soerp 0.9.6
Second Order Error Propagation
Overview
soerp is the Python implementation of the original Fortran code SOERP by N. D. Cox to apply a secondorder 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 submodule. 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 secondorder automatic differentiation (install this first).
 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 3rd8th 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 builtin convenience constructor (typically the easiest if you can):
>>> x = N(10, 1)
A Simple Example
Now let’s walk through an example of a threepart assembly stackup:
>>> 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))
We can now see the results of the calculations in two ways:
The usual print statement (or simply the object if in a terminal):
>>> Z # "print" is optional at the commandline uv(1176.45, 99699.6822917, 0.708013052944, 6.16324345127)
The describe class method that explains briefly what the values are:
>>> Z.describe() SOERP Uncertain Value: > Mean................... 1176.45 > Variance............... 99699.6822917 > Skewness Coefficient... 0.708013052944 > Kurtosis Coefficient... 6.16324345127
Distribution Moments
The eight moments of any input variable (and four of any output variable) can be accessed using the moments class method, as in:
>>> 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]
Correlations
Statistical correlations are correctly handled, even after calculations have taken place:
>>> x1  x1 0.0 >>> square = x1**2 >>> square  x1*x1 0.0
Derivatives
Derivatives with respect to original variables are calculated via the ad package and are accessed using the intuitive class methods:
>>> 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
When we need multiple derivatives at a time, we can use the gradient and hessian class methods:
>>> 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]]
Error Components/Variance Contributions
Another useful feature is available through the error_components class method that has various ways of representing the first and secondorder variance components:
>>> 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%
Advanced Example
Here’s a slightly more advanced example, estimating the statistical properties of volumetric gas flow through an orifice meter:
>>> 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
This seems to indicate that even though there are products, divisions, and the usage of sqrt, the result resembles a normal distribution (i.e., Q ~ N(1331, 7.63), where the standard deviation = sqrt(58.2) = 7.63).
Main Features
Transparent calculations with derivatives automatically calculated. No or little modification to existing code required.
Basic NumPy support without modification. Vectorized calculations builtin to the ad package.
Nearly all standard math module functions supported through the soerp.umath submodule. If you think a function is in there, it probably is.
Nearly all derivatives calculated analytically using ad functionality.
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) : Lognormal distribution
 Chi2(k) : Chisquared distribution
 F(d1, d2) : Fdistribution
 Tri(a, b, c) : Triangular distribution
 T(v) : Tdistribution
 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! (If you use options 3 or 4 below, this should be done automatically.)
You have several easy, convenient options to install the soerp package (administrative privileges may be required)
Download the package files below, unzip to any directory, and run:
$ [sudo] python setup.py install
Simply copy the unzipped soerpXYZ directory to any other location that python can find it and rename it soerp.
If setuptools is installed, run:
$ [sudo] easy_install [upgrade] soerp
If pip is installed, run:
$ [sudo] pip install [upgrade] soerp
Uninstallation
To remove the package, there are really two good ways to do this:
Go to the folder sitepackages or distpackages and simply delete the folder soerp and soerpXYZegginfo.
If pip is installed, run:
$ [sudo] pip uninstall soerp
See Also
 uncertainties : Firstorder error propagation
 mcerp : Realtime latinhypercube samplingbased 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 firstorder error propagation), from which many inspiring ideas (like maintaining object correlations, etc.) are reused 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 firstorder uncertainty analysis.
References
 N.D. Cox, 1979, Tolerance Analysis by Computer, Journal of Quality Technology, Vol. 11, No. 2, pp. 8087
File  Type  Py Version  Uploaded on  Size  

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 Author: Abraham Lee
 Home Page: https://github.com/tisimst/soerp
 Keywords: uncertainty analysis,uncertainties,error propagation,second order,derivative,statistics,method of moments,distribution
 License: BSD License

Categories
 Development Status :: 5  Production/Stable
 Intended Audience :: Education
 Intended Audience :: Science/Research
 License :: OSI Approved :: BSD License
 Operating System :: OS Independent
 Programming Language :: Python
 Programming Language :: Python :: 2.6
 Programming Language :: Python :: 2.7
 Programming Language :: Python :: 3.0
 Programming Language :: Python :: 3.1
 Programming Language :: Python :: 3.2
 Programming Language :: Python :: 3.3
 Topic :: Education
 Topic :: Scientific/Engineering
 Topic :: Scientific/Engineering :: Mathematics
 Topic :: Scientific/Engineering :: Physics
 Topic :: Software Development
 Topic :: Software Development :: Libraries
 Topic :: Software Development :: Libraries :: Python Modules
 Topic :: Utilities
 Package Index Owner: tisimst.myopenid.com
 DOAP record: soerp0.9.6.xml