I have a set of data and I want to compare which line describes it best (polynomials of different orders, exponential or logarithmic).
I use Python and Numpy and for polynomial fitting there is a function polyfit()
. But I found no such functions for exponential and logarithmic fitting.
Are there any? Or how to solve it otherwise?
This question is related to
python
numpy
scipy
curve-fitting
linear-regression
Well I guess you can always use:
np.log --> natural log
np.log10 --> base 10
np.log2 --> base 2
Slightly modifying IanVS's answer:
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
def func(x, a, b, c):
#return a * np.exp(-b * x) + c
return a * np.log(b * x) + c
x = np.linspace(1,5,50) # changed boundary conditions to avoid division by 0
y = func(x, 2.5, 1.3, 0.5)
yn = y + 0.2*np.random.normal(size=len(x))
popt, pcov = curve_fit(func, x, yn)
plt.figure()
plt.plot(x, yn, 'ko', label="Original Noised Data")
plt.plot(x, func(x, *popt), 'r-', label="Fitted Curve")
plt.legend()
plt.show()
This results in the following graph:
I was having some trouble with this so let me be very explicit so noobs like me can understand.
Lets say that we have a data file or something like that
# -*- coding: utf-8 -*-
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
import numpy as np
import sympy as sym
"""
Generate some data, let's imagine that you already have this.
"""
x = np.linspace(0, 3, 50)
y = np.exp(x)
"""
Plot your data
"""
plt.plot(x, y, 'ro',label="Original Data")
"""
brutal force to avoid errors
"""
x = np.array(x, dtype=float) #transform your data in a numpy array of floats
y = np.array(y, dtype=float) #so the curve_fit can work
"""
create a function to fit with your data. a, b, c and d are the coefficients
that curve_fit will calculate for you.
In this part you need to guess and/or use mathematical knowledge to find
a function that resembles your data
"""
def func(x, a, b, c, d):
return a*x**3 + b*x**2 +c*x + d
"""
make the curve_fit
"""
popt, pcov = curve_fit(func, x, y)
"""
The result is:
popt[0] = a , popt[1] = b, popt[2] = c and popt[3] = d of the function,
so f(x) = popt[0]*x**3 + popt[1]*x**2 + popt[2]*x + popt[3].
"""
print "a = %s , b = %s, c = %s, d = %s" % (popt[0], popt[1], popt[2], popt[3])
"""
Use sympy to generate the latex sintax of the function
"""
xs = sym.Symbol('\lambda')
tex = sym.latex(func(xs,*popt)).replace('$', '')
plt.title(r'$f(\lambda)= %s$' %(tex),fontsize=16)
"""
Print the coefficients and plot the funcion.
"""
plt.plot(x, func(x, *popt), label="Fitted Curve") #same as line above \/
#plt.plot(x, popt[0]*x**3 + popt[1]*x**2 + popt[2]*x + popt[3], label="Fitted Curve")
plt.legend(loc='upper left')
plt.show()
the result is: a = 0.849195983017 , b = -1.18101681765, c = 2.24061176543, d = 0.816643894816
We demonstrate features of lmfit
while solving both problems.
Given
import lmfit
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
np.random.seed(123)
# General Functions
def func_log(x, a, b, c):
"""Return values from a general log function."""
return a * np.log(b * x) + c
# Data
x_samp = np.linspace(1, 5, 50)
_noise = np.random.normal(size=len(x_samp), scale=0.06)
y_samp = 2.5 * np.exp(1.2 * x_samp) + 0.7 + _noise
y_samp2 = 2.5 * np.log(1.2 * x_samp) + 0.7 + _noise
Code
Approach 1 - lmfit
Model
Fit exponential data
regressor = lmfit.models.ExponentialModel() # 1
initial_guess = dict(amplitude=1, decay=-1) # 2
results = regressor.fit(y_samp, x=x_samp, **initial_guess)
y_fit = results.best_fit
plt.plot(x_samp, y_samp, "o", label="Data")
plt.plot(x_samp, y_fit, "k--", label="Fit")
plt.legend()
Approach 2 - Custom Model
Fit log data
regressor = lmfit.Model(func_log) # 1
initial_guess = dict(a=1, b=.1, c=.1) # 2
results = regressor.fit(y_samp2, x=x_samp, **initial_guess)
y_fit = results.best_fit
plt.plot(x_samp, y_samp2, "o", label="Data")
plt.plot(x_samp, y_fit, "k--", label="Fit")
plt.legend()
Details
You can determine the inferred parameters from the regressor object. Example:
regressor.param_names
# ['decay', 'amplitude']
To make predictions, use the ModelResult.eval()
method.
model = results.eval
y_pred = model(x=np.array([1.5]))
Note: the ExponentialModel()
follows a decay function, which accepts two parameters, one of which is negative.
See also ExponentialGaussianModel()
, which accepts more parameters.
Install the library via > pip install lmfit
.
Here's a linearization option on simple data that uses tools from scikit learn.
