# Temporal Multimodal Data-Processing Algorithms Based on Algebraic System of Aggregates

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Approach and Requirements for Temporal Multimodal Data Processing

- Multimodality means that the object is determined by a collection of data of a different nature; the logic of presenting and processing data about the object depends on the qualitative and quantitative composition of the data set.
- Temporality means that the elements of data collection are ordered by the time of their receiving; the order of following individual elements of the data sequence affects the result of processing the entire set of data of the object.

- Presenting a data set as a structure of semantically interconnected elements.
- Considering the sequence of data set elements when performing logical operations on them.
- Reordering elements of the data set.

## 4. Basics Notions of ASA

**Definition**

**1.**

- An aggregate is a complex mathematical object, all of whose components are ordered;
- Elements of the tuples can be individual values or tuples of values, and the tuples of values can consist of both values of the same type and values of different types, while each tuple contains elements belonging to one set.

Listing 1. Interface of a data structure for representing an aggregate. |

Aggregate structure:collection ← ordered collection of tuples length ← number of tuples in the aggregate Initialize Aggregatedefine length of the aggregate create collection of size length initialize tuples in collection Insertion (entity) ← add an entity, i.e., a set of values from different tuplesExtraction (entity) ← delete the specific entityTupleInsertion (position) ← add a new tuple to the specific position in the aggregateTupleExtraction (position) ← delete the specific tupleAscendingSorting (primaryTuple) ← sort tuples in the aggregate by the defined primary tuple in asceding orderDecendingSorting (primaryTuple) ← sort tuples in the aggregate by the defined primary tuple in descending orderSingling (primatyTuple) ← remove all non-unique entities by the defined primary tupleSetsOrdering (orderOfTuples) ← reorder tuples by the defined order |

**Definition**

**2.**

**1-1**, represents the element ${a}_{1}^{1,1}$ that belongs to the set ${M}_{1}$ from the definition of ${A}_{1}$ and the green circle, which contains the numbers

**2-1**, represents the element ${a}_{1}^{2,3}$ that belongs to the set ${M}_{3}$ from the definition of ${A}_{2}$.

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

## 5. Algorithms of Operations on Aggregates

#### 5.1. Logical Operations

- 1.
- If ${A}_{1}\doteqdot {A}_{2}$, then aggregates ${A}_{1}$ and ${A}_{2}$ are defined as$${A}_{1}=\u27e6{M}_{1},{M}_{2},\dots ,{M}_{N}\mid \langle {a}_{1}^{1,1},{a}_{2}^{1,1},\dots ,{a}_{{n}_{1}^{1}}^{1,1}\rangle ,\langle {a}_{1}^{1,2},{a}_{2}^{1,2},\dots ,{a}_{{n}_{2}^{1}}^{1,2}\rangle ,\dots ,\langle {a}_{1}^{1,N},{a}_{2}^{1,N},\dots ,{a}_{{n}_{N}^{1}}^{1,N}\rangle \u27e7,$$$${A}_{2}=\u27e6{M}_{1},{M}_{2},\dots ,{M}_{N}\mid \langle {a}_{1}^{2,1},{a}_{2}^{2,1},\dots ,{a}_{{n}_{1}^{2}}^{2,1}\rangle ,\langle {a}_{1}^{2,2},{a}_{2}^{2,2},\dots ,{a}_{{n}_{2}^{2}}^{2,2}\rangle ,\dots ,\langle {a}_{1}^{2,N},{a}_{2}^{2,N},\dots ,{a}_{{n}_{N}^{2}}^{2,N}\rangle \u27e7,$$$$\begin{array}{c}{R}_{\cup}={A}_{1}\cup {A}_{2}=\u27e6{M}_{1},{M}_{2},\dots ,{M}_{N}\mid \langle {a}_{1}^{1,1},{a}_{2}^{1,1},\dots ,{a}_{{n}_{1}^{1}}^{1,1},{a}_{1}^{2,1},{a}_{2}^{2,1},\dots ,{a}_{{n}_{1}^{2}}^{2,1}\rangle ,\\ \langle {a}_{1}^{1,2},{a}_{2}^{1,2},\dots ,{a}_{{n}_{2}^{1}}^{1,2},{a}_{1}^{2,2},{a}_{2}^{2,2},\dots ,{a}_{{n}_{2}^{2}}^{2,2}\rangle ,\dots ,\langle {a}_{1}^{1,N},{a}_{2}^{1,N},\dots ,{a}_{{n}_{N}^{1}}^{1,N},{a}_{1}^{2,N},{a}_{2}^{2,N},\dots ,{a}_{{n}_{N}^{2}}^{2,N}\rangle \u27e7.\end{array}$$

