MATLAB: How to decompose following gps time series ?? Please help me

Question

The time series contain position of a gps site from 2007-2011. Here the first column is the time and second column is the position and third one is the standard error.

 2007.00137      1.56954  0.00367
 2007.00411      1.57119  0.00357
 2007.00685      1.57185  0.00357
 2007.00959      1.56980  0.00354
 2007.01233      1.57311  0.00358
 2007.01507      1.57543  0.00352
 2007.01781      1.57261  0.00389
 2007.02055      1.57557  0.00432
 2007.02329      1.57508  0.00374
 2007.02603      1.57176  0.00379
 2007.02877      1.57425  0.00370
 2007.03150      1.57342  0.00367
 2007.03424      1.57477  0.00367
 2007.03698      1.57507  0.00348
 2007.03972      1.57431  0.00368
 2007.04246      1.57365  0.00356
 2007.04520      1.57356  0.00429
 2007.04794      1.57394  0.00373
 2007.05068      1.56943  0.00377
 2007.05342      1.57371  0.00395
 2007.05616      1.57291  0.00379
 2007.05890      1.57486  0.00370
 2007.06164      1.57595  0.00351
 2007.06438      1.57601  0.00333
 2007.06712      1.57461  0.00343
 2007.06986      1.57349  0.00345
 2007.07260      1.57378  0.00385
 2007.07534      1.57531  0.00376
 2007.07808      1.57553  0.00380
 2007.08082      1.57588  0.00354
 2007.08356      1.57471  0.00349
 2007.08630      1.57656  0.00356
 2007.08904      1.57867  0.00335
 2007.09178      1.57809  0.00337
 2007.09452      1.58016  0.00376
 2007.09726      1.57677  0.00349
 2007.10000      1.57834  0.00339
 2007.10274      1.57807  0.00334
 2007.10548      1.57783  0.00342
 2007.10822      1.58044  0.00342
 2007.11096      1.57999  0.00338
 2007.11644      1.58517  0.00342
 2007.11918      1.58198  0.00332
 2007.12192      1.58058  0.00330
 2007.12466      1.58050  0.00344
 2007.12740      1.58114  0.00323
 2007.13014      1.58252  0.00344
 2007.13287      1.58243  0.00337
 2007.13561      1.58281  0.00328
 2007.13835      1.58017  0.00333
 2007.14109      1.58194  0.00325
 2007.14383      1.58304  0.00324
 2007.14657      1.58323  0.00330
 2007.14931      1.58240  0.00329
 2007.15205      1.58007  0.00337
 2007.15479      1.58450  0.00363
 2007.15753      1.58209  0.00342
 2007.16027      1.58307  0.00326
 2007.16301      1.58021  0.00344
 2007.16575      1.57952  0.00343
 2007.16849      1.58394  0.00346
 2007.17123      1.58361  0.00334
 2007.17397      1.58183  0.00341
 2007.17671      1.58343  0.00355
 2007.17945      1.58263  0.00338
 2007.18219      1.58189  0.00336
 2007.18493      1.58322  0.00337
 2007.18767      1.58113  0.00349
 2007.19041      1.58400  0.00340
 2007.19315      1.58279  0.00332
 2007.19589      1.58315  0.00340
 2007.20137      1.58288  0.00340
 2007.20411      1.58428  0.00346
 2007.20685      1.58110  0.00340
 2007.20959      1.58349  0.00378
 2007.21233      1.58411  0.00353
 2007.21507      1.58312  0.00356
 2007.21781      1.58388  0.00339
 2007.22055      1.58321  0.00347
 2007.22329      1.58510  0.00331
 2007.22603      1.58362  0.00340
 2007.22877      1.58334  0.00347
 2007.23150      1.58527  0.00351
 2007.23424      1.58493  0.00335
 2007.23698      1.58397  0.00357
 2007.23972      1.58276  0.00353
 2007.24246      1.58275  0.00339
 2007.24520      1.58519  0.00357
 2007.24794      1.58216  0.00364
 2007.25068      1.58406  0.00342
 2007.25342      1.58464  0.00355
 2007.25616      1.58829  0.00362
 2007.25890      1.58328  0.00341
 2007.26164      1.58600  0.00344
 2007.26438      1.58468  0.00354
 2007.26712      1.58463  0.00349
 2007.26986      1.58587  0.00355
 2007.27260      1.58601  0.00366
 2007.27534      1.58374  0.00377
 2007.27808      1.58775  0.00425
 2007.28082      1.58570  0.00432
 2007.28356      1.58758  0.00415
 2007.28630      1.58663  0.00363
 2007.28904      1.58658  0.00363
 2007.29178      1.58803  0.00401
 2007.29452      1.58743  0.00417
 2007.29726      1.59811  0.00545
 2007.30000      1.59489  0.00455
 2007.30274      1.59898  0.00488
 2007.30548      1.59727  0.00483
 2007.30822      1.59115  0.00359
 2007.31096      1.58812  0.00385
 2007.31370      1.58738  0.00363
 2007.31644      1.58902  0.00352
 2007.31918      1.58878  0.00367
 2007.32192      1.58849  0.00375
 2007.32466      1.58652  0.00361
 2007.32740      1.58845  0.00401
 2007.33014      1.58779  0.00402
 2007.33287      1.58765  0.00433
 2007.33561      1.58918  0.00426
 2007.33835      1.58582  0.00351
 2007.34109      1.58803  0.00379
 2007.34383      1.58853  0.00401
 2007.34657      1.58881  0.00358
 2007.34931      1.58729  0.00347
