Polynomial Curve Fitting
The polyfit function is a general purpose curve fitter used to model
the non-linear relationship between two random variables.
The polyfit function is passed x- and y-axes and fits a smooth curve to the data.
If only a single array is provided it is treated as the y-axis and a sequence is generated
for the x-axis.
The polyfit function also has a parameter the specifies the degree of the polynomial. The higher
the degree the more curves that can be modeled.
The example below uses the polyfit function to fit a curve to an array using
a 3 degree polynomial. The fitted curve is then subtracted from the original curve. The output
shows the error between the fitted curve and the original curve, known as the residuals.
The output also includes the sum-of-squares of the residuals which provides a measure
of how large the error is.
let(echo="residuals, sumSqError",
y=array(0, 1, 2, 3, 4, 5.7, 6, 7, 6, 5, 5, 3, 2, 1, 0),
curve=polyfit(y, 3),
residuals=ebeSubtract(y, curve),
sumSqError=sumSq(residuals))When this expression is sent to the /stream handler it
responds with:
{
"result-set": {
"docs": [
{
"residuals": [
0.5886274509803899,
-0.0746078431372561,
-0.49492135315664765,
-0.6689571213100631,
-0.5933591898297781,
0.4352283990519288,
0.32016160310277897,
1.1647963800904968,
0.272488687782805,
-0.3534055160525744,
0.2904697263520779,
-0.7925296272355089,
-0.5990476190476182,
-0.12572829131652274,
0.6307843137254909
],
"sumSqError": 4.7294282482223595
},
{
"EOF": true,
"RESPONSE_TIME": 0
}
]
}
}In the next example the curve is fit using a 5 degree polynomial. Notice that the curve is fit closer, shown by the smaller residuals and lower value for the sum-of-squares of the residuals. This is because the higher polynomial produced a closer fit.
let(echo="residuals, sumSqError",
y=array(0, 1, 2, 3, 4, 5.7, 6, 7, 6, 5, 5, 3, 2, 1, 0),
curve=polyfit(y, 5),
residuals=ebeSubtract(y, curve),
sumSqError=sumSq(residuals))When this expression is sent to the /stream handler it
responds with:
{
"result-set": {
"docs": [
{
"residuals": [
-0.12337461300309674,
0.22708978328173413,
0.12266015718028167,
-0.16502738747320755,
-0.41142804563857105,
0.2603044014808713,
-0.12128970101106162,
0.6234168308471704,
-0.1754692675745293,
-0.5379689969473249,
0.4651616185671843,
-0.288175756132409,
0.027970945463215102,
0.18699690402476687,
-0.09086687306501587
],
"sumSqError": 1.413089480179252
},
{
"EOF": true,
"RESPONSE_TIME": 0
}
]
}
}Prediction, Derivatives and Integrals
The polyfit function returns a function that can be used with the predict
function.
In the example below the x-axis is included for clarity.
The polyfit function returns a function for the fitted curve.
The predict function is then used to predict a value along the curve, in this
case the prediction is made for the x value of 5.
let(x=array(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14),
y=array(0, 1, 2, 3, 4, 5.7, 6, 7, 6, 5, 5, 3, 2, 1, 0),
curve=polyfit(x, y, 5),
p=predict(curve, 5))When this expression is sent to the /stream handler it
responds with:
{
"result-set": {
"docs": [
{
"p": 5.439695598519129
},
{
"EOF": true,
"RESPONSE_TIME": 0
}
]
}
}The derivative and integrate functions can be used to compute the derivative
and integrals for the fitted
curve. The example below demonstrates how to compute a derivative
for the fitted curve.
let(x=array(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14),
y=array(0, 1, 2, 3, 4, 5.7, 6, 7, 6, 5, 5, 3, 2, 1, 0),
curve=polyfit(x, y, 5),
d=derivative(curve))When this expression is sent to the /stream handler it
responds with:
{
"result-set": {
"docs": [
{
"d": [
0.3198918573686361,
0.9261492094077225,
1.2374272373653175,
1.30051359631081,
1.1628032287629813,
0.8722983646900058,
0.47760852150945,
0.02795050408827482,
-0.42685159525716865,
-0.8363663967611356,
-1.1495552332084857,
-1.3147721499346892,
-1.2797639048258267,
-0.9916699683185771,
-0.3970225234002308
]
},
{
"EOF": true,
"RESPONSE_TIME": 0
}
]
}
}Gaussian Curve Fitting
The gaussfit function fits a smooth curve through a gaussian peak.
This is shown in the example below.
let(x=array(0,1,2,3,4,5,6,7,8,9, 10),
y=array(4,55,1200,3028,12000,18422,13328,6426,1696,239,20),
f=gaussfit(x, y))When this expression is sent to the /stream handler it
responds with:
{
"result-set": {
"docs": [
{
"f": [
2.81764431935644,
61.157417979413424,
684.2328985468831,
3945.9411154167447,
11729.758936952656,
17972.951897338007,
14195.201949425435,
5779.03836032222,
1212.7224502169634,
131.17742331530349,
7.3138931735866946
]
},
{
"EOF": true,
"RESPONSE_TIME": 0
}
]
}
}Like the polyfit function, the gaussfit function returns a function that can
be used directly by the predict, derivative and integrate functions.
The example below demonstrates how to compute an integral for a fitted gaussian curve.
let(x=array(0,1,2,3,4,5,6,7,8,9, 10),
y=array(4,55,1200,3028,12000,18422,13328,6426,1696,239,20),
f=gaussfit(x, y),
i=integrate(f, 0, 5))When this expression is sent to the /stream handler it
responds with:
{
"result-set": {
"docs": [
{
"i": 25261.666789766092
},
{
"EOF": true,
"RESPONSE_TIME": 3
}
]
}
}