What function fits my data...?

Hello, I am a bit unsure if this is the correct place to ask my question, but I'll try anyways:

I am doing laser triangulation, and I recorded the position of a laser point on the camera CMOS chip vs. a displacement.

I have a series of measurement points. y is the distance in mm from the camera (with an unknown offset), and x is the position of the laser point on the camera sensor (in pixels).

y=[0	5.97623500000000	11.9524700000000	17.9287050000000	23.9049400000000	29.8811750000000	35.8574100000000	41.8336450000000	47.8098800000000	53.7861150000000	59.7623500000000	65.7385850000000	71.7148200000000	77.6910550000000	83.6672900000000	89.6435250000000	95.6197600000000	101.595995000000	107.572230000000	113.548465000000	119.524700000000	125.500935000000	131.477170000000	137.453405000000	143.429640000000	149.405875000000	155.382110000000	161.358345000000	167.334580000000	173.310815000000	179.287050000000	185.263285000000	191.239520000000	197.215755000000	203.191990000000	209.168225000000	215.144460000000	221.120695000000	227.096930000000	233.073165000000	239.049400000000	245.025635000000	251.001870000000	256.978105000000	262.954340000000	268.930575000000	274.906810000000	280.883045000000	286.859280000000	292.835515000000	298.811750000000	304.787985000000	310.764220000000	316.740455000000	322.716690000000	328.692925000000	334.669160000000	340.645395000000	346.621630000000	352.597865000000	358.574100000000	364.550335000000	370.526570000000	376.502805000000	382.479040000000	388.455275000000	394.431510000000	400.407745000000	406.383980000000	412.360215000000	418.336450000000	424.312685000000	430.288920000000	436.265155000000	442.241390000000	448.217625000000	454.193860000000	460.170095000000	466.146330000000	472.122565000000	478.098800000000	484.075035000000	490.051270000000	496.027505000000	502.003740000000	507.979975000000	513.956210000000	519.932445000000	525.908680000000	531.884915000000	537.861150000000	543.837385000000	549.813620000000	555.789855000000	561.766090000000	567.742325000000	573.718560000000	579.694795000000	585.671030000000	591.647265000000	597.623500000000	603.599735000000	609.575970000000	615.552205000000	621.528440000000	627.504675000000	633.480910000000	639.457145000000	645.433380000000	651.409615000000	657.385850000000	663.362085000000	669.338320000000	675.314555000000	681.290790000000	687.267025000000	693.243260000000	699.219495000000	705.195730000000	711.171965000000	717.148200000000	723.124435000000	729.100670000000	735.076905000000	741.053140000000	747.029375000000	753.005610000000	758.981845000000	764.958080000000	770.934315000000	776.910550000000	782.886785000000	788.863020000000	794.839255000000	800.815490000000	806.791725000000	812.767960000000	818.744195000000	824.720430000000	830.696665000000	836.672900000000	842.649135000000	848.625370000000	854.601605000000	860.577840000000	866.554075000000	872.530310000000	878.506545000000	884.482780000000	890.459015000000	896.435250000000	902.411485000000	908.387720000000	914.363955000000	920.340190000000	926.316425000000	932.292660000000	938.268895000000	944.245130000000	950.221365000000	956.197600000000	962.173835000000	968.150070000000	974.126305000000	980.102540000000	986.078775000000	992.055010000000	998.031245000000	1004.00748000000	1009.98371500000	1015.95995000000	1021.93618500000	1027.91242000000	1033.88865500000	1039.86489000000	1045.84112500000	1051.81736000000	1057.79359500000	1063.76983000000	1069.74606500000	1075.72230000000	1081.69853500000	1087.67477000000	1093.65100500000	1099.62724000000	1105.60347500000	1111.57971000000	1117.55594500000	1123.53218000000	1129.50841500000	1135.48465000000	1141.46088500000	1147.43712000000	1153.41335500000	1159.38959000000	1165.36582500000	1171.34206000000	1177.31829500000	1183.29453000000	1189.27076500000	1195.24700000000];
x=[599.879427792916	669.403805496829	734.507982583454	793.957914338920	852.338173481085	906.812156862745	962.431716265797	1014.30095873281	1064.53915662651	1114.12381382738	1159.26240808824	1202.22931442080	1244.72289742343	1284.70789865872	1322.19885177453	1361.76739245884	1399.00976138829	1434.70044543430	1468.93040501997	1502.48020954598	1535.47037484885	1566.52059004302	1596.13613550816	1624.76059050064	1652.78768020970	1680.14513981358	1705.64774281806	1731.69461914745	1757.35486160397	1782.05118961788	1805.83234859675	1828.90676691729	1851.14908256881	1873.04921875000	1894.24110671937	1915.63857374392	1935.67927631579	1955.28571428571	1974.13945578231	1992.89205526770	2010.82620087336	2028.57193605684	2046.45855855856	2063.26118721461	2079.57953488372	2095.16353383459	2111.10516252390	2126.29990262902	2141.41878669276	2156.40556660040	2171.01210898083	2185.42959183673	2199.31714876033	2212.54214360042	2225.86933614331	2238.86552828175	2251.58817204301	2264.08487486398	2276.11378555799	2288.05016722408	2300.15919282511	2311.73356009070	2322.75834292290	2334.04608294931	2344.63647058824	2355.32621589561	2365.45748502994	2375.81961259080	2385.68466257669	2395.58478802993	2405.51194968553	2415.19211195929	2424.50709677419	2433.85039370079	2442.55978975033	2451.94101876676	2460.32479784367	2469.04377564980	2477.02200825310	2485.08495821727	2493.23809523810	2501.61276595745	2509.21325648415	2516.91159420290	2524.55964653903	2532.43387815750	2539.51274362819	2547.00455235205	2554.03703703704	2561.36939721793	2568.30685358255	2575.32176656151	2581.71860095390	2588.80317460317	2595.33386327504	2601.73003194888	2608.33006535948	2614.38398692810	2620.22795341098	2626.29950083195	2632.18686868687	2638.30034129693	2644.08233276158	2650.09230769231	2655.41166380789	2661.23529411765	2666.65391304348	2672.29100529101	2677.79322638146	2683.43963963964	2688.31588447653	2693.77396021700	2699.09854014599	2703.78727272727	2708.82935779817	2713.87316176471	2718.63401109057	2723.46728971963	2727.65485074627	2732.76603773585	2737.45419847328	2741.57442748092	2746.32882011605	2750.87719298246	2755.07677165354	2759.44356435644	2763.74749498998	2767.91919191919	2772.09572301426	2775.99168399168	2780.33333333333	2784.10351966874	2788.31524008351	2792.32008368201	2796.25847457627	2800.12903225806	2803.79175704989	2807.75599128540	2811.42825607064	2815.39606126915	2818.85176991150	2822.74610244989	2826.41610738255	2829.72045454545	2833.31292517007	2836.69107551487	2839.91685912240	2843.21658986175	2846.66744730679	2849.82159624413	2853.04716981132	2856.49408983452	2859.42553191489	2862.79616306954	2865.72901678657	2868.73557692308	2872.10436893204	2874.89603960396	2877.98756218905	2881.00997506234	2883.96491228070	2887.29591836735	2889.96212121212	2892.77577319588	2895.44559585492	2898.41361256545	2901.01827676240	2903.77248677249	2906.46010638298	2909.29946524064	2911.84239130435	2914.61788617886	2916.79781420765	2919.80874316940	2922.29972752044	2924.94535519126	2927.51366120219	2929.94505494506	2932.45257452575	2935.03814713896	2937.62534435262	2939.83888888889	2942.13407821229	2944.81232492997	2947.33519553073	2949.63352272727	2952.15492957747	2954.37005649718	2956.65168539326	2959.01424501425	2961.13960113960	2963.19428571429	2965.62500000000	2967.67638483965	2969.80289855072	2971.62170087977	2974.12865497076	2976.33333333333	2978.15339233038	2980.20238095238	2982.32748538012];

This is the calibration measurement, I need to find a suitable function for this dataset, because later, I need to extrapolate the data. I have successfully tried smoothing splines (in Python, because I don't have the Curve fitting toolbox), and they work ok for the above dataset. But this dataset is only a simulation, the real experiment will have more noise and smoothing splines might fail.

