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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..

0 Comments

Chris Martin

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2 Answers

Answer by Jurgen on 27 Nov 2012
Accepted answer

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?

3 Comments

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
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Jurgen
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.

4 Comments

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

Image Analyst

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