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# How to use Maximum Likelihood estimation technique to asses time correlated noise in the time series. I want to use MLE to characterize the amplitude of stochastic processes, please help me

Asked by Chris Martin on 17 Jan 2013

How to use Maximum Likelihood estimation technique to asses time correlated noise in the time series. I want to use MLE to characterize the amplitude of stochastic processes please help me

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## 1 Answer

Answer by Shashank on 17 Jan 2013

MLE is used to fit a distribution, and in your situation I suppose a noise model. A least squares might work as well for normally distributed noise or errors but the fit statistics may be way off and you may have to use a HAC estimator to get an unbiased estimation of the fit statistics. To start with use the Durbin-Watson test (dwtest) to check for serial correlation in the residuals. Now to your question, is there a specific error model you are interested in? What is the distribution of you errors? is it normal or another distribution? MATLAB has mle to fit distributions based on data, as long as you can specify what distribution you want.

## 4 Comments

Shashank on 17 Jan 2013

Sure, but you will have to define your distribution, do you already have a distribution assumption for the noise? if you do you then its as simple as plugging it into MLE function along with your data. I am not still unsure how to help you, any more information you could provide? about the data, noise model, maybe equations? anything more?

Chris Martin on 17 Jan 2013

MY data is GPS time series, the equation which governs the time series is y(t)=yl(t)+ys(t)+yn(t), where yl is the linear trend origination from long tectnoic movement, ys is the seasonal signal and yn is the noise. I am giving the time series below:

``` 2005.10274      4.09545  0.00220
2005.10548      4.09536  0.00208
2005.10822      4.09617  0.00205
2005.11096      4.09598  0.00210
2005.11370      4.09613  0.00209
2005.11644      4.09542  0.00246
2005.11918      4.09971  0.00272
2005.12192      4.09823  0.00217
2005.12466      4.09820  0.00219
2005.12740      4.09889  0.00229
2005.13014      4.09800  0.00275
2005.13287      4.09814  0.00238
2005.13561      4.10149  0.00278
2005.13835      4.10014  0.00310
2005.14109      4.09741  0.00245
2005.14383      4.09702  0.00228
2005.14657      4.09793  0.00231
2005.14931      4.09854  0.00229
2005.15205      4.09778  0.00219
2005.15479      4.09807  0.00236
2005.15753      4.09898  0.00219
2005.16027      4.09919  0.00214
2005.16301      4.10132  0.00242
2005.16575      4.09928  0.00220
2005.16849      4.10011  0.00247
2005.17123      4.09893  0.00223
2005.17397      4.09844  0.00251
2005.17671      4.10027  0.00248
2005.17945      4.09816  0.00276
2005.18219      4.09862  0.00248
2005.18493      4.09730  0.00348
2005.20137      4.10211  0.00346
2005.20411      4.10057  0.00301
2005.20685      4.10015  0.00261
2005.20959      4.09938  0.00219
2005.21233      4.10160  0.00257
2005.21507      4.10149  0.00298
2005.21781      4.09779  0.00279
2005.22055      4.10304  0.00396
2005.22329      4.10156  0.00358
2005.22603      4.10307  0.00312
2005.22877      4.10139  0.00283
2005.23150      4.10112  0.00249
2005.23424      4.10368  0.00240
2005.23698      4.10373  0.00423
2005.23972      4.09897  0.00321
2005.24246      4.10522  0.00304
2005.24520      4.09993  0.00329
2005.24794      4.10410  0.00308
2005.25068      4.10149  0.00267
2005.25342      4.10057  0.00329
2005.25890      4.10206  0.00234
2005.26164      4.10517  0.00316
2005.26438      4.10357  0.00390
2005.26712      4.10267  0.00345
2005.26986      4.10598  0.00410
2005.27260      4.11013  0.00496
