200 lines
6.7 KiB
Python
200 lines
6.7 KiB
Python
import numpy as np
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from numpy import poly1d
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from scipy.special import beta
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# The following code was used to generate the Pade coefficients for the
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# Tukey Lambda variance function. Version 0.17 of mpmath was used.
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#---------------------------------------------------------------------------
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# import mpmath as mp
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#
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# mp.mp.dps = 60
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#
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# one = mp.mpf(1)
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# two = mp.mpf(2)
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#
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# def mpvar(lam):
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# if lam == 0:
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# v = mp.pi**2 / three
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# else:
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# v = (two / lam**2) * (one / (one + two*lam) -
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# mp.beta(lam + one, lam + one))
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# return v
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#
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# t = mp.taylor(mpvar, 0, 8)
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# p, q = mp.pade(t, 4, 4)
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# print("p =", [mp.fp.mpf(c) for c in p])
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# print("q =", [mp.fp.mpf(c) for c in q])
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#---------------------------------------------------------------------------
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# Pade coefficients for the Tukey Lambda variance function.
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_tukeylambda_var_pc = [3.289868133696453, 0.7306125098871127,
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-0.5370742306855439, 0.17292046290190008,
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-0.02371146284628187]
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_tukeylambda_var_qc = [1.0, 3.683605511659861, 4.184152498888124,
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1.7660926747377275, 0.2643989311168465]
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# numpy.poly1d instances for the numerator and denominator of the
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# Pade approximation to the Tukey Lambda variance.
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_tukeylambda_var_p = poly1d(_tukeylambda_var_pc[::-1])
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_tukeylambda_var_q = poly1d(_tukeylambda_var_qc[::-1])
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def tukeylambda_variance(lam):
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"""Variance of the Tukey Lambda distribution.
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Parameters
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----------
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lam : array_like
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The lambda values at which to compute the variance.
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Returns
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-------
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v : ndarray
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The variance. For lam < -0.5, the variance is not defined, so
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np.nan is returned. For lam = 0.5, np.inf is returned.
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Notes
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-----
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In an interval around lambda=0, this function uses the [4,4] Pade
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approximation to compute the variance. Otherwise it uses the standard
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formula (https://en.wikipedia.org/wiki/Tukey_lambda_distribution). The
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Pade approximation is used because the standard formula has a removable
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discontinuity at lambda = 0, and does not produce accurate numerical
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results near lambda = 0.
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"""
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lam = np.asarray(lam)
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shp = lam.shape
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lam = np.atleast_1d(lam).astype(np.float64)
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# For absolute values of lam less than threshold, use the Pade
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# approximation.
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threshold = 0.075
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# Play games with masks to implement the conditional evaluation of
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# the distribution.
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# lambda < -0.5: var = nan
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low_mask = lam < -0.5
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# lambda == -0.5: var = inf
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neghalf_mask = lam == -0.5
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# abs(lambda) < threshold: use Pade approximation
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small_mask = np.abs(lam) < threshold
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# else the "regular" case: use the explicit formula.
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reg_mask = ~(low_mask | neghalf_mask | small_mask)
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# Get the 'lam' values for the cases where they are needed.
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small = lam[small_mask]
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reg = lam[reg_mask]
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# Compute the function for each case.
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v = np.empty_like(lam)
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v[low_mask] = np.nan
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v[neghalf_mask] = np.inf
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if small.size > 0:
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# Use the Pade approximation near lambda = 0.
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v[small_mask] = _tukeylambda_var_p(small) / _tukeylambda_var_q(small)
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if reg.size > 0:
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v[reg_mask] = (2.0 / reg**2) * (1.0 / (1.0 + 2 * reg) -
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beta(reg + 1, reg + 1))
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v.shape = shp
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return v
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# The following code was used to generate the Pade coefficients for the
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# Tukey Lambda kurtosis function. Version 0.17 of mpmath was used.
