def chi2ind_sample_size(starting_n,
effect_size,
dof,
power=0.8,
sig_level=0.05):
"""
Tested against reference in NCSS PASS manual:
https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/Chi-Square_Tests.pdf
(Page 7)
"""
n = starting_n # initialisation
sim_power = 0
while sim_power < power:
n += 1
ncp = (effect_size**2) * n
chi2_null = scs.chi2(df=dof, loc=0, scale=1)
chi2_alt = scs.ncx2(df=dof, nc=ncp)
cv = chi2_null.ppf(1 - sig_level)
sim_power = 1 - chi2_alt.cdf(cv)
print('ncp: ', ncp)
print('Critical chi2: ', cv)
print('Actual Power: ', sim_power)
return np.ceil(n)
def perform_kstest(k, lamda, theta, X_0, T, simulated):
c = (2 * k) / ((1 - np.exp(-k * T)) * theta**2)
df = 4 * k * lamda / theta**2
nc = 2 * c * X_0 * np.exp(-k * T)
rv = ncx2(df, nc, scale=1 / (2 * c))
cdf = lambda x: rv.cdf(x)
return kstest(simulated, cdf=cdf)
def eval_log_lik(t, vt, bet=1, mu=1, sig=1):
c = 2 * bet / (sig**2 * (1 - np.exp(-bet * np.diff(t))))
zt = 2 * c * vt[1:]
nc = 2 * np.exp(-bet * np.diff(t)) * c * vt[:-1]
df = 4 * bet * mu / sig**2
return (ncx2(df, nc).logpdf(zt) + np.log(2 * c)).sum()
def pdf(self,E,Ei=0):
""" calculates decision problem densities under H0 and H1
Parameters
----------
E : float
Legitimate Signal Energy
Ei : float
Interferer Signal Energy
See Also
--------
scipy.stats.chi2
scipy.stats.ncx2
"""
self.E = E
self.Ei = Ei
# H0 : Noise only (no interferer)
# central chi square
if Ei==0:
self.pH0 = st.chi2(self.order,
scale=self.scale)
else:
# H0 : Noise + interference
# non central chi square
self.pH0 = st.ncx2(self.order,
scale=self.scale,
nc=Ei/(self.scale))
# H1 : Signal + interference + Noise
# non central chi square
self.pH1 = st.ncx2(self.order,
scale=self.scale,
nc=(E+Ei)/(self.scale))
def construct_chisq_rv(lmb, mu=None, bill=None, dist=None, dim=None):
msg = 'Must have mu and bill, or distance between mu and bill and number of dimensions'
if dist is None:
assert (mu is not None and bill is not None), msg
delta = np.array(mu) - np.array(bill)
nc = np.sum(delta**2)
dof = len(mu)
else:
assert (dim is not None), msg
nc = dist**2
dof = dim
rv = ncx2(dof, nc)
return rv, dof, nc
def show_probplot(k, lamda, theta, X_0, T, simulated):
c = (2 * k) / ((1 - np.exp(-k * T)) * theta**2)
df = 4 * k * lamda / theta**2
nc = 2 * c * X_0 * np.exp(-k * T)
rv = ncx2(df, nc, scale=1 / (2 * c))
x, y = probplot(simulated, dist=rv, fit=False)
plt.plot(x, y, "bo")
plt.title("Probability Plot")
plt.xlabel("Theoretical quantiles")
plt.ylabel("Ordered Values")
x = np.linspace(min(x[0], y[0]), max(x[-1], y[-1]), 2)
plt.plot(x, x, "k--")
plt.gca().set_aspect("equal")
def pdf(self, E, Ei=0):
""" calculates decision problem densities under H0 and H1
Parameters
----------
E : float
Legitimate Signal Energy
Ei : float
Interferer Signal Energy
See Also
--------
scipy.stats.chi2
scipy.stats.ncx2
"""
self.E = E
self.Ei = Ei
# H0 : Noise only (no interferer)
# central chi square
if Ei == 0:
self.pH0 = st.chi2(self.order, scale=self.scale)
else:
# H0 : Noise + interference
# non central chi square
self.pH0 = st.ncx2(self.order,
scale=self.scale,
nc=Ei / (self.scale))
# H1 : Signal + interference + Noise
# non central chi square
self.pH1 = st.ncx2(self.order,
scale=self.scale,
nc=(E + Ei) / (self.scale))
def _sample(w, dof, loc, n=1):
"""
Sample from 𝑋.
Where
𝑋 = w[0]⋅χ²(dof[0], loc[1]) + w[1]⋅χ²(dof[1], loc[1]) + ...
