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).

• 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)

### A Simple Example

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:

1. The usual print statement (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)
2. 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 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).

## 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:

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)

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: $ [sudo] easy_install [--upgrade] soerp
4. If pip is installed, run:

$[sudo] pip install [--upgrade] soerp ## Uninstallation To remove the package, there are really two good ways to do this: 1. Go to the folder site-packages or dist-packages and simply delete the folder soerp and soerp-XYZ-egg-info. 2. If pip is installed, run: $ [sudo] pip uninstall soerp

• 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

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