http://imagej.273.s1.nabble.com/curve-fitting-examples-tp5022163p5022166.html
How can you know that the missing data is correctly interpolated. Did
Am 07.05.19 um 12:48 schrieb Rainer M. Engel:
> Good day Herbie,
>
> you are right about that. Some 1-2 years ago, but right now I'm refining
> my code and I have a better overview now, since the underlying data
> retrieval is more robust.
>
>> more an art than hard science
>
> I think I tended to 'misuse' it, even though it worked in 90% of the
> cases without the need to set filtering windows. So I wanted to check on
> the limits again.
>
> One thought:
> The data I get is very stable and missing data positions are marked to
> be filled. Right now I'm linear interpolating these, but by this these
> positions have the same "weight" like the former analysed ones, which
> then tend to be more deviating and hence will be smoothed stronger. This
> is one reason why I came back and tested the fitting methods, which are
> doing a nice job at least in some regions..
>
> Thank you for your fast reply and thoughts on this..
>
> Regards,
> Rainer
>
>
> Am 07.05.2019 um 12:06 schrieb Herbie:
>> Good day Rainer,
>>
>> if I remember correctly you've discussed a similar topic before...
>>
>> Please understand that curve fitting or general regression, to be more
>> precise, needs a "null hypothesis" or "model", i.e. you need to know the
>> type of function that underlies your data. If you don't have an idea
>> about the data generating process, you are lost or the results become
>> arbitrary.
>>
>> There are very few exceptions from this rule and two of them are
>> interpolation and smoothing (they are no regression techniques).
>> Interpolation is well-defined if the data is correctly sampled from a
>> continuous process. Smoothing and related operations are less
>> well-defined and they are more an art than hard science.
>>
>> Please keep this in mind and consider the proper approaches:
>>
>> Curve fitting provided by ImageJ is based on "general regression" and
>> requires a "null hypothesis" or "model". The most general "model" is a
>> polynomial of high degree but I don't recommend to use it without
>> profound reason.
>>
>> If your data doesn't fit a mathematically formulated "model", then
>> "general regression" according to an arbitrary chosen "model"-function
>> is like a shot in the dark.
>>
>> I fear that the "general regression"-approach is not what you looking for.
>>
>> Regards
>>
>> Herbie
>>
>> :::::::::::::::::::::::::::::::::::::::::::::
>> Am 07.05.19 um 11:40 schrieb Rainer M. Engel:
>>> Hello everyone,
>>>
>>> I retrieve some data, which can have missing data points even in the
>>> beginning or at the end. So what I needed was both; interpolation as
>>> well as extrapolation, to fill these positions.
>>>
>>> Typically the deviation over the data is not that huge and a straight
>>> fitting works well under this circumstance. Otherwise on more
>>> fluctuating/arbitrary data I got no satisfying results with any of the
>>> available fitting functions (see example makro below).
>>>
>>> Maybe my expectations are wrong about this and I thought that there is a
>>> way to adjusted methods in how close a resulting fitting would be
>>> applied. So I used median/mean as prefilter-methods.
>>>
>>> What wonders me is that sometimes a certain method works very well. In
>>> my case Rodbard, Error or a Gaussian. But sometimes, like in the given
>>> example, it makes no sense to use these at all.
>>>
>>> Is this typical or is my data strange :)
>>>
>>> Regards,
>>> Rainer
>>>
>>>
>>>
>>>
>>> // START (makro) #############################################
>>> // Curve Fitting Demo
>>> //
>>> // This macro demonstates how to use the Fit.* functions,
>>> // which were added to the macro language in v1.41k.
