interval analysis

Intervals play a central role in the probability bounds analysis and has been discussed in the context of computational functional analysis. It is known for making rigourous computations in terms of three kinds of errors in numerical analysis: rounding errors, truncation errors, and input errors.

Interval arithmetic

Hint

the key point of the definitions of basic arithmetic operations between intervals is computing with intervals is computing with sets.

Interval arithmetic operations can be defined as :

\[ X \odot Y = \{ x \odot y : x \in X, y \in Y \} \]

where \(\odot\) stands for the elementary binary operations such as addition or product etc.

A key consideration of the propagation of interval objects is the dependency issue, which hinders the naive uses of interval arithmetic in many problems as it often yields inflated interval outputs. The image set under a real-valued function mapping \(f\) as \(x\) varies through a given interval \([X]\) (or simply \(X\)) can be defined as:

Note

Square backets are used to visually hint the nature of an Interval typed variable. In Python, square brackets suggest a list datatype which is ubiquitous, as such, in PyUncertainNumber we provide a parser for easy creation of interval objects with lists.

\[ f(X) = \{ f(x): x \in X \} \]

It should be noted that when interval arithmetic is naively used in computations, it may not necessary yield the best-possible (or sharpest) range. A useful interval extension is the mean value form

\[ f(X) \subseteq F_{MV}(X) = f(m) + \sum_{i=1}^{n} D_{i}F(X) (X_{i} - m_{i}) \]

where \(X\) denotes an interval vector and \(m\) is the midpoint vector while \(D_{i}F\) be an interval extension of the first derivatives \(\partial f / \partial x_{i}\).

Refer to Propagation page for additional methods (e.g. vertex method, interval substitution) for interval propagation.

handling with measurement uncertainty

Naturally, intervals serve as an intuitive representations for measurement error. Engineers often report measurement incertitude in the form of \([m \pm w]\). Many statistical models arise based on the interval statistics. As an example, similar to using a probability distribution to characterise precise data, we can employ a p-box to characterise interval-valued data. Generalisation of empirical cumulative distribution function can also be intuitively made. In addition, Kolmogorov–Smirnov bounds can also be generalised to derive confidence limits for imprecise data.

\[ [\overline{F}(x), \underline{F}(x)] = \big[ \min(1, \hat{F}_{L}(x) + D_{N}^{\alpha}), \ \max(0, \hat{F}_{R}(x) - D_{N}^{\alpha}) \big] \]

PyUncertainNumber provides a straghtforward syntax in charactersing interval-valued data.

characterisation