def rf(n,df1,df2,ncp=0):
"""
Calculates the quantile function of the F-distribution
"""
from scipy.stats import f,ncf
if ncp==0:
result=f.rvs(size=n,dfn=df1,dfd=df2,loc=0,scale=1)
else:
result=ncf.rvs(size=n,dfn=df1,dfd=df2,nc=ncp,loc=0,scale=1)
return result
def rvs(t,delta_L, df1=100, df2=100):
random_part= f.rvs(df1,df2)
add=delta_L+t
non_random_part=0
if add>1:
non_random_part=add
else:
non_random_part=exp(add-1)
print random_part, 1/random_part
return t+random_part*(non_random_part-t)
# Display the probability density function (``pdf``): x = np.linspace(f.ppf(0.01, dfn, dfd), f.ppf(0.99, dfn, dfd), 100) ax.plot(x, f.pdf(x, dfn, dfd), 'r-', lw=5, alpha=0.6, label='f 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 = f(dfn, dfd) ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') # Check accuracy of ``cdf`` and ``ppf``: vals = f.ppf([0.001, 0.5, 0.999], dfn, dfd) np.allclose([0.001, 0.5, 0.999], f.cdf(vals, dfn, dfd)) # True # Generate random numbers: r = f.rvs(dfn, dfd, size=1000) # And compare the histogram: ax.hist(r, density=True, histtype='stepfilled', alpha=0.2) ax.legend(loc='best', frameon=False) plt.show()
def bootstrap(a,
f=None,
b=100,
method="balanced",
family=None,
strata=None,
smooth=False,
random_state=None):
"""
Calculate function values from bootstrap samples or
optionally return bootstrap samples themselves
Parameters
----------
a : array-like
Original sample
f : callable or None
Function to be bootstrapped
b : int
Number of bootstrap samples
method : string
* 'ordinary'
* 'balanced'
* 'parametric'
family : string or None
* 'gaussian'
* 't'
* 'laplace'
* 'logistic'
* 'F'
* 'gamma'
* 'log-normal'
* 'inverse-gaussian'
* 'pareto'
* 'beta'
* 'poisson'
strata : array-like or None
Stratification labels, ignored when method
is parametric
smooth : boolean
Whether or not to add noise to bootstrap
samples, ignored when method is parametric
random_state : int or None
Random number seed
Returns
-------
y | X : np.array
Function applied to each bootstrap sample
or bootstrap samples if f is None
"""
np.random.seed(random_state)
a = np.asarray(a)
n = len(a)
# stratification not meaningful for parametric sampling
if strata is not None and (method != "parametric"):
strata = np.asarray(strata)
if len(strata) != len(a):
raise ValueError("a and strata must have" " the same length")
# recursively call bootstrap without stratification
# on the different strata
masks = [strata == x for x in np.unique(strata)]
boot_strata = [
bootstrap(a=a[m],
f=None,
b=b,
method=method,
strata=None,
random_state=random_state) for m in masks
]
# concatenate resampled strata along first column axis
X = np.concatenate(boot_strata, axis=1)
else:
if method == "ordinary":
# i.i.d. sampling from ecdf of a
X = np.reshape(a[np.random.choice(range(a.shape[0]),
a.shape[0] * b)],
newshape=(b, ) + a.shape)
elif method == "balanced":
# permute b concatenated copies of a
r = np.reshape([a] * b, newshape=(b * a.shape[0], ) + a.shape[1:])
X = np.reshape(r[np.random.permutation(range(r.shape[0]))],
newshape=(b, ) + a.shape)
elif method == "parametric":
if len(a.shape) > 1:
raise ValueError("a must be one-dimensional")
# fit parameters by maximum likelihood and sample
if family == "gaussian":
theta = norm.fit(a)
arr = norm.rvs(size=n * b,
loc=theta[0],
scale=theta[1],
random_state=random_state)
elif family == "t":