2005.27534      4.10229  0.00407
2005.27808      4.10341  0.00303
2005.28082      4.10247  0.00275
2005.28356      4.10513  0.00286
2005.28630      4.10494  0.00337
2005.28904      4.10370  0.00327
2005.29178      4.10104  0.00437
2005.29452      4.10274  0.00375
2005.29726      4.10152  0.00304
2005.30000      4.10538  0.00374
2005.30274      4.10502  0.00269
2005.30548      4.10301  0.00314
2005.30822      4.10429  0.00289
2005.31096      4.10465  0.00338
2005.31370      4.10479  0.00367
2005.31644      4.10138  0.00456
2005.31918      4.10326  0.00360
2005.32192      4.09906  0.00308
2005.32466      4.10260  0.00362
2005.32740      4.09836  0.00261
2005.33014      4.10237  0.00296
2005.33287      4.10147  0.00285
2005.33561      4.10085  0.00363
2005.33835      4.09766  0.00334
2005.34109      4.10119  0.00292
2005.34383      4.09967  0.00294
2005.34657      4.10336  0.00291
2005.34931      4.09844  0.00271
2005.35205      4.10176  0.00308
2005.35479      4.10043  0.00420
2005.35753      4.09916  0.00328
2005.36027      4.10588  0.00378
2005.36301      4.10510  0.00389
2005.36575      4.10071  0.00372
2005.36849      4.10378  0.00309
2005.37123      4.10389  0.00340
2005.37397      4.10348  0.00280
2005.37671      4.10369  0.00340
2005.37945      4.10277  0.00323
2005.38219      4.10494  0.00295
2005.38493      4.10544  0.00288
2005.38767      4.10343  0.00253
2005.39041      4.10602  0.00268
2005.39315      4.10594  0.00306
2005.39589      4.10699  0.00353
2005.39863      4.10586  0.00317
2005.40137      4.10352  0.00349
2005.40411      4.10355  0.00407
2005.40685      4.10518  0.00353
2005.40959      4.10672  0.00260
2005.41233      4.10463  0.00331
2005.41507      4.10456  0.00372
2005.41781      4.10280  0.00362
2005.42055      4.10774  0.00345
2005.42329      4.10100  0.01067
2005.42603      4.10529  0.00279
2005.42877      4.10760  0.00280
2005.43150      4.10816  0.00277
2005.43424      4.10526  0.00278
2005.43698      4.10609  0.00298
2005.43972      4.10313  0.00304
2005.44246      4.10300  0.00317
2005.44520      4.10427  0.00396
2005.44794      4.10368  0.00275
2005.45068      4.10500  0.00351
2005.45342      4.10498  0.00462
2005.45616      4.10387  0.00316
2005.45890      4.10560  0.00270
2005.46164      4.10445  0.00298
2005.46438      4.10554  0.00304
2005.46712      4.10405  0.00312
2005.46986      4.10783  0.00258
2005.47260      4.10753  0.00268
2005.47534      4.10439  0.00271
2005.47808      4.10429  0.00280
2005.48082      4.10518  0.00264
2005.48356      4.10730  0.00291
2005.48630      4.10270  0.00278
2005.48904      4.10448  0.00307
2005.49178      4.10684  0.00407
2005.49452      4.10504  0.00311
2005.49726      4.10364  0.00318
2005.50000      4.10577  0.00259
2005.50274      4.10526  0.00301
2005.50548      4.10902  0.00298
2005.50822      4.10692  0.00259
2005.51096      4.10984  0.00285
2005.51370      4.10874  0.00292
2005.51644      4.10414  0.00303
2005.51918      4.10585  0.00265
2005.52192      4.10745  0.00254
2005.52466      4.10559  0.00248
2005.52740      4.10853  0.00249
2005.53014      4.10845  0.00256
2005.53287      4.10613  0.00250
2005.53561      4.10607  0.00244
2005.53835      4.10627  0.00233
2005.54109      4.10897  0.00241
2005.54383      4.10794  0.00249
2005.54657      4.10611  0.00263
2005.54931      4.11164  0.00271
2005.55205      4.10668  0.00280
2005.55479      4.10612  0.00279
2005.55753      4.10588  0.00269
2005.56027      4.10681  0.00250
2005.56301      4.10635  0.00311
2005.56575      4.10742  0.00282
2005.56849      4.10595  0.00261
2005.57123      4.10671  0.00258
2005.57397      4.10731  0.00277
2005.57671      4.11049  0.00315
2005.57945      4.10874  0.00278
2005.58219      4.10649  0.00242
2005.58493      4.11129  0.00265
