Fitting a correlation function to my data

Question

I have some spatial data and I want to fit a correlation function may be exponential or gaussian to my data. I could calculate the correlation for different pairs of my spatial point to get some empirical correlation function. What is the best way to fit some standard correlation functions to my empirical one.

I tried to use the gstat package to estimate the variogram. However, I had some issues. I generated some synthetic data with the given correlation matrix. I mean I generated some samples from a multivariate gaussian distribution with the given correlation matrix.

I am assuming it to be homogeneous. My first row of correlation matrix is something like

1   0.683429646106259   0.437789623184770   0.262915008717623   0.196106083531309   0.149045509783630   0.150509368156993   0.0707581842359845  0.0946249108754618  0.0778169918490679

So it kind of decreases with distance. The range is somewhere around at distance 10 when the correlation reaches approx 0.05.

Now when I tried to use the gstat package. I had to use the following data.

"x" "y" "data"
1 1 0.44168
1 2 0.099929
1 3 0.026909
1 4 -0.041947
1 5 -0.24288
1 6 0.78454
1 7 0.16761
1 8 -0.086277
1 9 0.023661
1 10 -0.26036
2 1 0.71725
2 2 -0.14143
2 3 -1.0092
2 4 -0.28496
2 5 -0.17743
2 6 0.22874
2 7 -0.94645
2 8 -0.14165
2 9 0.23027
2 10 0.50463
3 1 -1.1799
3 2 0.31035
3 3 0.83902
3 4 0.057151
3 5 -0.77414
3 6 -0.57698
3 7 0.034864
3 8 0.37105
3 9 -0.017573
3 10 -0.80297
4 1 -1.1183
4 2 0.04495
4 3 -0.43193
4 4 -0.32199
4 5 -0.30001
4 6 0.032265
4 7 0.76654
4 8 0.047539
4 9 -0.70933
4 10 -0.39096
5 1 0.8237
5 2 0.48326
5 3 -0.18275
5 4 -0.2011
5 5 -0.45547
5 6 0.056931
5 7 0.38949
5 8 -0.29351
5 9 -0.82567
5 10 -0.6183
6 1 1.2794
6 2 0.55406
6 3 1.0454
6 4 -0.80348
6 5 0.10936
6 6 -1.5121
6 7 0.86288
6 8 0.18994
6 9 -0.019555
6 10 -0.68697
7 1 1.7539
7 2 1.2916
7 3 0.41383
7 4 -0.24399
7 5 -0.023723
7 6 -1.0232
7 7 0.92783
7 8 0.64965
7 9 0.95411
7 10 0.19481
8 1 0.67385
8 2 1.1123
8 3 0.96762
8 4 0.69418
8 5 -0.025346
8 6 -0.69347
8 7 0.78425
8 8 1.082
8 9 1.1929
8 10 1.3393
9 1 0.87725
9 2 0.30799
9 3 -0.14401
9 4 -0.24109
9 5 0.3923
9 6 1.1385
9 7 1.1368
9 8 0.13259
9 9 0.86029
9 10 0.62073
10 1 -0.043159
10 2 0.13516
10 3 0.44809
10 4 0.1643
10 5 0.66301
10 6 1.9096
10 7 1.1722
10 8 1.1109
10 9 1.1154
10 10 1.0472

This is from just one sample. I thought of providing it more samples but I had to repeat the coordinates and it gave me rubbish variogram like this

However, I tried with just one sample and this is what I got for fitting a model variogram

  model      psill    range
1   Nug 0.06875421 0.000000
2   Exp 0.48353408 1.931677

I definitely cannot take these params for correlogram. In the case of correlation function, range is the distance at which the correlation is 0.05. If I look at my actual correlation function its approx at distance 10. So it doesn't match at all. Is it something wrong with the way I calculated the variogram or something else. I followed the tutorial here.

I am not sure but I think I am not supposed to get the nugget either. I added no noise when I sampled from the gaussian. Is it because I have less samples?