theta = t.fit(a, fscale=1)
arr = t.rvs(size=n * b,
df=theta[0],
loc=theta[1],
scale=theta[2],
random_state=random_state)
elif family == "laplace":
theta = laplace.fit(a)
arr = laplace.rvs(size=n * b,
loc=theta[0],
scale=theta[1],
random_state=random_state)
elif family == "logistic":
theta = logistic.fit(a)
arr = logistic.rvs(size=n * b,
loc=theta[0],
scale=theta[1],
random_state=random_state)
elif family == "F":
theta = F.fit(a, floc=0, fscale=1)
arr = F.rvs(size=n * b,
dfn=theta[0],
dfd=theta[1],
loc=theta[2],
scale=theta[3],
random_state=random_state)
elif family == "gamma":
theta = gamma.fit(a, floc=0)
arr = gamma.rvs(size=n * b,
a=theta[0],
loc=theta[1],
scale=theta[2],
random_state=random_state)
elif family == "log-normal":
theta = lognorm.fit(a, floc=0)
arr = lognorm.rvs(size=n * b,
s=theta[0],
loc=theta[1],
scale=theta[2],
random_state=random_state)
elif family == "inverse-gaussian":
theta = invgauss.fit(a, floc=0)
arr = invgauss.rvs(size=n * b,
mu=theta[0],
loc=theta[1],
scale=theta[2],
random_state=random_state)
elif family == "pareto":
theta = pareto.fit(a, floc=0)
arr = pareto.rvs(size=n * b,
b=theta[0],
loc=theta[1],
scale=theta[2],
random_state=random_state)
elif family == "beta":
theta = beta.fit(a)
arr = beta.rvs(size=n * b,
a=theta[0],
b=theta[1],
loc=theta[2],
scale=theta[3],
random_state=random_state)
elif family == "poisson":
theta = np.mean(a)
arr = poisson.rvs(size=n * b,
mu=theta,
random_state=random_state)
else:
raise ValueError("Invalid family")
X = np.reshape(arr, newshape=(b, n))
else:
raise ValueError("method must be either 'ordinary'"
" , 'balanced', or 'parametric',"
" '{method}' was supplied".format(method=method))
# samples are already smooth in the parametric case
if smooth and (method != "parametric"):
X += np.random.normal(size=X.shape, scale=1 / np.sqrt(n))
if f is None:
return X
else:
return np.asarray([f(x) for x in X])
from scipy.stats import f
##
# discussion items
# 1. Show histogram of all distributions
def plot_sample_hist(sample, title):
plt.figure()
plt.title(title)
plt.hist(sample)
sample = norm.rvs(size=1000)
plot_sample_hist(sample, 'normal distribution')
sample = expon.rvs(size=1000)
plot_sample_hist(sample, 'exponential distribution')
sample = binom.rvs(10, 0.5, size=1000)
plot_sample_hist(sample, 'binomial distribution')
sample = chi2.rvs(10, size=1000)
plot_sample_hist(sample, 'chi-square distribution')
sample = t.rvs(10, size=1000)
plot_sample_hist(sample, 't distribution')
sample = f.rvs(10, 20, size=1000)
plot_sample_hist(sample, 'f distribution')
# w_dist = uniform()
w_dist = f(dfn=500, dfd=10) # Underlying distribution function
w_pctiles = np.linspace(
10**(-10), 1 - 10**(-10),
num=n) # Percentiles for which a probability will be calculated
f_w = w_dist.pdf(w_dist.ppf(w_pctiles)) # value of the pdf for each percentile
f_w = f_w / sum(f_w) # Discretized pdf (i.e. this sums to one)
# Set up discount factor
beta = .97
# Set up value of unemployment
c = 12000
# Set up value function initial guess
V_0 = np.sort(f.rvs(dfn=2, dfd=80, size=n)) * wages
# Specify number of iterations
T = 100
# Get value functions and reservations wages over time
V, W = V_iter(V_0, wages, beta, T)
# Change to figures directory
cd(mdir + fdir)
# Set up value function plot
fig, ax = plt.subplots(figsize=(8.5 - margin, (8.5 - margin) * figratio))
# Plot the initial guess
ax.plot(wages,