2005.58767      4.10699  0.00303
2005.59041      4.10875  0.00259
2005.59315      4.10785  0.00265
2005.59589      4.10714  0.00258
2005.59863      4.10772  0.00251
2005.60137      4.10669  0.00275
2005.60411      4.10799  0.00258
2005.60685      4.10712  0.00267
2005.60959      4.10774  0.00258
2005.61233      4.10836  0.00249
2005.61507      4.10960  0.00319
2005.61781      4.11177  0.00274
2005.62055      4.11125  0.00317
2005.62329      4.10753  0.00240
2005.62603      4.10862  0.00251
2005.62877      4.10859  0.00270
2005.63150      4.10684  0.00292
2005.63424      4.10799  0.00292
2005.63698      4.10874  0.00278
2005.63972      4.10934  0.00267
2005.64246      4.11033  0.00308
2005.64520      4.11309  0.00291
2005.64794      4.10747  0.00264
2005.65068      4.10581  0.00325
2005.65342      4.10647  0.00335
2005.65616      4.10607  0.00321
2005.65890      4.10862  0.00363
2005.66164      4.11129  0.00308
2005.66438      4.10802  0.00250
2005.66712      4.10939  0.00266
2005.66986      4.11200  0.00313
2005.67260      4.10943  0.00273
2005.67534      4.10756  0.00270
2005.67808      4.10795  0.00230
2005.68082      4.10901  0.00236
2005.68356      4.10729  0.00254
2005.68630      4.10764  0.00279
2005.68904      4.10823  0.00268
2005.69178      4.10932  0.00256
2005.69452      4.11077  0.00271
2005.69726      4.10934  0.00253
2005.70000      4.11048  0.00282
2005.70274      4.10977  0.00253
2005.70548      4.11002  0.00266
2005.70822      4.10922  0.00272
2005.71096      4.10991  0.00229
2005.71370      4.10965  0.00241
2005.71644      4.11050  0.00244
2005.71918      4.10968  0.00252
2005.72192      4.10970  0.00246
2005.72466      4.11007  0.00238
2005.72740      4.10698  0.00282
2005.73014      4.11300  0.00275
2005.73287      4.10849  0.00263
2005.73561      4.10883  0.00264
2005.73835      4.10837  0.00274
2005.74109      4.11009  0.00262
2005.74383      4.10850  0.00285
2005.74657      4.11092  0.00356
2005.74931      4.11059  0.00250
2005.75205      4.10912  0.00243
2005.75479      4.11143  0.00240
2005.75753      4.10913  0.00285
2005.76027      4.11125  0.00273
2005.76301      4.11064  0.00279
2005.76575      4.11052  0.00309
2005.76849      4.11014  0.00278
2005.77123      4.11014  0.00289
2005.77397      4.11042  0.00297
2005.77671      4.11181  0.00273
2005.77945      4.11266  0.00310
2005.78219      4.11269  0.00318
2005.78493      4.11150  0.00253
2005.78767      4.11280  0.00261
2005.79041      4.11317  0.00285
2005.79315      4.11311  0.00278
2005.79589      4.11313  0.00237
2005.79863      4.11367  0.00252
2005.80137      4.11380  0.00232
2005.80411      4.11348  0.00222
2005.80685      4.11354  0.00282
2005.80959      4.11255  0.00277
2005.81233      4.11187  0.00217
2005.81507      4.11241  0.00241
2005.81781      4.11254  0.00231
2005.82055      4.11100  0.00271
2005.82329      4.11277  0.00227
2005.82603      4.11159  0.00230
2005.82877      4.11025  0.00247
2005.83150      4.11204  0.00299
2005.83424      4.11282  0.00244
2005.83698      4.11209  0.00230
2005.83972      4.11176  0.00220
2005.84246      4.11267  0.00275
2005.84520      4.11383  0.00244
2005.84794      4.11183  0.00235
2005.85068      4.10868  0.00245
2005.85342      4.11237  0.00237
2005.85616      4.11061  0.00243
2005.85890      4.10971  0.00246
2005.86164      4.11215  0.00230
2005.86438      4.11522  0.00230
2005.86712      4.11395  0.00230
2005.86986      4.11453  0.00216
2005.87260      4.11535  0.00213
2005.87534      4.11574  0.00213
2005.87808      4.11464  0.00239
2005.88082      4.11461  0.00226
2005.88356      4.11389  0.00232
2005.88630      4.11555  0.00224
2005.88904      4.11473  0.00243
2005.89178      4.11377  0.00237
2005.89452      4.11513  0.00230
2005.89726      4.11527  0.00232
2005.90000      4.11732  0.00217
