0117 0.0154 0.0216 0.0040 -0.0300 -0.0318 -0.0229 -0.0362 -0.0474 -0.0274 -0.0012 -0.0211 -0.0429 -0.0340 -0.0335 -0.0348 -0.0177 0.0033 0.0205 0.0538 0.0716 0.0563 0.0309 0.0001 -0.0277 -0.0340 -0.0046 0.0234 0.0210 0.0222 0.0207 -0.0016 -0.0101 -0.0446 -0.0516 0.0136 0.0163 -0.0027 0.0137 -0.0074 -0.0198 0.0089 0.0229 0.0092 -0.0115 -0.0232 -0.0177 0.0109 0.0331 0.0443 0.0507 0.0238 0.0062 0.0012 0.0091 0.0487 0.0591 0.0255 0.0181 0.0406 0.0251 -0.0155 -0.0369 -0.0143 0.0162 0.0193 0.0180 0.0255 0.0357 0.0305 0.0162 0.0320 0.0648 0.0575 0.0111 -0.0218 0.0120 0.0438 0.0296 0.0072 -0.0049 -0.0041 -0.0268 -0.0574 -0.0512 -0.0070 0.0199 0.0090 0.0003 -0.0082 -0.0080 -0.0042 -0.0209 -0.0149 0.0111 0.0155 0.0028 -0.0052 0.0202 0.0408 0.0312 0.0095 -0.0067 0.0092 0.0333 0.0349 0.0139 -0.0123 -0.0284 -0.0333 -0.0327 -0.0154 -0.0113 -0.0134 0.0192 0.0126 -0.0260 -0.0163 -0.0167 -0.0280 0.0003 -0.0072 -0.0228 0.0308 0.0556 0.0265 0.0456 0.0433 -0.0213 -0.0508 -0.0393 -0.0180 -0.0005 -0.0002 -0.0260 -0.0359 -0.0211 -0.0184 -0.0157 -0.0087 -0.0054 -0.0005 -0.0006 -0.0268 -0.0467 -0.0363 -0.0133 0.0015 0.0069 0.0329 0.0490 0.0259 0.0001 r=-0.5 r = -0.5000 r=-0.5, theta1=(1+sqrt(1-4*r^2))/(2*r), theta2=(1-sqrt(1-4*r^2))/(2*r) r = -0.5000 theta1 = -1 theta2 = -1 x1=z(2:n+1)+theta1*z(1:n); x2=z(2:n+1)+theta2*z(1:n); acf(x1,figure) ans = 1.0000 -0.5338 0.0378 0.0240 -0.0338 0.0220 -0.0396 0.0167 0.0172 -0.0013 0.0044 -0.0436 0.0348 0.0158 -0.0428 0.0165 -0.0008 0.0125 0.0080 -0.0021 -0.0533 0.0560 -0.0058 -0.0082 0.0154 -0.0427 0.0440 -0.0365 0.0341 -0.0109 -0.0096 0.0196 -0.0362 0.0463 -0.0599 0.0661 -0.0398 0.0109 -0.0022 0.0179 -0.0293 -0.0022 0.0293 -0.0453 0.0478 0.0052 -0.0415 0.0325 -0.0180 -0.0030 0.0309 -0.0461 0.0200 0.0029 -0.0098 -0.0029 0.0223 0.0411 -0.1001 0.0629 0.0115 -0.0137 -0.0655 0.1045 -0.0472 -0.0341 0.0333 -0.0224 0.0437 -0.0336 0.0125 0.0070 -0.0056 0.0038 -0.0413 0.0621 -0.0423 -0.0002 0.0262 -0.0227 0.0446 -0.0654 0.0224 0.0154 -0.0059 -0.0105 0.0411 -0.0835 0.1179 -0.0816 0.0331 -0.0465 0.0536 -0.0410 0.0288 -0.0246 0.0142 0.0059 -0.0323 0.0299 -0.0181 