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#---------------------------------------------------------------------------
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# import mpmath as mp
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#
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# mp.mp.dps = 60
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#
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# one = mp.mpf(1)
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# two = mp.mpf(2)
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# three = mp.mpf(3)
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# four = mp.mpf(4)
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#
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# def mpkurt(lam):
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# if lam == 0:
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# k = mp.mpf(6)/5
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# else:
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# numer = (one/(four*lam+one) - four*mp.beta(three*lam+one, lam+one) +
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# three*mp.beta(two*lam+one, two*lam+one))
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# denom = two*(one/(two*lam+one) - mp.beta(lam+one,lam+one))**2
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# k = numer / denom - three
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# return k
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#
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# # There is a bug in mpmath 0.17: when we use the 'method' keyword of the
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# # taylor function and we request a degree 9 Taylor polynomial, we actually
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# # get degree 8.
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# t = mp.taylor(mpkurt, 0, 9, method='quad', radius=0.01)
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# t = [mp.chop(c, tol=1e-15) for c in t]
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# p, q = mp.pade(t, 4, 4)
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# print("p =", [mp.fp.mpf(c) for c in p])
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# print("q =", [mp.fp.mpf(c) for c in q])
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#---------------------------------------------------------------------------
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# Pade coefficients for the Tukey Lambda kurtosis function.
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_tukeylambda_kurt_pc = [1.2, -5.853465139719495, -22.653447381131077,
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0.20601184383406815, 4.59796302262789]
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_tukeylambda_kurt_qc = [1.0, 7.171149192233599, 12.96663094361842,
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0.43075235247853005, -2.789746758009912]
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# numpy.poly1d instances for the numerator and denominator of the
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# Pade approximation to the Tukey Lambda kurtosis.
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_tukeylambda_kurt_p = poly1d(_tukeylambda_kurt_pc[::-1])
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_tukeylambda_kurt_q = poly1d(_tukeylambda_kurt_qc[::-1])
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def tukeylambda_kurtosis(lam):
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"""Kurtosis of the Tukey Lambda distribution.
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Parameters
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----------
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lam : array_like
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The lambda values at which to compute the variance.
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Returns
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-------
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v : ndarray
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The variance. For lam < -0.25, the variance is not defined, so
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np.nan is returned. For lam = 0.25, np.inf is returned.
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"""
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lam = np.asarray(lam)
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shp = lam.shape
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lam = np.atleast_1d(lam).astype(np.float64)
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# For absolute values of lam less than threshold, use the Pade
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# approximation.
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threshold = 0.055
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# Use masks to implement the conditional evaluation of the kurtosis.
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# lambda < -0.25: kurtosis = nan
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low_mask = lam < -0.25
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# lambda == -0.25: kurtosis = inf
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negqrtr_mask = lam == -0.25
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# lambda near 0: use Pade approximation
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small_mask = np.abs(lam) < threshold
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# else the "regular" case: use the explicit formula.
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reg_mask = ~(low_mask | negqrtr_mask | small_mask)
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# Get the 'lam' values for the cases where they are needed.
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small = lam[small_mask]
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reg = lam[reg_mask]
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# Compute the function for each case.
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k = np.empty_like(lam)
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k[low_mask] = np.nan
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k[negqrtr_mask] = np.inf
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if small.size > 0:
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k[small_mask] = _tukeylambda_kurt_p(small) / _tukeylambda_kurt_q(small)
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if reg.size > 0:
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numer = (1.0 / (4 * reg + 1) - 4 * beta(3 * reg + 1, reg + 1) +
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3 * beta(2 * reg + 1, 2 * reg + 1))
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denom = 2 * (1.0/(2 * reg + 1) - beta(reg + 1, reg + 1))**2
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k[reg_mask] = numer / denom - 3
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# The return value will be a numpy array; resetting the shape ensures that
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# if `lam` was a scalar, the return value is a 0-d array.
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k.shape = shp
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return k
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