2005.90274      4.12105  0.00220
2005.90548      4.11698  0.00218
2005.90822      4.11680  0.00228
2005.91096      4.11627  0.00216
2005.91370      4.11607  0.00215
2005.91644      4.11557  0.00218
2005.91918      4.11870  0.00216
2005.92192      4.11756  0.00214
2005.92466      4.11605  0.00224
2005.92740      4.11677  0.00221
2005.93014      4.11689  0.00208
2005.93287      4.11757  0.00215
2005.93561      4.11859  0.00214
2005.93835      4.12318  0.00226
2005.94109      4.12160  0.00236
2005.94383      4.11909  0.00217
2005.94657      4.11879  0.00216
2005.94931      4.11870  0.00225
2005.95205      4.11725  0.00220
2005.95479      4.11810  0.00221
2005.95753      4.11909  0.00224
2005.96027      4.11729  0.00219
2005.96301      4.11676  0.00221
2005.96575      4.11794  0.00216
2005.96849      4.11895  0.00222
2005.97123      4.11680  0.00233
2005.97397      4.11676  0.00244
2005.97671      4.11752  0.00228
2005.97945      4.11780  0.00217
2005.98219      4.11819  0.00219
2005.98493      4.11821  0.00225
2005.98767      4.11947  0.00222
2005.99041      4.11651  0.00233
2005.99315      4.11599  0.00221
2005.99589      4.11607  0.00216
2005.99863      4.11688  0.00209
2006.00137      4.11699  0.00205
2006.00411      4.11653  0.00214
2006.00685      4.11574  0.00213
2006.00959      4.11806  0.00215
2006.01233      4.11936  0.00224
2006.01507      4.11837  0.00222
2006.01781      4.11605  0.00227
2006.02055      4.12354  0.00236
2006.02329      4.12688  0.00232
2006.02603      4.12758  0.00229
2006.02877      4.13006  0.00332
2006.04520      4.11826  0.00280
2006.04794      4.11942  0.00218
2006.05068      4.12038  0.00211
2006.05342      4.12102  0.00213
2006.05616      4.12192  0.00212
2006.05890      4.12102  0.00224
2006.06164      4.12100  0.00215
2006.06438      4.12215  0.00216
2006.06712      4.12062  0.00216
2006.06986      4.12232  0.00251
2006.07260      4.12194  0.00217
2006.07534      4.12181  0.00220
2006.07808      4.12231  0.00237
2006.08082      4.12086  0.00211
2006.08356      4.12222  0.00204
2006.08630      4.12317  0.00207
2006.08904      4.12346  0.00211
2006.09178      4.12399  0.00211
2006.09726      4.12215  0.00226
2006.10000      4.12159  0.00287
2006.10274      4.12363  0.00225
2006.10548      4.12349  0.00223
2006.10822      4.12245  0.00305
2006.11096      4.12118  0.00225
2006.11370      4.12125  0.00215
2006.11644      4.12234  0.00220
2006.11918      4.12371  0.00220
2006.12192      4.12169  0.00225
2006.12466      4.12504  0.00307
2006.12740      4.12362  0.00256
2006.13014      4.12402  0.00274
2006.13287      4.12324  0.00217
2006.13561      4.12507  0.00210
2006.13835      4.12631  0.00223
2006.14109      4.12516  0.00250
2006.14383      4.12600  0.00241
2006.14657      4.12513  0.00217
2006.14931      4.12497  0.00241
2006.15205      4.12554  0.00233
2006.15479      4.12582  0.00237
2006.15753      4.12452  0.00247
2006.16027      4.12452  0.00350
2006.20959      4.12704  0.00263
2006.21233      4.12483  0.00244
2006.21507      4.12586  0.00227
2006.21781      4.12592  0.00235
2006.22055      4.12710  0.00301
2006.22329      4.12663  0.00258
2006.22603      4.12648  0.00233
2006.22877      4.12630  0.00232
2006.23150      4.12760  0.00246
2006.23424      4.12929  0.00248
2006.23698      4.12801  0.00269
2006.23972      4.12809  0.00250
2006.24246      4.12731  0.00245
2006.24520      4.12751  0.00232
2006.24794      4.12627  0.00280
2006.25068      4.12611  0.00350
2006.25342      4.12751  0.00299
2006.25616      4.12614  0.00379
2006.25890      4.12681  0.00366
2006.26164      4.12806  0.00362
2006.26438      4.12965  0.00288
2006.26712      4.12416  0.00572
2006.27534      4.12984  0.00345
2006.27808      4.12758  0.00351
2006.28082      4.12857  0.00351
