Given the following inputs and outputs to a function, how would I determine an algorithm to fit?
f( 1 ) = 1
f( 2 ) = 2
f( 3 ) = 3
f( 4 ) = 2
f( 5 ) = 4
f( 6 ) = 6
f( 7 ) = 3
f( 8 ) = 4
f( 9 ) = 14
f( 10 ) = 9
f( 11 ) = 18
f( 12 ) = 12
f( 13 ) = 12
f( 14 ) = 42
f( 15 ) = 15
f( 16 ) = 4
f( 17 ) = 24
f( 18 ) = 40
f( 19 ) = 60
f( 20 ) = 51
f( 21 ) = 24
f( 22 ) = 30
f( 23 ) = 130
f( 24 ) = 90
f( 25 ) = 60
f( 26 ) = 72
f( 27 ) = 140
f( 28 ) = 42
f( 29 ) = 156
f( 30 ) = 48
f( 31 ) = 15
f( 32 ) = 12
f( 33 ) = 168
f( 34 ) = 48
f( 35 ) = 414
f( 36 ) = 483
f( 37 ) = 33
f( 38 ) = 36
f( 39 ) = 660
f( 40 ) = 39
f( 41 ) = 380
f( 42 ) = 672
f( 43 ) = 437
f( 44 ) = 364
f( 45 ) = 403
f( 46 ) = 315
f( 47 ) = 550
f( 48 ) = 48
f( 49 ) = 48
f( 50 ) = 369
f( 51 ) = 1950
f( 52 ) = 481
f( 53 ) = 280
f( 54 ) = 180
f( 55 ) = 90
f( 56 ) = 56
f( 57 ) = 144
f( 58 ) = 756
f( 59 ) = 252
f( 60 ) = 4278
f( 61 ) = 4350
f( 62 ) = 816
f( 63 ) = 105
f( 64 ) = 12
f( 65 ) = 432
f( 66 ) = 98
f( 67 ) = 120
f( 68 ) = 1924
f( 69 ) = 186
f( 70 ) = 64
f( 71 ) = 8736
f( 72 ) = 4004
f( 73 ) = 351
f( 74 ) = 552
f( 75 ) = 32604
f( 76 ) = 720
f( 77 ) = 77
f( 78 ) = 496
f( 79 ) = 16740
f( 80 ) = 6783
f( 81 ) = 130
f( 82 ) = 72
f( 83 ) = 1680
f( 84 ) = 330
f( 85 ) = 228
f( 86 ) = 152
f( 87 ) = 87
f( 88 ) = 240
f( 89 ) = 711
f( 90 ) = 6930
f( 91 ) = 1344
f( 92 ) = 1947
f( 93 ) = 495
f( 94 ) = 1680
f( 95 ) = 30240
f( 96 ) = 96
f( 97 ) = 25200
f( 98 ) = 24486
f( 99 ) = 22440
f( 100 ) = 540
f( 101 ) = 1566
f( 102 ) = 279
f( 103 ) = 24024
f( 104 ) = 5040
f( 105 ) = 1197
f( 106 ) = 336
f( 107 ) = 48450
f( 108 ) = 707
f( 109 ) = 153972
f( 110 ) = 3009
f( 111 ) = 1958
f( 112 ) = 3068
f( 113 ) = 2366298
f( 114 ) = 37240
f( 115 ) = 5628
f( 116 ) = 1674
f( 117 ) = 4635
f( 118 ) = 34099
f( 119 ) = 40194
f( 120 ) = 48300
f( 121 ) = 43680
f( 122 ) = 13992
f( 123 ) = 2568
f( 124 ) = 4368
f( 125 ) = 234
f( 126 ) = 1680
f( 127 ) = 105
f( 128 ) = 24
f( 129 ) = 5320
f( 130 ) = 4664
f( 131 ) = 8760
f( 132 ) = 145860
f( 133 ) = 1361122
f( 134 ) = 630
f( 135 ) = 12078
f( 136 ) = 135
f( 137 ) = 17160
f( 138 ) = 445536
f( 139 ) = 6612
f( 140 ) = 15960
f( 141 ) = 99540
f( 142 ) = 34632
f( 143 ) = 14430
f( 144 ) = 57810
f( 145 ) = 39240
f( 146 ) = 24252
f( 147 ) = 1008
f( 148 ) = 144
f( 149 ) = 54054
f( 150 ) = 171380
f( 151 ) = 3444
f( 152 ) = 37076
f( 153 ) = 1320
f( 154 ) = 1188810
f( 155 ) = 3013
f( 156 ) = 91560
f( 157 ) = 239666
f( 158 ) = 720720
f( 159 ) = 11589240
f( 160 ) = 1260
f( 161 ) = 264040
f( 162 ) = 294
f( 163 ) = 1740
