TopolearnAreas, probabilities, expectations and totals over time are integrals with no closed form.
scipy.integrate handles two situations.
You have a function. integrate.quad(f, a, b) adaptively samples f until the estimated error is below tolerance. It returns (value, abs_error_estimate). Infinite limits work: quad(f, 0, np.inf). Extra parameters go in args=(...). For trickier integrands, points=[...] marks known kinks or discontinuities, and limit raises the number of subintervals; an IntegrationWarning means the result may be inaccurate. For multiple dimensions there are dblquad, tplquad and nquad, and quad_vec integrates vector-valued functions.
You have samples. When y is measured at points x (sensor readings, a sampled curve), use:
integrate.trapezoid(y, x): straight lines between points,integrate.simpson(y, x=x): parabolic segments, more accurate for smooth data,integrate.cumulative_trapezoid(y, x, initial=0): the running integral, for example distance from velocity.
The old names trapz, simps and cumtrapz were removed; use the new ones.
quad is accurate for smooth one-dimensional integrands. For high-dimensional integrals, Monte Carlo methods (scipy.stats.qmc helps generate low-discrepancy points) scale better.
Example
import numpy as np
from scipy import integrate
val, err = integrate.quad(np.sin, 0, np.pi)
print(round(val, 6), err < 1e-10) # 2.0 True
# Gaussian integral over the whole real line
val, _ = integrate.quad(lambda x: np.exp(-x ** 2), -np.inf, np.inf)
print(round(val, 6), round(np.sqrt(np.pi), 6)) # 1.772454 1.772454
t = np.linspace(0, 10, 11) # seconds
v = 3.0 * t # velocity, m/s
print(integrate.trapezoid(v, t)) # 150.0 (distance in m)
print(integrate.cumulative_trapezoid(v, t, initial=0)[:4]) # [ 0. 1.5 6. 13.5]Remember
quad(f, a, b)returns (value, error estimate) and accepts infinite limits- Use
trapezoid,simpson,cumulative_trapezoidfor sampled data - Watch for
IntegrationWarning; mark discontinuities withpoints=
Watch out for
- Calling the removed
trapz/simpsnames from old tutorials.