2006.28356      4.13161  0.00293
2006.28630      4.12988  0.00306
2006.28904      4.12827  0.00292
2006.29178      4.12715  0.00289
2006.29452      4.12612  0.00253
2006.29726      4.12835  0.00254
2006.30000      4.12754  0.00289
2006.30274      4.12921  0.00271
2006.30548      4.12862  0.00246
2006.30822      4.12970  0.00241
2006.31096      4.12907  0.00268
2006.31370      4.13277  0.00329
2006.31918      4.13131  0.00244
2006.32192      4.13014  0.00266
2006.32466      4.12885  0.00249
2006.32740      4.12962  0.00239
2006.33014      4.12786  0.00281
2006.33287      4.12904  0.00275
2006.33561      4.12808  0.00298
2006.33835      4.13034  0.00354
2006.34109      4.13001  0.00327
2006.34383      4.13042  0.00266
2006.34657      4.13153  0.00247
2006.34931      4.13211  0.00256
2006.35205      4.13206  0.00276
2006.35479      4.13546  0.00335
2006.35753      4.13148  0.00280
2006.36027      4.13213  0.00398
2006.36301      4.13363  0.00265
2006.36575      4.13327  0.00253
2006.36849      4.12991  0.00300
2006.37123      4.13215  0.00297
2006.37397      4.12956  0.00325
2006.37671      4.13370  0.00302
2006.37945      4.13351  0.00338
2006.38219      4.12916  0.00299
2006.38493      4.13108  0.00323
2006.38767      4.13301  0.00247
2006.39041      4.13271  0.00273
2006.39315      4.13351  0.00254
2006.39589      4.13176  0.00291
2006.39863      4.13452  0.00248
2006.40137      4.13227  0.00252
2006.40411      4.13140  0.00252
2006.40685      4.13312  0.00312
2006.40959      4.13293  0.00302
2006.41233      4.12820  0.00322
2006.41507      4.13055  0.00249
2006.41781      4.13272  0.00265
2006.42055      4.13049  0.00253
2006.42329      4.12985  0.00272
2006.42603      4.13370  0.00281
2006.42877      4.13339  0.00282
2006.43150      4.13288  0.00314
2006.43424      4.13254  0.00382
2006.43698      4.13406  0.00265
2006.43972      4.13330  0.00254
2006.44246      4.13160  0.00321
2006.44520      4.13296  0.00454
2006.44794      4.13181  0.00370
2006.45068      4.13114  0.00344
2006.45342      4.13500  0.00320
2006.45616      4.13533  0.00320
2006.45890      4.13381  0.00265
2006.46164      4.13402  0.00265
2006.46438      4.13318  0.00285
2006.46712      4.13448  0.00280
2006.46986      4.13393  0.00254
2006.47260      4.13186  0.00352
2006.47534      4.13223  0.00270
2006.47808      4.13455  0.00259
2006.48082      4.13376  0.00253
2006.48356      4.13226  0.00248
2006.48630      4.13219  0.00269
2006.48904      4.13328  0.00253
2006.49178      4.13538  0.00264
2006.49452      4.13460  0.00281
2006.49726      4.13463  0.00258
2006.50000      4.13441  0.00293
2006.50274      4.13687  0.00301
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2009.34657      4.21355  0.00337
2009.34931      4.21280  0.00357
2009.35205      4.21795  0.00364
2009.35479      4.21717  0.00295
2009.35753      4.21408  0.00241
2009.36027      4.21683  0.00262
2009.36301      4.21464  0.00281
2009.36575      4.21602  0.00249
2009.36849      4.21730  0.00251
2009.37123      4.21380  0.00279
2009.37397      4.21662  0.00277
2009.37671      4.21660  0.00276
2009.37945      4.21636  0.00271
2009.38219      4.21706  0.00275
2009.38493      4.21604  0.00314
2009.38767      4.21687  0.00295
2009.39041      4.21585  0.00287
2009.39315      4.21514  0.00273
2009.39589      4.21731  0.00258
2009.39863      4.21644  0.00265
2009.40137      4.21819  0.00275
2009.40411      4.21490  0.00309
2009.40685      4.21865  0.00292
2009.40959      4.21513  0.00347
2009.41233      4.21813  0.00290
2009.41507      4.21579  0.00313
2009.41781      4.21476  0.00278
2009.42055      4.21862  0.00259
2009.42329      4.21736  0.00270
2009.42603      4.21595  0.00286
2009.42877      4.21537  0.00278
2009.43150      4.21662  0.00296