When I try to use all the samples, I get something like this as my empirical variogram which is not correct.

Here is my data

"x" "y" "aod"
1 1 -0.75488
1 2 -0.78846
1 3 -0.96701
1 4 0.21699
1 5 0.52046
1 6 0.28913
1 7 0.12993
1 8 0.37072
1 9 0.58311
1 10 -0.16038
2 1 -0.2117
2 2 -1.2046
2 3 -1.0581
2 4 -0.67419
2 5 1.0385
2 6 1.1686
2 7 0.14002
2 8 -0.16956
2 9 0.25615
2 10 0.21616
3 1 1.0365
3 2 -1.3079
3 3 -0.51374
3 4 -0.35841
3 5 0.25237
3 6 0.97418
3 7 -0.10198
3 8 -0.21238
3 9 -0.39775
3 10 -0.93731
4 1 0.50214
4 2 -0.21804
4 3 -0.63425
4 4 0.68123
4 5 1.695
4 6 1.6203
4 7 -0.0013243
4 8 0.15842
4 9 0.29372
4 10 -1.7714
5 1 0.66179
5 2 -1.0807
5 3 -0.65248
5 4 0.15395
5 5 0.30281
5 6 -0.075867
5 7 0.31152
5 8 -1.2481
5 9 -0.8156
5 10 -1.3282
6 1 0.20887
6 2 -0.316
6 3 0.62447
6 4 -0.073839
6 5 -0.34457
6 6 -0.047924
6 7 0.044455
6 8 0.40695
6 9 -0.030344
6 10 -0.078803
7 1 -0.43136
7 2 0.46374
7 3 0.76833
7 4 1.1933
7 5 -0.72779
7 6 0.68372
7 7 0.3142
7 8 -0.78668
7 9 -0.41529
7 10 -1.1691
8 1 0.12466
8 2 0.26433
8 3 0.68538
8 4 0.55875
8 5 0.52583
8 6 -0.127
8 7 0.050293
8 8 -0.18494
8 9 0.25105
8 10 -0.84661
9 1 -0.22072
9 2 -0.11841
9 3 1.0484
9 4 1.2102
9 5 0.77315
9 6 0.76342
9 7 0.45916
9 8 0.0039676
9 9 0.014329
9 10 -0.64462
10 1 0.80869
10 2 -0.35877
10 3 0.73614
10 4 -0.4413
10 5 0.73746
10 6 0.31207
10 7 0.14122
10 8 1.3785
10 9 -0.89216
10 10 0.095206
1 1 1.2769
1 2 1.0405
1 3 -0.0031401
1 4 0.047758
1 5 -0.74042
1 6 -0.69732
1 7 -0.041547
1 8 -0.22316
1 9 0.066819
1 10 -0.010506
2 1 0.96758
2 2 0.31139
2 3 -0.17409
2 4 0.25964
2 5 -0.85022
2 6 -0.71892
2 7 -0.75265
2 8 -1.1433
2 9 -1.4629
2 10 -1.5973
3 1 0.092515
3 2 -0.22445
3 3 0.15134
3 4 -0.46512
3 5 -0.06476
3 6 -0.23128
3 7 -1.3388
3 8 -0.97593
3 9 -1.448
3 10 -0.43345
4 1 0.61175
4 2 0.61255
4 3 -0.10015
4 4 0.33106
4 5 0.34014
4 6 -0.23341
4 7 -0.86573
4 8 -1.2642
4 9 -0.62796
4 10 -1.1511
5 1 0.31983
5 2 0.53389
5 3 -0.76606
5 4 -0.73465
5 5 0.84788
5 6 0.47175
5 7 0.53173
5 8 -0.11972
5 9 0.21588
5 10 -0.46452
6 1 -0.7283
6 2 -0.70479
6 3 -1.1243
6 4 -0.15782
6 5 0.17425
6 6 -0.26241
6 7 1.1181
6 8 -0.15918
6 9 -0.69752
6 10 -0.68888
7 1 -0.56861
