How do I calculate the uncertainties in linearly interpolated values from a given tabulated function?
I am just coming back into the fold after a bit of a hiatus, and am having trouble jumpstarting my brain.
I have a tabulated function $(x_i, y_i, e_i)$, where $x_i$ is on an irregular grid and $e_i$ is the uncertainty in the value of the $y_i$. The datapoints a uncorrelated.
For various reasons, I need to carry out a linear interpolation to put these values onto a regular grid. I have no problems in calculating the interpolated values, but I can't seem to find a way to calculate the uncertainty in the interpolated value. The methods I remember fail to reproduce the tabulated uncertainties (if $x=x_i$), and seem generally clunky.
So, my question:
How do I calculate the uncertainties in linearly interpolated values from a given tabulated function?
Bonus: How to calculate covariances too…
(If you need it to illustrate your point) Some sample data:
xi yi ei x y e
1.0 7.0 2 1.0
1.9 8.1 3 2.0
3.1 9.0 4 --> 3.0
4.0 10.3 5 4.0
4.9 11.0 6 5.0
6.2 12.5 7 6.0
Edit: The real data and analysis:
I have x-ray diffraction patterns collected from the same sample with the detector in two (or more) positions, and as such, the irregular x grid for both datasets is different.
I want to add the two (or more) patterns together. The purpose of the addition is to obtain higher intensity, and thus, a higher confidence in the final intensities. The data will then be modelled by the Rietveld method.
In each dataset, there is ~20000 points, of which I've reproduced an indicative subset.
What would be the best method to put all datasets onto a common grid to enable the intensities to be added (taking into account the measurement uncertainties)?
Note:
Most (all) implementations of the Rietveld method assume that the intensities are uncorrelated, which is why I'd also like to be able to calculate the correlations that I'm introducing. At this point in time, I'm going to ignore intensity correlation in my analyses, but I would like to minimise it if possible.
x1 y1 e1 x2 y2 e2 x y e
26.699982 257.611938 16.819771 26.699972 206.448425 15.027238 26.7
26.703737 220.754547 15.610549 26.703726 231.364349 15.949701 26.70375
26.707493 222.360214 15.620609 26.707479 246.007446 16.460543 26.7075
26.711248 237.475983 16.173454 26.711233 228.180939 15.793293 26.71125
26.715004 240.830673 16.305054 26.714987 232.759888 16.082907 26.715
26.71876 223.493912 15.632176 26.718739 261.551666 17.050766 26.71875
26.722515 262.328674 17.005249 26.722492 225.026382 15.707023 26.7225
26.726271 228.291 15.732837 26.726246 231.585678 16.034586 26.72625
26.730026 248.281403 16.418236 26.73 241.725464 16.333258 26.73
26.733782 231.120026 15.864223 26.733753 231.883682 16.055712 26.73375
26.737537 239.188919 16.191217 26.737507 242.005142 16.454927 26.7375
26.741293 239.229172 15.99611 26.741261 255.124924 16.882158 26.74125
26.745049 202.675842 14.854559 26.745012 262.2164 17.161072 26.745
26.748804 239.148224 16.121639 26.748766 244.202606 16.573368 26.74875
26.752562 191.03067 14.338655 26.75252 254.840759 16.829659 26.7525
26.756317 236.928757 16.135286 26.756273 232.351868 16.019405 26.75625
26.760073 220.250137 15.574051 26.760027 255.123459 16.750168 26.76
26.763828 237.463058 16.071577 26.763781 232.629074 16.038975 26.76375
26.767584 226.210571 15.687851 26.767534 243.375717 16.346483 26.7675
26.771339 258.652649 16.857044 26.771288 225.64296 15.716341 26.77125
26.775095 254.923645 16.736721 26.775042 237.500946 16.209282 26.775
26.778852 278.621552 17.502522 26.778795 236.081726 16.144192 26.77875
26.782608 209.142868 15.011824 26.782549 247.611633 16.537083 26.7825
26.786364 258.206909 16.859592 26.786303 211.21933 15.235976 26.78625
26.790119 229.081314 15.720976 26.790056 246.538696 16.564161 26.79
26.793875 257.760193 16.797421 26.79381 234.845978 16.230515 26.79375