2009.43424      4.21381  0.00466
2009.43698      4.21917  0.00362
2009.43972      4.21679  0.00403
2009.44246      4.21705  0.00336
2009.44520      4.21504  0.00285
2009.44794      4.21574  0.00342
2009.45068      4.21871  0.00326
2009.45342      4.21808  0.00288
2009.45616      4.21683  0.00277
2009.45890      4.21817  0.00295
2009.46164      4.21687  0.00263
2009.46438      4.21809  0.00270
2009.46712      4.21980  0.00299
2009.46986      4.21775  0.00294
2009.47260      4.22014  0.00312
2009.47534      4.21993  0.00272
2009.47808      4.21832  0.00248
2009.48082      4.21928  0.00271
2009.48356      4.21927  0.00289
2009.48630      4.21803  0.00248
2009.48904      4.21889  0.00266
2009.49178      4.21657  0.00310
2009.49452      4.21679  0.00254
2009.49726      4.21962  0.00254
2009.50000      4.21927  0.00249
2009.50274      4.21684  0.00258
2009.50548      4.22049  0.00253
2009.50822      4.21618  0.00302
2009.51096      4.21880  0.00266
2009.51370      4.21606  0.00268
2009.51644      4.22004  0.00295
2009.51918      4.21700  0.00457
2009.52192      4.21825  0.00255
2009.52466      4.21986  0.00247
2009.52740      4.21606  0.00321
2009.53014      4.21650  0.00299
2009.53287      4.21508  0.00276
2009.53561      4.21579  0.00306
2009.53835      4.21648  0.00318
2009.54109      4.21803  0.00308
2009.54383      4.21635  0.00260
2009.54657      4.21529  0.00261
2009.54931      4.21674  0.00307
2009.55205      4.22056  0.00279
2009.55479      4.21569  0.00284
2009.55753      4.21946  0.00249
2009.56027      4.21833  0.00262
2009.56301      4.21885  0.00280
2009.56575      4.21744  0.00238
2009.56849      4.21720  0.00260
2009.57123      4.21561  0.00241
2009.57397      4.21649  0.00248
2009.57671      4.21926  0.00264
2009.57945      4.21622  0.00280
2009.58219      4.21537  0.00279
2009.58493      4.21995  0.00256
2009.58767      4.21842  0.00294
2009.59041      4.22001  0.00323
2009.59315      4.21725  0.00265
2009.59589      4.21749  0.00274
2009.59863      4.21758  0.00286
2009.60137      4.21978  0.00300
2009.60411      4.21817  0.00301
2009.60685      4.21705  0.00291
2009.60959      4.21674  0.00268
2009.61233      4.21786  0.00291
2009.61507      4.21847  0.00250
2009.61781      4.21826  0.00261
2009.62055      4.21924  0.00269
2009.62329      4.21730  0.00258
2009.62603      4.21665  0.00257
2009.62877      4.21880  0.00274
2009.63150      4.21794  0.00251
2009.63424      4.21814  0.00253
2009.63698      4.21685  0.00276
2009.63972      4.21744  0.00263
2009.64246      4.21715  0.00242
2009.64520      4.21928  0.00241
2009.64794      4.21815  0.00264
2009.65068      4.21679  0.00293
2009.65342      4.21813  0.00259
2009.65616      4.21837  0.00247
2009.65890      4.21553  0.00259
2009.66164      4.22324  0.00290
2009.66438      4.21857  0.00280
2009.66712      4.21990  0.00284
2009.66986      4.21997  0.00257
2009.67260      4.22210  0.00317
2009.67534      4.21588  0.00288
2009.67808      4.21821  0.00287
2009.68082      4.21721  0.00295
2009.68356      4.21789  0.00312
2009.68630      4.21999  0.00248
2009.68904      4.22012  0.00244
2009.69178      4.21881  0.00248
2009.69452      4.22078  0.00236
2009.69726      4.21998  0.00256
2009.70000      4.22221  0.00280
2009.70274      4.22947  0.00341
2009.70548      4.22126  0.00297
2009.70822      4.21993  0.00264
2009.71370      4.22160  0.00273
2009.71644      4.22250  0.00257
2009.71918      4.22157  0.00339
2009.72192      4.22572  0.00372
2009.72466      4.22117  0.00346
2009.72740      4.21658  0.00294
2009.73014      4.22233  0.00262
2009.73287      4.22331  0.00262
2009.73561      4.22184  0.00260
2009.73835      4.22225  0.00274
2009.74109      4.22232  0.00255
2009.74383      4.22349  0.00253
2009.74657      4.22204  0.00250
2009.74931      4.22240  0.00269
