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# How to apply Maximum likelihood estimation technique to characterize the amplitude of noise in a given time series

Asked by Chris Martin on 27 Nov 2012

I have got the power spectral density of a time series. Now I want to know the amount of noises present in the time series using MLE. Please help me out..

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Answer by Jurgen on 27 Nov 2012

Im just guessing here, but since the least square fit of your time function (inverse fourier?) is a maximum likelihood estimator of your signal model (assuming gaussian noise), then the sum of residuals should be a measure for the amount of noise?

Chris Martin on 27 Nov 2012

I want to know the steps for obtaining the result if I am having a time series.I am new to matlab.

Jurgen on 27 Nov 2012

Also the noise is probably in the higher frequency part of the psd, I'm afraid I don't know what kind of signal you're working with.

Chris Martin on 28 Nov 2012

actually I am working with time series, for better communication I am attaching the series below: the first column is Time(year), second column is the signal and third column is the error

``` 2005.10274      5.66988  0.00223
2005.10548      5.67000  0.00215
2005.10822      5.66938  0.00212
2005.11096      5.67094  0.00218
2005.11370      5.67120  0.00219
2005.11644      5.66837  0.00256
2005.11918      5.66971  0.00288
2005.12192      5.66931  0.00227
2005.12466      5.66980  0.00231
2005.12740      5.66964  0.00242
2005.13014      5.66732  0.00295
2005.13287      5.66913  0.00251
2005.13561      5.67238  0.00292
2005.13835      5.67422  0.00332
2005.14109      5.67309  0.00257
2005.14383      5.67210  0.00240
2005.14657      5.67266  0.00246
2005.14931      5.67057  0.00240
2005.15205      5.67228  0.00226
2005.15479      5.67407  0.00247
2005.15753      5.67290  0.00232
2005.16027      5.67254  0.00224
2005.16301      5.67269  0.00268
2005.16575      5.67290  0.00230
2005.16849      5.67318  0.00260
2005.17123      5.67350  0.00236
2005.17397      5.67235  0.00264
2005.17671      5.67314  0.00261
2005.17945      5.67561  0.00292
2005.18219      5.67372  0.00261
2005.18493      5.66916  0.00441
2005.20137      5.67418  0.00355
2005.20411      5.67229  0.00331
2005.20685      5.67560  0.00273
2005.20959      5.67665  0.00231
2005.21233      5.67696  0.00271
2005.21507      5.67654  0.00317
2005.21781      5.67418  0.00296
2005.22055      5.67387  0.00428
2005.22329      5.67613  0.00379
2005.22603      5.67408  0.00322
2005.22877      5.67542  0.00303
2005.23150      5.67412  0.00254
2005.23424      5.67344  0.00258
2005.23698      5.67269  0.00432
2005.23972      5.67367  0.00339
2005.24246      5.67364  0.00308
2005.24520      5.67211  0.00348
2005.24794      5.67596  0.00319
2005.25068      5.67663  0.00292
2005.25342      5.67824  0.00337
2005.25890      5.67509  0.00241
2005.26164      5.67649  0.00342
2005.26438      5.67530  0.00403
2005.26712      5.67571  0.00391
2005.26986      5.68093  0.00435
2005.27260      5.68041  0.00556
2005.27534      5.67842  0.00413
2005.27808      5.68079  0.00336
2005.28082      5.68277  0.00299
2005.28356      5.67929  0.00332
2005.28630      5.67865  0.00373
2005.28904      5.67555  0.00343
2005.29178      5.67501  0.00492
2005.29452      5.68044  0.00421
2005.29726      5.68005  0.00341
2005.30000      5.68312  0.00403
2005.30274      5.67974  0.00297
2005.30548      5.67703  0.00346
