Table of Contents¶
1 Requirements
2 KL Functions
2.1 klGauss
2.2 klBern
3 Threshold functions
3.1 Threshold for GLR Bernoulli
KL Functions¶
klGauss¶
Generating some data:
[3]:
def random_Gaussian(sig2=0.25):
return sig2 * np.random.randn()
[4]:
%timeit (random_Gaussian(), random_Gaussian())
1.12 µs ± 74.1 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)
In pure Python:
[5]:
def klGauss(x: float, y: float, sig2=0.25) -> float:
return (x - y)**2 / (2 * sig2**x)
[6]:
%timeit klGauss(random_Gaussian(), random_Gaussian())
1.37 µs ± 54.3 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)
With numba:
[7]:
@numba.jit(nopython=True)
def klGauss_numba(x: float, y: float, sig2=0.25) -> float:
return (x - y)**2 / (2 * sig2**x)
[8]:
help(klGauss_numba)
Help on CPUDispatcher in module __main__:
klGauss_numba(x:float, y:float, sig2=0.25) -> float
[9]:
%timeit klGauss_numba(random_Gaussian(), random_Gaussian())
The slowest run took 13.64 times longer than the fastest. This could mean that an intermediate result is being cached.
4.87 µs ± 7.2 µs per loop (mean ± std. dev. of 7 runs, 1 loop each)
[10]:
%timeit klGauss_numba(random_Gaussian(), random_Gaussian())
1.43 µs ± 29.9 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)
[11]:
print(f"Speed up using Numba for klGauss was: {(1290-993)/(20300-993):.2g} faster!")
Speed up using Numba for klGauss was: 0.015 faster!
With Cython
[12]:
%%cython --annotate
def klGauss_cython(double x, double y, double sig2=0.25) -> double:
return (x - y)**2 / (2 * sig2**x)
[12]:
Generated by Cython 0.29.1
Yellow lines hint at Python interaction.
Click on a line that starts with a "+" to see the C code that Cython generated for it.
1:
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[13]:
help(klGauss_cython)
Help on built-in function klGauss_cython in module _cython_magic_49fb421fab37e1decb48fdb471e97e6b:
klGauss_cython(...)
[14]:
%timeit klGauss_cython(random_Gaussian(), random_Gaussian())
1.21 µs ± 50.7 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)
[15]:
print(f"Speed up using Cython for klGauss was: {(1290-993)/(1100-993):.2g} faster!")
Speed up using Cython for klGauss was: 2.8 faster!
klBern¶
Generating some data:
[16]:
def random_Bern():
return np.random.random()
[17]:
%timeit (random_Bern(), random_Bern())
786 ns ± 30.3 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)
In pure Python:
[18]:
from math import log
def klBern(x: float, y: float) -> float:
x = max(1e-7, min(1 - 1e-7, x))
x = max(1e-7, min(1 - 1e-7, x))
return x * log(x/y) + (1-x) * log((1-x)/(1-y))
[19]:
%timeit klBern(random_Bern(), random_Bern())
1.93 µs ± 96.3 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)
With numba:
[20]:
from math import log
@numba.jit(nopython=True)
def klBern_numba(x: float, y: float) -> float:
x = max(1e-7, min(1 - 1e-7, x))
x = max(1e-7, min(1 - 1e-7, x))
return x * log(x/y) + (1-x) * log((1-x)/(1-y))
[21]:
help(klBern_numba)
Help on CPUDispatcher in module __main__:
klBern_numba(x:float, y:float) -> float
[22]:
%timeit klBern_numba(random_Bern(), random_Bern())
1.27 µs ± 128 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)
[23]:
%timeit klBern_numba(random_Bern(), random_Bern())
1.14 µs ± 42.7 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)
[24]:
print(f"Speed up using Numba for klBern was: {(1740-753)/(996-753):.2g} faster!")
Speed up using Numba for klBern was: 4.1 faster!
