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Real-time latin-hypercube-sampling-based Monte Carlo Error Propagation

Project description


mcerp is an on-the-fly calculator for Monte Carlo methods that uses latin-hypercube sampling (see soerp for the Python implementation of the analytical second-order error propagation original Fortran code ‘SOERP’ by N. D. Cox) to perform non-order specific error propagation (or uncertainty analysis). The mcerp package allows you to easily and transparently track the effects of uncertainty through mathematical calculations. Advanced mathematical functions, similar to those in the standard math module can also be evaluated directly.

Due to the nature of random sampling techniques, calculation results will vary from session to session (but be consistent within the session) since new latin-hypercube samples are only generated when variables are newly defined or re-defined. By default, each variable uses 10,000 latin-hypercube samples that are sufficiently random. The number of samples can be changed by assigning an integer value to the mcerp.npts object (typically, values between 1,000 and 1,000,000 are sufficiently large to ensure small errors in the resulting statistics). This should only be changed prior to performing calculations since all subsequent calculations assume that each input has the same number of sampled points (this could be changed through resampling, I suppose…).

In order to correctly use mcerp, knowledge of the distributions from the scipy.stats sub-module is required. The result of all calculations generates a mean, variance, and standardized skewness and kurtosis coefficients.

What’s New In This Release

  • New correlation features (see examples below):
    • correlate: Impose a set of correlation coefficients on variables (if needed, this should be used before performing any calculations).
    • plotcorr: Show a scatterplot matrix of variables

Required Packages

  • NumPy : Numeric Python
  • SciPy : Scientific Python

Suggested Packages

  • Matplotlib : Python plotting library (required for mcerp.plot)

Basic examples

Let’s start with the main import:

>>> from mcerp import *  # N, U, Gamma, Beta, etc.

Now, we can construct many kinds of statistical distributions (both continuous and discrete). Here’s a basic example that involves a three-part assembly:

>>> x1 = N(24, 1) # normally distributed
>>> x2 = N(37, 4) # normally distributed
>>> x3 = Exp(2)  # exponentially distributed

The first four moments can be accessed at any time:

>>> x1.mean
>>> x1.var
>>> x1.skew
>>> x1.kurt
>>> x1.stats  # this returns all four as a list

Now we’ll compute the actual stack-up using normal mathematics and see what happens:

>>> Z = (x1*x2**2)/(15*(1.5 + x3))

We can see how the statistics of each of these distributions propagated through the calculations in two basic ways:

  1. Telling python to print the object:

    >>> Z  # Explicit "print" command not necessary at the command-line
    uv(1161.35518507, 116688.945979, 0.353867228823, 3.00238273799)
  2. Using the describe class method (provides a more detailed explanation):

    >>> Z.describe()
    MCERP Uncertain Value:
     > Mean...................  1161.35518507
     > Variance...............  116688.945979
     > Skewness Coefficient...  0.353867228823
     > Kurtosis Coefficient...  3.00238273799

Viewing the distribution

We can also plot the distributions using the plot class method:

>>> x1.plot()  # No inputs shows the distribution's PDF
>>>  # explicit 'show()' required to display to screen

and for the outputs:

>>> Z.plot()  # shows the Kernel Density Estimate (KDE) of the data
>>> Z.plot(hist=True)  # shows a histogram instead of a KDE

or both:

>>> Z.plot()
>>> Z.plot(hist=True)


Even correlations are correctly handled:

>>> x1 - x1

>>> Z*Z - Z**2

If it desired to force a set of variables to have certain correlations, which is not uncommon in real-life situations, this can be done with the correlate function (NOTE: this should be done BEFORE any calculations have taken place in order to work properly). For example, say we desire to impose a -0.75 correlation between x1 and x2 from the above example:

# The correlation coefficients before adjusting
>>> print correlation_matrix([x1, x2, x3])
[[ 1.          0.00558381  0.01268168]
 [ 0.00558381  1.          0.00250815]
 [ 0.01268168  0.00250815  1.        ]]

And here is what the actual data looks like in a matrix plot form:

Now we’ll apply the desired correlations (note that all diagonal elements should be 1.0):

# The desired correlation coefficient matrix
>>> c = np.array([[  1.0, -0.75, 0.0],
...               [-0.75,   1.0, 0.0],
...               [  0.0,   0.0, 1.0]])

# Apply the correlations into the samples (works in-place)
>>> correlate([x1, x2, x3], c)

# Show the new correlation coefficients
>>> print correlation_matrix([x1, x2, x3])
[[  1.00000000e+00  -7.50010477e-01   1.87057576e-03]
 [ -7.50010477e-01   1.00000000e+00   8.53061774e-04]
 [  1.87057576e-03   8.53061774e-04   1.00000000e+00]]

The correlation matrix is roughly what we expected within a few percent. This is what the newly correlated samples looks like:

Now that the inputs’ relations have been modified, let’s check how the output of our stack-up has changed:

# Z should now be a little different
>>> Z = (x1*x2**2)/(15*(1.5 + x3))
>>> Z.describe
MCERP Uncertain Value:
 > Mean...................  1153.710442
 > Variance...............  97123.3417748
 > Skewness Coefficient...  0.211835225063
 > Kurtosis Coefficient...  2.87618465139

We can also see what adding that correlation did: reduced the mean, reduced the variance, increased the skewness, increased the kurtosis.