f( 164 ) = 16170
f( 165 ) = 67284
f( 166 ) = 162
f( 167 ) = 1560
f( 168 ) = 9798
f( 169 ) = 74542
f( 170 ) = 62118
f( 171 ) = 508164
f( 172 ) = 578952
f( 173 ) = 156
f( 174 ) = 85470
f( 175 ) = 636
f( 176 ) = 2880
f( 177 ) = 6120
f( 178 ) = 1890
f( 179 ) = 98640
f( 180 ) = 329448
f( 181 ) = 1452990
f( 182 ) = 39480
f( 183 ) = 1833
f( 184 ) = 43000
f( 185 ) = 28210
f( 186 ) = 51414
f( 187 ) = 1358708
f( 188 ) = 4218
f( 189 ) = 83754
f( 190 ) = 146160
f( 191 ) = 9090
f( 192 ) = 870
f( 193 ) = 340
f( 194 ) = 168
f( 195 ) = 1979040
f( 196 ) = 251940
f( 197 ) = 168
f( 198 ) = 6020
f( 199 ) = 28200
f( 200 ) = 9006
f( 201 ) = 388
f( 202 ) = 3817632
f( 203 ) = 9975
f( 204 ) = 360
f( 205 ) = 35784
f( 206 ) = 9490
f( 207 ) = 1260
f( 208 ) = 138
f( 209 ) = 1911
f( 210 ) = 337554
f( 211 ) = 166152
f( 212 ) = 375648
f( 213 ) = 7182
f( 214 ) = 81788
f( 215 ) = 2604
f( 216 ) = 40455
f( 217 ) = 42432
f( 218 ) = 1115180
f( 219 ) = 406980
f( 220 ) = 85680
f( 221 ) = 8484
f( 222 ) = 10440
f( 223 ) = 5910
f( 224 ) = 12390
f( 225 ) = 224
f( 226 ) = 2520
f( 227 ) = 420420
f( 228 ) = 159120
f( 229 ) = 27192
f( 230 ) = 128520
f( 231 ) = 15660
f( 232 ) = 2772
f( 233 ) = 13095
f( 234 ) = 286440
f( 235 ) = 1789320
f( 236 ) = 175140
f( 237 ) = 8810490
f( 238 ) = 185650
f( 239 ) = 78672
f( 240 ) = 2055648
f( 241 ) = 3480
f( 242 ) = 27612420
f( 243 ) = 990
f( 244 ) = 21240
f( 245 ) = 769860
f( 246 ) = 4140
f( 247 ) = 63756
f( 248 ) = 28890
f( 249 ) = 3420
f( 250 ) = 11704
f( 251 ) = 1014600
f( 252 ) = 16560
f( 253 ) = 1226160
f( 254 ) = 55440
f( 255 ) = 315
f( 256 ) = 24
It's been a while since I've had to do it, but I recall a process for building polynomial functions based on input & output. I would appreciate help in remembering the process, and I would also be interested to know if there are any tools to help.
Best Answer
The first step is to look at the simple $XY$ plot of the data:
Whoa. What if we take the logarithm of $Y$?
This looks like a power of $X$ with a lot of noise. Averaging the ratio of $\ln\ln Y/\ln X$ gives $0.42$. Indeed, $\ln Y\approx X^{0.42}$ is a reasonable "smoothed" approximation.
But I don't think that curve-fitting is the way to go here. The nature of the function, and the fact that the argument runs from $1$ to $256$ suggest that some binary operations might be involved. And look, we have spikes at
Actually, I suspect we should subtract $1$ from $X$ to make it run from $0$ to $255$. Then these three spikes are at
Must be something about these chains of 0s and 1s.