With Cython
[25]:
%load_ext cython
The cython extension is already loaded. To reload it, use:
%reload_ext cython
[26]:
%%cython --annotate
from libc.math cimport log
def klBern_cython(double x, double y) -> double:
x = max(1e-7, min(1 - 1e-7, x))
x = max(1e-7, min(1 - 1e-7, x))
return x * log(x/y) + (1-x) * log((1-x)/(1-y))
[26]:
Generated by Cython 0.29.1
Yellow lines hint at Python interaction.
Click on a line that starts with a "+" to see the C code that Cython generated for it.
1: from libc.math cimport log
2:
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[27]:
help(klBern_cython)
Help on built-in function klBern_cython in module _cython_magic_0ce0e64bf037172e33037b12661fd5c4:
klBern_cython(...)
[28]:
%timeit klBern_cython(random_Bern(), random_Bern())
922 ns ± 36.6 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)
[29]:
print(f"Speed up using Cython for klBern was: {(1740-753)/(861-753):.2g} faster!")
Speed up using Cython for klBern was: 9.1 faster!
Threshold functions¶
Threshold for GLR Bernoulli¶
Generating some data:
[30]:
def random_t0_s_t_delta(min_t: int=100, max_t: int=1000) -> (int, int, int, float):
t0 = 0
t = np.random.randint(min_t, max_t + 1)
s = np.random.randint(t0, t)
delta = np.random.choice([0.1, 0.05, 0.01, 0.005, 0.001, max(0.0005, 1/t)])
return (t0, s, t, delta)
[31]:
%timeit random_t0_s_t_delta()
7.04 µs ± 148 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
In pure Python:
[32]:
def threshold(t0: int, s: int, t: int, delta: float) -> float:
return np.log((s - t0 + 1) * (t - s) / delta)
[33]:
%timeit threshold(*random_t0_s_t_delta())
10.2 µs ± 175 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
It’s way faster to use math.log instead of numpy.log (of course)!
[34]:
from math import log
def threshold2(t0: int, s: int, t: int, delta: float) -> float:
return log((s - t0 + 1) * (t - s) / delta)
[35]:
%timeit threshold2(*random_t0_s_t_delta())
7.93 µs ± 132 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
In numba:
[36]:
from math import log
@numba.jit(nopython=True)
def threshold_numba(t0: int, s: int, t: int, delta: float) -> float:
return log((s - t0 + 1) * (t - s) / delta)
[37]:
help(threshold_numba)
Help on CPUDispatcher in module __main__:
threshold_numba(t0:int, s:int, t:int, delta:float) -> float
[38]:
%timeit threshold_numba(*random_t0_s_t_delta())
7.57 µs ± 105 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
[39]:
print(f"Speed up using Cython for thresold was: {(7510-7200)/(7750-7200):.2g} faster!")
Speed up using Cython for thresold was: 0.56 faster!
In Cython:
[40]:
%%cython --annotate
from libc.math cimport log
cpdef double threshold_cython(int t0, int s, int t, double delta):
return log((s - t0 + 1) * (t - s) / delta)
[40]:
Generated by Cython 0.29.1
Yellow lines hint at Python interaction.
Click on a line that starts with a "+" to see the C code that Cython generated for it.
1: from libc.math cimport log
2:
+3: cpdef double threshold_cython(int t0, int s, int t, double delta):
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[41]:
%%cython --annotate
from libc.math cimport log
def threshold_cython2(int t0, int s, int t, double delta) -> double:
return log((s - t0 + 1) * (t - s) / delta)
[41]:
Generated by Cython 0.29.1
Yellow lines hint at Python interaction.
Click on a line that starts with a "+" to see the C code that Cython generated for it.
1: from libc.math cimport log
2:
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[42]:
help(threshold_cython)
Help on built-in function threshold_cython in module _cython_magic_4e48b0e25538ff2883ba5a92832943be:
threshold_cython(...)
[43]:
%timeit threshold_cython(*random_t0_s_t_delta())
7.24 µs ± 103 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
[44]:
%timeit threshold_cython2(*random_t0_s_t_delta())
8.8 µs ± 994 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
[45]:
print(f"Speed up using Cython for thresold was: {abs(7510-7200)/abs(7070-7200):.2g} faster!")
Speed up using Cython for thresold was: 2.4 faster!
[ ]: