Simulate Examples: Chan2 and AU2

This page documents two simulate runs using different Monte Carlo modes.

Note: Table values are rounded to 4 significant figures for readability. Very small/large values use scientific notation. Refer to the linked Excel/JSON artifacts for full precision.

• Example 1 — Chan2: plain Monte Carlo, no importance sampling.
• Example 2 — AU2: Monte Carlo with importance sampling (-i flag).

Both runs use the default profile.

Profile Customization

These examples use the default profile, but simulation behavior is configurable. See ../profile-reference.md.

• Monte Carlo controls: reliability_options.mc_n, mc_max_cv, mc_seed, mc_remove_oob.
• Importance sampling controls: run_configuration.mc_with_is (or -i), reliability_options.is_method, is_kde_bandwidth, is_fit_samples, is_mixture_components, is_mcmc_*.
• Report toggles: reporting_options.save_plots_to_pdf, save_plots_to_pickle, save_excel_summary.

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Example 1: Chan2 — Plain Monte Carlo

Run Context

Problem module:

• problems/problems2/Chan2Problem.py

Recorded result set:

• results/2026-03-16/10-52-46/Chan2-51baa.xlsx
• results/2026-03-16/10-52-46/Chan2-51baa.json
• results/2026-03-16/10-52-46/profile-51baa.yaml

For your own runs, use the same naming pattern under a different timestamped folder:

• results/<YYYY-MM-DD>/<HH-MM-SS>/<ProblemName>-<suffix>.xlsx
• results/<YYYY-MM-DD>/<HH-MM-SS>/<ProblemName>-<suffix>.json
• results/<YYYY-MM-DD>/<HH-MM-SS>/profile-<suffix>.yaml

Note: <YYYY-MM-DD>/<HH-MM-SS> and <suffix> are generated per run and will differ on your machine.

Profile and run mode from saved profile:

• Profile used: default
• run_type: simulate
• include_mc: true
• mc_with_is: false

Results path/ID convention used in this example:

• Results are saved under results/<YYYY-MM-DD>/<HH-MM-SS>/ (date and time subfolders).
• Each run gets a new random request ID; the exact ID changes every time the analysis is run.
• Output files include the last 5 digits of that request ID in their names.

Equivalent command shape:

python -m reliafy simulate <profile>

The simulate command without -i runs plain (crude) Monte Carlo only.

Problem File Used

Source: Chan, C. L. and Low, B. K., "Practical second-order reliability analysis applied to foundation engineering," International Journal for Numerical and Analytical Methods in Geomechanics, 2012, vol. 36, no. 11, p. 1387–1409, Problem 2, p. 1397. DOI: 10.1002/nag.1057

Chan2Problem.py defines:

• Stochastic variables: cp, fp, g, Ph, Pv — all Normal
• Correlation: full 5 × 5 matrix defined using the cor key (see note below)
• Deterministic variables: B = 5.0, L = 25.0, D = 1.8, h = 2.5
• LSFreturnsGradient: False, LSFreturnsHessian: False, LSFreturnsLandR: True
• Limit state returns load ($P_v / B'$) and resistance (bearing-capacity formula)

Using cor vs cor_list

Chan2Problem.py specifies the correlation matrix as a full n × n list of lists under the cor key. This is an alternative to cor_list, which accepts a flat list of [var1, var2, value] pairs. If neither cor nor cor_list is supplied, Reliafy assumes zero correlation for all variables. Either key may be used; the full matrix form is convenient when many off-diagonal correlations are non-zero.

"StochasticVariables": {
    "name": ["cp", "fp", "g", "Ph", "Pv"],
    "type": ["normal", "normal", "normal", "normal", "normal"],
    "cor": [
        [1.0, -0.5,  0.0,  0.0, 0.0],
        [-0.5, 1.0,  0.5,  0.0, 0.0],
        [0.0,  0.5,  1.0,  0.0, 0.0],
        [0.0,  0.0,  0.0,  1.0, 0.5],
        [0.0,  0.0,  0.0,  0.5, 1.0],
    ],
    "mean": [15.0, 25.0, 20.0, 400.0, 800.0],
    "std":  [4.5,  5.0,  2.0,  40.0,  80.0],
},

Because LSFreturnsLandR: True, the CLI generates a Load and Resistance Histogram in addition to the per-variable and LSF histograms.

Extracted Results Worksheet Tables

The tables below are transcribed from the Results worksheet in Chan2-51baa.xlsx.