0.0276 -0.0177 -0.0096 0.0324 0.0010 -0.0573 0.0576 -0.0159 -0.0144 0.0041 0.0394 -0.0603 0.0372 -0.0166 0.0069 -0.0064 0.0205 -0.0309 0.0089 0.0232 -0.0447 0.0267 0.0222 -0.0284 0.0167 -0.0317 0.0469 -0.0389 0.0068 0.0018 0.0208 -0.0083 0.0185 -0.0595 0.0403 0.0119 -0.0400 0.0272 -0.0108 0.0074 0.0079 -0.0110 0.0122 -0.0133 0.0116 -0.0147 0.0264 -0.0496 0.0348 -0.0305 0.0278 0.0039 -0.0003 -0.0126 0.0052 0.0035 -0.0026 -0.0060 0.0176 -0.0146 0.0136 -0.0161 0.0084 -0.0114 0.0172 -0.0118 -0.0151 0.0305 -0.0214 0.0067 0.0124 -0.0107 0.0043 0.0044 -0.0166 -0.0098 0.0193 0.0284 -0.0346 -0.0114 0.0138 0.0216 -0.0440 0.0409 -0.0264 0.0274 -0.0287 0.0059 0.0301 -0.0325 0.0296 -0.0396 0.0311 -0.0133 -0.0157 0.0132 -0.0063 0.0280 0.0045 -0.0554 0.0665 -0.0560 0.0423 -0.0767 0.0727 -0.0140 -0.0024 0.0153 -0.0236 0.0178 -0.0036 -0.0290 0.0430 -0.0243 0.0183 -0.0038 -0.0465 0.0665 -0.0457 0.0073 0.0256 -0.0237 -0.0237 0.0550 -0.0286 0.0006 0.0155 -0.0167 -0.0091 0.0326 -0.0218 0.0143 -0.0409 0.0567 -0.0489 0.0164 -0.0034 0.0270 -0.0196 0.0015 -0.0044 0.0231 -0.0287 0.0258 -0.0364 0.0048 0.0525 -0.0532 0.0233 -0.0268 0.0391 -0.0359 0.0376 -0.0286 0.0042 -0.0002 0.0336 -0.0593 0.0195 0.0228 -0.0247 0.0088 -0.0089 0.0213 -0.0185 0.0559 -0.0964 0.0832 -0.0369 0.0176 -0.0360 0.0139 0.0174 -0.0285 0.0279 -0.0102 0.0055 0.0356 -0.0867 0.0285 0.0393 -0.0253 0.0063 0.0149 -0.0305 0.0175 0.0016 -0.0290 0.0256 0.0097 -0.0210 0.0007 0.0171 -0.0095 0.0077 -0.0129 0.0166 -0.0201 0.0280 -0.0369 0.0317 -0.0115 -0.0009 -0.0111 0.0180 -0.0112 -0.0122 0.0259 -0.0278 0.0539 -0.0746 0.0732 -0.0494 0.0110 0.0020 0.0106 -0.0279 0.0202 -0.0146 0.0395 -0.0454 0.0416 -0.0311 0.0072 0.0107 -0.0214 0.0230 -0.0367 0.0686 -0.0828 0.0600 -0.0284 0.0136 0.0004 -0.0313 0.0406 -0.0111 -0.0067 -0.0075 0.0144 0.0195 -0.0550 0.0558 -0.0499 0.0459 -0.0234 -0.0153 0.0323 -0.0299 0.0497 -0.0547 0.0303 -0.0161 0.0223 -0.0299 0.0086 0.0062 0.0218 -0.0483 0.0463 -0.0363 0.0281 -0.0361 0.0406 -0.0423 0.0287 -0.0004 0.0036 0.0027 -0.0039 0.0022 -0.0204 0.0054 0.0099 0.0028 -0.0187 0.0369 -0.0362 0.0227 0.0155 -0.0787 0.0746 -0.0122 -0.0298 0.0388 -0.0129 -0.0207 0.0160 0.0022 0.0048 -0.0049 -0.0023 -0.0067 -0.0068 0.0262 -0.0358 0.0500 -0.0328 0.0063 0.0083 -0.0346 0.0308 0.0003 0.0102 -0.0455 0.0528 -0.0245 0.0195 -0.0319 0.0029 0.0190 -0.0154 0.0159 -0.0245 0.0306 -0.0218 0.0213 -0.0494 0.0622 -0.0369 0.0477 -0.0716 0.0332 0.0071 -0.0051 0.0106 -0.0249 0.0261 -0.0055 -0.0068 -0.0159 0.0031 0.0259 -0.0207 0.0135 -0.0061 -0.0093 0.0211 -0.0140 -0.0144 0.0235 -0.0123 0.0173 -0.0268 0.0056 0.0197 -0.0171 0.0261 -0.0405 0.0258 -0.0181 0.0308 -0.0226 0.0194 -0.0255 0.0207 -0.0213 0.0061 0.0213 -0.0428 0.0312 0.0167 -0.0343 0.0073 0.0290 -0.0558 0.0462 -0.0028 -0.0333 0.0047 0.0509 -0.0560 0.0161 0.0430 -0.0435 0.0114 -0.0175 0.0146 -0.0085 0.0182 -0.0039 -0.0246 0.0303 -0.0250 0.0201 -0.0192 0.0213 -0.0250 0.0334 -0.0175 -0.0037 -0.0036 -0.0005 0.0120 -0.0148 -0.0018 0.0273 -0.0160 0.0073 r=0.4, theta1=(1+sqrt(1-4*r^2))/(2*r), theta2=(1-sqrt(1-4*r^2))/(2*r) r = 0.4000 theta1 = 2.0000 theta2 = 0.5000 acf(x1,figure) ans = 1.0000 -0.5338 0.0378 0.0240 -0.0338 0.0220 -0.0396 0.0167 0.0172 -0.0013 0.0044 -0.0436 0.0348 0.0158 -0.0428 0.0165 -0.0008 0.0125 0.0080 -0.0021 -0.0533 0.0560 -0.0058 -0.0082 0.0154 -0.0427 0.0440 -0.0365 0.0341 -0.0109 -0.0096 0.0196 -0.0362 0.0463 -0.0599 0.0661 -0.0398 0.0109 -0.0022 0.0179 -0.0293 -0.0022 0.0293 -0.0453 0.0478 0.0052 -0.0415 0.0325 -0.0180 -0.0030 0.0309 -0.0461 0.0200 0.0029 -0.0098 -0.0029 0.0223 0.0411 -0.1001 0.0629 0.0115 -0.0137 -0.0655 0.1045 -0.0472 -0.0341 0.0333 -0.0224 0.0437 -0.0336 0.0125 0.0070 -0.0056 0.0038 -0.0413 0.0621 -0.0423 -0.0002 0.0262 -0.0227 0.0446 -0.0654 0.0224 0.0154 -0.0059 -0.0105 0.0411 -0.0835 0.1179 -0.0816 0.0331 -0.0465 0.0536 -0.0410 0.0288 -0.0246 0.0142 0.0059 -0.0323 0.0299 -0.0181 0.0276 -0.0177 -0.0096 0.0324 0.0010 -0.0573 0.0576 -0.0159 -0.0144 0.0041 0.0394 -0.0603 0.0372 -0.0166 0.0069 -0.0064 0.0205 -0.0309 0.0089 0.0232 -0.0447 0.0267 0.0222 -0.0284 0.0167 -0.0317 0.0469 -0.0389 0.0068 0.0018 0.0208 -0.0083 0.0185 -0.0595 0.0403 0.0119 -0.0400 0.0272 -0.0108 0.0074 0.0079 -0.0110 0.0122 -0.0133 0.0116 -0.0147 0.0264 -0.0496 0.0348 -0.0305 0.0278 0.0039 -0.0003 -0.0126 0.0052 0.0035 -0.0026 -0.0060 0.0176 -0.0146 0.0136 -0.0161 0.0084 -0.0114 0.0172 -0.0118 -0.0151 0.0305 -0.0214 0.0067 0.0124 -0.0107 0.0043 0.0044 -0.0166 -0.0098 0.0193 0.0284 -0.0346 -0.0114 0.0138 0.0216 -0.0440 0.0409 -0.0264 0.0274 -0.0287 0.0059 0.0301 -0.0325 0.0296 -0.0396 0.0311 -0.0133 -0.0157 0.0132 -0.0063 0.0280 0.0045 -0.0554 0.0665 -0.0560 0.0423 -0.0767 0.0727 -0.0140 -0.0024 0.0153 -0.0236 0.0178 -0.0036 -0.0290 0.0430 -0.0243 0.0183 -0.0038 -0.0465 0.0665 -0.0457 0.0073 0.0256 -0.0237 -0.0237 0.0550 -0.0286 0.0006 0.0155 -0.0167 -0.0091 0.0326 -0.0218 0.0143 -0.0409 0.0567 -0.0489 0.0164 -0.0034 0.0270 -0.0196 0.0015 -0.0044 0.0231 -0.0287 0.0258 -0.0364 0.0048 0.0525 -0.0532 0.0233 -0.0268 0.0391 -0.0359 0.0376 -0.0286 0.0042 -0.0002 0.0336 -0.0593 0.0195 0.0228 -0.0247 0.0088 -0.0089 0.0213 -0.0185 0.0559 -0.0964 0.0832 -0.0369 0.0176 -0.0360 0.0139 0.0174 -0.0285 0.0279 -0.0102 0.0055 0.0356 -0.0867 0.0285 0.0393 -0.0253 0.0063 0.0149 -0.0305 0.0175 0.0016 -0.0290 0.0256 0.0097 -0.0210 0.0007 0.0171 -0.0095 0.0077 -0.0129 0.0166 -0.0201 0.0280 -0.0369 0.0317 -0.0115 -0.0009 -0.0111 0.0180 -0.0112 -0.0122 0.0259 -0.0278 0.0539 -0.0746 0.0732 -0.0494 0.0110 0.0020 0.0106 -0.0279 0.0202 -0.0146 0.0395 -0.0454 0.0416 -0.0311 0.0072 0.0107 -0.0214 0.0230 -0.0367 0.0686 -0.0828 0.0600 -0.0284 0.0136 0.0004 -0.0313 0.0406 -0.0111 -0.0067 -0.0075 0.0144 0.0195 -0.0550 0.0558 -0.0499 0.0459 -0.0234 -0.0153 0.0323 -0.0299 0.0497 -0.0547 0.0303 -0.0161 0.0223 -0.0299 0.0086 0.0062 0.0218 -0.0483 0.0463 -0.0363 0.0281 -0.0361 0.0406 -0.0423 0.0287 -0.0004 0.0036 0.0027 -0.0039 0.0022 -0.0204 0.0054 0.0099 0.0028 -0.0187 0.0369 -0.0362 0.0227 0.0155 -0.0787 0.0746 -0.0122 -0.0298 0.0388 -0.0129 -0.0207 0.0160 0.0022 0.0048 -0.0049 -0.0023 -0.0067 -0.0068 0.0262 -0.0358 0.0500 -0.0328 0.0063 0.0083 -0.0346 0.0308 0.0003 0.0102 -0.0455 0.0528 -0.0245 0.0195 -0.0319 0.0029 0.0190 -0.0154 0.0159 -0.0245 0.0306 -0.0218 0.0213 -0.0494 0.0622 -0.0369 0.0477 -0.0716 0.0332 0.0071 -0.0051 0.0106 -0.0249 