26.797632 233.179077 16.043262 26.797564 239.260361 16.297928 26.7975
26.801388 241.209442 16.297544 26.801317 262.272491 17.26512 26.80125
26.805143 240.669388 16.293869 26.805071 240.155869 16.292097 26.805
26.808899 287.645905 17.957613 26.808825 262.359009 16.943207 26.80875
26.812656 244.147873 16.433184 26.812578 259.774536 16.899658 26.8125
26.816412 255.581009 16.748405 26.816332 216.269989 15.321012 26.81625
26.820168 243.794861 16.375322 26.820087 238.458344 16.207962 26.82
26.823923 220.18042 15.498717 26.823841 216.359161 15.257962 26.82375
26.827681 238.572983 16.115116 26.827595 232.603851 16.037193 26.8275
26.831436 240.484207 16.147575 26.831348 264.862213 17.206285 26.83125
26.835192 261.019531 16.854525 26.835102 245.935654 16.422411 26.835
26.838949 232.71814 15.875105 26.838856 230.69754 16.041531 26.83875
26.842705 244.266739 16.374409 26.842609 245.167053 16.469751 26.8425
26.84646 270.787384 17.307497 26.846365 253.443481 16.735092 26.84625
26.850216 282.155518 17.636786 26.850119 277.531006 17.620699 26.85
26.853973 265.355927 17.173996 26.853872 259.03952 16.980762 26.85375
26.857729 257.106873 16.916601 26.857626 242.682373 16.365713 26.8575
26.861486 287.789215 17.998287 26.86138 255.16008 16.818199 26.86125
26.865242 261.124512 16.957371 26.865135 252.704788 16.718046 26.865
26.868998 257.830231 16.834381 26.868889 243.115891 16.262362 26.86875
26.872755 276.640381 17.562687 26.872643 253.842941 16.795219 26.8725
26.876511 262.03244 16.95348 26.876396 231.254578 15.941954 26.87625
26.880266 268.417633 17.215582 26.880152 234.510742 16.206791 26.88
26.884024 281.486755 17.655949 26.883905 272.454926 17.481216 26.88375
26.887779 235.749084 15.985992 26.887659 286.539307 17.886707 26.8875
26.891537 235.417145 15.996305 26.891413 249.974762 16.565758 26.89125
26.895292 242.421432 16.314156 26.895168 264.592438 17.155731 26.895
26.899048 263.415894 17.077515 26.898922 264.821533 17.236376 26.89875
26.902805 248.896179 16.558359 26.902676 287.547363 17.982763 26.9025
26.906561 254.223572 16.722179 26.906431 250.62146 16.74309 26.90625
26.910318 264.850677 17.205517 26.910185 252.432877 16.799511 26.91
26.914074 268.862457 17.212996 26.913939 288.753265 17.934757 26.91375
26.917831 284.015381 17.787331 26.917694 266.463898 17.182816 26.9175
26.921587 285.391663 17.782124 26.921448 262.388794 17.172642 26.92125
26.925344 256.088654 16.847933 26.925203 282.458008 17.656187 26.925
26.9291 278.193542 17.568636 26.928957 264.536438 17.119614 26.92875
26.932858 261.607147 16.925268 26.932711 273.392395 17.510523 26.9325
26.936613 293.501556 18.081835 26.936466 241.179169 16.363207 26.93625
26.940371 281.078217 17.598589 26.94022 296.551788 18.337276 26.94
26.944126 277.295685 17.448036 26.943975 264.610352 17.222281 26.94375
26.947884 275.656372 17.435543 26.947729 268.674225 17.29664 26.9475
26.951639 280.630981 17.632053 26.951485 267.253937 17.170872 26.95125
26.955397 310.859436 18.560068 26.955238 295.029663 18.20949 26.955
26.959152 309.720245 18.579161 26.958992 284.24057 17.89658 26.95875
26.96291 254.522202 16.709709 26.962748 319.454041 18.907776 26.9625
26.966667 328.244995 19.204264 26.966501 272.877869 17.509052 26.96625
26.970423 301.559174 18.193857 26.970257 292.79657 18.161213 26.97
26.97418 286.889801 17.815968 26.97401 311.167603 18.728683 26.97375
26.977936 340.261688 19.487011 26.977766 299.764618 18.262012 26.9775
26.981693 296.298218 18.075247 26.98152 344.325958 19.784254 26.98125
26.985451 301.822418 18.299801 26.985275 322.565887 19.127018 26.985
26.989206 304.588318 18.381588 26.989031 288.446136 18.008892 26.98875
26.992964 302.583008 18.287107 26.992785 292.30899 18.098988 26.9925
26.996719 309.682159 18.487858 26.99654 310.9328 18.74452 26.99625
27.000477 296.918793 18.144386 27.000294 290.236755 18.092058 27
27.004234 328.477325 19.130945 27.004049 341.49704 19.674849 27.00375