2009.75205      4.22372  0.00270
2009.75479      4.22151  0.00262
2009.75753      4.22255  0.00246
2009.76027      4.22223  0.00241
2009.76301      4.22377  0.00242
2009.76575      4.22179  0.00262
2009.76849      4.22164  0.00300
2009.77123      4.22293  0.00331
2009.77397      4.22347  0.00324
2009.77671      4.22297  0.00272
2009.77945      4.22272  0.00277
2009.78219      4.22321  0.00252
2009.78493      4.22449  0.00261
2009.78767      4.22479  0.00294
2009.79041      4.22234  0.00287
2009.79315      4.22338  0.00290
2009.79589      4.22442  0.00254
2009.79863      4.22461  0.00277
2009.80137      4.22366  0.00286
2009.80411      4.22262  0.00241
2009.80685      4.22481  0.00270
2009.80959      4.22483  0.00342
2009.81233      4.22374  0.00283
2009.81507      4.22454  0.00249
2009.81781      4.22193  0.00249
2009.82055      4.22242  0.00252
2009.82329      4.22492  0.00261
2009.82603      4.22420  0.00240
2009.82877      4.22446  0.00243
2009.83150      4.22487  0.00230
2009.83424      4.22442  0.00232
2009.83698      4.22439  0.00230
2009.83972      4.22478  0.00237
2009.84246      4.22423  0.00239
2009.84520      4.22254  0.00245
2009.84794      4.22409  0.00232
2009.85068      4.23067  0.00244
2009.85342      4.22901  0.00255
2009.85616      4.22663  0.00261
2009.85890      4.22487  0.00287
2009.86164      4.22534  0.00246
2009.86438      4.22312  0.00248
2009.86712      4.22481  0.00238
2009.86986      4.22454  0.00243
2009.87260      4.22420  0.00282
2009.87534      4.22284  0.00287
2009.87808      4.20700  0.00495
2009.88082      4.22552  0.00244
2009.88356      4.22554  0.00250
2009.88630      4.22571  0.00254
2009.88904      4.22495  0.00224
2009.89178      4.22387  0.00224
2009.89452      4.22551  0.00229
2009.89726      4.22496  0.00234
2009.90000      4.22484  0.00222
2009.90274      4.22554  0.00227
2009.90548      4.22592  0.00226
2009.90822      4.22605  0.00215
2009.91096      4.22573  0.00223
2009.91370      4.22656  0.00228
2009.91644      4.22681  0.00222
2009.91918      4.22747  0.00223
2009.92192      4.22591  0.00216
2009.92466      4.22613  0.00216
2009.92740      4.22485  0.00226
2009.93014      4.22671  0.00232
2009.93287      4.22500  0.00270
2009.93561      4.22521  0.00260
2009.93835      4.22618  0.00263
2009.94109      4.22684  0.00258
2009.94383      4.22744  0.00265
2009.94657      4.22586  0.00257
2009.94931      4.22736  0.00260
2009.95205      4.22678  0.00260
2009.95479      4.22769  0.00253
2009.95753      4.22786  0.00241
2009.96027      4.22986  0.00219
2009.96301      4.22776  0.00226
2009.96575      4.22677  0.00238
2009.96849      4.22702  0.00216
2009.97123      4.22680  0.00215
2009.97397      4.22740  0.00220
2009.97671      4.22857  0.00227
2009.97945      4.22789  0.00245
2009.98219      4.22836  0.00223
2009.98493      4.22813  0.00216
2009.98767      4.22771  0.00218
2009.99041      4.22739  0.00217
2009.99315      4.22755  0.00238
2009.99589      4.22637  0.00259
2009.99863      4.22539  0.00260
2010.00137      4.22623  0.00251
2010.00411      4.22532  0.00250
2010.00685      4.22680  0.00248
2010.00959      4.22798  0.00223
2010.01233      4.23021  0.00220
2010.01507      4.22843  0.00221
2010.01781      4.22925  0.00219
2010.02055      4.23070  0.00218
2010.02329      4.23228  0.00229
2010.02603      4.23098  0.00222
2010.02877      4.23261  0.00221
2010.03150      4.22880  0.00219
2010.03424      4.22700  0.00230
2010.03698      4.22800  0.00226
2010.03972      4.22947  0.00224
2010.04246      4.22996  0.00229
2010.04520      4.22910  0.00259
2010.04794      4.22991  0.00235
2010.05068      4.23047  0.00279
2010.05342      4.22906  0.00226
2010.05616      4.22968  0.00228
2010.05890      4.22932  0.00218
2010.06164      4.22988  0.00217