2005.30822      5.68163  0.00317
2005.31096      5.68257  0.00374
2005.31370      5.68381  0.00401
2005.31644      5.68238  0.00518
2005.31918      5.67832  0.00426
2005.32192      5.67892  0.00335
2005.32466      5.67860  0.00404
2005.32740      5.67876  0.00289
2005.33014      5.68020  0.00332
2005.33287      5.68102  0.00322
2005.33561      5.67564  0.00403
2005.33835      5.68552  0.00373
2005.34109      5.68625  0.00327
2005.34383      5.68460  0.00330
2005.34657      5.68716  0.00324
2005.34931      5.68736  0.00297
2005.35205      5.68760  0.00349
2005.35479      5.68684  0.00473
2005.35753      5.68635  0.00372
2005.36027      5.68208  0.00428
2005.36301      5.68338  0.00450
2005.36575      5.68490  0.00415
2005.36849      5.68400  0.00345
2005.37123      5.68521  0.00381
2005.37397      5.68020  0.00302
2005.37671      5.67989  0.00362
2005.37945      5.68081  0.00351
2005.38219      5.68094  0.00321
2005.38493      5.68330  0.00309
2005.38767      5.68203  0.00274
2005.39041      5.68146  0.00292
2005.39315      5.68326  0.00343
2005.39589      5.68224  0.00385
2005.39863      5.68190  0.00339
2005.40137      5.68054  0.00383
2005.40411      5.68425  0.00440
2005.40685      5.68432  0.00387
2005.40959      5.68259  0.00288
2005.41233      5.68214  0.00356
2005.41507      5.68092  0.00411
2005.41781      5.68514  0.00404
2005.42055      5.68541  0.00385
2005.42329      5.68629  0.01202
2005.42603      5.68311  0.00305
2005.42877      5.68436  0.00307
2005.43150      5.68359  0.00300
2005.43424      5.68321  0.00306
2005.43698      5.68452  0.00337
2005.43972      5.68596  0.00338
2005.44246      5.68531  0.00359
2005.44520      5.68241  0.00440
2005.44794      5.68367  0.00305
2005.45068      5.68796  0.00379
2005.45342      5.68152  0.00510
2005.45616      5.68589  0.00354
2005.45890      5.68373  0.00296
2005.46164      5.68398  0.00325
2005.46438      5.68736  0.00333
2005.46712      5.68708  0.00342
2005.46986      5.68360  0.00285
2005.47260      5.68540  0.00291
2005.47534      5.68800  0.00295
2005.47808      5.68727  0.00316
2005.48082      5.68502  0.00306
2005.48356      5.68468  0.00338
2005.48630      5.68951  0.00309
2005.48904      5.69401  0.00338
2005.49178      5.69348  0.00527
2005.49452      5.69488  0.00344
2005.49726      5.69379  0.00351
2005.50000      5.68803  0.00284
2005.50274      5.68773  0.00332
2005.50548      5.68492  0.00337
2005.50822      5.68885  0.00286
2005.51096      5.68989  0.00324
2005.51370      5.68904  0.00331
2005.51644      5.68881  0.00341
2005.51918      5.68866  0.00295
2005.52192      5.68757  0.00279
2005.52466      5.68811  0.00273
2005.52740      5.68909  0.00277
2005.53014      5.68848  0.00284
2005.53287      5.68637  0.00273
2005.53561      5.68934  0.00265
2005.53835      5.68711  0.00254
2005.54109      5.68806  0.00261
2005.54383      5.68812  0.00271
2005.54657      5.68986  0.00291
2005.54931      5.68569  0.00294
2005.55205      5.68731  0.00309
2005.55479      5.68875  0.00304
2005.55753      5.68833  0.00293
2005.56027      5.68895  0.00273
2005.56301      5.68777  0.00347
2005.56575      5.68940  0.00305
2005.56849      5.68879  0.00285
2005.57123      5.68785  0.00281
2005.57397      5.69192  0.00302
2005.57671      5.69193  0.00352
2005.57945      5.68949  0.00305
2005.58219      5.68843  0.00264
2005.58493      5.69079  0.00290
2005.58767      5.69032  0.00331
2005.59041      5.68685  0.00284
2005.59315      5.69136  0.00292