If any exceptions appear during this operation, it is likely because the correlation matrix is either not symmetric, not positive-definite, or both.

Advanced Example

Here’s a slightly more advanced engineering example, showing how the random effects of input parameters propagates through the calculation of the volumetric gas flow through an orifice meter:

>>> import mcerp.umath as umath  # sin, exp, sqrt, etc.
>>> H = N(64, 0.5)
>>> M = N(16, 0.1)
>>> P = N(361, 2)
>>> t = N(165, 0.5)
>>> C = 38.4
>>> Q = C*umath.sqrt((520*H*P)/(M*(t + 460)))
>>> Q.describe()
MCERP Uncertain Value:
 > Mean...................  1330.9997362
 > Variance...............  57.5497899824
 > Skewness Coefficient...  0.0229295468388
 > Kurtosis Coefficient...  2.99662898689

Interestingly enough, even though the calculations involve multiplication, division, and a square-root, the result appears to be very close to a Normal (Gaussian) distribution (Q ~ N(1331, 7.6) where 7.6 = sqrt(58.2)).

Using Distributions

Since all of the variables in mcerp are statistical distributions, they are created using the scipy.stats distributions. There are also some convenience constructors that should make defining a distribution easier, though it’s not necessary to use them. See the source code of the UncertainVariable class for info that describes how to construct many of the most common statistical continuous and discrete distributions using the scipy.stats distributions.

Here are the equivalent constructors that I’ve found to be easier to use (the location, scale, and shape parameters are described in the respective Wikipedia pages):

Continuous Distributions
N(mu, sigma) Normal distribution
U(a, b) Uniform distribution
Exp(lamda, [mu]) Exponential distribution
Gamma(k, theta) Gamma distribution
Beta(alpha, beta, [a, b]) Beta distribution
LogN(mu, sigma) Log-normal distribution
X2(k) Chi-squared distribution
F(d1, d2) F-distribution
Tri(a, b, c) Triangular distribution
T(v) T-distribution
Weib(lamda, k) Weibull distribution
Discrete Distributions
Bern(p) Bernoulli distribution
B(n, p) Binomial distribution
G(p) Geometric distribution
H(M, n, N) Hypergeometric distribution
Pois(lamda) Poisson distribution

For example, the following constructions are equivalent:

# explicitly calling out the scipy.stats distribution
>>> import scipy.stats as ss
>>> x = uv(ss.norm(loc=10, scale=1))

# using a built-in constructor
>>> x = N(10, 1)

From my experience, the first option can be tedious and difficult to work with, but you make the choice. That’s why there are options!

Main Features

  1. Transparent calculations. No or little modification to existing code required.
  2. Basic NumPy support without modification. (I haven’t done extensive testing, so please let me know if you encounter bugs.)
  3. Advanced mathematical functions supported through the mcerp.umath sub-module. If you think a function is in there, it probably is. If it isn’t, please request it!
  4. Easy statistical distribution constructors. The location, scale, and shape parameters follow the notation in the respective Wikipedia articles.
  5. Correlation enforcement and variable sample visualization capabilities.


Make sure you have the SciPy and NumPy packages installed! This package won’t work without them.

You have several easy, convenient options to install the mcerp package (administrative privileges may be required)

  1. Simply copy the unzipped mcerp-XYZ directory to any other location that python can find it and rename it mcerp.

  2. From the command-line, do one of the following:

    1. Manually download the package files below, unzip to any directory, and run:

      $ [sudo] python install
    2. If setuptools is installed, run:

      $ [sudo] easy_install --upgrade mcerp
    3. If pip is installed, run:

      $ [sudo] pip install --upgrade mcerp

Python 3

To use this package with Python 3.x, you will need to run the 2to3 conversion tool at the command-line using the following syntax while in the unzipped mcerp directory:

$ 2to3 -w .

This should take care of the main changes required. Then, run:

$ python3 install

If bugs continue to pop up, please email the author.

See also


Please send feature requests, bug reports, or feedback to Abraham Lee.

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