Smoothing, differentiating and integrating curves

A single Prism analysis smooths a curves and also (optionally) converts the resulting curve to its derivative or integral.

Note: Prism cannot do symbolic algebra or calculus. If you give Prism a series of XY points that define a curve, it can compute the numerical derivative of that series of points. But if you give Prism an equation, it cannot compute a new equation that defines the derivative.

The first derivative is the steepness of the curve at every X value. The derivative is positive when the curve heads uphill and is negative when the curve heads downhill. The derivative equals zero at peaks and troughs in the curve. After calculating the numerical derivative, Prism can smooth the results, if you choose.

The integral is the cumulative area under the curve. The integral at any value X equals the area of the curve for all values less than X.

This analysis integrates a curve, resulting in another curve showing cumulative area. Another Prism analysis computes a single value for the area under the curve.

Prism uses the trapezoid ruleto integrate curves. The X values of the results are the same as the X values of the data you are analyzing. The first Y value of the results equals a value you specify (usually 0.0). For other rows, the resulting Y value equals the previous result plus the area added to the curve by adding this point. This area equals the difference between X values times the average of the previous and this Y value.

If you import a curve from an instrument, you may wish to smooth the data to improve the appearance of a graph. Since you lose data when you smooth a curve, you should not smooth a curve prior to nonlinear regression or other analyses. Smoothing is not a method of data analysis, but is purely a way to create a more attractive graph.

Prism gives you two ways to adjust the smoothness of the curve. You choose the number of neighboring points to average and the 'order' of the smoothing polynomial. Since the only goal of smoothing is to make the curve look better, you can simply try a few settings until you like the appearance of the results. If the settings are too high, you lose some peaks which get smoothed away. If the settings are too low, the curve is not smooth enough. The right balance is subjective -- use trial and error.