2005.59589      5.68996  0.00280
2005.59863      5.69130  0.00271
2005.60137      5.69030  0.00298
2005.60411      5.68862  0.00289
2005.60685      5.68859  0.00283
2005.60959      5.68720  0.00275
2005.61233      5.69041  0.00267
2005.61507      5.68517  0.00375
2005.61781      5.68699  0.00294
2005.62055      5.68852  0.00347
2005.62329      5.68940  0.00259
2005.62603      5.68907  0.00277
2005.62877      5.68666  0.00291
2005.63150      5.68903  0.00316
2005.63424      5.68906  0.00318
2005.63698      5.68851  0.00291
2005.63972      5.68933  0.00287
2005.64246      5.68957  0.00333
2005.64520      5.68834  0.00315
2005.64794      5.69267  0.00285
2005.65068      5.69051  0.00356
2005.65342      5.69188  0.00365
2005.65616      5.69059  0.00346
2005.65890      5.68935  0.00395
2005.66164      5.69177  0.00334
2005.66438      5.69174  0.00267
2005.66712      5.68968  0.00291
2005.66986      5.68872  0.00339
2005.67260      5.69019  0.00295
2005.67534      5.69347  0.00291
2005.67808      5.68962  0.00247
2005.68082      5.69193  0.00257
2005.68356      5.69361  0.00277
2005.68630      5.69443  0.00303
2005.68904      5.69257  0.00291
2005.69178      5.68937  0.00277
2005.69452      5.69163  0.00303
2005.69726      5.69129  0.00281
2005.70000      5.69119  0.00313
2005.70274      5.69402  0.00274
2005.70548      5.69167  0.00291
2005.70822      5.69047  0.00300
2005.71096      5.69219  0.00249
2005.71370      5.68986  0.00266
2005.71644      5.69025  0.00263
2005.71918      5.69218  0.00276
2005.72192      5.69043  0.00267
2005.72466      5.69215  0.00256
2005.72740      5.69223  0.00308
2005.73014      5.69394  0.00302
2005.73287      5.69578  0.00286
2005.73561      5.69412  0.00291
2005.73835      5.69362  0.00295
2005.74109      5.69331  0.00288
2005.74383      5.69462  0.00310
2005.74657      5.69244  0.00384
2005.74931      5.69306  0.00272
2005.75205      5.69480  0.00261
2005.75479      5.69638  0.00258
2005.75753      5.69633  0.00312
2005.76027      5.69692  0.00295
2005.76301      5.69525  0.00310
2005.76575      5.69656  0.00341
2005.76849      5.69548  0.00301
2005.77123      5.69927  0.00314
2005.77397      5.69536  0.00326
2005.77671      5.69424  0.00307
2005.77945      5.69404  0.00343
2005.78219      5.69360  0.00345
2005.78493      5.69580  0.00278
2005.78767      5.69844  0.00289
2005.79041      5.69607  0.00307
2005.79315      5.69430  0.00299
2005.79589      5.69707  0.00259
2005.79863      5.69822  0.00278
2005.80137      5.69785  0.00253
2005.80411      5.70007  0.00242
2005.80685      5.69851  0.00309
2005.80959      5.69672  0.00304
2005.81233      5.69909  0.00232
2005.81507      5.69731  0.00259
2005.81781      5.69865  0.00249
2005.82055      5.70158  0.00293
2005.82329      5.69714  0.00246
2005.82603      5.69935  0.00249
2005.82877      5.70134  0.00266
2005.83150      5.69735  0.00330
2005.83424      5.69842  0.00264
2005.83698      5.69894  0.00252
2005.83972      5.69989  0.00240
2005.84246      5.70115  0.00301
2005.84520      5.69809  0.00273
2005.84794      5.69697  0.00256
2005.85068      5.69776  0.00273
2005.85342      5.69784  0.00269
2005.85616      5.69957  0.00274
2005.85890      5.69739  0.00278
2005.86164      5.69695  0.00258
2005.86438      5.69402  0.00257
2005.86712      5.70031  0.00256
2005.86986      5.69875  0.00239
2005.87260      5.69962  0.00237
2005.87534      5.69797  0.00236
2005.87808      5.69796  0.00267
2005.88082      5.69920  0.00252
