Given an array of x and y values, the following code will calculate a regression curve for these data points.
# calculate polynomial
z = np.polyfit(x, y, 5)
f = np.poly1d(z)
# calculate new x's and y's
x_new = np.linspace(x[0], x[-1], 50)
y_new = f(x_new)
plt.plot(x,y,'o', x_new, y_new)
plt.xlim([x[0]-1, x[-1] + 1 ])
plt.show()
How can I use the above to derive the actual equation for this curve?
If you want to show the equation, you can use sympy to output latex:
from sympy import S, symbols, printing
from matplotlib import pyplot as plt
import numpy as np
x=np.linspace(0,1,100)
y=np.sin(2 * np.pi * x)
p = np.polyfit(x, y, 5)
f = np.poly1d(p)
# calculate new x's and y's
x_new = np.linspace(x[0], x[-1], 50)
y_new = f(x_new)
x = symbols("x")
poly = sum(S("{:6.2f}".format(v))*x**i for i, v in enumerate(p[::-1]))
eq_latex = printing.latex(poly)
plt.plot(x_new, y_new, label="${}$".format(eq_latex))
plt.legend(fontsize="small")
plt.show()
the result:
Construct a simple example:
In [94]: x=np.linspace(0,1,100)
In [95]: y=2*x**3-3*x**2+x-1
In [96]: z=np.polyfit(x,y,3)
In [97]: z
Out[97]: array([ 2., -3., 1., -1.])
The z coefficients correspond to the [2,-3,1,-1] I used to construct y.
In [98]: f=np.poly1d(z)
In [99]: f
Out[99]: poly1d([ 2., -3., 1., -1.])
The str, or print, string for f is a representation of this polynomial equation. But it's the z coeff that defines the equation.
In [100]: print(f)
3 2
2 x - 3 x + 1 x - 1
In [101]: str(f)
Out[101]: ' 3 2\n2 x - 3 x + 1 x - 1'
What else do you mean by 'actual equation'?
polyval will evaluate f at a specific set of x. Thus to recreate y, use polyval(f,x):
In [107]: np.allclose(np.polyval(f,x),y)
Out[107]: True
If you only want to see the equation on your screen to get an impression about the equation, you only need to add the below line:
print(f)
Here comes more details:
polyfit returns a vector of coefficients of the polynomial fit. poly1d takes this vector and make a polynomial function out of it.
For example (from Numpy documentation for poly1d ):
p = np.poly1d([1, 2, 3])
>>> print(np.poly1d(p))
2
1 x + 2 x + 3
Or, build a string in a loop (this one expects coefficients for increasing order terms, so use np.flip(np.polyfit()), and pass the output to the following function.
def polyname(poly):
name = f"$y = {round(poly[0],2)}$"
for i in range(1, len(poly)):
if i == 1:
if poly[i] >= 0:
name = name + f"$ + {round(poly[i],2)}x$"
else:
name = name + f"$ - {abs(round(poly[i],2))}x$"
else:
if poly[i] >= 0:
name = name + f"$ + {round(poly[i],2)}x^{{{i}}}$"
else:
name = name + f"$ - {abs(round(poly[i],2))}x^{{{i}}}$"
return name