7 2 -1.2355
7 3 -0.6948
7 4 0.58936
7 5 0.056865
7 6 0.64748
7 7 -0.58011
7 8 0.070476
7 9 -0.92674
7 10 0.24241
8 1 -0.46411
8 2 -0.44081
8 3 0.54728
8 4 1.5323
8 5 -0.05007
8 6 0.98995
8 7 -0.183
8 8 -0.91211
8 9 0.00081732
8 10 -0.18632
9 1 -0.032806
9 2 0.52061
9 3 1.4889
9 4 0.56336
9 5 -0.18504
9 6 -0.79679
9 7 0.089379
9 8 0.069477
9 9 -0.224
9 10 0.35521
10 1 -0.89194
10 2 0.13515
10 3 0.6474
10 4 0.88063
10 5 -0.68164
10 6 -0.075601
10 7 -0.63008
10 8 0.168
10 9 -0.84589
10 10 0.46715
1 1 0.98109
1 2 1.5317
1 3 0.46533
1 4 0.12393
1 5 -1.5359
1 6 -0.68207
1 7 -1.6462
1 8 -1.4625
1 9 -1.9343
1 10 -2.3259
2 1 -0.30806
2 2 0.15317
2 3 -0.12825
2 4 0.43423
2 5 -0.71407
2 6 -0.11921
2 7 0.019226
2 8 0.35049
2 9 -1.4383
2 10 -1.5044
3 1 -0.12857
3 2 0.488
3 3 0.53606
3 4 1.0531
3 5 -0.29948
3 6 0.23948
3 7 -0.13268
3 8 -0.24308
3 9 -0.95118
3 10 -0.78948
4 1 -0.11342
4 2 -1.0059
4 3 -0.42124
4 4 -0.34023
4 5 0.078659
4 6 0.88902
4 7 0.23341
4 8 -0.15338
4 9 -0.59989
4 10 -0.89918
5 1 0.16374
5 2 -0.26299
5 3 0.77992
5 4 1.0819
5 5 0.7238
5 6 -0.42438
5 7 -0.19448
5 8 0.08083
5 9 0.48362
5 10 -0.11019
6 1 0.4394
6 2 0.040646
6 3 0.10204
6 4 0.23335
6 5 0.85101
6 6 -0.15969
6 7 -0.11772
6 8 1.4047
6 9 1.591
6 10 0.91408
7 1 0.15331
7 2 -0.61206
7 3 -0.47657
7 4 0.71106
7 5 0.0081887
7 6 -0.040958
7 7 1.0739
7 8 1.175
7 9 1.2231
7 10 0.42095
8 1 0.63768
8 2 0.22313
8 3 0.10201
8 4 0.91221
8 5 -0.2826
8 6 0.53168
8 7 0.043632
8 8 0.12458
8 9 1.1582
8 10 1.036
9 1 -0.83108
9 2 0.31008
9 3 -0.47923
9 4 -0.39486
9 5 0.51966
9 6 0.024501
9 7 0.5772
9 8 -0.18336
9 9 0.51455
9 10 0.99499
10 1 -0.40089
10 2 0.32306
10 3 -0.3433
10 4 0.0047221
10 5 -0.21988
10 6 -0.87257
10 7 -0.22983
10 8 -1.0848
10 9 0.23949
10 10 -0.53886
1 1 -0.89999
1 2 -0.60765
1 3 -1.3965
1 4 -1.1678
1 5 0.50079
1 6 0.7848
1 7 0.31866
1 8 0.33285
1 9 -0.057325
1 10 0.91442
2 1 -1.6385
2 2 -0.66963
2 3 -1.1245
2 4 -0.14442
2 5 0.98717
2 6 0.17885
2 7 0.077735
2 8 -0.22889
2 9 -0.38144
2 10 -0.07872
3 1 -0.77031
3 2 -0.24496
3 3 -0.76457
3 4 -0.74676
3 5 -0.50832
3 6 0.78217
3 7 0.62673
3 8 0.39
3 9 0.065218
3 10 0.488
4 1 0.73499
4 2 -0.22049
4 3 -0.59947
