how to use Savitzky-Golay smooth coefficient to calculate derivatives

c++ smoothing derivative
Lin Song Yang picture Lin Song Yang · Aug 26, 2010 · Viewed 7.3k times · Source

Savitzky-Golay smoothing filter can be used to calculate the coefficients so as to calculate the smoothed y-values by applying the coefficients to the adjacent values. The smoothed curve looks great.

According to the papers, the coefficients can also be used to calculate the derivatives up to 5th order. The coefficients calculation parameter ld would need to be set to the order of derivatives. For the first derivative, the appropriate setting is ld=1, and the value of the derivative is the accumulated sum divided by the sampling interval h.

My question is: how to use the obtained coefficients to calculate the accumulated sum? how is the derivative calculated? any sample code?

Answer

Lin Song Yang picture Lin Song Yang · Aug 27, 2010

To calculate the derivatives using Savitzky-Golay smoothing filter, the polynomial coefficients computation has a parameter b, the value b[derivative] must be set to 1.0, the array be will be used in the LU decomposition call.

The key to get derivatives right is to understand the polynomial formula: Y = a0 + a1 * z + a2 * z^2 + ... + ak * z^k. The values a0, a1, a2, ..., ak are actually the smoothed values within the moving window, z = (x - x0) / h, for the centre point of the moving window, we can assume z = 0 since x = x0.

Therefore, in the derivative calculations: 

dY/dx = a1/h; and d2Y/dx2 = 2a2/h^2.

Where a1, a2 are the smoothed values of y by using the coefficients calculated on the corresponding derivatives.

Related questions

Easiest way to convert int to string in C++

What is the easiest way to convert from int to equivalent string in C++. I am aware of two methods. Is there any easier way? (1) int a = 10; char *intStr = itoa(a); string str = string(intStr); (2) int a = 10; stringstream ss; ss &…

Read More
The Definitive C++ Book Guide and List

This question attempts to collect the few pearls among the dozens of bad C++ books that are published every year. Unlike many other programming languages, which are often picked up on the go from tutorials found on the Internet, few …

Read More
How do I iterate over the words of a string?

I'm trying to iterate over the words of a string. The string can be assumed to be composed of words separated by whitespace. Note that I'm not interested in C string functions or that kind of character manipulation/access. Also, …

Read More