Smoothing small data with a second-order quadratic curve

Question

I am doing a certain analysis of the signals, and I need a method that would smooth out this distribution curve in the form of a bell. The moving average method does not give the desired results. I want to keep the minimum / maximum and overall shape of my fitted curve intact, but eliminate the inconsistencies in the sample.

In short: if you were to get a dataset modeling a simple quadratic curve, what statistical smoothing method would you recommend?

If possible, please refer to the implementation, library, or framework.

Thank you SO!

Edit : some useful data

(Possible signal graph)

The dark square is my “fitted” curve of the light connected data points.

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Answer 1

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y = exp(-squared(x-mean)/variance)

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Answer 2

#include <iostream>
#include <math.h>

struct WeightedData 
{
double x;
double y;
double weight;
};

void findQuadraticFactors(WeightedData *data, double &a, double &b, double &c, unsigned int const datasize)
{
double w1 = 0.0;
double wx = 0.0, wx2 = 0.0, wx3 = 0.0, wx4 = 0.0;
double wy = 0.0, wyx = 0.0, wyx2 = 0.0;
double tmpx, tmpy;
double den;

for (unsigned int i = 0; i < datasize; ++i) 
    {
    double x = data[i].x;
    double y = data[i].y;
    double w = data[i].weight;  

    w1 += w;
    tmpx = w * x;
    wx += tmpx;
    tmpx *= x;
    wx2 += tmpx;
    tmpx *= x;
    wx3 += tmpx;
    tmpx *= x;
    wx4 += tmpx;
    tmpy = w * y;
    wy += tmpy;
    tmpy *= x;
    wyx += tmpy;
    tmpy *= x;
    wyx2 += tmpy;
    }

den = wx2 * wx2 * wx2 - 2.0 * wx3 * wx2 * wx + wx4 * wx * wx + wx3 * wx3 * w1 - wx4 * wx2 * w1;
if (den == 0.0) 
    {
    a = 0.0;
    b = 0.0;
    c = 0.0;
    }
else    
    {
    a = (wx * wx * wyx2 - wx2 * w1 * wyx2 - wx2 * wx * wyx + wx3 * w1 * wyx + wx2 * wx2 * wy - wx3 * wx * wy) / den;
    b = (-wx2 * wx * wyx2 + wx3 * w1 * wyx2 + wx2 * wx2 * wyx - wx4 * w1 * wyx - wx3 * wx2 * wy + wx4 * wx * wy) / den;
    c = (wx2 * wx2 * wyx2 - wx3 * wx * wyx2 - wx3 * wx2 * wyx + wx4 * wx * wyx + wx3 * wx3 * wy - wx4 * wx2 * wy) / den;
    }

}

double findY(double const a, double const b, double const c, double const x)
{       
return a * x * x + b * x + c; 
};




int main(int argc, char* argv[])
{
WeightedData data[9];
data[0].weight=1; data[0].x=1; data[0].y=-52.0; 
data[1].weight=1; data[1].x=2; data[1].y=-48.0; 
data[2].weight=1; data[2].x=3; data[2].y=-43.0; 
data[3].weight=1; data[3].x=4; data[3].y=-44.0; 
data[4].weight=1; data[4].x=5; data[4].y=-35.0; 
data[5].weight=1; data[5].x=6; data[5].y=-31.0; 
data[6].weight=1; data[6].x=7; data[6].y=-32.0; 
data[7].weight=1; data[7].x=8; data[7].y=-43.0; 
data[8].weight=1; data[8].x=9; data[8].y=-52.0; 

double a=0.0, b=0.0, c=0.0;
findQuadraticFactors(data, a, b, c, 9);
std::cout << " x \t y" << std::endl;
for (int i=0; i<9; ++i)
    {
    std::cout << " " << data[i].x << ", " << findY(a,b,c,data[i].x) << std::endl;
    }
}

Answer 3

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Answer 4

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y[0] = x[0];
for (i = 1; i < len; ++i)
    y[i] = a * x[i] + (1.0 - a) * y[i - 1];

In this case, x [] is your input, and y [] is your filtered output. The coefficient a is a value from 0 to 1 that you must adjust. The value of a 1 reproduces the input, and the cutoff frequency decreases as the a approaches 0.