27.00799 305.255676 18.363239 27.007803 324.09375 19.100769 27.0075
27.011747 331.636841 19.262068 27.011559 349.302277 19.932892 27.01125
27.015505 298.813782 18.232843 27.015314 340.791534 19.633039 27.015
27.01926 377.82962 20.703222 27.019068 333.971771 19.401537 27.01875
27.023018 337.940399 19.406948 27.022823 321.540894 18.975531 27.0225
27.026775 315.944 18.782553 27.026579 351.45636 19.887695 27.02625
27.030531 329.575897 19.110327 27.030333 332.281525 19.417547 27.03
27.034288 395.013153 21.12104 27.034088 354.798248 20.02552 27.03375
27.038046 376.693756 20.584587 27.037844 355.299011 20.054609 27.0375
27.041801 377.918793 20.545406 27.041597 371.191742 20.603811 27.04125
27.045559 379.556 20.583424 27.045353 378.1474 20.748695 27.045
27.049316 353.320923 19.883623 27.049109 359.688446 20.224152 27.04875
27.053074 384.733154 20.791872 27.052862 367.917542 20.361286 27.0525
27.056829 381.299683 20.57424 27.056618 381.987885 20.719294 27.05625
27.060587 394.974365 21.1723 27.060373 396.737946 21.296721 27.06
27.064344 410.550293 21.496117 27.064127 367.995148 20.421124 27.06375
27.068102 398.530121 21.26141 27.067883 382.928436 20.908066 27.0675
27.071857 432.425446 22.100449 27.071638 421.565887 22.011845 27.07125
27.075615 467.759308 23.259808 27.075394 462.320862 23.155806 27.075
27.079372 455.897797 22.65077 27.079149 435.147186 22.45047 27.07875
27.08313 450.400574 22.790157 27.082903 429.493591 22.332462 27.0825
27.086887 452.224335 22.708344 27.086658 487.865631 23.960541 27.08625
27.090643 457.796967 22.896755 27.090414 466.144714 23.226973 27.09
27.0944 525.631104 24.670202 27.09417 475.891571 23.476278 27.09375
27.098158 503.192535 24.212145 27.097923 507.236694 24.462635 27.0975
27.101915 517.572388 24.4188 27.101679 488.699615 23.682343 27.10125
27.105673 496.762482 23.964514 27.105434 486.923187 23.739136 27.105
27.109428 532.518188 24.957394 27.10919 541.927673 25.270697 27.10875
27.113186 533.502869 24.934303 27.112946 509.530853 24.190039 27.1125
27.116943 628.999268 27.041285 27.116701 575.867737 26.024132 27.11625
27.120701 587.537964 26.205851 27.120457 573.836609 25.906281 27.12
27.124458 641.46051 27.382612 27.12421 622.094727 27.276428 27.12375
27.128216 598.144653 26.295834 27.127966 594.514221 26.573986 27.1275
27.131973 646.872864 27.515984 27.131721 682.880066 28.333942 27.13125
27.135731 685.346069 28.378246 27.135477 715.428162 29.170425 27.135
27.139486 705.060852 28.710655 27.139233 701.500671 29.095297 27.13875
27.143244 742.444946 29.799131 27.142988 743.038147 29.763405 27.1425
27.147001 739.384155 29.611042 27.146744 742.684509 29.910776 27.14625
27.150759 756.785889 30.014313 27.150499 813.66803 31.310173 27.15
27.154516 811.989807 31.013382 27.154255 857.058044 32.132923 27.15375
27.158274 862.963013 32.065449 27.15801 805.47345 31.155424 27.1575
27.162031 859.406433 32.093735 27.161766 875.511963 32.665203 27.16125
27.165789 895.645935 32.995667 27.165522 915.463867 33.07769 27.165
27.169546 969.672363 34.077866 27.169277 996.215515 34.755962 27.16875
27.173304 1032.02478 35.365906 27.173033 996.287354 34.484081 27.1725
27.177061 1101.285034 36.482513 27.176788 1097.823242 36.574898 27.17625
27.180819 1089.692505 36.372944 27.180544 1172.005981 38.15443 27.18
27.184576 1233.372925 39.165882 27.184299 1213.508423 38.770805 27.18375
27.188334 1286.500488 40.216381 27.188055 1316.095947 40.561798 27.1875
27.192091 1351.022949 41.322506 27.191811 1444.984009 42.750305 27.19125
27.195848 1410.998779 42.263809 27.195566 1474.066528 43.096645 27.195
27.199606 1605.868286 45.268463 27.199322 1485.535522 43.290718 27.19875
27.203363 1670.843506 46.438526 27.203077 1600.005127 44.899033 27.2025
27.207121 1852.570068 48.931038 27.206833 1712.454712 46.573402 27.20625
27.210878 1949.015503 50.506145 27.210588 1849.947876 48.792233 27.21
