62 lines
1.7 KiB
Python
62 lines
1.7 KiB
Python
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"""
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Created on Sat Mar 13 07:56:22 2010
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Author: josef-pktd
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"""
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import sympy as sy
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def pdf(x, mu, sigma):
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"""Return the probability density function as an expression in x"""
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#x = sy.sympify(x)
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return 1/(sigma*sy.sqrt(2*sy.pi)) * sy.exp(-(x-mu)**2 / (2*sigma**2))
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def cdf(x, mu, sigma):
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"""Return the cumulative density function as an expression in x"""
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#x = sy.sympify(x)
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return (1+sy.erf((x-mu)/(sigma*sy.sqrt(2))))/2
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mu = sy.Symbol('mu')
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sigma = sy.Symbol('sigma')
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sigma2 = sy.Symbol('sigma2')
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x = sy.Symbol('x')
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y = sy.Symbol('y')
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df = sy.Symbol('df')
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s = sy.Symbol('s')
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dldxnorm = sy.log(pdf(x, mu,sigma)).diff(x)
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print(sy.simplify(dldxnorm))
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print(sy.diff(sy.log(sy.gamma((s+1)/2)),s))
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print(sy.diff((df+1)/2. * sy.log(1+df/(df-2)), df))
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#standard t distribution, not verified
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tllf1 = sy.log(sy.gamma((df+1)/2.)) - sy.log(sy.gamma(df/2.)) - 0.5*sy.log((df)*sy.pi)
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tllf2 = (df+1.)/2. * sy.log(1. + (y-mu)**2/(df)/sigma2) + 0.5 * sy.log(sigma2)
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tllf2std = (df+1.)/2. * sy.log(1. + y**2/df) + 0.5
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tllf = tllf1 - tllf2
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print(tllf1.diff(df))
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print(tllf2.diff(y))
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dlddf = (tllf1-tllf2).diff(df)
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print(dlddf)
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print(sy.cse(dlddf))
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print('\n derivative of loglike of t distribution wrt df')
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for k,v in sy.cse(dlddf)[0]:
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print(k, '=', v)
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print(sy.cse(dlddf)[1][0])
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print('\nstandard t distribution, dll_df, dll_dy')
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tllfstd = tllf1 - tllf2std
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print(tllfstd.diff(df))
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print(tllfstd.diff(y))
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print('\n')
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print(dlddf.subs(dict(y=1,mu=1,sigma2=1.5,df=10.0001)))
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print(dlddf.subs(dict(y=1,mu=1,sigma2=1.5,df=10.0001)).evalf())
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# Note: derivatives of nested function does not work in sympy
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# at least not higher order derivatives (second or larger)
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# looks like print(failure
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