OTL Circuit Function#

import numpy as np
import matplotlib.pyplot as plt
import uqtestfuns as uqtf

The OTL circuit test function is a six-dimensional scalar-valued function. The function has been used as a test function in metamodeling exercises [BAS07] and sensitivity analysis [Moo10]. In [Moo10], a 20-dimensional variant was used for sensitivity analysis by introducing 14 additional inert input variables.

Test function instance#

To create a default instance of the OTL circuit test function:

my_testfun = uqtf.OTLCircuit()

Check if it has been correctly instantiated:

print(my_testfun)

Name              : OTLCircuit
Spatial dimension : 6
Description       : Output transformerless (OTL) circuit model from Ben-Ari and Steinberg (2007)

Description#

The OTL circuit function computes the mid-point voltage of an output transformerless (OTL) push-pull circuit using the following analytical formula:

\[\begin{split} \begin{align} \mathcal{M}(\boldsymbol{x}) & = \frac{(V_{b1} + 0.74) \beta (R_{c2} + 9)}{\beta (R_{c2} + 9) + R_f} + \frac{11.35 R_f}{\beta (R_{c2} + 9) + R_f} + \frac{0.74 R_f \beta (R_{c2} + 9)}{(\beta (R_{c2} + 9) + R_f) R_{c1}}, \\ V_{b1} & = \frac{12 R_{b2}}{R_{b1} + R_{b2}}, \\ \end{align} \end{split}\]

where \(\boldsymbol{x} = \{ R_{b1}, R_{b2}, R_f, R_{c1}, R_{c2}, \beta \}\) is the six-dimensional vector of input variables further defined below.

Probabilistic input#

Two probabilistic input model specifications for the OTL circuit function are available as shown in the table below.

No.

Keyword

Source

1.

BenAri2007 (default)

[BAS07]

2.

Moon2010

[Moo10]

The default selection, based on [BAS07], contains six input variables given as independent uniform random variables with specified ranges shown in the table below.

my_testfun.prob_input

Name: OTLCircuit-BenAri2007

Spatial Dimension: 6

Description: Probabilistic input model for the OTL Circuit function from Ben-Ari and Steinberg (2007).

Marginals:

No. Name Distribution Parameters Description
1 Rb1 uniform [ 50. 150.] Resistance b1 [kOhm]
2 Rb2 uniform [25. 70.] Resistance b2 [kOhm]
3 Rf uniform [0.5 3. ] Resistance f [kOhm]
4 Rc1 uniform [1.2 2.5] Resistance c1 [kOhm]
5 Rc2 uniform [0.25 1.2 ] Resistance c2 [kOhm]
6 beta uniform [ 50. 300.] Current gain [A]

Copulas: None

Note

In [Moo10], 14 additional inert independent input variables are introduced (totaling 20 input variables); these input variables, being inert, do not affect the output of the function.

To create an instance of the OTL circuit test function with the probabilistic input specified in [Moo10], pass the corresponding keyword ("Moon2010") to the parameter (prob_input_selection):

my_testfun = uqtf.OTLCircuit(prob_input_selection="Moon2010")

Reference results#

This section provides several reference results of typical UQ analyses involving the test function.

Sample histogram#

Shown below is the histogram of the output based on \(100'000\) random points:

np.random.seed(42)
xx_test = my_testfun.prob_input.get_sample(100000)
yy_test = my_testfun(xx_test)

plt.hist(yy_test, bins="auto", color="#8da0cb");
plt.grid();
plt.ylabel("Counts [-]");
plt.xlabel("$\mathcal{M}(\mathbf{X})$");
plt.gcf().set_dpi(150);
../_images/otl-circuit_9_0.png

Moments estimation#

Shown below is the convergence of a direct Monte-Carlo estimation of the output mean and variance with increasing sample sizes.