 2007.35205      1.58868  0.00339
 2007.35479      1.58904  0.00347
 2007.35753      1.58876  0.00344
 2007.36027      1.58783  0.00333
 2007.36301      1.58732  0.00341
 2007.36575      1.58983  0.00366
 2007.36849      1.58935  0.00361
 2007.37123      1.58959  0.00334
 2007.37397      1.58994  0.00363
 2007.37671      1.58925  0.00373
 2007.37945      1.58886  0.00392
 2007.38219      1.59304  0.00390
 2007.38493      1.59038  0.00346
 2007.38767      1.59130  0.00356
 2007.39041      1.59295  0.00417
 2007.39315      1.59323  0.00663
 2007.39589      1.59401  0.00383
 2007.39863      1.59246  0.00367
 2007.40137      1.58997  0.00412
 2007.40411      1.59189  0.00371
 2007.40685      1.59275  0.00354
 2007.40959      1.59013  0.00365
 2007.41233      1.58903  0.00383
 2007.41507      1.58927  0.00390
 2007.41781      1.59369  0.00380
 2007.42055      1.59210  0.00398
 2007.42329      1.59364  0.00441
 2007.42603      1.59105  0.00390
 2007.42877      1.58943  0.00406
 2007.43150      1.59061  0.00386
 2007.43424      1.59123  0.00345
 2007.43698      1.59293  0.00345
 2007.43972      1.59075  0.00340
 2007.44246      1.59167  0.00372
 2007.44520      1.59399  0.00411
 2007.44794      1.59251  0.00379
 2007.45068      1.59278  0.00364
 2007.45342      1.59536  0.00360
 2007.45616      1.59191  0.00341
 2007.45890      1.59346  0.00348
 2007.46164      1.59145  0.00344
 2007.46438      1.59347  0.00346
 2007.46712      1.59182  0.00342
 2007.46986      1.59151  0.00364
 2007.47260      1.59280  0.00340
 2007.47534      1.59258  0.00358
 2007.47808      1.59525  0.00354
 2007.48082      1.59364  0.00340
 2007.48356      1.59824  0.00366
 2007.48630      1.59303  0.00356
 2007.48904      1.59833  0.00365
 2007.49178      1.59615  0.00379
 2007.49452      1.59639  0.00371
 2007.49726      1.59489  0.00360
 2007.50000      1.59473  0.00365
 2007.50274      1.59629  0.00366
 2007.50548      1.59446  0.00350
 2007.50822      1.59816  0.00378
 2007.51096      1.59618  0.00363
 2007.51370      1.59688  0.00361
 2007.51644      1.59788  0.00362
 2007.51918      1.59485  0.00358
 2007.52192      1.59528  0.00359
 2007.52466      1.59667  0.00351
 2007.52740      1.59618  0.00370
 2007.53014      1.59611  0.00377
 2007.53287      1.59711  0.00364
 2007.53561      1.59686  0.00354
 2007.53835      1.59507  0.00351
 2007.54109      1.59617  0.00382
 2007.54383      1.59661  0.00354
 2007.54657      1.60108  0.00417
 2007.54931      1.59467  0.00377
 2007.55205      1.59855  0.00366
 2007.55479      1.59723  0.00360
 2007.55753      1.59821  0.00360
 2007.56027      1.60016  0.00378
 2007.56301      1.60052  0.00366
 2007.56575      1.59465  0.00428
 2007.56849      1.59962  0.00371
 2007.57123      1.60099  0.00365
 2007.57397      1.59855  0.00364
 2007.57671      1.60001  0.00361
 2007.57945      1.59913  0.00372
 2007.58219      1.59623  0.00371
 2007.58493      1.59975  0.00357
 2007.58767      1.59974  0.00356
 2007.59041      1.59990  0.00388
 2007.59315      1.60074  0.00379
 2007.59589      1.60285  0.00376
 2007.59863      1.60303  0.00370
 2007.60137      1.60045  0.00393
 2007.60411      1.60128  0.00364
 2007.60685      1.60081  0.00374
 2007.60959      1.59954  0.00369
 2007.61233      1.60369  0.00373
 2007.61507      1.59816  0.00376
 2007.61781      1.59998  0.00361
 2007.62055      1.60186  0.00354
 2007.62329      1.60193  0.00360
 2007.62603      1.60148  0.00339
 2007.62877      1.60392  0.00373
 2007.63150      1.60806  0.00400
 2007.63424      1.60722  0.00388
 2007.63698      1.60381  0.00362
 2007.63972      1.60487  0.00356
 2007.64246      1.60162  0.00377
 2007.64520      1.60445  0.00365
 2007.64794      1.60376  0.00375
 2007.65068      1.60256  0.00351
 2007.65342      1.60354  0.00346
 2007.65616      1.60254  0.00341
 2007.65890      1.60350  0.00363
 2007.66164      1.60344  0.00346
 2007.66438      1.60195  0.00339
 2007.66712      1.60318  0.00333
 2007.66986      1.60373  0.00338
 2007.67260      1.60335  0.00330
 2007.67534      1.60462  0.00357
 2007.67808      1.60298  0.00347
 2007.68082      1.60553  0.00344
 2007.68356      1.60626  0.00337
 2007.68630      1.60230  0.00382
 2007.68904      1.60275  0.00401
 2007.69178      1.60604  0.00372
 2007.69452      1.60355  0.00409
 2007.69726      1.60613  0.00451
 2007.70000      1.60517  0.00416
 2007.70274      1.60389  0.00401
 2007.70548      1.60265  0.00380
 2007.70822      1.60608  0.00399
 2007.71096      1.60367  0.00395
 2007.71370      1.60339  0.00425
