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I have taken an equation from an article which was fitted on x being 1 (kg) at a minimum, the equation is as follows

log10(y) = -0.619 + 0.812 * log10(x)^1.171

But now I would like to use the equation to calculate values of x when x<1, but if x<1 than log10(x) is <0 so this doesn't work. I don't have the original data so I can't refit the function. How can I modify the equation so values of x can be < 1?

original graph from article

Herman Toothrot
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  • It is not meaningful for $x<1$. Near $x=1$ you have $y \approx 0.2404363$. If you plot $y$ against $x$ for $x \ge 1$, it looks close to linear, though it seems to me $\frac{dy}{dx} \to 0$ as $x \to 1^{+}$, so really anything could happen for smaller $x$ if you try to extrapolate down. I would be interested in what motivated this model in the original article, rather than something more natural: $1$kg looks like an artificial minimum mass. – Henry Jul 20 '21 at 14:40
  • @Henry I guess they didn't have enough data in that region to fit a model. I have posted the original graph. – Herman Toothrot Jul 20 '21 at 15:05
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    Now you have included the graph, there is even less reason to justify extrapolating below $10^0$ with that complicated function. For the ruminants the log-log graph is close to linear with a slope close to $1$ so perhaps you might sensibly use something like $y \approx 0.19 x$. For the hindgut animals, the difference between Porcupine and Rock Hydrax is so far off the blue line that extrapolating downwards is unlikely to be helpful in any way – Henry Jul 20 '21 at 15:18

1 Answers1

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There's no answer to your question. There are an infinite number of possible ways to change the function to work for a larger domain (where x < 1), especially without the original data.

It's the same type of question of "what number follows after 1 2 4 7" - there are infinitely many patterns/equations that would satisfy this start.

naslundx
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  • Thanks, that's what I thought. Is there anyway that I could do a linear extrapolation so that the blue curve in the graph can be plotted for x < 1? – Herman Toothrot Jul 20 '21 at 15:15