0.0261 -0.0055 -0.0068 -0.0159 0.0031 0.0259 -0.0207 0.0135 -0.0061 -0.0093 0.0211 -0.0140 -0.0144 0.0235 -0.0123 0.0173 -0.0268 0.0056 0.0197 -0.0171 0.0261 -0.0405 0.0258 -0.0181 0.0308 -0.0226 0.0194 -0.0255 0.0207 -0.0213 0.0061 0.0213 -0.0428 0.0312 0.0167 -0.0343 0.0073 0.0290 -0.0558 0.0462 -0.0028 -0.0333 0.0047 0.0509 -0.0560 0.0161 0.0430 -0.0435 0.0114 -0.0175 0.0146 -0.0085 0.0182 -0.0039 -0.0246 0.0303 -0.0250 0.0201 -0.0192 0.0213 -0.0250 0.0334 -0.0175 -0.0037 -0.0036 -0.0005 0.0120 -0.0148 -0.0018 0.0273 -0.0160 0.0073 acf(x2,figure); x1=z(2:n+1)+theta1*z(1:n); x2=z(2:n+1)+theta2*z(1:n); acf(x1,figure) ans = 1.0000 0.3911 0.0349 0.0315 -0.0259 -0.0503 -0.0590 -0.0158 0.0104 -0.0136 -0.0530 -0.0642 -0.0156 -0.0166 -0.0456 -0.0104 0.0270 0.0419 0.0243 -0.0195 -0.0208 0.0450 0.0419 0.0109 -0.0106 -0.0216 0.0076 0.0108 0.0240 0.0048 -0.0110 -0.0136 -0.0195 -0.0045 -0.0056 0.0246 -0.0006 -0.0072 -0.0016 -0.0154 -0.0399 -0.0157 0.0130 0.0198 0.0576 0.0278 -0.0191 -0.0098 -0.0183 -0.0136 -0.0133 -0.0368 -0.0059 0.0192 0.0298 0.0610 0.0882 0.0351 -0.0395 0.0097 0.0005 -0.0741 -0.0732 -0.0105 -0.0746 -0.0981 -0.0373 0.0007 0.0205 -0.0053 -0.0070 -0.0126 -0.0393 -0.0632 -0.0543 -0.0130 -0.0334 -0.0166 0.0146 0.0095 -0.0080 -0.0457 -0.0006 0.0398 0.0334 0.0350 0.0292 0.0158 0.0445 -0.0370 -0.0726 -0.0697 -0.0418 -0.0539 -0.0488 -0.0493 -0.0338 -0.0324 -0.0284 0.0140 0.0362 0.0475 0.0306 0.0359 0.0490 0.0005 -0.0288 0.0136 -0.0039 -0.0244 -0.0039 -0.0130 -0.0587 -0.0413 -0.0376 -0.0357 -0.0290 -0.0317 -0.0458 -0.0194 -0.0059 -0.0119 0.0350 0.0445 0.0084 -0.0008 0.0051 0.0248 0.0055 0.0230 0.0581 0.0657 0.0342 -0.0133 -0.0420 0.0064 0.0088 -0.0146 0.0140 0.0268 0.0337 0.0277 0.0081 -0.0054 -0.0239 -0.0344 -0.0460 -0.0590 -0.0739 -0.0355 -0.0032 0.0411 0.0489 0.0246 0.0089 0.0160 0.0197 0.0141 0.0164 0.0185 0.0009 -0.0124 -0.0279 -0.0280 -0.0224 -0.0147 -0.0233 -0.0090 0.0203 0.0160 0.0257 0.0298 0.0109 0.0008 -0.0130 -0.0264 -0.0003 0.0448 0.0335 -0.0246 -0.0240 0.0184 0.0228 0.0065 0.0310 0.0291 0.0254 0.0101 0.0272 0.0304 -0.0074 -0.0267 -0.0476 -0.0361 -0.0460 -0.0405 -0.0016 0.0276 0.0352 -0.0152 -0.0520 -0.0310 -0.0569 -0.0565 -0.0328 0.0558 0.0675 0.0497 0.0349 0.0088 0.0060 -0.0109 -0.0121 0.0157 -0.0022 -0.0204 -0.0540 -0.0556 -0.0118 -0.0317 -0.0165 0.0002 -0.0233 0.0025 0.0516 0.0282 0.0185 0.0183 -0.0013 0.0090 0.0253 -0.0011 -0.0208 -0.0175 0.0123 -0.0032 0.0229 0.0594 0.0660 0.0347 0.0206 0.0250 0.0180 -0.0112 -0.0207 -0.0303 0.0118 0.0297 -0.0231 -0.0263 -0.0117 0.0072 -0.0041 -0.0046 -0.0299 -0.0294 -0.0141 -0.0307 -0.0686 -0.0247 0.0094 -0.0010 0.0146 0.0404 0.0593 0.0559 0.0262 -0.0247 0.0113 -0.0164 -0.0512 -0.0620 -0.0234 0.0014 0.0024 0.0249 0.0214 0.0112 -0.0324 -0.0826 -0.0026 0.0575 0.0310 0.0191 0.0025 -0.0250 -0.0146 -0.0168 -0.0123 0.0304 0.0347 0.0139 0.0285 0.0462 0.0341 0.0215 0.0122 0.0091 -0.0007 -0.0052 -0.0209 -0.0094 -0.0247 -0.0418 -0.0370 -0.0229 -0.0293 -0.0165 0.0099 0.0149 0.0115 -0.0195 -0.0000 -0.0303 -0.0270 -0.0070 -0.0095 -0.0116 0.0184 0.0415 0.0452 0.0158 0.0137 -0.0114 -0.0125 -0.0114 -0.0202 -0.0130 -0.0060 0.0016 -0.0345 -0.0042 -0.0071 -0.0123 -0.0227 -0.0184 0.0171 0.0024 -0.0165 -0.0061 0.0137 -0.0071 -0.0343 -0.0082 -0.0122 0.0034 -0.0162 -0.0120 0.0172 0.0162 0.0089 -0.0327 -0.0251 -0.0222 -0.0299 -0.0456 -0.0234 -0.0004 -0.0164 -0.0434 -0.0254 -0.0340 -0.0281 -0.0195 0.0074 0.0139 0.0509 0.0640 0.0508 0.0279 -0.0002 -0.0244 -0.0325 -0.0036 0.0220 0.0189 0.0179 0.0225 -0.0054 -0.0066 -0.0381 -0.0546 0.0200 0.0132 -0.0055 0.0164 -0.0080 -0.0199 0.0097 0.0205 0.0087 -0.0107 -0.0210 -0.0166 0.0089 0.0325 0.0360 0.0505 0.0178 0.0062 0.0021 0.0044 0.0468 0.0529 0.0240 0.0113 0.0419 0.0198 -0.0119 -0.0362 -0.0124 0.0164 0.0156 0.0180 0.0202 0.0353 0.0251 0.0166 0.0234 0.0645 0.0474 0.0149 -0.0271 0.0142 0.0399 0.0259 0.0077 -0.0072 -0.0009 -0.0247 -0.0521 -0.0474 -0.0061 0.0205 0.0058 0.0018 -0.0081 -0.0083 -0.0014 -0.0202 -0.0147 0.0123 0.0127 0.0043 -0.0076 0.0185 0.0384 0.0261 0.0114 -0.0104 0.0111 0.0277 0.0345 0.0101 -0.0090 -0.0280 -0.0275 -0.0315 -0.0131 -0.0078 -0.0166 0.0205 0.0131 -0.0269 -0.0138 -0.0117 -0.0309 0.0051 -0.0067 -0.0239 0.0280 0.0551 0.0178 0.0424 0.0432 -0.0238 -0.0441 -0.0370 -0.0145 -0.0015 0.0016 -0.0237 -0.0348 -0.0157 -0.0191 -0.0118 -0.0099 -0.0023 -0.0031 0.0030 -0.0258 -0.0422 -0.0328 -0.0119 0.0027 0.0048 0.0291 0.0465 0.0215 0.0008 acf(x2,figure) ans = 1.0000 0.3911 0.0348 0.0314 -0.0259 -0.0503 -0.0591 -0.0158 0.0105 -0.0131 -0.0527 -0.0639 -0.0151 -0.0169 -0.0452 -0.0106 0.0270 0.0423 0.0243 -0.0195 -0.0209 0.0450 0.0419 0.0107 -0.0108 -0.0216 0.0079 0.0107 0.0239 0.0049 -0.0111 -0.0139 -0.0194 -0.0042 -0.0057 0.0247 -0.0008 -0.0073 -0.0017 -0.0151 -0.0397 -0.0157 0.0130 0.0199 0.0576 0.0278 -0.0191 -0.0101 -0.0186 -0.0135 -0.0134 -0.0366 -0.0060 0.0189 0.0299 0.0610 0.0881 0.0353 -0.0396 0.0096 0.0003 -0.0741 -0.0732 -0.0106 -0.0745 -0.0983 -0.0373 0.0010 0.0206 -0.0053 -0.0069 -0.0127 -0.0392 -0.0629 -0.0543 -0.0134 -0.0336 -0.0162 0.0148 0.0092 -0.0079 -0.0457 -0.0005 0.0398 0.0334 0.0353 0.0291 0.0158 0.0444 -0.0370 -0.0730 -0.0697 -0.0418 -0.0539 -0.0490 -0.0494 -0.0336 -0.0323 -0.0283 0.0144 0.0364 0.0475 0.0304 0.0362 0.0492 0.0004 -0.0290 0.0136 -0.0041 -0.0242 -0.0038 -0.0131 -0.0586 -0.0411 -0.0374 -0.0361 -0.0291 -0.0315 -0.0459 -0.0194 -0.0058 -0.0119 0.0351 0.0446 0.0084 -0.0010 0.0051 0.0251 0.0053 0.0231 0.0580 0.0656 0.0340 -0.0134 -0.0419 0.0060 0.0086 -0.0146 0.0135 0.0273 0.0339 0.0279 0.0081 -0.0054 -0.0240 -0.0345 -0.0461 -0.0592 -0.0736 -0.0356 -0.0033 0.0411 0.0488 0.0246 0.0088 0.0160 0.0197 0.0140 0.0161 0.0185 0.0008 -0.0122 -0.0280 -0.0277 -0.0226 -0.0146 -0.0235 -0.0093 0.0204 0.0158 0.0259 0.0297 0.0107 0.0006 -0.0130 -0.0267 -0.0002 0.0449 0.0335 -0.0248 -0.0239 0.0187 0.0229 0.0069 0.0316 0.0294 0.0253 0.0101 0.0271 0.0304 -0.0075 -0.0268 -0.0474 -0.0363 -0.0461 -0.0404 -0.0016 0.0278 0.0350 -0.0152 -0.0519 -0.0310 -0.0569 -0.0566 -0.0328 0.0563 0.0675 0.0498 0.0352 0.0087 0.0060 -0.0114 -0.0121 0.0157 -0.0021 -0.0206 -0.0539 -0.0553 -0.0119 -0.0316 -0.0164 0.0001 -0.0230 0.0026 0.0517 0.0286 0.0185 0.0185 -0.0013 0.0090 0.0251 -0.0013 -0.0207 -0.0175 0.0120 -0.0031 0.0230 0.0591 0.0663 0.0347 0.0211 0.0252 0.0179 -0.0107 -0.0209 -0.0305 0.0120 0.0298 -0.0230 -0.0261 -0.0113 