4 4 1.237
4 5 0.95703
4 6 -0.024532
4 7 0.86404
4 8 0.44883
4 9 0.1068
4 10 0.35538
5 1 -0.39373
5 2 0.72013
5 3 0.95173
5 4 0.5653
5 5 1.0025
5 6 -0.1623
5 7 -0.33946
5 8 0.77085
5 9 -0.34686
5 10 0.84909
6 1 0.45291
6 2 1.6368
6 3 0.49003
6 4 1.044
6 5 1.1733
6 6 0.96909
6 7 1.17
6 8 0.789
6 9 0.95848
6 10 0.6586
7 1 0.92626
7 2 1.336
7 3 1.0835
7 4 0.27797
7 5 0.96565
7 6 0.67386
7 7 0.13545
7 8 0.33533
7 9 1.9113
7 10 0.72334
8 1 0.62559
8 2 0.95157
8 3 1.2696
8 4 1.01
8 5 0.75892
8 6 1.1327
8 7 1.0734
8 8 1.5875
8 9 0.27577
8 10 0.52487
9 1 1.3763
9 2 1.1715
9 3 0.33185
9 4 0.7078
9 5 1.7302
9 6 1.6211
9 7 1.2113
9 8 0.28632
9 9 1.4362
9 10 0.05508
10 1 1.6734
10 2 2.0098
10 3 1.3373
10 4 1.9932
10 5 1.6035
10 6 2.3012
10 7 1.4291
10 8 1.1221
10 9 0.66535
10 10 0.92711
1 1 -0.99997
1 2 0.022719
1 3 -0.31203
1 4 0.18867
1 5 0.92715
1 6 1.1044
1 7 0.76145
1 8 -0.17697
1 9 0.25073
1 10 0.31083
2 1 -0.66634
2 2 -1.0159
2 3 -0.72899
2 4 0.32125
2 5 0.54591
2 6 1.1202
2 7 0.64981
2 8 0.03742
2 9 0.77014
2 10 0.012899
3 1 -1.2878
3 2 -1.0884
3 3 -1.0713
3 4 0.22714
3 5 0.36626
3 6 0.20953
3 7 0.14857
3 8 1.3116
3 9 1.2593
3 10 0.65079
4 1 -0.80787
4 2 -0.048305
4 3 -0.87667
4 4 -0.24298
4 5 0.055799
4 6 0.46683
4 7 -0.48911
4 8 -1.1424
4 9 -0.38136
4 10 -1.1061
5 1 -0.27704
5 2 -0.13177
5 3 -0.53845
5 4 -1.1699
5 5 -0.32735
5 6 -0.42001
5 7 0.19617
5 8 0.42689
5 9 -0.13028
5 10 -0.19488
6 1 -0.38319
6 2 0.1336
6 3 -0.49355
6 4 0.14656
6 5 -0.80513
6 6 0.65367
6 7 0.2069
6 8 0.13485
6 9 -0.26875
6 10 0.34008
7 1 -0.62224
7 2 -0.71156
7 3 0.40706
7 4 -0.51364
7 5 -0.045004
7 6 -1.1103
7 7 -1.0858
7 8 0.11086
7 9 -0.97742
7 10 -0.014179
8 1 -1.631
8 2 0.25644
8 3 -0.07203
8 4 0.019954
8 5 -0.78594
8 6 -1.2284
8 7 -0.59174
8 8 -0.29216
8 9 -0.087709
8 10 0.033502
9 1 -1.2977
9 2 -0.18927
9 3 0.67041
9 4 -1.6209
9 5 -0.28613
9 6 -0.025997
9 7 -0.30618
9 8 -1.2943
9 9 -1.587
9 10 -1.9803
10 1 0.46204
10 2 0.074855
10 3 -0.2965
10 4 -0.14154
10 5 -1.1861
10 6 -0.47409
10 7 0.46038
10 8 -0.34689
10 9 -0.80103
10 10 -1.8803
1 1 1.3411
1 2 1.4873
1 3 0.94437
1 4 1.7851
1 5 0.080725
1 6 0.41006
1 7 0.41028
1 8 -0.12254
1 9 0.26331
1 10 -0.42256
