The justification for your choice of model is essentially dependent upon the data itself. A mathematical model is only as good as its underlying assumptions, and the question you asked is how you justify your choice of model, which is the assumption you made about the data. There is no real mathematical way to prove one model is better than another. In this case you could compare R squared values and pick the best one, or perhaps you know some property of the data which you believe to be more important than a R squared value, and you base your choice on that. For example, if I was trying to plot a curve to fit wind tunnel drag data I would choose a model with an X^2 in it, because I happen to already know one property of drag is that it is quadratic with respect to angle of attack. It may be possible to get a model with lower error than an X^2 model for my particular data, but I already know the relationship, which is in essence a property of the data, so I know that a model with lower error is simply overfitting the relationship, or fitting noise in the data. Good question, but there's no real good answer in general.

Quote from Jack

How can I justify my choice of model? I doubt "I graphed them all in excel and the power equation was the best fit" will be a sufficient answer. We can rule out the linear equation because, well, it's a line and the plotted points aren't in a straight line, but how do I prove that the equation cannot be the second (exponential) one?

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A photographer is taking portraits and he knows that there is a relationship between the distance (x) in metres from the light sources, and the intensity (y), on a surface x metres from the light source.

Irradiance. Specifically the units associated with it, W/m², where W = watts (power, or joules/s). Since your watts don't change (at least from the information given), you can say W is constant, so you can let W + c₁ = a.* As the given variable is distance, you can let m = x. And finally, -2 + c₂ = n.*

From this, there's an inverse relationship of the intensity to a constant (a) times variable distance (x) with a fixed power (n).

*Here, c₁ and c₂ are some arbitrary fudge factor. They allow the problem to not be as straightforward as going to google, finding the formula for what you want, and saying "yep, dat's the equation", while still allowing it to fit the bill. It could also be multiplication of the fudge factor, but addition is simpler and usually (for me) harder to get rid of in fractions.

That's the mathematical way of saying "it makes sense if you take a bit of knowledge from outside of the problem and apply it". To be fair, I put the data into Excel, and let the data speak for itself. I didn't bother to think about the rationale until you asked, though I knew light intensity was in an inverse relationship to distance from last year's physics.

Simply put, though, there are some applications you cannot do when fitting data. If you know it's supposed to be one type of line, it's the equivalent of breaking a commandment (to me) to use a different fit for better data. For example, most graphs I make straight from data, before I torture and manipulate it, has an R² of roughly .95-.98. While I could put it on a polynomial fit with 6 polynomials and get R² of 1, this is very bad statistical sinning, which is what Vrael was warning against.

I got my dad to have a look at this and the conclusion we came to is a) is ruled out because it's a straight line, and b) is ruled out because no matter what number you put in for N, the line still crosses the y axis. Because the x axis is the distance from the light source and the paper, x will never equal 0, so the only possible conclusion is that the graph is c). Which my dad knew anyway because of his work with irradiation, which uses the same graphs and such, but the difficulty was in proving it was c). Does that sound like a reasonable "proof" to everyone here?

The values don't change at a constant rate. It isn't linear.

The values don't change at a consistent percentage. It isn't exponential.

The values change at a decreasing percentage as they approach the x-axis. That's the behavior of a power function.

You don't even need a graph to answer this.