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

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..

Answer 1

[Jurgen]

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]

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

[Jurgen]

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]

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

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 2009.63424      5.85207  0.00274
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 2010.17397      5.87919  0.00277
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 2010.17945      5.88207  0.00270
 2010.18219      5.88376  0.00283
 2010.18493      5.87982  0.00274
 2010.18767      5.88189  0.00264
 2010.19041      5.88095  0.00292
 2010.19315      5.88464  0.00268
 2010.19589      5.88123  0.00280
 2010.19863      5.88069  0.00316
 2010.20137      5.88297  0.00308
 2010.20411      5.88365  0.00277
 2010.20685      5.88304  0.00260
 2010.20959      5.88029  0.00270
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 2010.21507      5.88265  0.00247
 2010.21781      5.88145  0.00288
 2010.22055      5.88232  0.00287
 2010.22329      5.88335  0.00365
 2010.22603      5.88538  0.00337
 2010.22877      5.88330  0.00342
 2010.23150      5.88692  0.00317
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 2010.23698      5.88223  0.00395
 2010.23972      5.88350  0.00315
 2010.24246      5.88772  0.00425
 2010.24520      5.88234  0.00527
 2010.24794      5.88201  0.00386
 2010.25068      5.88238  0.00405
 2010.25342      5.88074  0.00465
 2010.25616      5.88026  0.00398
 2010.25890      5.88057  0.00379
 2010.26164      5.88529  0.00345
 2010.26438      5.88238  0.00278
 2010.26712      5.88079  0.00302
 2010.26986      5.88360  0.00345
 2010.27260      5.88604  0.00353
 2010.27534      5.88025  0.00359
 2010.27808      5.88364  0.00367
 2010.28082      5.88454  0.00382
 2010.28356      5.88653  0.00377
 2010.28630      5.88271  0.00324
 2010.28904      5.88327  0.00278
 2010.29178      5.88272  0.00310
 2010.29452      5.88212  0.00314
 2010.29726      5.88365  0.00403
 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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 2010.33014      5.88113  0.00271
 2010.33287      5.88390  0.00285
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 2010.34383      5.88525  0.00278
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 2010.35205      5.88791  0.00334
 2010.35479      5.88150  0.00375
 2010.35753      5.88330  0.00533
 2010.36027      5.88857  0.00566
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 2010.36849      5.88925  0.00375
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 2010.37397      5.88717  0.00310
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 2010.37945      5.88578  0.00315
 2010.38219      5.88765  0.00391
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 2010.38767      5.88553  0.00293
 2010.39041      5.88622  0.00321
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 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 2

[Image Analyst]

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.

[Image Analyst]

Another option: http://www.mathworks.com/matlabcentral/fileexchange/16683-estimatenoise

[Chris Martin]

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

[Matt J]

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]

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