- 2.
- If ${A}_{1}\doteq {A}_{2}$, then aggregates ${A}_{1}$ and ${A}_{2}$ are defined as$$\begin{array}{c}{A}_{1}=\u27e6{M}_{1},{M}_{2}^{1},\dots ,{M}_{k},\dots ,{M}_{{N}^{1}}^{1}\mid \langle {a}_{1}^{1,1},{a}_{2}^{1,1},\dots ,{a}_{{n}_{1}^{1}}^{1,1}\rangle ,\langle {a}_{1}^{1,2},{a}_{2}^{1,2},\dots ,{a}_{{n}_{2}^{1}}^{1,2}\rangle ,\dots ,\langle {a}_{1}^{1,k},{a}_{2}^{1,k},\dots ,{a}_{{n}_{k}^{1}}^{1,k}\rangle ,\dots ,\\ \langle {a}_{1}^{1,{N}^{1}},{a}_{2}^{1,{N}^{1}},\dots ,{a}_{{n}_{{N}^{1}}^{1}}^{1,{N}^{1}}\rangle \u27e7,\end{array}$$$$\begin{array}{c}{A}_{2}=\u27e6{M}_{1},{M}_{2}^{2},\dots ,{M}_{k},\dots ,{M}_{{N}^{2}}^{2}\mid \langle {a}_{1}^{2,1},{a}_{2}^{2,1},\dots ,{a}_{{n}_{1}^{2}}^{2,1}\rangle ,\langle {a}_{1}^{2,2},{a}_{2}^{2,2},\dots ,{a}_{{n}_{2}^{2}}^{2,2}\rangle ,\dots ,\langle {a}_{1}^{2,k},{a}_{2}^{2,k},\dots ,{a}_{{n}_{k}^{2}}^{2,k}\rangle ,\dots ,\\ \langle {a}_{1}^{2,{N}^{2}},{a}_{2}^{2,{N}^{2}},\dots ,{a}_{{n}_{{N}^{2}}^{2}}^{2,{N}^{2}}\rangle \u27e7,\end{array}$$

