Erroneous: Tracking Statistical Dependencies, Calculating Better Errors

Erroneous has multiple components

• A token system for tracking statistical dependencies: Data Correlation Tokens (DCTs)
  • A specification for tokens
  • Token readers and writers
• Tools for making use of this tracking data, and providing related functionality
  • Calculating standard errors
  • Stochastic error calculations
  • A basic symbolic calculator
  • Abstract classes for projects that wish to leverage DCTs

The Problem

Example

The problem erroneous solves is best illustrated by a problem that occurs with the standard approach to propagating errors, which is to calculate a new variance whenever you perform a calculation.

Consider two variables, $a$ and $b$ with standard deviations $\sigma_a$ and $\sigma_b$. Add these together to get $x = a+b$ and the calculated standard deviation of $x$ using the standard error propagation rules will be $\sigma_x = \sqrt{\sigma_a^2 + \sigma_b^2}$.

All good up to this point.

Now lets take $b$ away from $x$.

$y = x - b$

Again, we have an error propagation formula, $\sigma_y = \sqrt{\sigma_x^2 + \sigma_b^2}$. But this calculation gives a standard deviation of $y$ as:

$\sigma_y = \sqrt{\sigma_x^2 + \sigma_b^2} = \sqrt{\sigma_a^2 + 2\sigma_b^2}$.

But we know this is wrong, because $y = a$, so $\sigma_y$ should be equal to $\sigma_a$!

Why does this happen?

Fundamentally it is because $b$ and $x$ are not statistically independent and out calculations did not account for this.

$a$ and $b$ were statistically independent and the error propagation rule was correct, but $x$ depends on $b$ (being the sum of $a$ and $b$). Because of this wrong formula was used because our calculation forgot that $x$ was dependent on $b$.

A Solution

You only need to save a token

Example, with the two $a$ and $b$ variables from before, lets make two derived datasets:

$p = a - b$ which will have a variance of $\sigma_a^2 + \sigma_b^2$ $q = b - a$ which will have a variance of $\sigma_a^2 + \sigma_b^2$

Now we can save these datasets, throwing away the original data, but saving the DCT with them, then if we load them we can calculate difference, which should be zero

$d = p - q$ and find that $\sigma_d = 0$

WOHOOO!

A Standard

Scope: DCTs are intended to be used on data represents functional relationships, i.e. data which specifies (non-unique) output values for a list of (unique) input values, with errors and other statistical moments (the latter being a more experimental future path).

Data Integrity

Guiding Principles

• It should not try to do too much. Nothing can do all maths. It cannot represent all maths.