Prof. Dr. Vladik Kreinovich,
University of Texas at El Paso
Title: How to represent and process uncertainty in practical situations: towards a feasible combination of probabilistic, interval, and fuzzy uncertainty.
In the very beginning of the data processing process, as raw data, we have measurement results and expert estimates. Measurements are never 100% accurate, there is usually a difference
between the measurement result and the (unknown) actual value of the measured quantity. This difference is known as the measurement error. For a measurement result, sometimes, we know the probability distribution of the measurement error, but often, all we know is the upper bound on the absolute value of the measurement error – in which case, we have interval uncertainty.
For expert estimates, we can have different upper bounds with different degrees of confidence – which forms what is known as a
fuzzy number.
In a few cases, we directly deal with raw data, but usually, when we process data, we deal with the results of some previous data processing, data processing that typically uses results with all
three types of uncertainty. So, to describe both inputs and outputs of data processing, we need to take all these three types of uncertainty into account.
In many cases, measurement errors are small, so we can safely ignore terms which are quadratic and higher order with respect to these errors – and only take linear terms into account. In this
case, the resulting uncertainty can be described as the sum of the three types of uncertainty – which can be therefore processed separately.
This is how we envision a number of the future: a tuple consisting of the numerical result of data processing plus the three uncertainty measures: upper bound for the interval part of the
uncertainty, standard deviation (for example) for the probabilistic part, and an interval (for example) corresponding to the triangular form of the fuzzy uncertainty part.
But this is not all: the same raw data values are often used to compute different inputs to our data processing algorithm. In this case, even when all original measurement errors are independent, there is a correlation between the inputs. So, to properly describe uncertainty, it is not enough to describe the numbers themselves, it is important to also describe relations between different numbers. In the probabilistic case, this can be described by correlation. In the interval case, it can be proven that the simplest way to describe such relation is by an ellipsoid – so for every two numbers, we describe the parameters of the corresponding ellipse. For the fuzzy part, it is also reasonable to use the corresponding ellipse. In the linearized case, this enables us, e.g., to avoid overestimation – the main problem of the traditional interval computation techniques.
Now we have a more adequate picture of numbers of the future: numbers plus relations between pairs of numbers. This is similar to the 1960s transition in physics. Before that, we had a hierarchical model of matter: molecules consist of atoms, atoms consist of elementary particles – and we can separate atoms, separate particles, and study the properties of each component. In the modern physics, e.g., a proton consists of quarks, but quarks cannot be separated, we always need to take into account their interaction. Similarly, we cannot reduce the description of uncertainty by simply describing uncertainty of each number, we need to take relation into account.
Another analogy is quantum computing, where it is crucial to use – and take into account – entanglement between states of individual objects. And in the future, when quantum computing
becomes ubiquitous, we will need to add the fourth uncertainty component to the description of umbers and their relation – a component corresponding to quantum uncertainty.
And, of course, in situations when measurement errors are larger andquadratic terms can no longer be ignored, we need to take into account relation between different types of uncertainty – as described, e.g., by p-boxes and related techniques.
Additional information
The workshop will take place in the institutes library, room 116, on Tuesday, June 9, 2026. The workshop will also be available via Webex online meeting. If you want to participate online please contact Niklas Winnwisser.