2010.06438      4.22964  0.00212
2010.06712      4.22978  0.00226
2010.06986      4.22887  0.00221
2010.07260      4.22957  0.00219
2010.07808      4.22864  0.00223
2010.08082      4.23074  0.00217
2010.08356      4.23087  0.00214
2010.08904      4.23054  0.00224
2010.09178      4.23082  0.00223
2010.09452      4.22963  0.00218
2010.09726      4.23036  0.00217
2010.10000      4.22955  0.00223
2010.10274      4.22875  0.00213
2010.10548      4.22934  0.00219
2010.10822      4.23025  0.00225
2010.11096      4.23292  0.00228
2010.11370      4.23590  0.00236
2010.11918      4.23580  0.00252
2010.12192      4.23360  0.00264
2010.12740      4.23607  0.00239
2010.13014      4.23450  0.00241
2010.13561      4.23299  0.00235
2010.13835      4.23400  0.00237
2010.14109      4.23384  0.00214
2010.14383      4.23398  0.00226
2010.14657      4.23454  0.00243
2010.14931      4.23374  0.00253
2010.15205      4.23468  0.00248
2010.15479      4.23394  0.00235
2010.15753      4.23511  0.00226
2010.16027      4.23580  0.00221
2010.16301      4.23544  0.00237
2010.16575      4.23408  0.00307
2010.16849      4.23377  0.00270
2010.17123      4.23411  0.00242
2010.17397      4.23488  0.00244
2010.17671      4.23567  0.00261
2010.17945      4.23552  0.00240
2010.18219      4.23518  0.00245
2010.18493      4.23863  0.00242
2010.18767      4.23552  0.00231
2010.19041      4.23677  0.00256
2010.19315      4.23806  0.00231
2010.19589      4.23849  0.00240
2010.19863      4.23918  0.00269
2010.20137      4.24009  0.00268
2010.20411      4.23813  0.00244
2010.20685      4.23783  0.00228
2010.20959      4.23848  0.00234
2010.21233      4.23756  0.00232
2010.21507      4.23838  0.00219
2010.21781      4.23648  0.00245
2010.22055      4.23680  0.00250
2010.22329      4.23754  0.00308
2010.22603      4.23869  0.00293
2010.22877      4.24069  0.00294
2010.23150      4.23758  0.00275
2010.23424      4.23815  0.00355
2010.23698      4.24319  0.00336
2010.23972      4.24030  0.00269
2010.24246      4.24093  0.00367
2010.24520      4.23970  0.00450
2010.24794      4.23772  0.00330
2010.25068      4.23617  0.00346
2010.25342      4.24180  0.00397
2010.25616      4.24238  0.00344
2010.25890      4.23842  0.00328
2010.26164      4.24107  0.00295
2010.26438      4.24192  0.00241
2010.26712      4.24158  0.00265
2010.26986      4.24251  0.00294
2010.27260      4.23897  0.00306
2010.27534      4.24120  0.00311
2010.27808      4.24344  0.00318
2010.28082      4.24052  0.00329
2010.28356      4.23537  0.00328
2010.28630      4.24350  0.00279
2010.28904      4.23929  0.00249
2010.29178      4.24290  0.00271
2010.29452      4.24401  0.00274
2010.29726      4.24488  0.00349
2010.30000      4.23938  0.00301
2010.30274      4.23901  0.00351
2010.30548      4.23991  0.00329
2010.30822      4.24150  0.00350
2010.31096      4.24510  0.00371
2010.31370      4.24437  0.00365
2010.31644      4.24390  0.00357
2010.31918      4.24265  0.00259
2010.32192      4.24071  0.00249
2010.32466      4.24064  0.00280
2010.33014      4.23988  0.00242
2010.33287      4.24411  0.00255
2010.33561      4.24049  0.00266
2010.33835      4.24281  0.00310
2010.34109      4.24229  0.00266
2010.34383      4.24255  0.00251
2010.34657      4.24129  0.00330
2010.34931      4.24401  0.00269
2010.35205      4.24167  0.00299
2010.35479      4.24044  0.00331
2010.35753      4.24066  0.00474
2010.36027      4.23824  0.00494
2010.36575      4.23963  0.00478
2010.36849      4.24281  0.00334
2010.37123      4.24171  0.00330
2010.37397      4.24408  0.00281
2010.37671      4.24032  0.00268
2010.37945      4.24417  0.00286
2010.38219      4.24101  0.00348
2010.38493      4.24263  0.00350
```
Chris Martin on 19 Jan 2013

please some one help me

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