2005.88356      5.69987  0.00256
2005.88630      5.70079  0.00250
2005.88904      5.70077  0.00275
2005.89178      5.70003  0.00262
2005.89452      5.70040  0.00254
2005.89726      5.69965  0.00260
2005.90000      5.69967  0.00242
2005.90274      5.69865  0.00246
2005.90548      5.70045  0.00238
2005.90822      5.70082  0.00244
2005.91096      5.69970  0.00235
2005.91370      5.70112  0.00236
2005.91644      5.70188  0.00236
2005.91918      5.70220  0.00238
2005.92192      5.70200  0.00237
2005.92466      5.70206  0.00248
2005.92740      5.70135  0.00243
2005.93014      5.70211  0.00229
2005.93287      5.70271  0.00238
2005.93561      5.69987  0.00232
2005.93835      5.70080  0.00248
2005.94109      5.70006  0.00258
2005.94383      5.70181  0.00237
2005.94657      5.70048  0.00236
2005.94931      5.70131  0.00239
2005.95205      5.70422  0.00233
2005.95479      5.70271  0.00239
2005.95753      5.70380  0.00239
2005.96027      5.70321  0.00232
2005.96301      5.70353  0.00231
2005.96575      5.70363  0.00231
2005.96849      5.70448  0.00241
2005.97123      5.70610  0.00259
2005.97397      5.70314  0.00265
2005.97671      5.70520  0.00250
2005.97945      5.70534  0.00238
2005.98219      5.70438  0.00234
2005.98493      5.70698  0.00245
2005.98767      5.70363  0.00240
2005.99041      5.70530  0.00249
2005.99315      5.70503  0.00228
2005.99589      5.70562  0.00230
2005.99863      5.70683  0.00226
2006.00137      5.70562  0.00220
2006.00411      5.70454  0.00223
2006.00685      5.70533  0.00230
2006.00959      5.70512  0.00226
2006.01233      5.70498  0.00237
2006.01507      5.70365  0.00235
2006.01781      5.70383  0.00243
2006.02055      5.70445  0.00246
2006.02329      5.70269  0.00238
2006.02603      5.70391  0.00239
2006.02877      5.70296  0.00405
2006.04520      5.70818  0.00288
2006.04794      5.70867  0.00233
2006.05068      5.70778  0.00226
2006.05342      5.70659  0.00225
2006.05616      5.70884  0.00224
2006.05890      5.70846  0.00239
2006.06164      5.70923  0.00229
2006.06438      5.70948  0.00228
2006.06712      5.71115  0.00229
2006.06986      5.71435  0.00269
2006.07260      5.70987  0.00230
2006.07534      5.70914  0.00230
2006.07808      5.71107  0.00250
2006.08082      5.70837  0.00222
2006.08356      5.70880  0.00215
2006.08630      5.70778  0.00217
2006.08904      5.70742  0.00222
2006.09178      5.70798  0.00225
2006.09726      5.71323  0.00243
2006.10000      5.71646  0.00329
2006.10274      5.70980  0.00239
2006.10548      5.71086  0.00236
2006.10822      5.71987  0.00363
2006.11096      5.71045  0.00236
2006.11370      5.71127  0.00227
2006.11644      5.70893  0.00235
2006.11918      5.71074  0.00233
2006.12192      5.70962  0.00239
2006.12466      5.70955  0.00331
2006.12740      5.71242  0.00278
2006.13014      5.71150  0.00293
2006.13287      5.71029  0.00232
2006.13561      5.70924  0.00223
2006.13835      5.70960  0.00241
2006.14109      5.71023  0.00270
2006.14383      5.71044  0.00261
2006.14657      5.71252  0.00235
2006.14931      5.71104  0.00257
2006.15205      5.71125  0.00248
2006.15479      5.71229  0.00253
2006.15753      5.71098  0.00266
2006.16027      5.71231  0.00437
2006.20959      5.71563  0.00291
2006.21233      5.71422  0.00264
2006.21507      5.71565  0.00253
2006.21781      5.71408  0.00257
2006.22055      5.71474  0.00350
2006.22329      5.71549  0.00300
2006.22603      5.71431  0.00273
2006.22877      5.71686  0.00263
2006.23150      5.71348  0.00282
2006.23424      5.71706  0.00285