- 3.
- If ${A}_{1}\circeq {A}_{2}$, then aggregates ${A}_{1}$ and ${A}_{2}$ are defined as$${A}_{1}=\u27e6{M}_{1}^{1},{M}_{2}^{1},\dots ,{M}_{{N}^{1}}^{1}\mid \langle {a}_{1}^{1,1},{a}_{2}^{1,1},\dots ,{a}_{{n}_{1}^{1}}^{1,1}\rangle ,\langle {a}_{1}^{1,2},{a}_{2}^{1,2},\dots ,{a}_{{n}_{2}^{1}}^{1,2}\rangle ,\dots ,\langle {a}_{1}^{1,{N}^{1}},{a}_{2}^{1,{N}^{1}},\dots ,{a}_{{n}_{{N}^{1}}^{1}}^{1,{N}^{1}}\rangle \u27e7,$$$${A}_{2}=\u27e6{M}_{1}^{2},{M}_{2}^{2},\dots ,{M}_{{N}^{2}}^{2}\mid \langle {a}_{1}^{2,1},{a}_{2}^{2,1},\dots ,{a}_{{n}_{1}^{2}}^{2,1}\rangle ,\langle {a}_{1}^{2,2},{a}_{2}^{2,2},\dots ,{a}_{{n}_{2}^{2}}^{2,2}\rangle ,\dots ,\langle {a}_{1}^{2,{N}^{2}},{a}_{2}^{2,{N}^{2}},\dots ,{a}_{{n}_{{N}^{2}}^{2}}^{2,{N}^{2}}\rangle \u27e7,$$$$\begin{array}{c}{R}_{\cup}={A}_{1}\cup {A}_{2}=\u27e6{M}_{1}^{1},{M}_{2}^{1},\dots ,{M}_{{N}^{1}}^{1},{M}_{1}^{2},{M}_{2}^{2},\dots ,{M}_{{N}^{2}}^{2}\mid \langle {a}_{1}^{1,1},{a}_{2}^{1,1},\dots ,{a}_{{n}_{1}^{1}}^{1,1}\rangle ,\\ \begin{array}{c}\langle {a}_{1}^{1,2},{a}_{2}^{1,2},\dots ,{a}_{{n}_{2}^{1}}^{1,2}\rangle ,\dots ,\langle {a}_{1}^{1,{N}^{1}},{a}_{2}^{1,{N}^{1}},\dots ,{a}_{{n}_{{N}^{1}}^{1}}^{1,{N}^{1}}\rangle ,\langle {a}_{1}^{2,1},{a}_{2}^{2,1},\dots ,{a}_{{n}_{1}^{2}}^{2,1}\rangle ,\\ \langle {a}_{1}^{2,2},{a}_{2}^{2,2},\dots ,{a}_{{n}_{2}^{2}}^{2,2}\rangle ,\dots ,\langle {a}_{1}^{2,{N}^{2}},{a}_{2}^{2,{N}^{2}},\dots ,{a}_{{n}_{{N}^{2}}^{2}}^{2,{N}^{2}}\rangle \u27e7.\end{array}\end{array}$$

Listing 2: Algorithm of the union operation for two aggregates. |

Input: aggregates ${A}_{1}$ and ${A}_{2}$Output: aggregate $R$if ${A}_{1}\doteqdot {A}_{2}$ thenfor $i=1:N$ dofor ${k}_{i}=1:{n}_{i}^{1}$do ${r}_{{k}_{i}}={a}_{{k}_{i}}^{1,i}$for ${k}_{i}=\left({n}_{i}^{1}+1\right):\left({n}_{i}^{1}+{n}_{i}^{2}\right)$do ${r}_{{k}_{i}}={a}_{{k}_{i}}^{2,i}$endelse if ${A}_{1}\circeq {A}_{2}$ thenfor $i=1:{N}^{1}$ do $\langle {r}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}}=\langle {a}_{{k}_{i}}^{1,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}$for $i=\left({N}^{1}+1\right):\left({N}^{1}+{N}^{2}\right)$ do $\langle {r}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}}=\langle {a}_{{k}_{i}}^{2,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}$elsefor $i=1:\left({N}^{1}+{N}^{2}\right)$ do if $\langle {a}_{{k}_{i}}^{1,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}$ and $\langle {a}_{{k}_{i}}^{2,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}$ $\in {M}_{i}$ thenfor ${k}_{i}=1:{n}_{i}^{1}$do ${r}_{{k}_{i}}={a}_{{k}_{i}}^{1,i}$for ${k}_{i}=\left({n}_{i}^{1}+1\right):\left({n}_{i}^{1}+{n}_{i}^{2}\right)$do ${r}_{{k}_{i}}={a}_{{k}_{i}}^{2,i}$else$\langle {b}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}=\langle {a}_{{k}_{i}}^{2,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}$ endendfor $i=\left({N}^{1}+{N}^{2}+1\right):\left({N}^{1}+{N}^{2}+{N}^{b}\right)\phantom{\rule{0ex}{0ex}}\langle {r}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}=\langle {b}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}$endend |

