Calculating the area under a curve given a set of coordinates, without knowing the function

Question

I have one list of 100 numbers as height for Y axis, and as length for X axis: 1 to 100 with a constant step of 5. I need to calculate the Area that it is included by the curve of the (x,y) points, and the X axis, using rectangles and Scipy. Do I have to find the function of this curve? or not? ... almost all the examples I have read are about a specific equation for the Y axis. In my case there is no equation, just data from a list. The classic solution is to add or the Y points and multiple by the step X distance... using Scipy any idea?

Please, can anyone recommend any book which focusing on numerical (finite elementary) methods, using Scipy and Numpy? ...

Gudrunguelderrose

Answer 1

The numpy and scipy libraries include the composite trapezoidal (numpy.trapz) and Simpson's (scipy.integrate.simpson) rules.

Here's a simple example. In both trapz and simpson, the argument dx=5 indicates that the spacing of the data along the x axis is 5 units.

import numpy as np
from scipy.integrate import simpson
from numpy import trapz


# The y values.  A numpy array is used here,
# but a python list could also be used.
y = np.array([5, 20, 4, 18, 19, 18, 7, 4])

# Compute the area using the composite trapezoidal rule.
area = trapz(y, dx=5)
print("area =", area)

# Compute the area using the composite Simpson's rule.
area = simpson(y, dx=5)
print("area =", area)

Output:

area = 452.5
area = 460.0

Wolk

that's great! ... Both answers help me to understand and solve any questions I had. I would like to ask something relative... Do you recommend to use arrays and not list? is something that help the user ? or the logic and speed of the algorithm? – Gudrunguelderrose

The first thing trapz and simps functions do is convert the y argument into a numpy array, so it doesn't really matter. You might look at your code that generates the y values, and see if that would benefit from the use of additional numpy or scipy functions. If so, y would already be an array when you passed it to simps. – Wolk

which one these two methods are more accurate? – Extraterritorial

Both are accurate – Glory

Answer 2

If you have sklearn installed, a simple alternative is to use sklearn.metrics.auc

This computes the area under the curve using the trapezoidal rule given arbitrary x, and y array

import numpy as np
from sklearn.metrics import auc

dx = 5
xx = np.arange(1,100,dx)
yy = np.arange(1,100,dx)

print('computed AUC using sklearn.metrics.auc: {}'.format(auc(xx,yy)))
print('computed AUC using np.trapz: {}'.format(np.trapz(yy, dx = dx)))

both output the same area: 4607.5

the advantage of sklearn.metrics.auc is that it can accept arbitrarily-spaced 'x' array, just make sure it is ascending otherwise the results will be incorrect

Soileau

Answer 3

You can use Simpsons rule or the Trapezium rule to calculate the area under a graph given a table of y-values at a regular interval.

Python script that calculates Simpsons rule:

def integrate(y_vals, h):
    i = 1
    total = y_vals[0] + y_vals[-1]
    for y in y_vals[1:-1]:
        if i % 2 == 0:
            total += 2 * y
        else:
            total += 4 * y
        i += 1
    return total * (h / 3.0)

h is the offset (or gap) between y values, and y_vals is an array of well, y values.

Example (In same file as above function):

y_values = [13, 45.3, 12, 1, 476, 0]
interval = 1.2
area = integrate(y_values, interval)
print("The area is", area)

Imperial

I'm not sure.. it could be really tricky finding the equation of a line, especially if you don't know the type of curve it is (exponential, parabola, etc) – Imperial

THANK you ... I really appreciate your help... just y_vals is array ? or my Y data list (H[i]) ? Is better to use arrays, not a list? do recommend to change my list to array? and about h, "h is the x-interval between y values" ? .. little help on this... on the wiki example say: """f=function, a=initial value, b=end value, n=number of intervals of size h, n must be even""" h = float(b - a) / n .. is the same h? so is the distance between each step? – Gudrunguelderrose

Yes, h is the interval between each step. y_vals can be anything that can be iterated in a for loop. I just always use arrays because they are easy to use. – Imperial

... so the y_vals can be list or array that defined in previous part of the algorithm ? in my case the list is defined as H.... do I have to insert a for loop for the def integrate? – Gudrunguelderrose

What if the data is not equally spaced out? – Dilatometer

What if is a time series? I mean, x-axis is time instead of int values – Debauchery

Time increments happen periodically. Begins at zero and proceeds incrementally. Any other reason why you raised this question? – Phrenic