scipy integrate.quad return an incorrect value

Question

i use scipy integrate.quad to calc cdf of normal distribution:

def nor(delta, mu, x):
    return 1 / (math.sqrt(2 * math.pi) * delta) * np.exp(-np.square(x - mu) / (2 * np.square(delta)))


delta = 0.1
mu = 0
t = np.arange(4.0, 10.0, 1)
nor_int = lambda t: integrate.quad(lambda x: nor(delta, mu, x), -np.inf, t)
nor_int_vec = np.vectorize(nor_int)

s = nor_int_vec(t)
for i in zip(s[0],s[1]): 
    print i

while it print as follows:

(1.0000000000000002, 1.2506543424265854e-08)
(1.9563704110140217e-11, 3.5403445591955275e-11)
(1.0000000000001916, 1.2616577562700088e-08)
(1.0842532749783998e-34, 1.9621183122960244e-34)
(4.234531567162006e-09, 7.753407284370446e-09)
(1.0000000000001334, 1.757986959115912e-10)

for some x, it return a value approximate to zero, it should be return 1.
can somebody tell me what is wrong?

Asked By: Ronaldinho

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Source

Answer 1

Same reason as in why does quad return both zeros when integrating a simple Gaussian pdf at a very small variance? but seeing as I can’t mark it as a duplicate, here goes:

You are integrating a function with tight localization (at scale delta) over a very large (in fact infinite) interval. The integration routine can simply miss the part of the interval where the function is substantially different from 0, judging it to be 0 instead. Some guidance is required. The parameter points can be used to this effect (see the linked question) but since quad over an infinite interval does not support it, the interval has to be manually split, like so:

for t in range(4, 10):
    int1 = integrate.quad(lambda x: nor(delta, mu, x), -np.inf, mu - 10*delta)[0] 
    int2 = integrate.quad(lambda x: nor(delta, mu, x), mu - 10*delta, t)[0] 
    print(int1 + int2)

This prints 1 or nearly 1 every time. I picked mu-10*delta as a point to split on, figuring most of the function lies to the right of it, no matter what mu and delta are.

Notes:

Use np.sqrt etc; there is usually no reason for put math functions in NumPy code. The NumPy versions are available and are vectorized.
Applying np.vectorize to quad is not doing anything besides making the code longer and slightly harder to read. Use a normal Python loop or list comprehension. See NumPy vectorization with integration

Answered By: user6655984

scipy integrate.quad return an incorrect value

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