Listing 3: Algorithm of the intersection operation for two aggregates. |

Input: aggregates ${A}_{1}$ and ${A}_{2}$Output: aggregate $R$if ${A}_{1}\doteqdot {A}_{2}$ thenfor $i=1:N$ dofor ${k}_{i}=1:{n}_{i}^{1}$doif ${a}_{{k}_{i}}^{1,i}\in {A}_{2}^{i}\phantom{\rule{0ex}{0ex}}{r}_{{k}_{i}}={a}_{{k}_{i}}^{1,i}$endendfor ${k}_{i}=\left({n}_{i}^{1}+1\right):\left({n}_{i}^{1}+{n}_{i}^{2}\right)$doif ${a}_{{k}_{i}}^{2,i}\in {A}_{1}^{i}\phantom{\rule{0ex}{0ex}}{r}_{{k}_{i}}={a}_{{k}_{i}}^{2,i}$endendendelse if ${A}_{1}\circeq {A}_{2}$ then$\langle {r}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}=\varnothing $ elsefor $i=1:\left({N}^{1}+{N}^{2}\right)$ do if $\langle {a}_{{k}_{i}}^{1,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}$ and $\langle {a}_{{k}_{i}}^{2,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}$ $\in {M}_{i}$ thenfor ${k}_{i}=1:{n}_{i}^{1}$doif $\begin{array}{c}{a}_{{k}_{i}}^{1,i}\in {A}_{2}^{i}\\ {r}_{{k}_{i}}={a}_{{k}_{i}}^{1,i}\end{array}$endendfor ${k}_{i}=\left({n}_{i}^{1}+1\right):\left({n}_{i}^{1}+{n}_{i}^{2}\right)$doif ${a}_{{k}_{i}}^{2,i}\in {A}_{1}^{i}\phantom{\rule{0ex}{0ex}}{r}_{{k}_{i}}={a}_{{k}_{i}}^{2,i}$endendendend |

Listing 4: Algorithm of the exclusive intersection operation for two aggregates. |

Input: aggregates ${A}_{1}$ and ${A}_{2}$Output: aggregate $R$if ${A}_{1}\doteqdot {A}_{2}$ thenfor $i=1:N$ dofor ${k}_{i}=1:{n}_{i}^{1}$doif ${a}_{{k}_{i}}^{1,i}\in {A}_{2}^{i}\phantom{\rule{0ex}{0ex}}{r}_{{k}_{i}}={a}_{{k}_{i}}^{1,i}$endendendelse if ${A}_{1}\circeq {A}_{2}$ then${r}_{{k}_{i}}{}_{{k}_{i}=1}^{{n}_{i}^{1}}=\varnothing $ elsefor $i=1:{N}^{1}$ do if $\langle {a}_{{k}_{i}}^{1,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}$ and $\langle {a}_{{k}_{i}}^{2,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}$ $\in {M}_{i}$ thenfor ${k}_{i}=1:{n}_{i}^{1}$doif ${a}_{{k}_{i}}^{1,i}\in {A}_{2}^{i}\phantom{\rule{0ex}{0ex}}{r}_{{k}_{i}}={a}_{{k}_{i}}^{1,i}$endendendend |

Listing 5: Algorithm of the difference operation for two aggregates. |

Input: aggregates ${A}_{1}$ and ${A}_{2}$Output: aggregate $R$if ${A}_{1}\doteqdot {A}_{2}$ thenfor $i=1:N$ dofor ${k}_{i}=1:{n}_{i}^{1}$doif ${a}_{{k}_{i}}^{1,i}\notin {A}_{2}^{i}\phantom{\rule{0ex}{0ex}}{r}_{{k}_{i}}={a}_{{k}_{i}}^{1,i}$endendendelse if ${A}_{1}\circeq {A}_{2}$ thenfor $i=1:N$ do$\langle {r}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}=\langle {a}_{{k}_{i}}^{1,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}$ endelsefor $i=1:{N}^{1}$ do if $\langle {a}_{{k}_{i}}^{1,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}$ and $\langle {a}_{{k}_{i}}^{2,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}$ $\in {M}_{i}$ thenfor ${k}_{i}=1:{n}_{i}^{1}$doif ${a}_{{k}_{i}}^{1,i}\notin {A}_{2}^{i}\phantom{\rule{0ex}{0ex}}{r}_{{k}_{i}}={a}_{{k}_{i}}^{1,i}$endendelse$\langle {r}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}=\langle {a}_{{k}_{i}}^{1,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}$ endend |