# --- Compute the mean and variance estimate
np.random.seed(42)
sample_sizes = np.array([1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7], dtype=int)
mean_estimates = np.empty(len(sample_sizes))
var_estimates = np.empty(len(sample_sizes))

for i, sample_size in enumerate(sample_sizes):
    xx_test = my_testfun.prob_input.get_sample(sample_size)
    yy_test = my_testfun(xx_test)
    mean_estimates[i] = np.mean(yy_test)
    var_estimates[i] = np.var(yy_test)

# --- Compute the error associated with the estimates
mean_estimates_errors = np.sqrt(var_estimates) / np.sqrt(np.array(sample_sizes))
var_estimates_errors = var_estimates * np.sqrt(2 / (np.array(sample_sizes) - 1))

# --- Do the plot
fig, ax_1 = plt.subplots(figsize=(6,4))

ax_1.errorbar(
    sample_sizes,
    mean_estimates,
    yerr=mean_estimates_errors,
    marker="o",
    color="#66c2a5",
    label="Mean",
)
ax_1.set_xlabel("Sample size")
ax_1.set_ylabel("Output mean estimate")
ax_1.set_xscale("log");
ax_2 = ax_1.twinx()
ax_2.errorbar(
    sample_sizes + 1,
    var_estimates,
    yerr=var_estimates_errors,
    marker="o",
    color="#fc8d62",
    label="Variance",
)
ax_2.set_ylabel("Output variance estimate")

# Add the two plots together to have a common legend
ln_1, labels_1 = ax_1.get_legend_handles_labels()
ln_2, labels_2 = ax_2.get_legend_handles_labels()
ax_2.legend(ln_1 + ln_2, labels_1 + labels_2, loc=0)

plt.grid()
fig.set_dpi(150)
../_images/otl-circuit_11_0.png

The tabulated results for is shown below.

from tabulate import tabulate

# --- Compile data row-wise
outputs = []
for (
    sample_size,
    mean_estimate,
    mean_estimate_error,
    var_estimate,
    var_estimate_error,
) in zip(
    sample_sizes,
    mean_estimates,
    mean_estimates_errors,
    var_estimates,
    var_estimates_errors,
):
    outputs += [
        [
            sample_size,
            mean_estimate,
            mean_estimate_error,
            var_estimate,
            var_estimate_error,
            "Monte-Carlo",
        ],
    ]

header_names = [
    "Sample size",
    "Mean",
    "Mean error",
    "Variance",
    "Variance error",
    "Remark",
]

tabulate(
    outputs,
    headers=header_names,
    floatfmt=(".1e", ".4f", ".4e", ".4f", ".4e", "s"),
    tablefmt="html",
    stralign="center",
    numalign="center",
)
Sample size Mean Mean error Variance Variance error Remark
1.0e+01 5.5320 2.9861e-01 0.8917 4.2035e-01 Monte-Carlo
1.0e+02 5.3385 1.1612e-01 1.3485 1.9166e-01 Monte-Carlo
1.0e+03 5.4424 3.7023e-02 1.3707 6.1330e-02 Monte-Carlo
1.0e+04 5.4333 1.1514e-02 1.3258 1.8750e-02 Monte-Carlo
1.0e+05 5.4376 3.6184e-03 1.3093 5.8553e-03 Monte-Carlo
1.0e+06 5.4345 1.1434e-03 1.3073 1.8488e-03 Monte-Carlo
1.0e+07 5.4343 3.6159e-04 1.3075 5.8471e-04 Monte-Carlo

References#

BAS07(1,2,3)

Einat Neumann Ben-Ari and David M. Steinberg. Modeling data from computer experiments: an empirical comparison of kriging with MARS and projection pursuit regression. Quality Engineering, 19(4):327–338, 2007. doi:10.1080/08982110701580930.

Moo10(1,2,3,4,5)

Hyejung Moon. Design and analysis of computer experiments for screening input variables. PhD thesis, Ohio State University, Ohio, 2010. URL: http://rave.ohiolink.edu/etdc/view?acc_num=osu1275422248.