27.214636 1991.649902 51.212296 27.214344 2024.669678 51.460781 27.21375
27.218393 2165.216553 53.421818 27.2181 2088.17041 51.730793 27.2175
27.222151 2337.598877 55.876877 27.221855 2346.668457 55.977787 27.22125
27.225908 2473.485596 57.56147 27.225611 2412.149414 56.592995 27.225
27.229666 2797.586426 61.769291 27.229368 2696.092773 60.324032 27.22875
27.233423 2991.792969 64.599525 27.233124 2909.843506 63.37001 27.2325
27.237181 3118.806885 65.786011 27.236879 3074.491699 65.597015 27.23625
27.240938 3504.134521 70.3414 27.240635 3375.719727 68.693611 27.24
27.244696 3891.235352 75.364098 27.24439 3758.651611 73.219566 27.24375
27.248453 4244.597168 79.501091 27.248146 4025.467773 76.340263 27.2475
27.252211 4531.950195 82.043549 27.251902 4374.370605 80.199883 27.25125
27.255968 4912.705566 85.931183 27.255659 4829.439941 84.864685 27.255
27.259726 5583.304688 93.659035 27.259415 5185.368652 88.253494 27.25875
27.263483 5801.212402 94.912148 27.26317 5837.462402 95.271805 27.2625
27.267242 6527.44873 101.581688 27.266926 6164.637207 97.746201 27.26625
27.271 7149.898438 107.501686 27.270681 6996.065918 105.987755 27.27
27.274757 8024.799805 115.727562 27.274439 7717.926758 113.278625 27.27375
27.278515 8782.509766 122.771156 27.278194 8513.098633 120.068474 27.2775
27.282272 9776.30957 130.636856 27.28195 9582.835938 129.320023 27.28125
27.28603 10684.06055 138.121933 27.285706 10629.82422 138.458023 27.285
27.289787 12160.97461 150.254181 27.289463 11498.96484 145.095963 27.28875
27.293545 13230.85352 159.008743 27.293219 13194.12207 157.515976 27.2925
27.297304 14882.88086 170.719299 27.296974 14356.03418 166.620728 27.29625
27.301062 16311.61523 180.528198 27.30073 15988.86719 178.297455 27.3
27.304819 18470.22266 196.541138 27.304487 17430.61523 188.006866 27.30375
27.308577 18905.74023 199.37529 27.308243 18763.6582 196.852402 27.3075
27.312334 19843.95898 206.008362 27.311998 19107.41602 199.18042 27.31125
27.316092 19446.24805 202.544922 27.315756 19581.70898 202.482391 27.315
27.319851 18449.27148 195.371994 27.319511 18548.38477 196.500046 27.31875
27.323608 17099.03516 185.706924 27.323267 17395.10156 188.679016 27.3225
27.327366 15468.32129 174.745087 27.327023 15864.07031 177.236633 27.32625
27.331123 13734.60449 161.846298 27.33078 14618.6084 169.099823 27.33
27.334881 12631.53125 152.229172 27.334536 12794.08008 155.524338 27.33375
27.33864 11271.24707 142.581253 27.338293 11787.24023 148.055161 27.3375
27.342398 10512.77539 136.916977 27.342049 10371.50879 135.465714 27.34125
27.346155 9237.021484 126.318268 27.345804 9684.5 129.716415 27.345
27.349913 8405.818359 119.341942 27.349562 8767.294922 122.610458 27.34875
27.353672 7818.925781 113.856117 27.353317 8091.36377 115.909096 27.3525
27.35743 7115.508301 108.37809 27.357073 7147.862305 107.73645 27.35625
27.361187 6415.062012 100.436485 27.36083 6645.275391 103.097359 27.36
27.364944 5803.602051 95.109955 27.364586 5926.142578 95.61367 27.36375
27.368704 5287.605957 89.370728 27.368343 5482.560059 91.383064 27.3675
27.372461 4909.004395 85.33609 27.372099 4844.45459 85.257843 27.37125
27.376219 4367.385742 79.370476 27.375854 4488.047852 81.663086 27.375
27.379976 3993.688721 75.518738 27.379612 4184.442383 78.276047 27.37875
27.383736 3681.97876 72.029762 27.383368 3660.422119 72.288017 27.3825
27.387493 3425.320801 69.451126 27.387125 3500.459961 70.925537 27.38625
27.391251 3095.907471 65.610405 27.390881 3144.00415 66.149261 27.39
27.39501 2904.301758 63.035709 27.394638 2957.160156 64.107269 27.39375
27.398767 2597.830811 59.3172 27.398394 2628.188721 59.741573 27.3975
27.402525 2454.955566 57.157295 27.402151 2406.833252 56.985153 27.40125
27.406284 2210.20874 54.052696 27.405907 2295.345947 55.378689 27.405
27.410042 2074.173096 52.329163 27.409664 2135.306641 52.890289 27.40875
27.413799 1886.714233 49.559029 27.41342 1919.739868 50.086132 27.4125