"""
samples = []
for _ in range(n):
x = 0.0
for i, (d, l) in enumerate(zip(dof, loc)):
x += w[i] * ncx2(d, l).rvs()
samples.append(x)
return asarray(samples)
def power_confidence_limits(preal, n=1, c=0.95):
"""Confidence limits on power, given a (theoretical) signal power.
This is to be used when we *expect* a given power (e.g. from the pulsed
fraction measured in previous observations) and we want to know the
range of values the measured power could take to a given confidence level.
Adapted from Vaughan et al. 1994, noting that, after appropriate
normalization of the spectral stats, the distribution of powers in the PDS
and the Z^2_n searches is always described by a noncentral chi squared
distribution.
Parameters
----------
preal: float
The theoretical signal-generated value of power
Other Parameters
----------------
n: int
The number of summed powers to obtain the result. It can be multiple
harmonics of the PDS, adjacent bins in a PDS summed to collect all the
power in a QPO, or the n in Z^2_n
c: float
The confidence level (e.g. 0.95=95%)
Returns
-------
pmeas: [float, float]
The upper and lower confidence interval (a, 1-a) on the measured power
Examples
--------
>>> cl = power_confidence_limits(150, c=0.84)
>>> np.allclose(cl, [127, 176], atol=1)
True
"""
rv = stats.ncx2(2 * n, preal)
return rv.ppf([1 - c, c])
def isf(x):
rv = stats.ncx2(2 * n, x)
return rv.ppf(c)
def ppf(x):
rv = stats.ncx2(2 * n, x)
return rv.ppf(1-c)
-1.8475789041433829e-22,
36120900.670255348)
income_model_dict['nakagami'] = st.nakagami(0.10038339454419823,
-3.0390927147076284e-22,
33062195.426077582)
income_model_dict['exponweib'] = st.exponweib(-3.5157658448986489,
0.44492833350419714,
-15427.454196748848,
2440.0278856175246)
drivingdistance_model_dict = ct.OrderedDict()
drivingdistance_model_dict['nakagami'] = st.nakagami(0.11928581143831021,
14.999999999999996,
41.404620910360876)
drivingdistance_model_dict['ncx2'] = st.ncx2(0.30254190304723211,
1.1286538320791935,
14.999999999999998,
8.7361471573932192)
drivingdistance_model_dict['chi'] = st.chi(0.47882729877571095,
14.999999999999996,
44.218301183844645)
drivingdistance_model_dict['recipinvgauss'] = st.recipinvgauss(
2447246.0546641815, 14.999999999994969, 31.072009722580802)
drivingdistance_model_dict['f'] = st.f(0.85798489720127036, 4.1904554804436929,
14.99998319939356, 21.366492843433996)
drivingduration_model_dict = ct.OrderedDict()
drivingduration_model_dict['betaprime'] = st.betaprime(2.576282082814398,
9.7247974165209996,
9.1193851632305201,
261.3457987967214)
drivingduration_model_dict['exponweib'] = st.exponweib(2.6443841639764942,
def mod_liu_corrected(q, w):
r"""Joint significance of statistics derived from chi2-squared distributions.
Parameters
----------
q : float
Test statistics.
w : array_like
Weights of the linear combination.
Returns
-------
float
Estimated p-value.
Ref
-------
Liu, Huan, Yongqiang Tang, and Hao Helen Zhang. "A new chi-square
approximation to the distribution of non-negative definite quadratic forms
in non-central normal variables."
Computational Statistics & Data Analysis 53.4 (2009): 853-856.
"""
q = asarray(q, float)
if not all(isfinite(atleast_1d(q))):
raise ValueError("There are non-finite values in `q`.")
w = asarray(w, float)
if not all(isfinite(atleast_1d(w))):
raise ValueError("There are non-finite values in `w`.")
d = sum(w)
w /= d
c1 = sum(w)
c2 = sum(w**2)
c3 = sum(w**3)
c4 = sum(w**4)
s1 = c3 / (c2**(3 / 2))
s2 = c4 / c2**2
muQ = c1
sigmaQ = sqrt(2 * c2)
if s1**2 > s2:
a = 1 / (s1 - sqrt(s1**2 - s2))
delta = s1 * a**3 - a**2
l = a**2 - 2 * delta
if l < 0:
raise RuntimeError("This term cannot be negative.")
