Multivariate time series are a very common non-tabular type of data. In many practical cases, multivariate time series encode real-world situations that include temporal information, and, recently, machine learning from datasets of multivariate time series has become a very active area of research. Modal symbolic learning has shown itself to be a serious alternative to sub-symbolic methods such as neural networks for non-tabular data; when applied to multivariate time series, modal symbolic learning makes use propositional temporal logic such as interval temporal logic. In special cases, however, multivariate time series display an internal structure that propositional modal logics are unable to capture. In this paper, we propose a first-order extension of propositional interval temporal logic, we describe its syntax and semantics, and we study its expressive power in relationship with such special cases of multivariate time series.
essentially based on propositional logic, with very few exceptions, and consequently limited to
tabular data. In the past few years, however, modal symbolic learning has been proposed as
a generalization of symbolic learning to modal logic, and applied to non-tabular data. Modal
logics have the ability of capturing a greater fraction of the internal structure of the instances
of a non-tabular dataset, and by enriching intelligent symbolic models, such as decision trees,
with the possibility of learning modal logic rules, one is able to extract such structure, as well as
to learn and express interesting and complex patterns. In the temporal case, for example, modal
symbolic learning is instantiated to temporal symbolic learning, and learning from temporal
datasets is accomplished with, for instance, temporal decision trees [
In [
In special cases, multivariate time series may display a richer internal structure than the one that can be captured with propositional HS. Two prominent examples of this situation are audio and electroencephalogram signals. In both cases, pre-processing of the original signal (in the first case, the single sound power expressed in , in the second case the electric power of each electrode expressed in ) produces several temporal variables (in the first case audio frequencies, in the second one electric frequencies) which are naturally ordered and not mutually independent; we call such temporal variables adjacent. In this paper, we propose a simple first-order extension of HS that allows us to capture patterns of temporal datasets of multivariate time series with adjacent variables, by encompassing the possibility of comparing the natural order between temporal variables as well as the possibility of relating the behaviour of two variables in an interval of time. 2. First-Order Interval Temporal Logic for Adjacent Variables The language of the First-Order Modal Logic for Time Intervals for Adjacent Variables (FOHSa, for short) encompasses a numerable set of first-order variables 1, 2, . . ., a set of unary function symbols 1, 2, . . ., arbitrary real constants, the set of comparison operators <, ≤ , =, ≥ , and >, standard Boolean operators, and unary interval temporal logic operators, one for each Allen’s relation between any two intervals on a linear order, namely ⟨⟩ (meets), ⟨⟩ (later), ⟨⟩ Modality
Definition ⟨⟩ (after) ⟨⟩ (later) ⟨⟩ (begins) ⟨⟩ (ends) ⟨⟩ (during) ⟨⟩ (overlaps) (overlaps), ⟨⟩ (during), ⟨⟩ (begins), ⟨⟩ (ends), and their inverse ones (if ⟨⟩ is an interval operator, ⟨⟩ denotes its inverse one). Formulas of FOHSa are built using the following ::= ::= ::= ( ) | ( ) | ◁▷ | ◁▷ | ¬ |
∨ | ⟨⟩ | ∀, where is a temporal variable, is a first-order variable, is a function, ◁▷∈ {<, ≤ , =, ≥ , >}, ∈ R, and ∈ {, , , , , , , , , , , }.
Intuitively, a FOHSa formula is interpreted on a multivariate time series; towards a precise
definition of the truth relation, though, a few definitions are necessary. First, a multivariate time
series = {1, . . . , } is said to be adjacent variables if the set of time series {1, . . . , }
is linearly ordered by a relation <; the equality relation = is defined in the standard way.
Now, given the set D (the domain of all time series in ), we say that I(D) is the set of all
intervals that can be built on D, that is I(D) = {[, ] | , ∈ D, < }. Moreover, let
turn, a set = { | ∈ N+}, and each element of a set ∈ is a function : R
ℱ = {1, . . . , } be a set of feature extraction function templates, where each template is, in
→ R. A
set represents the interpretation of a function symbol ; since the value ( ) is computed on
an interval [, ], the specific function that must be used to compute it depends on the quantity
− + 1; for example, if is the (generalized) mean of a set of reals, computing the mean of
on the interval [
A typical audio signal presents a spectrum of frequencies that range roughly from 20 to 20,000. In audio signal processing of a sample, it is customary to extract its spectral representation, facilitating their interpretation in terms of audio frequencies. To this end, the most widespread adopted technique is known as extracting the Mel-Frequency Cepstral Coeficients
(MFCC) [
In time series processing, a set of feature extraction functions can be identified from the current literature (in most cases features have been presented as specific to a problem, but they often re-occur in diferent application areas). Among many others, examples of properties that can be expressed in FOHSa in the above domains include: there exists a frequency for which it is always true that, during every interval when its mean value is lower than there is an interval in which its maximum value is greater than ′ and for every frequency it is true that if in an interval its mean value is lower than then in some future interval there is a higher frequency whose mean value is greater than ′.
We presented FOHSa, a novel logical system specifically designed to express non-propositional temporal interval properties that could be of interest in several applicative areas and diferent contexts including, for example, symbolic machine learning.