1. Differenciation with Regression Equations

I'm asked to find the Reaction Rate when t = 120 seconds (which involves differentiation) for a Generic Chemical Equation. The rate of a chemical reaction is defined as:

R = dC/dt (Differentiation of Concentration with respect to time)

I'm given a set of 11 points (x= time, y= Concentration) which I need to plot in order to find a general trend for them.

After plotting all points and observing the resulting graph, I tried performing multiple different Regression Analyses in order to find a Function (C(t)) that best fits the plotted points.

For each of the Functions I obtained from each individual Regression...

1) I derived each one

And

2) Evaluated each resulting derivative for t = 120, in order to obtain the Reaction Rate for each case.

Finally, I compared each result (Reaction Rate) with the Literature Value, and calculated an error percentage for every case.

These were my results:

My question is:

Many sources (including my book) claim that a High-Degree Polynomial Regression (Cubed Regression and above) is not appropriate for describing a Physical Phenomenon, even when the Regression Fit is perfect (R=1).

However, based on the table above, the most accurate answer I got was obtained through a Quartic Regression, while the rest had considerably higher error percentages.

Am I missing something or is it just an exception? Any clarification would be much appreciated!

2. Re: Differenciation with Regression Equations

What's going on is that you have noisy data.

With a high enough degree polynomial you can fit a curve to any set of data.

But physical processes rarely have any more than the cube of some parameter in their model. Nature just doesn't use higher degree polynomials.

So pursuing a lower error by using increasing order curve fits is chasing ghosts.

You'll find those higher order fits are extremely sensitive to noise. Change one data point slightly and the fitted curve changes radically.

3. Re: Differenciation with Regression Equations

Hmm I understand that the Function obtained through a High-Degree Polynomial Regression isn't ideal for representing the global behavior of a Natural Phenomenon; however, isn't it more accurate for determining an Instantaneous Magnitude such as Instantaneous Rate of Reaction?

I'm asking this because I've tried repeating the same method on different problems (points given, Instantaneous Reaction Rates asked), and the most accurate Instantaneous Rate Answers I get are always the ones corresponding to the Highest-Degree Polynomial Regression Equation.