问题
I have 44 pairs of values, the following:
X Y X Y 1 1303 23 1471979 2 2689 24 1855942 3 4373 25 2339735 4 7421 26 3096779 5 10037 27 3903252 6 13333 28 5153666 7 20665 29 6199765 8 26849 30 8185063 9 37305 31 10314552 10 47879 32 13588513 11 65572 33 17122961 12 89127 34 21576366 13 106217 35 27187657 14 152379 36 35747356 15 193512 37 45043166 16 244886 38 56755887 17 309618 39 71513915 18 414190 40 93863574 19 552058 41 118269663 20 660106 42 149021335 21 925396 43 187768443 22 1108885 44 246053390
I put them in Excel and the graph that is produced, looks like it gives an exponential formula
When I tried to calculate the Y values with the produced formula, the results had big differencies from the original ones. I assume that the exponential formula must have an extra coefficient and it's full format must be the following:
Is there a way to calculate the c coefficient in Excel?
回答1:
I am guessing you could approximate using the exponential trend formula. This will give you the top curve. You can difference the curves to work out an approximate c value.
Exponential Trend- Equation
Equation: y = c *e ^(b * x)
c: =EXP(INDEX(LINEST(LN(y),x),1,2))
b: =INDEX(LINEST(LN(y),x),1)
See here for an example with your data:
And plot
More info here:
http://spreadsheetpage.com/index.php/tip/chart_trendline_formulas/
And see the ExcelIsFun Youtube series on Forecasting and Trends.
回答2:
The asymptotic part of the curve is biased by the lower x values which aren't consistent with the formula Y=a*exp(bx).
See below it is more visible when you apply ln() to the values and use a linear regression :
Adding an offset Y=ab^x-c won't cut it (or at least I couldn't do it)
Just for fun I tried an approximation using Qtiplot : A near perfect approximation for your function with a Boltzmann expression :
function: A2+(A1-A2)/(1+exp((x-x0)/dx))
A1 = 2.2184780190041e+04 +/- 1.5722449570412e+05
A2 =8.1115013418524e+11 +/- 6.6413291414825e+13
dx = 4.1166909249630e+00 +/- 5.6368166087880e-02
x0 = 7.7386674788554e+01 +/- 3.3758286906650e+02
Which you are free to simplify/approximate.
Regarding your question :
there is no built-in exponential regression tool to fit your data. Using excel you could perform the regression above following the methodology (diffenrentiating and solving equations) but it defeats the purpose.
also, if you want the asymptotic part to fit better, you can skip the first x values (start at x=8 for instance) and use the built in exponential regression in xl.
回答3:
You have to use an X-Y plot when using the graph trendline. If you select any other kind of graph, the trendline equation will be wrong.
回答4:
Simply ask Excel to give you the parameters with, say, 20 or 30 decimal points. We just did it and it makes a difference, particularly when x is high (in our case, around 2000).
来源：https://stackoverflow.com/questions/48767395/how-to-find-coefficients-of-exponential-formula-from-given-pairs-of-numbers