Given
import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import FunctionTransformer
np.random.seed(123)
# General Functions
def func_exp(x, a, b, c):
"""Return values from a general exponential function."""
return a * np.exp(b * x) + c
def func_log(x, a, b, c):
"""Return values from a general log function."""
return a * np.log(b * x) + c
# Helper
def generate_data(func, *args, jitter=0):
"""Return a tuple of arrays with random data along a general function."""
xs = np.linspace(1, 5, 50)
ys = func(xs, *args)
noise = jitter * np.random.normal(size=len(xs)) + jitter
xs = xs.reshape(-1, 1) # xs[:, np.newaxis]
ys = (ys + noise).reshape(-1, 1)
return xs, ys
transformer = FunctionTransformer(np.log, validate=True)
Code
Fit exponential data
# Data
x_samp, y_samp = generate_data(func_exp, 2.5, 1.2, 0.7, jitter=3)
y_trans = transformer.fit_transform(y_samp) # 1
# Regression
regressor = LinearRegression()
results = regressor.fit(x_samp, y_trans) # 2
model = results.predict
y_fit = model(x_samp)
# Visualization
plt.scatter(x_samp, y_samp)
plt.plot(x_samp, np.exp(y_fit), "k--", label="Fit") # 3
plt.title("Exponential Fit")
Fit log data
# Data
x_samp, y_samp = generate_data(func_log, 2.5, 1.2, 0.7, jitter=0.15)
x_trans = transformer.fit_transform(x_samp) # 1
# Regression
regressor = LinearRegression()
results = regressor.fit(x_trans, y_samp) # 2
model = results.predict
y_fit = model(x_trans)
# Visualization
plt.scatter(x_samp, y_samp)
plt.plot(x_samp, y_fit, "k--", label="Fit") # 3
plt.title("Logarithmic Fit")
Details
General Steps
x
, y
or both)np.exp()
) and fit to original dataAssuming our data follows an exponential trend, a general equation+ may be:
We can linearize the latter equation (e.g. y = intercept + slope * x) by taking the log:
Given a linearized equation++ and the regression parameters, we could calculate:
A
via intercept (ln(A)
)B
via slope (B
)Summary of Linearization Techniques
Relationship | Example | General Eqn. | Altered Var. | Linearized Eqn.
-------------|------------|----------------------|----------------|------------------------------------------
Linear | x | y = B * x + C | - | y = C + B * x
Logarithmic | log(x) | y = A * log(B*x) + C | log(x) | y = C + A * (log(B) + log(x))
Exponential | 2**x, e**x | y = A * exp(B*x) + C | log(y) | log(y-C) = log(A) + B * x
Power | x**2 | y = B * x**N + C | log(x), log(y) | log(y-C) = log(B) + N * log(x)
+Note: linearizing exponential functions works best when the noise is small and C=0. Use with caution.
++Note: while altering x data helps linearize exponential data, altering y data helps linearize log data.
Wolfram has a closed form solution for fitting an exponential. They also have similar solutions for fitting a logarithmic and power law.
I found this to work better than scipy's curve_fit. Especially when you don't have data "near zero". Here is an example:
import numpy as np
import matplotlib.pyplot as plt
# Fit the function y = A * exp(B * x) to the data
# returns (A, B)
# From: https://mathworld.wolfram.com/LeastSquaresFittingExponential.html
def fit_exp(xs, ys):
S_x2_y = 0.0
S_y_lny = 0.0
S_x_y = 0.0
S_x_y_lny = 0.0
S_y = 0.0
for (x,y) in zip(xs, ys):
S_x2_y += x * x * y
S_y_lny += y * np.log(y)
S_x_y += x * y
S_x_y_lny += x * y * np.log(y)
S_y += y
#end
a = (S_x2_y * S_y_lny - S_x_y * S_x_y_lny) / (S_y * S_x2_y - S_x_y * S_x_y)
b = (S_y * S_x_y_lny - S_x_y * S_y_lny) / (S_y * S_x2_y - S_x_y * S_x_y)
return (np.exp(a), b)
xs = [33, 34, 35, 36, 37, 38, 39, 40, 41, 42]
ys = [3187, 3545, 4045, 4447, 4872, 5660, 5983, 6254, 6681, 7206]
(A, B) = fit_exp(xs, ys)
plt.figure()
plt.plot(xs, ys, 'o-', label='Raw Data')
plt.plot(xs, [A * np.exp(B *x) for x in xs], 'o-', label='Fit')
plt.title('Exponential Fit Test')
plt.xlabel('X')
plt.ylabel('Y')
plt.legend(loc='best')
plt.tight_layout()
plt.show()
You can also fit a set of a data to whatever function you like using curve_fit
from scipy.optimize
. For example if you want to fit an exponential function (from the documentation):
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
def func(x, a, b, c):
return a * np.exp(-b * x) + c
x = np.linspace(0,4,50)
y = func(x, 2.5, 1.3, 0.5)
yn = y + 0.2*np.random.normal(size=len(x))
popt, pcov = curve_fit(func, x, yn)
And then if you want to plot, you could do:
plt.figure()
plt.plot(x, yn, 'ko', label="Original Noised Data")
plt.plot(x, func(x, *popt), 'r-', label="Fitted Curve")
plt.legend()
plt.show()
(Note: the *
in front of popt
when you plot will expand out the terms into the a
, b
, and c
that func
is expecting.)
Source: Stackoverflow.com