Header Information

Field	Value
Problem	Chan2
Request ID	9614c856e5bb41f4b96bd9b1d4051baa
Run time	00 min 25.62 sec

Deterministic Variables

var_name	value
B	5
L	25
D	1.8
h	2.5

Stochastic Variables Definition

var_name	var_type	mean	std	param1	param2
cp	Normal	15	4.5	15	4.5
fp	Normal	25	5	25	5
g	Normal	20	2	20	2
Ph	Normal	400	40	400	40
Pv	Normal	800	80	800	80

Monte Carlo Results

beta	pf	cv	max_cv	size	%_removed	cycles	auto_size	mc_with_is
1.5409	0.06167	0.002252	0.05	3,000,000	0.000733	3	True	False

Monte Carlo Variable Statistics and Correlations

The sampled statistics confirm the input distributions and correlation structure.

var_name	mean	std	%_oob	cor(cp)	cor(fp)	cor(g)	cor(Ph)	cor(Pv)
cp	14.9993	4.4999	0	1.0000	−0.4999	0.0003	0.0003	0.0008
fp	24.9972	4.9994	0	−0.4999	1.0000	0.5003	−0.0008	0.0000
g	19.9997	2.0012	0	0.0003	0.5003	1.0000	0.0001	0.0008
Ph	400.035	39.985	0	0.0003	−0.0008	0.0001	1.0000	0.5006
Pv	799.993	79.962	0	0.0008	0.0000	0.0008	0.5006	1.0000

Notes Reported by Reliafy

1. Validation: Stochastic variables definition and limit state function validation required 1 function call.
2. Monte Carlo: Completed 3 cycles with 1.00e+06 samples per cycle.
3. Monte Carlo: Detected 22 NaN values in the limit state function out of 3.00e+06 samples. Review the list of code warnings and update the limit state function to address their source.

Interpretation Snapshot

• beta = 1.541, pf = 6.17% — a relatively low reliability index for a bearing-capacity problem with correlated soil parameters.
• The coefficient of variation (cv = 0.00225) is well below max_cv = 0.05, indicating high MC precision with 3 million samples.
• The sampled correlation matrix closely matches the targets: cor(cp, fp) ≈ −0.500, cor(fp, g) ≈ 0.500, cor(Ph, Pv) ≈ 0.501.
• A small number of NaN results (22 out of 3,000,000) were detected. These typically arise from geometric degeneration in the bearing-capacity formula (e.g., B' ≤ 0 due to large eccentricity). They do not invalidate the result at this sample size.

Generated Figures

The PDF result file for this run is saved as results/2026-03-16/10-52-46/Chan2-51baa.pdf. That PDF is composed of vector-based pages, rendered below at 2× resolution from each page.

Figure 1: MC Histogram — cp

Figure 2: MC Histogram — fp

Figure 3: MC Histogram — g

Figure 4: MC Histogram — Ph

Figure 5: MC Histogram — Pv

Figure 6: Histogram of Limit State Function Values

Figure 7: Load and Resistance Histogram

Generated because LSFreturnsLandR: True in the problem file.

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Example 2: AU2 — Monte Carlo with Importance Sampling

Run Context

Problem module:

• problems/AU2Problem.py

Recorded result set:

• results/2026-03-16/11-06-54/AU2-42d54.xlsx
• results/2026-03-16/11-06-54/AU2-42d54.json
• results/2026-03-16/11-06-54/profile-42d54.yaml

For your own runs, use the same naming pattern under a different timestamped folder:

• results/<YYYY-MM-DD>/<HH-MM-SS>/<ProblemName>-<suffix>.xlsx
• results/<YYYY-MM-DD>/<HH-MM-SS>/<ProblemName>-<suffix>.json
• results/<YYYY-MM-DD>/<HH-MM-SS>/profile-<suffix>.yaml

Note: <YYYY-MM-DD>/<HH-MM-SS> and <suffix> are generated per run and will differ on your machine.

Profile and run mode from saved profile:

• Profile used: default
• run_type: simulate
• include_mc: true
• mc_with_is: true

Equivalent command shape:

python -m reliafy simulate <profile> -i

The -i flag enables importance sampling (IS), which drastically reduces the number of samples needed to estimate very small failure probabilities by concentrating samples near the failure region.

Problem File Used

Source: Au, S. K. and Beck, J. L., "A new adaptive importance sampling scheme for reliability calculations," Structural Safety, vol. 21, no. 2, 1999, pp. 135–158. Available at: https://people.duke.edu/~hpgavin/risk/Au-1999.pdf

AU2Problem.py defines:

• Stochastic variables: T1, T2 — both standard Normal (μ = 0, σ = 1)
• Deterministic variables: c = 5
• LSFreturnsGradient: False, LSFreturnsHessian: False, LSFreturnsLandR: False
• Compound limit state: $g = \min(g_1, g_2)$

$$ g_1 = c - 1 - T_2 + e^{-T_1^2/10} + \left(\frac{T_1}{5}\right)^4 $$

$$ g_2 = \frac{c^2}{2} - T_1 T_2 $$

• ISplot key defined — required for IS diagnostics plots

ISplot key

The ISplot key works like LSFplot and RFADplot: it specifies the axis variables and plot extents used to render the IS proposal distribution diagnostics. When running with -i, the CLI uses ISplot to produce the IS Proposal Diagnostics figures (both u-space and x-space views). If ISplot is absent, the diagnostics plots will not be generated.