 2007.72192      1.60874  0.00387
 2007.72740      1.60610  0.00389
 2007.73014      1.60665  0.00349
 2007.73287      1.60502  0.00348
 2007.73561      1.60480  0.00340
 2007.73835      1.60702  0.00357
 2007.74109      1.60748  0.00350
 2007.74383      1.60625  0.00382
 2007.74657      1.60606  0.00353
 2007.74931      1.60450  0.00374
 2007.75205      1.60780  0.00375
 2007.75479      1.60385  0.00339
 2007.75753      1.60631  0.00361
 2007.76027      1.60524  0.00336
 2007.76301      1.60562  0.00348
 2007.76575      1.60577  0.00363
 2007.76849      1.60590  0.00371
 2007.77123      1.60484  0.00334
 2007.77397      1.60533  0.00417
 2007.77671      1.60855  0.00380
 2007.77945      1.60537  0.00384
 2007.78219      1.60667  0.00335
 2007.78493      1.60488  0.00342
 2007.78767      1.60557  0.00333
 2007.79041      1.60803  0.00357
 2007.79315      1.60953  0.00413
 2007.79589      1.60780  0.00459
 2007.79863      1.60817  0.00379
 2007.80685      1.60808  0.00437
 2007.80959      1.60806  0.00420
 2007.81233      1.61256  0.00478
 2007.81507      1.60760  0.00382
 2007.81781      1.60678  0.00385
 2007.82055      1.60939  0.00378
 2007.82329      1.60686  0.00361
 2007.82603      1.60997  0.00410
 2007.82877      1.60885  0.00361
 2007.83150      1.60795  0.00381
 2007.83424      1.60850  0.00381
 2007.83698      1.60851  0.00377
 2007.83972      1.60936  0.00376
 2007.84246      1.60898  0.00341
 2007.84520      1.60831  0.00365
 2007.84794      1.60653  0.00417
 2007.85068      1.60618  0.00367
 2007.85342      1.61567  0.00473
 2007.85616      1.61283  0.00449
 2007.85890      1.61026  0.00351
 2007.86164      1.61067  0.00354
 2007.86438      1.60912  0.00344
 2007.86712      1.60767  0.00357
 2007.86986      1.60889  0.00330
 2007.87260      1.61001  0.00337
 2007.87534      1.60901  0.00344
 2007.87808      1.60869  0.00355
 2007.88082      1.61099  0.00355
 2007.88356      1.60973  0.00353
 2007.88630      1.60890  0.00336
 2007.88904      1.61131  0.00340
 2007.89178      1.60911  0.00339
 2007.89452      1.61097  0.00336
 2007.89726      1.60800  0.00335
 2007.90000      1.61080  0.00340
 2007.90274      1.61159  0.00343
 2007.90548      1.61050  0.00359
 2007.90822      1.61164  0.00339
 2007.91096      1.61018  0.00340
 2007.91370      1.61122  0.00349
 2007.91644      1.61151  0.00334
 2007.91918      1.61004  0.00345
 2007.92192      1.61044  0.00333
 2007.92466      1.60973  0.00357
 2007.92740      1.61039  0.00371
 2007.93014      1.61323  0.00358
 2007.93287      1.61033  0.00340
 2007.93561      1.61169  0.00342
 2007.93835      1.61039  0.00344
 2007.94109      1.60934  0.00348
 2007.94383      1.61211  0.00357
 2007.94657      1.61198  0.00342
 2007.94931      1.61118  0.00366
 2007.95205      1.61154  0.00383
 2007.95479      1.61057  0.00376
 2007.95753      1.61481  0.00353
 2007.96027      1.61429  0.00364
 2007.96301      1.61377  0.00366
 2007.96575      1.61414  0.00373
 2007.96849      1.62056  0.00420
 2007.97123      1.62199  0.00435
 2007.97397      1.62567  0.00425
 2007.97671      1.62579  0.00405
 2007.97945      1.62427  0.00413
 2007.98219      1.62607  0.00404
 2007.98493      1.62663  0.00393
 2007.98767      1.61194  0.00343
 2007.99041      1.61374  0.00342
 2007.99315      1.61521  0.00339
 2007.99589      1.61517  0.00346
 2007.99863      1.61565  0.00346
 2008.00136      1.61565  0.00339
 2008.00410      1.61551  0.00339
 2008.00683      1.61815  0.00346
 2008.00956      1.61741  0.00377
 2008.01229      1.61519  0.00363
 2008.01503      1.61738  0.00370
 2008.01776      1.61548  0.00355
 2008.02322      1.61909  0.00431
 2008.02595      1.62338  0.00450
 2008.02869      1.62386  0.00432
 2008.03142      1.62582  0.00428
 2008.03415      1.62383  0.00442
 2008.03962      1.62246  0.00435
 2008.04508      1.61848  0.00420
 2008.05054      1.62487  0.00453
 2008.05328      1.62261  0.00457
 2008.05601      1.62100  0.00473
 2008.05874      1.62344  0.00456
 2008.06147      1.62631  0.00428
 2008.06421      1.62949  0.00439
 2008.06694      1.62786  0.00415
 2008.06967      1.62955  0.00422
 2008.07240      1.62838  0.00417
 2008.07513      1.62530  0.00429
 2008.08060      1.62769  0.00423
 2008.08333      1.62998  0.00411
 2008.08606      1.62870  0.00418
 2008.08880      1.62762  0.00454
 2008.09153      1.63317  0.00459
 2008.09426      1.62738  0.00425
 2008.09699      1.62848  0.00455
 2008.09972      1.63007  0.00437
 2008.10246      1.62912  0.00410
 2008.10519      1.63193  0.00404