else:
delta = 0
l = 1 / (s1**2)
a = sqrt(l)
Q_norm = (q / d - muQ) / sigmaQ * sqrt(2 * l + 4 * delta) + (l + delta)
Qq = atleast_1d(ncx2(df=l, nc=delta).sf(Q_norm))[0]
return (Qq, muQ * d, sigmaQ * d, l)
def all_dists():
# dists param were taken from scipy.stats official
# documentaion examples
# Total - 89
return {
"alpha":
stats.alpha(a=3.57, loc=0.0, scale=1.0),
"anglit":
stats.anglit(loc=0.0, scale=1.0),
"arcsine":
stats.arcsine(loc=0.0, scale=1.0),
"beta":
stats.beta(a=2.31, b=0.627, loc=0.0, scale=1.0),
"betaprime":
stats.betaprime(a=5, b=6, loc=0.0, scale=1.0),
"bradford":
stats.bradford(c=0.299, loc=0.0, scale=1.0),
"burr":
stats.burr(c=10.5, d=4.3, loc=0.0, scale=1.0),
"cauchy":
stats.cauchy(loc=0.0, scale=1.0),
"chi":
stats.chi(df=78, loc=0.0, scale=1.0),
"chi2":
stats.chi2(df=55, loc=0.0, scale=1.0),
"cosine":
stats.cosine(loc=0.0, scale=1.0),
"dgamma":
stats.dgamma(a=1.1, loc=0.0, scale=1.0),
"dweibull":
stats.dweibull(c=2.07, loc=0.0, scale=1.0),
"erlang":
stats.erlang(a=2, loc=0.0, scale=1.0),
"expon":
stats.expon(loc=0.0, scale=1.0),
"exponnorm":
stats.exponnorm(K=1.5, loc=0.0, scale=1.0),
"exponweib":
stats.exponweib(a=2.89, c=1.95, loc=0.0, scale=1.0),
"exponpow":
stats.exponpow(b=2.7, loc=0.0, scale=1.0),
"f":
stats.f(dfn=29, dfd=18, loc=0.0, scale=1.0),
"fatiguelife":
stats.fatiguelife(c=29, loc=0.0, scale=1.0),
"fisk":
stats.fisk(c=3.09, loc=0.0, scale=1.0),
"foldcauchy":
stats.foldcauchy(c=4.72, loc=0.0, scale=1.0),
"foldnorm":
stats.foldnorm(c=1.95, loc=0.0, scale=1.0),
# "frechet_r": stats.frechet_r(c=1.89, loc=0.0, scale=1.0),
# "frechet_l": stats.frechet_l(c=3.63, loc=0.0, scale=1.0),
"genlogistic":
stats.genlogistic(c=0.412, loc=0.0, scale=1.0),
"genpareto":
stats.genpareto(c=0.1, loc=0.0, scale=1.0),
"gennorm":
stats.gennorm(beta=1.3, loc=0.0, scale=1.0),
"genexpon":
stats.genexpon(a=9.13, b=16.2, c=3.28, loc=0.0, scale=1.0),
"genextreme":
stats.genextreme(c=-0.1, loc=0.0, scale=1.0),
"gausshyper":
stats.gausshyper(a=13.8, b=3.12, c=2.51, z=5.18, loc=0.0, scale=1.0),
"gamma":
stats.gamma(a=1.99, loc=0.0, scale=1.0),
"gengamma":
stats.gengamma(a=4.42, c=-3.12, loc=0.0, scale=1.0),
"genhalflogistic":
stats.genhalflogistic(c=0.773, loc=0.0, scale=1.0),
"gilbrat":
stats.gilbrat(loc=0.0, scale=1.0),
"gompertz":
stats.gompertz(c=0.947, loc=0.0, scale=1.0),
"gumbel_r":
stats.gumbel_r(loc=0.0, scale=1.0),
"gumbel_l":
stats.gumbel_l(loc=0.0, scale=1.0),
"halfcauchy":
stats.halfcauchy(loc=0.0, scale=1.0),
"halflogistic":
stats.halflogistic(loc=0.0, scale=1.0),
"halfnorm":
stats.halfnorm(loc=0.0, scale=1.0),
"halfgennorm":
stats.halfgennorm(beta=0.675, loc=0.0, scale=1.0),
"hypsecant":
stats.hypsecant(loc=0.0, scale=1.0),
"invgamma":
stats.invgamma(a=4.07, loc=0.0, scale=1.0),
"invgauss":
stats.invgauss(mu=0.145, loc=0.0, scale=1.0),
"invweibull":
stats.invweibull(c=10.6, loc=0.0, scale=1.0),
"johnsonsb":
stats.johnsonsb(a=4.32, b=3.18, loc=0.0, scale=1.0),
"johnsonsu":
stats.johnsonsu(a=2.55, b=2.25, loc=0.0, scale=1.0),
"ksone":
stats.ksone(n=1e03, loc=0.0, scale=1.0),
"kstwobign":
stats.kstwobign(loc=0.0, scale=1.0),
"laplace":