2 1 1.2357
2 2 0.32614
2 3 -0.163
2 4 0.98255
2 5 0.72907
2 6 0.31425
2 7 0.45023
2 8 0.26328
2 9 0.029089
2 10 0.95292
3 1 1.073
3 2 1.3613
3 3 0.1275
3 4 0.2476
3 5 0.2768
3 6 0.10945
3 7 0.097306
3 8 -0.39726
3 9 0.33722
3 10 0.061228
4 1 0.48769
4 2 0.78546
4 3 0.84834
4 4 0.97745
4 5 0.42715
4 6 0.077498
4 7 0.01148
4 8 -0.37943
4 9 -1.2956
4 10 0.23022
5 1 1.1574
5 2 0.71203
5 3 1.0309
5 4 1.9035
5 5 0.66869
5 6 0.46976
5 7 -0.31242
5 8 -1.0098
5 9 -0.16017
5 10 -0.055089
6 1 -1.4935
6 2 0.0068701
6 3 0.76397
6 4 0.13412
6 5 -0.36506
6 6 -0.093885
6 7 -0.46345
6 8 0.45901
6 9 -0.3404
6 10 -0.45926
7 1 -0.86785
7 2 -0.1634
7 3 0.043536
7 4 0.62364
7 5 -0.38783
7 6 1.2379
7 7 0.36635
7 8 -0.27407
7 9 0.53955
7 10 0.26264
8 1 -0.49472
8 2 0.33707
8 3 0.34908
8 4 0.39176
8 5 0.49744
8 6 -0.024764
8 7 0.13152
8 8 0.83401
8 9 0.3669
8 10 0.65591
9 1 0.26119
9 2 0.96254
9 3 0.53369
9 4 -0.38361
9 5 -0.31952
9 6 -0.032861
9 7 0.42385
9 8 0.60828
9 9 0.94206
9 10 0.65124
10 1 0.58401
10 2 0.28137
10 3 0.75139
10 4 0.66865
10 5 -0.61473
10 6 0.14
10 7 -0.44916
10 8 -0.41535
10 9 0.072928
10 10 -0.34029
1 1 -1.5582
1 2 -1.9252
1 3 -1.3097
1 4 -2.6259
1 5 -0.86581
1 6 -1.8963
1 7 0.58395
1 8 0.48188
1 9 2.882
1 10 3.7548
2 1 -1.2683
2 2 -1.3114
2 3 -1.1344
2 4 -1.2372
2 5 -1.4385
2 6 -0.38154
2 7 0.21844
2 8 1.2685
2 9 2.0933
2 10 1.6789
3 1 -0.67296
3 2 -0.46436
3 3 -1.1013
3 4 -0.75914
3 5 -0.49573
3 6 -0.071953
3 7 -0.36423
3 8 0.2675
3 9 1.2086
3 10 1.4457
4 1 0.59226
4 2 0.35089
4 3 1.0304
4 4 0.91696
4 5 0.93319
4 6 0.2622
4 7 -0.46816
4 8 0.87504
4 9 0.73656
4 10 0.89449
5 1 0.74513
5 2 0.73003
5 3 1.0218
5 4 0.74639
5 5 0.70416
5 6 -0.29666
5 7 -0.74854
5 8 0.78541
5 9 0.42556
5 10 1.0588
6 1 0.019868
6 2 0.17096
6 3 0.31712
6 4 0.37095
6 5 0.51432
6 6 -0.67542
6 7 -1.0559
6 8 -0.27615
6 9 0.63227
6 10 1.8216
7 1 0.11227
7 2 1.2998
7 3 1.361
7 4 0.84188
7 5 -0.20492
7 6 -0.32484
7 7 -0.58076
7 8 -0.16101
7 9 0.80934
7 10 0.35618
8 1 -0.59692
8 2 1.0501
8 3 0.42325
8 4 -0.78226
8 5 -0.0093265
8 6 0.49451
8 7 -0.57139
8 8 -0.71568
8 9 -1.0949
8 10 -1.322
9 1 -0.3825
9 2 -0.43926
9 3 -0.45886
9 4 0.18823
9 5 0.36943
