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.
- ad : For first- and second-order automatic differentiation (install this first).
- NumPy : Numeric Python
- SciPy : Scientific Python (the nice distribution constructors require this)
- Matplotlib : Python plotting library
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
rvkwarg to input a distribution from thescipy.statsmodule:>>> 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)
Now let's walk through an example of a three-part assembly stack-up:
>>> 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
printstatement (or simply the object if in a terminal):>>> Z # "print" is optional at the command-line uv(1176.45, 99699.6822917, 0.708013052944, 6.16324345127)
The
describeclass 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
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]
Statistical correlations are correctly handled, even after calculations have taken place:
>>> x1 - x1 0.0 >>> square = x1**2 >>> square - x1*x1 0.0
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]]
Another useful feature is available through the error_components class method that has various ways of representing the first- and second-order 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%
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).
Transparent calculations with derivatives automatically calculated. No or little modification to existing code required.
Basic NumPy support without modification. Vectorized calculations built-in to the
adpackage.Nearly all standard math module functions supported through the
soerp.umathsub-module. If you think a function is in there, it probably is.Nearly all derivatives calculated analytically using
adfunctionality.Easy continuous distribution constructors:
N(mu, sigma): Normal distributionU(a, b): Uniform distributionExp(lamda, [mu]): Exponential distributionGamma(k, theta): Gamma distributionBeta(alpha, beta, [a, b]): Beta distributionLogN(mu, sigma): Log-normal distributionChi2(k): Chi-squared distributionF(d1, d2): F-distributionTri(a, b, c): Triangular distributionT(v): T-distributionWeib(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.
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
soerp-XYZdirectory to any other location that python can find it and rename itsoerp.If
setuptoolsis installed, run:$ [sudo] easy_install [--upgrade] soerp
If
pipis installed, run:$ [sudo] pip install [--upgrade] soerp
To remove the package, there are really two good ways to do this:
Go to the folder
site-packagesordist-packagesand simply delete the foldersoerpandsoerp-XYZ-egg-info.If
pipis installed, run:$ [sudo] pip uninstall soerp
- uncertainties : First-order error propagation
- mcerp : Real-time latin-hypercube sampling-based Monte Carlo error propagation
Please send feature requests, bug reports, or feedback to Abraham Lee.
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.
- N.D. Cox, 1979, Tolerance Analysis by Computer, Journal of Quality Technology, Vol. 11, No. 2, pp. 80-87
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