2006.23698      5.71713  0.00303
2006.23972      5.71241  0.00285
2006.24246      5.71432  0.00273
2006.24520      5.71498  0.00270
2006.24794      5.71453  0.00310
2006.25068      5.71713  0.00390
2006.25342      5.71694  0.00350
2006.25616      5.71820  0.00428
2006.25890      5.71594  0.00411
2006.26164      5.71598  0.00423
2006.26438      5.71696  0.00325
2006.26712      5.71736  0.00647
2006.27534      5.71747  0.00387
2006.27808      5.71798  0.00411
2006.28082      5.71954  0.00398
2006.28356      5.72300  0.00344
2006.28630      5.72222  0.00349
2006.28904      5.71689  0.00343
2006.29178      5.72105  0.00345
2006.29452      5.71843  0.00288
2006.29726      5.71927  0.00296
2006.30000      5.72000  0.00337
2006.30274      5.72112  0.00309
2006.30548      5.72019  0.00289
2006.30822      5.71895  0.00286
2006.31096      5.71959  0.00310
2006.31370      5.71762  0.00399
2006.31918      5.72104  0.00286
2006.32192      5.72191  0.00324
2006.32466      5.72042  0.00283
2006.32740      5.71978  0.00271
2006.33014      5.72011  0.00317
2006.33287      5.72282  0.00313
2006.33561      5.72238  0.00338
2006.33835      5.71805  0.00397
2006.34109      5.71922  0.00368
2006.34383      5.72264  0.00301
2006.34657      5.72278  0.00277
2006.34931      5.72276  0.00290
2006.35205      5.72164  0.00310
2006.35479      5.72034  0.00373
2006.35753      5.72218  0.00312
2006.36027      5.72393  0.00442
2006.36301      5.72397  0.00298
2006.36575      5.71923  0.00282
2006.36849      5.72697  0.00334
2006.37123      5.72832  0.00327
2006.37397      5.72701  0.00358
2006.37671      5.72787  0.00336
2006.37945      5.72825  0.00378
2006.38219      5.73221  0.00334
2006.38493      5.72971  0.00359
2006.38767      5.72193  0.00274
2006.39041      5.72402  0.00303
2006.39315      5.72315  0.00286
2006.39589      5.72015  0.00325
2006.39863      5.72276  0.00279
2006.40137      5.72610  0.00282
2006.40411      5.72388  0.00281
2006.40685      5.72417  0.00351
2006.40959      5.72277  0.00344
2006.41233      5.72170  0.00358
2006.41507      5.72315  0.00278
2006.41781      5.72352  0.00300
2006.42055      5.72555  0.00289
2006.42329      5.72150  0.00307
2006.42603      5.72103  0.00316
2006.42877      5.72199  0.00326
2006.43150      5.72191  0.00360
2006.43424      5.72215  0.00437
2006.43698      5.72584  0.00297
2006.43972      5.72584  0.00284
2006.44246      5.72647  0.00357
2006.44520      5.72624  0.00501
2006.44794      5.72784  0.00412
2006.45068      5.72487  0.00384
2006.45342      5.72675  0.00354
2006.45616      5.72392  0.00364
2006.45890      5.72399  0.00289
2006.46164      5.72565  0.00295
2006.46438      5.72691  0.00326
2006.46712      5.71815  0.00322
2006.46986      5.72442  0.00286
2006.47260      5.73238  0.00404
2006.47534      5.72724  0.00308
2006.47808      5.72567  0.00294
2006.48082      5.72488  0.00284
2006.48356      5.72714  0.00274
2006.48630      5.72443  0.00302
2006.48904      5.72474  0.00281
2006.49178      5.72598  0.00297
2006.49452      5.72705  0.00307
2006.49726      5.72840  0.00280
2006.50000      5.72358  0.00329
2006.50274      5.72289  0.00354
2006.50548      5.72410  0.00333
2006.50822      5.72287  0.00366
2006.51096      5.72150  0.00348
2006.51370      5.72352  0.00339
2006.51644      5.72375  0.00316
2006.51918      5.72517  0.00358
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2009.99315      5.87677  0.00263
2009.99589      5.88387  0.00281
2009.99863      5.88440  0.00282