            

            
            
            

    

        

	
    

        
Best Answer
    

    

								

								
OK, I suspected but then didn't check, that was dumb of me..
So with that in mind:
    resid = y-yfit;
    % find the DFT of the residuals
    residDFT = fft(resid);
    Fs = 1;
    freq = 0:Fs/length(resid):Fs/2;
    residDFT = residDFT(1:floor(length(resid)/2)+1);
    plot(freq(1:100),abs(residDFT(1:100)))
You'll see the biggest peak at a frequency of 0.002 cycles/day, which is a period of 500 days. Looks to me like the next most prominent peak is at 0.0059 cycles/day which is a period of almost 170 days (169.89).
I think if you fit those two frequencies using the Fourier coefficients along with the linear model fitted above, you'll have a very good fit to the data.
						    
						
		
    
        

    

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

								

								
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?
						    
						
MATLAB: I think no one will be able to answer this………

								

								
Your question is ill-defined:
You have not presented enough info.
1. Is the middle column the noisy signal?
2. Is the noise supposed to be
 a. additive?
 b. zero-mean?
 c. constant variance?
 d. Gaussian distributed?
If so, how was the variance measured? In particular, how many points were used to estimate each tabulated value?
						    
						
        
        

            
            
            
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Answer 1

OK, I suspected but then didn't check, that was dumb of me..

So with that in mind:

    resid = y-yfit;
    % find the DFT of the residuals
    residDFT = fft(resid);
    Fs = 1;
    freq = 0:Fs/length(resid):Fs/2;
    residDFT = residDFT(1:floor(length(resid)/2)+1);
    plot(freq(1:100),abs(residDFT(1:100)))

You'll see the biggest peak at a frequency of 0.002 cycles/day, which is a period of 500 days. Looks to me like the next most prominent peak is at 0.0059 cycles/day which is a period of almost 170 days (169.89).

I think if you fit those two frequencies using the Fourier coefficients along with the linear model fitted above, you'll have a very good fit to the data.