soerp 0.9.3

Second Order ERror Propagation

Overview

soerp is the Python implementation of the original Fortran code SOERP by N. D. Cox to apply a second-order analysis to error propagation (or uncertainty analysis). The soerp 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.

In order to correctly use soerp, the first eight statistical moments of the underlying distribution are required. These are the mean, variance, and then the standardized third through eighth moments. These can be input manually in the form of an array, but they can also be conveniently generated using either the nice constructors or directly by using the distributions from the scipy.stats sub-module. See the examples below for usage examples of both input methods. The result of all calculations generates a mean, variance, and standardized skewness and kurtosis coefficients.

Required Packages

• ad : For first- and second-order automatic differentiation (install this first).

Suggested Packages

• NumPy : Numeric Python
• SciPy : Scientific Python (the nice distribution constructors require this)
• Matplotlib : Python plotting library

Basic examples

Let's begin by importing all the available constructors:

>>> from soerp import *   # uv, N, U, Exp, etc.

Now, we can see that there are several equivalent ways to specify a statistical distribution, say a Normal distribution with a mean value of 10 and a standard deviation of 1:

• Manually input the first 8 moments (mean, variance, and 3rd-8th standardized central moments):

  >>> x = uv([10, 1, 0, 3, 0, 15, 0, 105])
  

• Use the rv kwarg to input a distribution from the scipy.stats module:

  >>> x = uv(rv=ss.norm(loc=10, scale=1))
  

• Use a built-in convenience constructor (typically the easiest if you can):

  >>> x = N(10, 1)
  

>>> x1 = N(24, 1)  # normally distributed
>>> x2 = N(37, 4)  # normally distributed
>>> x3 = Exp(2)  # exponentially distributed
>>> Z = (x1*x2**2)/(15*(1.5 + x3))

>>> x1.moments()
[24.0, 1.0, 0.0, 3.0000000000000053, 0.0, 15.000000000000004, 0.0, 105.0]
>>> Z.moments()
[1176.45, 99699.6822917, 0.708013052944, 6.16324345127]

>>> x1 - x1
0.0
>>> square = x1**2
>>> square - x1*x1
0.0

>>> Z.d(x1)  # dZ/dx1
45.63333333333333

>>> Z.d2(x2)  # d^2Z/dx2^2
1.6

>>> Z.d2c(x1, x3)  # d^2Z/dx1dx3 (order doesn't matter)
-22.816666666666666

>>> Z.gradient([x1, x2, x3])
[45.63333333333333, 59.199999999999996, -547.6]

>>> Z.hessian([x1, x2, x3])
[[0.0, 2.466666666666667, -22.816666666666666], [2.466666666666667, 1.6, -29.6], [-22.816666666666666, -29.6, 547.6]]

>>> Z.error_components(pprint=True)
COMPOSITE VARIABLE ERROR COMPONENTS
uv(37.0, 16.0, 0.0, 3.0) = 58202.9155556 or 58.378236%
uv(24.0, 1.0, 0.0, 3.0) = 2196.15170139 or 2.202767%
uv(0.5, 0.25, 2.0, 9.0) = -35665.8249653 or 35.773258%

>>> from soerp.umath import *  # 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()
SOERP Uncertain Value:
 > Mean...................  1330.99973939
 > Variance...............  58.210762839
 > Skewness Coefficient...  0.0109422068056
 > Kurtosis Coefficient...  3.00032693502

Main Features

1. Transparent calculations with derivatives automatically calculated. No or little modification to existing code required.

2. Basic NumPy support without modification. Vectorized calculations built-in to the ad package.

3. Nearly all standard math module functions supported through the soerp.umath sub-module. If you think a function is in there, it probably is.

4. Nearly all derivatives calculated analytically using ad functionality.

5. Easy continuous distribution constructors:

   • 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

   The location, scale, and shape parameters follow the notation in the respective Wikipedia articles. Discrete distributions are not recommended for use at this time. If you need discrete distributions, try the mcerp python package instead.

Installation

Make sure you install the ad package first!

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

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

   $ [sudo] python setup.py install
   

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

3. If setuptools is installed, run:

   $ easy_install --upgrade soerp
   

4. If pip is installed, run:

   $ pip install --upgrade soerp
   

$ 2to3 -w -f all *.py

See Also

• uncertainties : First-order error propagation
• mcerp : Real-time latin-hypercube sampling-based Monte Carlo error propagation

Contact

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

Acknowledgements

The author wishes to thank Eric O. LEBIGOT who first developed the uncertainties python package (for first-order error propagation), from which many inspiring ideas (like maintaining object correlations, etc.) are re-used and/or have been slightly evolved. If you don't need second order functionality, his package is an excellent alternative since it is optimized for first-order uncertainty analysis.

References

• N.D. Cox, 1979, Tolerance Analysis by Computer, Journal of Quality Technology, Vol. 11, No. 2, pp. 80-87