Truncated gaussian

[3]:

# Parameters
func_name = "Truncated_gaussian"
positive_prior = False


Description

[5]:

func.display()

• description: A truncated Gaussian function defined on the interval between the lower_bound (a) and upper_bound (b)
• formula: $\begin{split}f(x;\mu,\sigma,a,b)=\frac{\frac{1}{\sigma} \phi\left( \frac{x-\mu}{\sigma} \right)}{\Phi\left( \frac{b-\mu}{\sigma} \right) - \Phi\left( \frac{a-\mu}{\sigma} \right)}\\\phi\left(z\right)=\frac{1}{\sqrt{2 \pi}}\exp\left(-\frac{1}{2}z^2\right)\\\Phi\left(z\right)=\frac{1}{2}\left(1+erf\left(\frac{z}{\sqrt(2)}\right)\right)\end{split}$
• parameters:
• F:
• value: 1.0
• desc: Integral between -inf and +inf. Fix this to 1 to obtain a Normal distribution
• min_value: None
• max_value: None
• unit:
• is_normalization: False
• delta: 0.1
• free: True
• mu:
• value: 0.0
• desc: Central value
• min_value: None
• max_value: None
• unit:
• is_normalization: False
• delta: 0.1
• free: True
• sigma:
• value: 1.0
• desc: standard deviation
• min_value: 1e-12
• max_value: None
• unit:
• is_normalization: False
• delta: 0.1
• free: True
• lower_bound:
• value: -1.0
• desc: lower bound of gaussian, setting to -np.inf results in half normal distribution
• min_value: None
• max_value: None
• unit:
• is_normalization: False
• delta: 0.1
• free: True
• upper_bound:
• value: 1.0
• desc: upper bound of gaussian setting to np.inf results in half normal distribution
• min_value: None
• max_value: None
• unit:
• is_normalization: False
• delta: 0.1
• free: True

Shape

The shape of the function.

If this is not a photon model but a prior or linear function then ignore the units as these docs are auto-generated

[6]:

fig, ax = plt.subplots()

ax.plot(energy_grid, func(energy_grid), color=blue, lw=3)

ax.set_xlabel("x")
ax.set_ylabel("probability")

[6]:

Text(0, 0.5, 'probability')


Random Number Generation

This is how we can generate random numbers from the prior.

[7]:

u = np.random.uniform(0,1, size=5000)

draws = [func.from_unit_cube(x) for x in u]

fig, ax = plt.subplots()

ax.hist(draws, color=green, bins=50)

ax.set_xlabel("value")
ax.set_ylabel("N")

[7]:

Text(0, 0.5, 'N')