Apply kurtosis to a distribution in python

Question

I have a dataset which is in the format of

frequency, direction, normalised power spectral density, spread, skewness, kurtosis

I am able to visualise the distribution of a specific record using the code from the top answer in skew normal distribution in scipy but I am not sure how to apply a kurtosis value to a distribution?

from scipy import linspace
from scipy import pi,sqrt,exp
from scipy.special import erf
from pylab import plot,show

def pdf(factor, x):
    return (100*factor)/sqrt(2*pi) * exp(-x**2/2)

def cdf(x):
    return (1 + erf(x/sqrt(2))) / 2

def skew(x,e=0,w=1,a=0, norm_psd=1):
    t = (x-e) / w
    return 2 / w * pdf(norm_psd, t) * cdf(a*t)

n = 540
e = 341.9 # direction
w = 59.3 # spread
a = 3.3 # skew
k = 4.27 # kurtosis
n_psd = 0.5 # normalised power spectral density
x = linspace(-90, 450, n) 


p = skew(x, e, w, a, n_psd)
print max(p)
plot(x,p)
show()

Edit: I removed skew normal from my title as I don't think it is actually possible to apply a kurtosis value to the above distribution, I think a different distribution is necessary, as direction is involved a distribution from circular statistics may be more appropriate?

Thanks to the answer below I can apply kurtosis using the pdf_mvsk function demonstrated in the code below, unfortunately my skew values cause a negative y value, but the answer satisfies my question.

import numpy as np
import matplotlib.pyplot as plt
import statsmodels.sandbox.distributions.extras as extras

pdffunc = extras.pdf_mvsk([341.9, 59.3, 3.3, 4.27])
range = np.arange(0, 360, 0.1)
plt.plot(range, pdffunc(range))
plt.show()

Answer 1

If you have mean, standard deviation, skew and kurtosis, then you can build an approximately normal distribution with those moments using Gram-Charlier expansion.

I looked into this some time ago, scipy.stats had a function that was wrong and was removed.

I don't remember what the status of this is, since it was a long time ago that I put this in the statsmodels sandbox http://statsmodels.sourceforge.net/devel/generated/statsmodels.sandbox.distributions.extras.pdf_mvsk.html#statsmodels.sandbox.distributions.extras.pdf_mvsk