Does anyone have a tip for me....? Thanks!

[Baium]

What about polyfit/polyder/polyval function set?

[William Thielicke]

That gives some results, but they are worse than the smoothing spline in python. I am looking for a "physical function" that correctly explains my simulation, but I don't have a good idea :(

Answer 1

[Alex Sha]

The follow function is much simple, but more better:

y = x/(p1+p2*x)+p3
Root of Mean Square Error (RMSE): 0.313525722040886
Sum of Squared Residual: 19.757974054633
Correlation Coef. (R): 0.999999591246961
R-Square: 0.999999182494088
Adjusted R-Square: 0.999999174236453
Determination Coef. (DC): 0.99999918249409
Chi-Square: 0.0768861820713897
F-Statistic: 121099942.96929

Parameter	Best Estimate
----------	-------------
p1	16.9216329635141
p2	-0.00486625930554683
p3	-42.7872667344874

[Walter Roberson]

That is a nice fit. Good work.

[William Thielicke]

Very nice, thanks a ton!

[Walter Roberson]

At the moment I do not understand why this is not the equivalent of a ratio between two first degree polynomials. If I have not made a mistake (and I might have at this time of night) that formula should be equivalent to a formula (a*x + b)/(c*x + c*b/(a - 1)) but the fit is completely different.

[Alex Sha]

Hi，Walter, the formual of "y = x/(p1+p2*x)+p3" is sure to be equivalent to "y=(a*x + b)/(c*x + c*b/(a - 1))":

for y=(a*x + b)/(c*x + c*b/(a - 1)), the results are:

Root of Mean Square Error (RMSE): 0.313525722040911

Sum of Squared Residual: 19.7579740546361

Correlation Coef. (R): 0.999999591246962

R-Square: 0.99999918249409

Adjusted R-Square: 0.999999174236455

Determination Coef. (DC): 0.99999918249409

Chi-Square: 0.0768860578252577

F-Statistic: 121099942.964391

Parameter Best Estimate

---------- -------------

a 1.2082139354555

b -724.030425163212

c -0.00486625930282033

Answer 2

[Walter Roberson]

There are an infinite number of functions that explain your data perfectly to within rounding error. It would be the infinity of the real numbers if matlab supported indefinite precision; with finite precision it is possibly only the infinity of the integers.

Because of this, the probability that any one of them is correct is 1/infinity which is 0.

It is thus not possible to find the "right" function given only the data.

[William Thielicke]

OK, then I need to re-sentence my question: Does someone have an idea about a "reasonable" function that would fit the data. "Reasonable" in terms of "smallest amount of characters when writing down the function". I am optimistic, that this might also represent the "physically correct" equation. I have given details about the experiment above, it is laser-triangulation just like this: http://www.movimed.com/wp-content/uploads/2018/02/02_Laser_Triangulation_explained-1024x553.png

[Walter Roberson]

A degree 5 polynomial is not horrible if you permit centering an scaling.

Linear model Poly5:
     f(x) = p1*x^5 + p2*x^4 + p3*x^3 + p4*x^2 + p5*x + p6
       where x is normalized by mean 2422 and std 571.2
Coefficients (with 95% confidence bounds):
       p1 =       7.501  (7.015, 7.986)
       p2 =       54.75  (52.38, 57.11)
       p3 =       147.3  (144.5, 150)
       p4 =       223.6  (220.7, 226.6)
       p5 =       353.2  (350.6, 355.8)
       p6 =       424.8  (423.6, 426)

Goodness of fit:
  SSE: 4230
  R-square: 0.9998
  Adjusted R-square: 0.9998
  RMSE: 4.658

However if you do not center and scale then the curve fitting routines complain that it is badly conditioned.

[William Thielicke]

Thanks Walter, that looks already good, but especially for extrapolation, my feeling is that a polynomial might give surprising reults...