Pasco Science Workshop^TM

Curve Fitting

BHS -> Staff -> Mr. Stanbrough ->Pasco Interface Notes-> this page

• Linear Fit: Fits the data to y = a₁ + a₂x (where "y" is whatever is plotted on the vertical axis, and "x" is whatever is plotted on the horizontal axis. In other words, a₁ is the y-intercept of the line, and a₂ is the line's slope.)
• Logarithmic Fit: Fits the data to y = a₁ + a₂ln(a₃x + a₄) where "ln" is the natural (base e) logarithm function.
• Exponential Fit: Fits the data to y = a₁ + a₂e^{(a3x + a4)} where "e" is the base of the natural logarithm function.
• Power Fit: Fits the data to y = a₁ + a₂(x + a₃)^a4
• Polynomial Fit: Fits the data to y = a₁ + a₂x + a₃x² + a₄x³ + ...
• Sine Series Fit: Oh, my...

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Messing With the Best Fit Parameters

Often theory or intuition will tell you that a curve should meet certain conditions. For instance, if the velocity of an object is zero, the distance that it travels will be zero in any arbitrary time interval. Therefore, the graph of velocity vs. distance really should pass through the origin, but the "best fit" graph gives it some small y-intercept. What would the graph look like if you could insist that it passes through the origin, you wonder? Well, wonder no more - it's easy to find out.

This linear curve fit has a small y-intercept (a1). By clicking on its value, you can type in a new one and the other parameter(s) will be recalculated.
Notice that a small lock icon () appears to show that the value of a1 has been changed and "locked" by the user.

All you have to do is click the mouse on the value that you would like to change, and type a new value. The software will automatically recalculate the curve fit using your value. A small lock icon () appears next to the value(s) that you changed, signifying that the computer considers this value "locked" and will not change it.

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Helpful Hints:

Keep in mind that, as often as not, you aren't really looking for the curve that fits your data the best. What you usually want to know is: "Does this data fit, as nearly as I can tell, a particular theoretical model?".

Huh? Well, specifically, it is probably possible to find some 7th-degree polynomial that gives a lower chi-square error than a linear fit. Does this mean that the theoretical model that says that "x should be proportional to y" is wrong, and that actually, the 7th-degree polynomial is the "true" relationship between x and y? No, it doesn't.

There is an interesting (an powerful, and very useful) theorem in calculus (Taylor's Theorem) that says, more or less, that any continuous function can be approximated by a polynomial function over a restricted domain as closely as you like, as long as you are willing to calculate enough terms. That is why you can get very good curve fits with high-degree polynomials - not because that the high-degree polynomial models the physics of the situation.

The question that you should be thinking about (and answering) in this situation is "Does this data support the theoretical model, or not?", so in the example above, you should be thinking "Given the uncertainties in the data (which should be indicated by error bars, by the way...), does this graph support the notion that x is proportional to y, or not?