Conditional variable

In this example, we will demonstrate how to define a conditional variable. This is a variable which parameters depend on a value of another variable.

First, let’s import the necessary packages:

from probabilistic_library import DistributionType, ReliabilityMethod, ReliabilityProject, ConditionalValue
import matplotlib.pyplot as plt

Consider a variable \(h\) (kPa/m), which represents the pore water pressure in a clay layer of a levee. The variable follows a log-normal distribution, but its mean and standard deviation depend on the overtopping discharge \(q\) (l/s/m), which varies per day. We assume the following relationships between the overtopping discharge \(q\) and the mean and standard deviation of \(h\):

q = [0, 0.1, 1.0, 10.0, 30.0, 50.0, 75.0]
mu = [0, 1.42, 2.24, 3.63, 6.53, 6.31, 6.48]
sigma = [0, 0.32, 1.10, 1.32, 1.70, 2.15, 2.65]

plt.plot(q, mu, '-o', label="mean of h")
plt.plot(q, sigma, '-o', label="deviation of h")
plt.grid()
plt.legend()
plt.xlabel("q (l/s/m)")
plt.ylabel("Values (kPa/m)")

Text(0, 0.5, 'Values (kPa/m)')

Let’s assume that on a certain day the overtopping discharge \(q\) is normally distributed with a mean of \(50.0\) l/s/m and a standard deviation of \(1.0\) l/s/m. The goal is to find the probability that the pore water pressure \(h\) exceeds a level of \(8.0\) kPa/m:

\(\int P(h>8.0|q) \cdot f(q) dq\) and \(q \sim N(50.0, 1.0)\)

To compute this probability, we define the following limit state function, in which both \(h\) and \(q\) are included as inputs:

def pore_water_pressue(q, h, level):
    return level-h

Next, we define a reliability project that uses the above limit state function as its model:

project = ReliabilityProject()
project.model = pore_water_pressue

Then, we define the variable \(q\) and the exceedance level:

project.variables["q"].distribution = DistributionType.normal
project.variables["q"].mean = 50.0
project.variables["q"].deviation = 1.0

project.variables["level"].distribution = DistributionType.deterministic
project.variables["level"].mean = 8.0

The conditional variable \(h\) is defined as follows. It is necessary to specify the source of the variable - namely, the parameter \(q\) - and to define the relationship between the values of \(q\) and the parameters of \(h\). In the case of a conditional variable, the distribution parameters will be interpolated between the grid points. When extrapolating, the first and last points from the grid will be used.

project.variables["h"].distribution = DistributionType.log_normal
project.variables["h"].mean = 0
project.variables["h"].deviation = 1
project.variables["h"].conditional = True
project.variables["h"].conditional_source = "q"

for ii in range(0, len(q)):
    conditional = ConditionalValue()
    conditional.x = q[ii]
    conditional.mean = mu[ii]
    conditional.deviation = sigma[ii]
    project.variables["h"].conditional_values.append(conditional)

project.variables["h"].print()
project.variables["h"].plot()

Variable h:
  distribution = log_normal
Definition:
  conditional source = q
  q = [0.0, 0.1, 1.0, 10.0, 30.0, 50.0, 75.0]
  mean = [0.0, 1.42, 2.24, 3.63, 6.53, 6.31, 6.48]
  deviation = [0.0, 0.32, 1.1, 1.32, 1.7, 2.15, 2.65]

We use the reliability method form and run the calculations.

project.settings.reliability_method = ReliabilityMethod.form
project.settings.relaxation_factor = 0.75
project.settings.maximum_iterations = 50
project.settings.epsilon_beta = 0.01

project.run()

The reliability results are as follows:

project.design_point.print()

Reliability (FORM)
 Reliability index = 0.8817
 Probability of failure = 0.189
 Convergence = 0.0053 (converged)
 Model runs = 15
Alpha values:
 q: alpha = -0.0069, x = 50.0061
 h: alpha = -1.0, x = 7.9998
 level: alpha = 0.0, x = 8.0

On a different day, when the distribution of \(q\) is normal with a mean of \(20.0\) and a standard deviation of \(1.5\), the probability that the pore water pressure exceeds \(8.0\) kPa/m becomes:

project.variables["q"].distribution = DistributionType.normal
project.variables["q"].mean = 20.0
project.variables["q"].deviation = 1.5

project.run()

project.design_point.print()

Reliability (FORM)
 Reliability index = 1.7774
 Probability of failure = 0.0377
 Convergence = 0.0043 (converged)
 Model runs = 12
Alpha values:
 q: alpha = -0.1078, x = 20.2873
 h: alpha = -0.9942, x = 7.9998
 level: alpha = 0.0, x = 8.0