I have a dataset that I'm currently smoothing using a 6th order least-squares polynomial which models the initial sharp curve well but as the data tapers out some noise at the end causes a downwards curve at the end. It's probably easiest to show a plot of my current data followed by the results of fitting extrapolated by a few extra points:
I'm expecting the data to taper out as it moves along so my initial thoughts were to maybe to break the data into thirds and apply lower order polynomials as it goes along. But I don't have a good background in statistics and I'm guessing this problem has some elegant solution I'm not aware of. Because the dataset is fairly short I thought I'd include a copy in case anyone was inspired to demonstrate some techniques:
0 0.632696209
1 0.954847196
2 1.211956632
3 1.282479631
4 1.348572899
5 1.36480867
6 1.411463353
7 1.413356854
8 1.443252208
9 1.5086791
10 1.519056572
11 1.515427082
12 1.545547955
13 1.567970692
14 1.554558155
15 1.539432376
16 1.662380739
17 1.630363821
18 1.627295705
19 1.574494008
20 1.651035552
21 1.59834609
22 1.612328401
23 1.634990079
24 1.668922299
25 1.622188079
26 1.759410212
27 1.849993968
28 1.78653232
29 1.7872419
30 1.639521297
31 1.70151144
32 1.868248296
33 1.707963085
34 1.703480801
35 1.811524105
36 1.790136202
37 1.79970863
38 1.944272907
39 1.818103305
40 1.750744767
43 1.77700684
44 1.80647612
Best Answer
My equation search on your data turned up a simple two-parameter power equation, "y = a * pow(x, b)", with parameters a = 1.0769014059561925E+00 and b = 1.4153886539395866E-01 yielding R-squared = 0.789 and RMSE = 0.112