27.417559 1770.544434 47.881283 27.417177 1816.12146 48.338406 27.41625
27.421316 1557.283813 44.360882 27.420933 1684.497192 46.274689 27.42
27.425074 1503.644043 43.386501 27.42469 1542.882446 44.166553 27.42375
27.428833 1377.763306 41.332214 27.428446 1567.353882 44.635925 27.4275
27.43259 1353.757812 40.70454 27.432203 1396.887451 41.758747 27.43125
27.436348 1170.140991 37.721062 27.435959 1336.477661 40.869045 27.435
27.440107 1180.836548 38.067421 27.439716 1239.929199 39.100754 27.43875
27.443865 1147.78894 37.471424 27.443474 1160.866455 37.903599 27.4425
27.447622 1058.838013 35.651024 27.447229 1054.965942 35.629913 27.44625
27.451382 1044.922241 35.50663 27.450987 1013.544983 35.041023 27.45
27.455139 981.188721 34.571899 27.454742 927.535461 33.372013 27.45375
27.458899 875.704956 32.246819 27.4585 905.684937 32.86861 27.4575
27.462656 865.433899 32.310036 27.462257 917.381287 33.185966 27.46125
27.466413 885.555725 32.773746 27.466013 823.479797 31.281075 27.465
27.470173 834.432922 31.544697 27.46977 781.178345 30.347902 27.46875
27.47393 731.929443 29.260817 27.473526 759.411682 29.847685 27.4725
27.47769 698.259766 28.603771 27.477283 708.03717 28.836737 27.47625
27.481447 687.848328 28.403528 27.481041 705.581177 28.773079 27.48
27.485207 678.07135 28.106058 27.484797 652.00238 27.51049 27.48375
27.488964 610.292725 26.522755 27.488554 651.322815 27.606476 27.4875
27.492722 631.037537 27.175158 27.492311 624.520813 26.990871 27.49125
27.496481 523.445618 24.470995 27.496067 583.280762 26.098219 27.495
27.500238 582.989319 25.97337 27.499825 522.271545 24.460474 27.49875
27.503998 552.380798 25.282471 27.503582 555.14447 25.34499 27.5025
27.507755 547.555786 25.190542 27.507338 488.443329 23.549656 27.50625
27.511515 517.328491 24.430441 27.511095 532.32251 24.760672 27.51
27.515272 575.333679 25.930967 27.514853 497.951385 23.879543 27.51375
27.519032 512.204834 24.226938 27.518608 512.114502 24.152409 27.5175
27.522789 495.360992 23.918774 27.522366 456.453064 22.703745 27.52125
27.526548 466.525665 23.121805 27.526123 510.131897 24.172394 27.525
27.530306 457.033813 22.80759 27.529881 466.042755 23.023245 27.52875
27.534065 459.896362 22.758112 27.533636 419.527649 21.644688 27.5325
27.537823 461.733063 22.827629 27.537394 447.970917 22.637915 27.53625
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28.003355 236.759277 16.157446 28.00125
Best Answer
Typically the uncertainties from counting measurements are made using Poisson statistics. The standard deviation of such a distribution is just the square root of the counts. I am not sure why your uncertainties are slightly off from this.
Why do you need to interpolate the data at all? I am not too familiar with Reitveld refinement, but why don't you just merge the datasets? You gain no statistical power by adding together two data points. For example, even if you had two independent measurements at the same x value, there is nothing gained by averaging the counts other than making prettier plots of the diffraction data.
Finally, are the x positions actually different? I assume you ran the same scan twice. The slight differences in the x values in the data are either from the instrument having higher accuracy in measuring position than in its motors or it is just a result in always writing out the data to a precision higher than merited. Either way, treating the x-values as the same introduces very little affect on your uncertainties. If you treat the range of x-values as the uncertainty in the x-position and estimate this contribution to uncertainty in y by looking at the local slope, you'll see that it is ~3 orders of magnitude lower than the counting uncertainty.
Were I do this, I would merge the data and just have twice the data points. If for some reason you cannot do this, I would just take the average x-value as doing so introduces negligible error.