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2010.00959      5.87315  0.00239
2010.01233      5.86904  0.00235
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2010.21781      5.88145  0.00288
2010.22055      5.88232  0.00287
2010.22329      5.88335  0.00365
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2010.22877      5.88330  0.00342
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2010.23972      5.88350  0.00315
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2010.24520      5.88234  0.00527
2010.24794      5.88201  0.00386
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2010.25342      5.88074  0.00465
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2010.27260      5.88604  0.00353
2010.27534      5.88025  0.00359
2010.27808      5.88364  0.00367
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2010.28356      5.88653  0.00377
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2010.29452      5.88212  0.00314
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2010.30000      5.88807  0.00342
2010.30274      5.88167  0.00402
2010.30548      5.88549  0.00378
2010.30822      5.88242  0.00416
2010.31096      5.88095  0.00435
2010.31370      5.88584  0.00424
2010.31644      5.88214  0.00414
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2010.32466      5.88088  0.00321
2010.33014      5.88113  0.00271
2010.33287      5.88390  0.00285
2010.33561      5.88354  0.00301
2010.33835      5.88299  0.00358
2010.34109      5.88484  0.00291
2010.34383      5.88525  0.00278
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2010.34931      5.88524  0.00299
2010.35205      5.88791  0.00334
2010.35479      5.88150  0.00375
2010.35753      5.88330  0.00533
2010.36027      5.88857  0.00566
2010.36575      5.89457  0.00513
2010.36849      5.88925  0.00375
2010.37123      5.88867  0.00373
2010.37397      5.88717  0.00310
2010.37671      5.88667  0.00302
2010.37945      5.88578  0.00315
2010.38219      5.88765  0.00391
2010.38493      5.88387  0.00397
2010.38767      5.88553  0.00293
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2010.39589      5.88738  0.00341
2010.39863      5.88544  0.00270
2010.40411      5.88788  0.00318
2010.40959      5.88873  0.00324
2010.41233      5.88458  0.00427
2010.41507      5.89074  0.00415
2010.41781      5.88524  0.00372
2010.42055      5.88833  0.00313
2010.42329      5.88924  0.00320
2010.42603      5.88703  0.00429
2010.42877      5.88682  0.00592
2010.43150      5.89136  0.00362
2010.43424      5.88964  0.00317
2010.43698      5.88946  0.00322
2010.43972      5.88984  0.00397
2010.44246      5.89063  0.00322
2010.44520      5.89048  0.00369
2010.45068      5.89021  0.00335
2010.45342      5.89084  0.00319
2010.45616      5.89040
```
Answer by Image Analyst on 27 Nov 2012

Isn't the average a max likelihood estimator? So can't you just subtract your signal from the average signal in the time domain? You can use conv() if you want a sliding average.

Chris Martin on 31 Dec 2012

Give me details of steps how to determine the noise component using MLE

Matt J on 31 Dec 2012

I don't see why further estimation (MLE or otherwise) is necessary when you already have the PSD. The variance (i.e. noise) of the process is the sum/integral of the PSD, which boils down to the same thing as Jurgen's answer (which you accepted).

Chris Martin on 4 Jan 2013

Because I want to extract the noise component individually from the time series