>>>
>>> // modified to see some larger data sets
>>>
>>> x = newArray(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
>>> 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
>>> 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52,
>>> 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75,
>>> 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93,
>>> 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108,
>>> 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122,
>>> 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136,
>>> 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150,
>>> 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164,
>>> 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178,
>>> 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192,
>>> 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206,
>>> 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220,
>>> 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234,
>>> 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248,
>>> 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262,
>>> 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276,
>>> 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290,
>>> 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304,
>>> 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318,
>>> 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332,
>>> 333, 334, 335, 336, 337, 338, 339);
>>> y = newArray(96, 97, 98, 101, 101, 102, 101, 101, 101, 101, 101, 101,
>>> 100, 100, 101, 101, 102, 101, 100, 100, 101, 100, 99, 99, 100, 101, 100,
>>> 100, 101, 101, 100, 99, 99, 99, 99, 99, 99, 98, 99, 99, 98, 97, 96, 96,
>>> 96, 96, 95, 94, 95, 95, 94, 94, 93, 90, 89, 89, 88, 87, 87, 87, 87, 84,
>>> 85, 84, 85, 84, 83, 81, 81, 81, 80, 80, 78, 77, 77, 77, 76, 75, 75, 75,
>>> 75, 74, 73, 73, 73, 73, 72, 71, 71, 71, 70, 69, 68, 68, 68, 67, 66, 66,
>>> 65, 66, 65, 66, 64, 64, 64, 64, 64, 65, 65, 66, 66, 65, 65, 65, 65, 66,
>>> 66, 66, 67, 68, 68, 68, 68, 69, 69, 70, 69, 69, 69, 70, 70, 71, 70, 72,
>>> 72, 72, 71, 71, 72, 73, 73, 73, 73, 75, 75, 75, 74, 75, 75, 76, 76, 77,
>>> 76, 76, 77, 78, 77, 77, 77, 78, 79, 80, 79, 79, 80, 81, 80, 80, 81, 81,
>>> 81, 80, 81, 81, 82, 82, 82, 81, 82, 83, 83, 83, 82, 83, 83, 83, 82, 84,
>>> 84, 84, 84, 84, 83, 83, 84, 84, 83, 84, 84, 84, 83, 83, 83, 83, 85, 85,
>>> 84, 83, 84, 84, 84, 83, 83, 84, 84, 85, 84, 83, 83, 84, 83, 83, 83, 83,
>>> 84, 83, 83, 84, 84, 84, 84, 84, 83, 84, 84, 84, 83, 83, 83, 84, 84, 83,
>>> 83, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 85, 84, 85, 84, 83, 83, 84,
>>> 84, 84, 84, 85, 85, 84, 83, 84, 84, 83, 84, 83, 84, 85, 84, 84, 84, 84,
>>> 84, 85, 84, 83, 83, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 85, 84, 84,
>>> 84, 83, 83, 84, 84, 83, 83, 84, 84, 84, 83, 83, 84, 84, 84, 84, 83, 83,
>>> 84, 84, 83, 82, 83, 83, 83, 83, 83, 83, 84, 84, 83, 83, 83, 84, 84, 83,
>>> 83, 83, 83);
>>>
>>> doAllCurveFittings("raw", x, y); //pretty dense x-range
>>>
>>> x = newArray(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
>>> 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
>>> 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52,
>>> 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70,
>>> 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88,
>>> 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104,
>>> 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118,
>>> 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132,
>>> 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146,
>>> 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160,
>>> 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174,
>>> 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188,
>>> 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202,
>>> 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216,
>>> 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230,
>>> 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244,
>>> 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258,
>>> 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272,
>>> 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286,
>>> 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300,
>>> 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314,
>>> 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328,