stats.laplace(loc=0.0, scale=1.0),
"levy":
stats.levy(loc=0.0, scale=1.0),
"levy_l":
stats.levy_l(loc=0.0, scale=1.0),
"levy_stable":
stats.levy_stable(alpha=0.357, beta=-0.675, loc=0.0, scale=1.0),
"logistic":
stats.logistic(loc=0.0, scale=1.0),
"loggamma":
stats.loggamma(c=0.414, loc=0.0, scale=1.0),
"loglaplace":
stats.loglaplace(c=3.25, loc=0.0, scale=1.0),
"lognorm":
stats.lognorm(s=0.954, loc=0.0, scale=1.0),
"lomax":
stats.lomax(c=1.88, loc=0.0, scale=1.0),
"maxwell":
stats.maxwell(loc=0.0, scale=1.0),
"mielke":
stats.mielke(k=10.4, s=3.6, loc=0.0, scale=1.0),
"nakagami":
stats.nakagami(nu=4.97, loc=0.0, scale=1.0),
"ncx2":
stats.ncx2(df=21, nc=1.06, loc=0.0, scale=1.0),
"ncf":
stats.ncf(dfn=27, dfd=27, nc=0.416, loc=0.0, scale=1.0),
"nct":
stats.nct(df=14, nc=0.24, loc=0.0, scale=1.0),
"norm":
stats.norm(loc=0.0, scale=1.0),
"pareto":
stats.pareto(b=2.62, loc=0.0, scale=1.0),
"pearson3":
stats.pearson3(skew=0.1, loc=0.0, scale=1.0),
"powerlaw":
stats.powerlaw(a=1.66, loc=0.0, scale=1.0),
"powerlognorm":
stats.powerlognorm(c=2.14, s=0.446, loc=0.0, scale=1.0),
"powernorm":
stats.powernorm(c=4.45, loc=0.0, scale=1.0),
"rdist":
stats.rdist(c=0.9, loc=0.0, scale=1.0),
"reciprocal":
stats.reciprocal(a=0.00623, b=1.01, loc=0.0, scale=1.0),
"rayleigh":
stats.rayleigh(loc=0.0, scale=1.0),
"rice":
stats.rice(b=0.775, loc=0.0, scale=1.0),
"recipinvgauss":
stats.recipinvgauss(mu=0.63, loc=0.0, scale=1.0),
"semicircular":
stats.semicircular(loc=0.0, scale=1.0),
"t":
stats.t(df=2.74, loc=0.0, scale=1.0),
"triang":
stats.triang(c=0.158, loc=0.0, scale=1.0),
"truncexpon":
stats.truncexpon(b=4.69, loc=0.0, scale=1.0),
"truncnorm":
stats.truncnorm(a=0.1, b=2, loc=0.0, scale=1.0),
"tukeylambda":
stats.tukeylambda(lam=3.13, loc=0.0, scale=1.0),
"uniform":
stats.uniform(loc=0.0, scale=1.0),
"vonmises":
stats.vonmises(kappa=3.99, loc=0.0, scale=1.0),
"vonmises_line":
stats.vonmises_line(kappa=3.99, loc=0.0, scale=1.0),
"wald":
stats.wald(loc=0.0, scale=1.0),
"weibull_min":
stats.weibull_min(c=1.79, loc=0.0, scale=1.0),
"weibull_max":
stats.weibull_max(c=2.87, loc=0.0, scale=1.0),
"wrapcauchy":
stats.wrapcauchy(c=0.0311, loc=0.0, scale=1.0),
}
df, nc = 21, 1.06 mean, var, skew, kurt = ncx2.stats(df, nc, moments='mvsk') # Display the probability density function (``pdf``): x = np.linspace(ncx2.ppf(0.01, df, nc), ncx2.ppf(0.99, df, nc), 100) ax.plot(x, ncx2.pdf(x, df, nc), 'r-', lw=5, alpha=0.6, label='ncx2 pdf') # Alternatively, the distribution object can be called (as a function) # to fix the shape, location and scale parameters. This returns a "frozen" # RV object holding the given parameters fixed. # Freeze the distribution and display the frozen ``pdf``: rv = ncx2(df, nc) ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') # Check accuracy of ``cdf`` and ``ppf``: vals = ncx2.ppf([0.001, 0.5, 0.999], df, nc) np.allclose([0.001, 0.5, 0.999], ncx2.cdf(vals, df, nc)) # True # Generate random numbers: r = ncx2.rvs(df, nc, size=1000) # And compare the histogram: ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
def __init__(self, df, ncp):
self.dist_ = ncx2(df, ncp)