9 6 -0.01429
9 7 -0.9238
9 8 -0.81771
9 9 -0.58659
9 10 -0.56081
10 1 -1.4616
10 2 -0.83716
10 3 0.83614
10 4 0.28808
10 5 -0.12755
10 6 -1.0188
10 7 -1.2117
10 8 -0.21896
10 9 -1.3982
10 10 -1.9128
1 1 1.5017
1 2 1.1149
1 3 1.5846
1 4 0.49772
1 5 1.6667
1 6 -0.1268
1 7 0.12223
1 8 -0.17968
1 9 0.41108
1 10 1.5414
2 1 0.46015
2 2 -0.18267
2 3 0.18842
2 4 -0.073743
2 5 0.89561
2 6 -0.18944
2 7 0.094503
2 8 0.11414
2 9 0.75025
2 10 0.64978
3 1 0.020306
3 2 -0.25737
3 3 -1.135
3 4 -0.35354
3 5 -0.001421
3 6 -0.12285
3 7 -0.61034
3 8 -0.25232
3 9 0.005621
3 10 -1.3972
4 1 -0.76662
4 2 -0.49325
4 3 -0.76528
4 4 -2.3272
4 5 -0.32207
4 6 0.08859
4 7 0.31069
4 8 0.75689
4 9 -0.36381
4 10 0.10409
5 1 -0.80493
5 2 -0.90191
5 3 -0.64442
5 4 -1.4952
5 5 -0.32429
5 6 -0.017915
5 7 -0.30078
5 8 -0.090466
5 9 1.2656
5 10 0.74413
6 1 0.16902
6 2 0.052467
6 3 -0.23645
6 4 -0.19061
6 5 0.15529
6 6 -0.57483
6 7 -1.1003
6 8 -0.47875
6 9 0.17602
6 10 -0.70837
7 1 0.092632
7 2 -0.73462
7 3 -0.3454
7 4 -0.82274
7 5 -0.51007
7 6 -0.51348
7 7 -1.7752
7 8 -0.58945
7 9 0.30645
7 10 -0.50629
8 1 -0.15482
8 2 -0.4444
8 3 0.1943
8 4 -0.94296
8 5 -0.67503
8 6 -0.7453
8 7 -1.4938
8 8 -0.80872
8 9 -0.052597
8 10 -1.0797
9 1 -0.76072
9 2 0.034279
9 3 0.035288
9 4 -0.47851
9 5 -0.9365
9 6 -0.56425
9 7 -0.75692
9 8 -0.72401
9 9 -0.52698
9 10 -1.2094
10 1 -1.2616
10 2 -0.85022
10 3 -1.2871
10 4 0.61099
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Answer 1

Indeed the empirical correlations you mentioned will be the 'raw data' to which a parametric correlation function is fit. In spatial statistics this is referred to as the 'variogram'. As you suggest, you are able to: 1) pair the data; 2) put the pairs into distance bins; and 3) calculate correlations at each distance. This gives you empirical correlations as a function of distance - i.e. the empirical variogram. Now you have a scatterplot of points to which you can fit a Gaussian or exponential curve. It is common to try a few and see which one minimizes the error or the AIC/BIC, etc. See a basic example here which uses the R packages sp and gstat. In these packages the pairing/binning is handled for you.