>>> 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339);
>>> y = newArray(100.8147, 100.7783, 100.7480, 100.6839, 100.6290, 100.6147,
>>> 100.6335, 100.6794, 100.6647, 100.6252, 100.5897, 100.5647, 100.5397,
>>> 100.4647, 100.3897, 100.3647, 100.3647, 100.3147, 100.2897, 100.2897,
>>> 100.3147, 100.2897, 100.2147, 100.1647, 100.0897, 100.0147, 99.9147,
>>> 99.7897, 99.6647, 99.5897, 99.5397, 99.4147, 99.2647, 99.1147, 98.9397,
>>> 98.7147, 98.4397, 98.1897, 97.8897, 97.5897, 97.2647, 96.9647, 96.7147,
>>> 96.3897, 96.0730, 95.7147, 95.3397, 94.9480, 94.5897, 94.1897, 93.7147,
>>> 93.2397, 92.7897, 92.3397, 91.8897, 91.4147, 90.9647, 90.4397, 89.9647,
>>> 89.4397, 88.9647, 88.4647, 87.9147, 87.3647, 86.8064, 86.2397, 85.6647,
>>> 85.0814, 84.4397, 83.7647, 83.1647, 82.5897, 81.9897, 81.3897, 80.8147,
>>> 80.2397, 79.6647, 79.1147, 78.5147, 77.9397, 77.3147, 76.7397, 76.1897,
>>> 75.6397, 75.1147, 74.6147, 74.1147, 73.5647, 73.0397, 72.5647, 72.0647,
>>> 71.5397, 71.0147, 70.5397, 70.0147, 69.5397, 69.0397, 68.6397, 68.1647,
>>> 67.6897, 67.2397, 66.7897, 66.3897, 66.0897, 65.7897, 65.5397, 65.3147,
>>> 65.1397, 64.9897, 64.8647, 64.7397, 64.6897, 64.7147, 64.7147, 64.7897,
>>> 64.9147, 65.0397, 65.1397, 65.3647, 65.6147, 65.8647, 66.1647, 66.3897,
>>> 66.6147, 66.8397, 67.0647, 67.3147, 67.6147, 67.9147, 68.2647, 68.6147,
>>> 68.9147, 69.1647, 69.4397, 69.7147, 69.9647, 70.2397, 70.5147, 70.7647,
>>> 71.0897, 71.4147, 71.6897, 71.9897, 72.3147, 72.6147, 72.9147, 73.1897,
>>> 73.4647, 73.7147, 73.9147, 74.1647, 74.4397, 74.7147, 74.9897, 75.2397,
>>> 75.4647, 75.7397, 76.0647, 76.3397, 76.5397, 76.7897, 77.0647, 77.3397,
>>> 77.5397, 77.8147, 78.0647, 78.3147, 78.4897, 78.7397, 78.9897, 79.2397,
>>> 79.4647, 79.7147, 79.9397, 80.1897, 80.4647, 80.6897, 80.8397, 81.0147,
>>> 81.2147, 81.3647, 81.4897, 81.6147, 81.8397, 82.0147, 82.1647, 82.3147,
>>> 82.5147, 82.6397, 82.7647, 82.8647, 82.9647, 83.0647, 83.2147, 83.3147,
>>> 83.3647, 83.3647, 83.4147, 83.4897, 83.5147, 83.6147, 83.7397, 83.8397,
>>> 83.8147, 83.8147, 83.8147, 83.8397, 83.7897, 83.8147, 83.8397, 83.8397,
>>> 83.8897, 83.9147, 83.8647, 83.8397, 83.8397, 83.8397, 83.8397, 83.8397,
>>> 83.8397, 83.7897, 83.6897, 83.6147, 83.6397, 83.6397, 83.6397, 83.6147,
>>> 83.6647, 83.6397, 83.6397, 83.6647, 83.6147, 83.5647, 83.5647, 83.5397,
>>> 83.5397, 83.5897, 83.5647, 83.5647, 83.5897, 83.6147, 83.6647, 83.7147,
>>> 83.7147, 83.7147, 83.7397, 83.7647, 83.7897, 83.8397, 83.8897, 83.8897,
>>> 83.9397, 83.9897, 84.0147, 84.0397, 84.0647, 84.0897, 84.1647, 84.2147,
>>> 84.2647, 84.2897, 84.2897, 84.2897, 84.2897, 84.2897, 84.2397, 84.2397,
>>> 84.1897, 84.2147, 84.2147, 84.2397, 84.2397, 84.2897, 84.3397, 84.3897,
>>> 84.4397, 84.4397, 84.3897, 84.3647, 84.3397, 84.3147, 84.3397, 84.3647,
>>> 84.3897, 84.4147, 84.4647, 84.4897, 84.5397, 84.5397, 84.5397, 84.5397,
>>> 84.5147, 84.4897, 84.4397, 84.4147, 84.3647, 84.3647, 84.3897, 84.3647,
>>> 84.3647, 84.3397, 84.3147, 84.2647, 84.2147, 84.2147, 84.2147, 84.1897,
>>> 84.1647, 84.1147, 84.0647, 84.0397, 84.0397, 83.9897, 83.9647, 83.9397,
>>> 83.9147, 83.8647, 83.8397, 83.8397, 83.7897, 83.8147, 83.8147, 83.8147,
>>> 83.8147, 83.7647, 83.7147, 83.6647, 83.6397, 83.6647, 83.6647, 83.6647,
>>> 83.6252, 83.6369, 83.6794, 83.6960, 83.6814, 83.7004, 83.7224, 83.7480,
>>> 83.7783);
>>>
>>> doAllCurveFittings("pre-filtered", x, y); //full x-range
>>>
>>> function doAllCurveFittings(stackTitle, xpoints, ypoints) {
>>> // Do a straight line fit
>>> Fit.doFit("Straight Line", xpoints, ypoints);
>>> //print("a="+d2s(Fit.p(0),6)+", b="+d2s(Fit.p(1),6));
>>>
>>> // Do all possible fits, plot them and add the plots to a stack
>>> setBatchMode(true);
>>> for (i=0; i<Fit.nEquations; i++) {
>>> Fit.doFit(i, xpoints, ypoints);
>>> Fit.plot();
>>> if (i==0)
>>> stack = getImageID;
>>> else {
>>> run("Copy");
>>> close();
>>> selectImage(stack);
>>> run("Add Slice");
>>> run("Paste");
>>> }
>>> Fit.getEquation(i, name, formula);
>>> //print("");
>>> print("index: "+i+", "+name+ " ["+formula+"]");
>>> /*print(" R^2="+d2s(Fit.rSquared,3));
>>> for (j=0; j<Fit.nParams; j++)
>>> print(" p["+j+"]="+d2s(Fit.p(j),6));*/
>>> }
>>> setBatchMode(false);
>>> run("Select None");
>>> rename(stackTitle);
>>> }
>>> // END ##################################################
>>>
>>>
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