Line of best fit - for logarithmic-like dataset

[Phystudent91]

Hi all,

Firstly, hope I'm posting in the right place!

I'm working with a company that works with air filters. Without going into specifics etc, I have 4 data points:

| x | y |
|0mm|21%|
|150mm|59.6%|
|300mm|83%|
|550mm|90%|

where x is Filter Height and y is Filtration.
(Apologies for bad formatting - I looked but couldn't find the syntax anywhere for posting in a table!)


In the title, I've called it 'logarithmic-like' and what I mean by that is that it will not go above 100% (obviously) and looking at the data points, it's got a definite exponential fall off.

21% at 0mm I know sounds wierd, but it's a characteristic of the system which, through laziness and some contractual NDA stuff, I won't go in to.

I've used 3 different graph-drawing softwares (Excel, JMP 11 and graph) + WolframAlpha and have come to the conclusion that it isn't possible to alter the curve of a logarithmic plot enough to fit this data.

I've also tried flipping the data and an exponential won't fit - x^2 does, but doesn't asymptote at 100%, so isn't really accurate enough. Increasing the power just makes any graphical software peak between the 1st and second points and then rise to the rest.

TLDR;

I have this data set (above) and I'm attempting to find a line of best fit so that we can design a unit and "know" what filtration it will give. Can anyone help me with a way of finding a formula that fits or a piece of software that will?

Thanks in advance!

 

[mathman]

One method that might work is to add one more data point: x=1 billion, y = 100%.

 

[da_nang]

You could try modeling it with $y = 1 - (1 - a_1)e^{-a_2 x}$ and perform a nonlinear fit to estimate the parameters. And if you know that $f(0) = 0.21$ then you know one of the parameters, $y = 1 - 0.79e^{-a_2 x} , a_1 = 0.21$.

 

[haruspex]

You could try modeling it with $y = 1 - (1 - a_1)e^{-a_2 x}$ and perform a nonlinear fit

You could make it linear:
ln(1-y) = ln(1-a₁) - a₂x
Plot ln(1-y) against x.

 

[blue_raver22]

Hello,

Thank you for reaching out and providing the details of your data set. From what you have described, it seems like your data follows a logarithmic-like trend, where there is an exponential decrease in filtration as filter height increases. This is a common trend in many scientific and engineering applications.

In terms of finding a line of best fit for this type of data, there are a few options you can try. One approach is to transform your data so that it follows a linear trend. This can be done by taking the logarithm of both the x and y values, and then plotting the transformed data. This will give you a straight line that can be used to find the slope and intercept, which can then be used to create a logarithmic equation for your data.

Another option is to use a non-linear regression analysis, which takes into account the non-linear nature of your data. This can be done using software such as R, Python, or MATLAB, which have built-in functions for non-linear regression. These programs also allow you to customize the type of equation you want to fit to your data, so you can try different types of logarithmic functions until you find the one that best fits your data.

I hope this helps and good luck with your analysis! If you have any further questions, please don't hesitate to reach out.