pdf_multimode (L:int, lambdas:list, probs:list)
Computes the discrete PDF of multi-mode exponential of the form
\[ \Pr(L) = \sum_{i=1,2} \omega_i (1-e^{-1/\lambda_i}) e^{-(L-1)/\lambda_i} \, , \] where \(\omega\) is the probability of each mode and \(\lambda\) it’s scale.
| Type | Details | |
|---|---|---|
| L | int | Either int or array for which pdf is calculated |
| lambdas | list | Scales of each modes |
| probs | list | Probability weight of each mode |
| Returns | array | Array with probability of each L |
lambdas = np.array([2,15])
probs = np.array([0.99, 0.01])
L_max = 100
plt.loglog(np.arange(1, L_max),
pdf_multimode(L = np.arange(1, L_max), lambdas=lambdas, probs=probs)
)
plt.xlabel('L'); plt.ylabel('P(L)')Text(0, 0.5, 'P(L)')
pdf_powerlaw (L:float, beta:float=1)
Computes the discrete PDF of a powerlaw of the form \[ \Pr(L)\sim L^{-1-\mathrm{beta}}. \]
| Type | Default | Details | |
|---|---|---|---|
| L | float | Either int or array for which pdf is calculated | |
| beta | float | 1 | Exponent of the power law |
| Returns | array | Array with probability of each L |
plt.loglog(np.arange(1, 1000),
pdf_powerlaw(L = np.arange(1,1000), beta = 1)
)
plt.xlabel('L'); plt.ylabel('P(L)')Text(0, 0.5, 'P(L)')
pdf_discrete_sample (pdf_func:object, num_samples:int, **args_func)
Samples discrete values from a given PDF
| Type | Details | |
|---|---|---|
| pdf_func | object | Function generating the pdf |
| num_samples | int | Number of samples to create |
| args_func | ||
| Returns | array | Samples |
samples = pdf_discrete_sample(pdf_func = pdf_multimode,
num_samples=10000,
L = np.arange(1, L_max), lambdas=lambdas, probs=probs)
counts = np.bincount(samples)[1:]
plt.loglog(np.arange(1, len(counts)+1),
counts/counts.sum(), 'o',
label = 'Histogram samples')
plt.loglog(np.arange(1, L_max),
pdf_multimode(L = np.arange(1, L_max), lambdas=lambdas, probs=probs),
label = 'Theory'
)
plt.xlabel('L'); plt.ylabel('P(L)'); plt.legend()<matplotlib.legend.Legend>
get_policy_from_dist (n_max, func, renorm=True, **args_func)
Given a PDF of step lengths, calculates the corresponding policy
| Type | Default | Details | |
|---|---|---|---|
| n_max | Maximum counter n_max for which the policy is calculated | ||
| func | Function generating the pdf | ||
| renorm | bool | True | If true, we check whether the distribution has a boundary N, for which _n=N^Pr(L=nd) = 0 |
| args_func | |||
| Returns | array | Policy at each counter value |
L = 100
betas = np.linspace(0.1, 2, 10)
colors = plt.cm.plasma(np.linspace(0,1,len(betas)+2))
fig, ax = plt.subplots()
for beta, color in zip(betas, colors):
policy = get_policy_from_dist(n_max = L,
func = pdf_powerlaw,
beta = beta)
ax.plot(np.arange(2, L+1), policy[1:], c = color)
# Plot features
plt.setp(ax, xlabel =r'$n$', ylabel = r'$\pi(\uparrow|n)$', xscale = 'log')
cbar = fig.colorbar(plt.cm.ScalarMappable(norm= mcolors.Normalize(vmin=betas.min(),
vmax=betas.max()),
cmap=plt.cm.plasma),
ax = ax)
cbar.set_label(r'$\beta$')