"ISplot": {
    "x_var": "T1",
    "x_lim": [-5.0, 5.0],
    "y_var": "T2",
    "y_lim": [-5.0, 5.0],
},

Extracted Results Worksheet Tables

The tables below are transcribed from the Results worksheet in AU2-42d54.xlsx.

Header Information

Field	Value
Problem	AU2
Request ID	cfcb942355da4179b9b48eb4dfa42d54
Run time	00 min 08.93 sec

Deterministic Variables

var_name	value
c	5

Stochastic Variables Definition

var_name	var_type	mean	std	param1	param2
T1	Normal	0	1	0	1
T2	Normal	0	1	0	1

Monte Carlo Results (with Importance Sampling)

beta	pf	cv	max_cv	size	%_removed	cycles	auto_size	mc_with_is	method	bandwidth	fit_samples
4.7775	8.873e−7	0.02978	0.05	6,000	8.333%	3	True	True	kde	0.2817	2,000

Key IS parameters:

• method: kde — the IS proposal distribution was fitted as a kernel density estimate.
• bandwidth: 0.2817 — KDE bandwidth used for the proposal.
• fit_samples: 2,000 — number of samples drawn near the failure region to fit the proposal.

Monte Carlo Variable Statistics and Correlations (IS-weighted)

IS-weighted statistics

Because importance sampling re-weights each sample, the tabulated means and standard deviations reflect the proposal distribution, not the original input distributions. This is expected behavior.

var_name	mean	std	%_oob	cor(T1)	cor(T2)
T1	−0.2182	3.5806	0	1.0000	0.5670
T2	2.3820	4.2807	0	0.5670	1.0000

Notes Reported by Reliafy

1. Validation: Stochastic variables definition and limit state function validation required 4 function calls.
2. Monte Carlo: Creation of importance sampling proposal distribution required 109,980 calls to the limit state function.
3. Monte Carlo: Completed 3 cycles with 2.00e+03 samples per cycle.
4. Monte Carlo: 0.03% of T1 and 0.05% of T2 values were not finite (NaN, inf, complex) out of 6.00e+03 samples and were excluded from the Monte Carlo statistics.
5. Monte Carlo: Load and resistance histograms are not available because LSFreturnsLandR: False, so the limit state function returns empty load/resistance placeholders.

Interpretation Snapshot

• The failure probability is very small (pf ≈ 8.87 × 10⁻⁷, beta ≈ 4.78), making plain Monte Carlo impractical — it would require on the order of 10⁸ samples for reliable estimation. Importance sampling achieves good precision with only 6,000 weighted samples.
• The cv = 0.0298 is below max_cv = 0.05, confirming adequate IS precision for this run.
• The IS proposal construction required ~110,000 LSF evaluations to locate and characterize the failure region before the weighted sampling phase began.
• The IS-weighted variable statistics (T1 mean ≈ −0.22, T2 mean ≈ 2.38) clearly reflect bias toward the failure region, which is expected and correct.
• No load/resistance histogram is generated because LSFreturnsLandR: False, so L and R are returned as empty placeholders.

Generated Figures

The PDF result file for this run is saved as results/2026-03-16/11-06-54/AU2-42d54.pdf.

Figure 1: IS Proposal Diagnostics (u-space)

The KDE proposal distribution in standard-normal space. Contour lines show the proposal PDF; the failure region boundary is visible.

Figure 2: IS Proposal Diagnostics (x-space)

The same proposal distribution plotted in original variable space (T1, T2).

Figure 3: IS Histogram — T1

Figure 4: IS Histogram — T2

Figure 5: Limit State Function Histogram (Importance Sampling)

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Reproducing These Examples

1. Ensure the problem file is present at the path shown in the run context.
2. Run the appropriate command with the desired profile.

For Chan2 (plain MC):

python -m reliafy simulate <profile>

For AU2 (with importance sampling):

python -m reliafy simulate <profile> -i

1. Open the timestamped folder under results/<date>/<time>/.
2. Use *.xlsx (Results worksheet) for tabular summaries and *.json for machine-readable values.
3. For a Chan2-style run, the Load and Resistance Histogram requires LSFreturnsLandR: True and non-empty L, R outputs in the problem file.
4. For an AU2-style IS run, the IS Proposal Diagnostics figures require the ISplot key in the problem dict.