0.0073 -0.0040 -0.0047 -0.0298 -0.0294 -0.0140 -0.0309 -0.0684 -0.0245 0.0094 -0.0008 0.0149 0.0404 0.0595 0.0557 0.0261 -0.0247 0.0112 -0.0161 -0.0513 -0.0621 -0.0237 0.0017 0.0023 0.0250 0.0215 0.0111 -0.0326 -0.0828 -0.0026 0.0574 0.0309 0.0188 0.0026 -0.0244 -0.0148 -0.0166 -0.0125 0.0299 0.0346 0.0138 0.0290 0.0460 0.0344 0.0217 0.0121 0.0092 -0.0009 -0.0055 -0.0210 -0.0093 -0.0248 -0.0418 -0.0366 -0.0229 -0.0291 -0.0165 0.0099 0.0151 0.0116 -0.0193 0.0002 -0.0304 -0.0269 -0.0071 -0.0094 -0.0120 0.0184 0.0417 0.0450 0.0154 0.0138 -0.0114 -0.0125 -0.0114 -0.0201 -0.0129 -0.0058 0.0017 -0.0349 -0.0045 -0.0072 -0.0123 -0.0225 -0.0185 0.0168 0.0026 -0.0166 -0.0061 0.0137 -0.0071 -0.0344 -0.0084 -0.0119 0.0035 -0.0165 -0.0122 0.0172 0.0161 0.0087 -0.0325 -0.0254 -0.0221 -0.0301 -0.0455 -0.0238 -0.0004 -0.0167 -0.0435 -0.0255 -0.0336 -0.0281 -0.0198 0.0072 0.0137 0.0513 0.0640 0.0505 0.0280 -0.0005 -0.0246 -0.0326 -0.0034 0.0219 0.0191 0.0178 0.0223 -0.0051 -0.0066 -0.0383 -0.0544 0.0201 0.0133 -0.0057 0.0164 -0.0079 -0.0200 0.0096 0.0208 0.0088 -0.0109 -0.0209 -0.0166 0.0093 0.0323 0.0356 0.0508 0.0178 0.0062 0.0018 0.0044 0.0468 0.0528 0.0238 0.0113 0.0420 0.0198 -0.0116 -0.0365 -0.0125 0.0166 0.0157 0.0175 0.0202 0.0351 0.0248 0.0169 0.0233 0.0646 0.0475 0.0151 -0.0270 0.0144 0.0399 0.0258 0.0074 -0.0069 -0.0009 -0.0244 -0.0520 -0.0475 -0.0058 0.0205 0.0059 0.0016 -0.0078 -0.0080 -0.0016 -0.0202 -0.0150 0.0125 0.0125 0.0044 -0.0074 0.0188 0.0387 0.0260 0.0111 -0.0103 0.0107 0.0279 0.0345 0.0099 -0.0090 -0.0281 -0.0276 -0.0316 -0.0131 -0.0079 -0.0164 0.0205 0.0129 -0.0269 -0.0138 -0.0120 -0.0309 0.0052 -0.0068 -0.0238 0.0281 0.0551 0.0177 0.0425 0.0433 -0.0236 -0.0442 -0.0370 -0.0147 -0.0012 0.0019 -0.0236 -0.0346 -0.0156 -0.0190 -0.0121 -0.0097 -0.0028 -0.0031 0.0030 -0.0258 -0.0422 -0.0328 -0.0120 0.0026 0.0044 0.0294 0.0468 0.0213 0.0009 acf(x2,figure); r=-0.4, theta1=(1+sqrt(1-4*r^2))/(2*r), theta2=(1-sqrt(1-4*r^2))/(2*r) r = -0.4000 theta1 = -2.0000 theta2 = -0.5000 x1=z(2:n+1)+theta1*z(1:n); x2=z(2:n+1)+theta2*z(1:n); acf(x1,figure); acf(x2,figure); close all exit