Listing 6: Algorithm of the symmetric difference operation for two aggregates. |

Input: aggregates ${A}_{1}$ and ${A}_{2}$Output: aggregate $R$if ${A}_{1}\doteqdot {A}_{2}$ thenfor $i=1:N$ dofor ${k}_{i}=1:{n}_{i}^{1}$doif ${a}_{{k}_{i}}^{1,i}\notin {A}_{2}^{i}\phantom{\rule{0ex}{0ex}}{r}_{{k}_{i}}={a}_{{k}_{i}}^{1,i}$endendfor ${k}_{i}=\left({n}_{i}^{1}+1\right):\left({n}_{i}^{1}+{n}_{i}^{2}\right)$doif ${a}_{{k}_{i}}^{2,i}\notin {A}_{1}^{i}\phantom{\rule{0ex}{0ex}}{r}_{{k}_{i}}={a}_{{k}_{i}}^{2,i}$endendendelse if ${A}_{1}\circeq {A}_{2}$ thenfor $i=1:{N}^{1}$ do$\langle {r}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}=\langle {a}_{{k}_{i}}^{1,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}$ endfor $i=1:\left({N}^{1}+1\right):\left({N}^{1}+{N}^{2}\right)$do$\langle {r}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}=\langle {a}_{{k}_{i}}^{2,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}$ endelsefor $i=1:\left({N}^{1}+{N}^{2}\right)$ do if $\langle {a}_{{k}_{i}}^{1,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}$ and $\langle {a}_{{k}_{i}}^{2,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}$ $\in {M}_{i}$ thenfor ${k}_{i}=1:{n}_{i}^{1}$doif ${a}_{{k}_{i}}^{1,i}\notin {A}_{2}^{i}\phantom{\rule{0ex}{0ex}}{r}_{{k}_{i}}={a}_{{k}_{i}}^{1,i}$endendfor ${k}_{i}=1:{n}_{i}^{2}$doif ${a}_{{k}_{i}}^{2,i}\notin {A}_{1}^{i}\phantom{\rule{0ex}{0ex}}{r}_{{k}_{i}}={a}_{{k}_{i}}^{2,i}$endendelse$\langle {r}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}=\langle {a}_{{k}_{i}}^{1,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{1}}\phantom{\rule{0ex}{0ex}}\langle {b}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}=\langle {a}_{{k}_{i}}^{2,i}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}$ endendfor $i=\left({N}^{1}+1\right):\left({N}^{1}+{N}^{2}\right)$ do$\langle {r}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}=\langle {b}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}^{2}}$ endend |

#### 5.2. Ordering Operations

Listing 7. Algorithm of the sets ordering operation. |

Input: aggregates $A$ and ${A}_{T}$Output: aggregate $R$while not ordered do$j=1$ for $i=1:N$ doif $\langle {a}_{{k}_{i}}^{i}\rangle {}_{{k}_{i}=1}^{{n}_{i}}\in {M}_{j}^{T}$ then$\langle {r}_{{k}_{i}}\rangle {}_{{k}_{i}=1}^{{n}_{i}}=\langle {a}_{{k}_{i}}^{i}\rangle {}_{{k}_{i}=1}^{{n}_{i}}\phantom{\rule{0ex}{0ex}}j=j+1$ if $j>N$ thensets are orderedendendendend |

**Theorem**

**1.**

**Proof of Theorem**

**1.**

- $\widehat{{A}_{1}}={A}_{1}\models {A}_{2}$;
- $\widehat{{A}_{2}}={A}_{2}\models {A}_{1}$.

**Remark**

**1.**

Listing 8: Algorithm of the ascending sorting operation. |

Input: aggregate $A$ and primary tuple $\langle {a}_{{k}_{p}}^{p}\rangle {}_{{k}_{p}=1}^{{n}_{p}}$Output: aggregate $R$for ${k}_{p}=1:{n}_{p}$ dofor ${k}_{m}=1:\left({n}_{p}-k\right)$ doswapped is falseif ${a}_{{k}_{m}}^{p}>{a}_{{k}_{m}+1}^{p}$ dofor $i=1:N$ do${r}_{{k}_{m}}^{i}={a}_{{k}_{m}+1}^{i}\phantom{\rule{0ex}{0ex}}{r}_{{k}_{m}+1}^{i}={a}_{{k}_{m}}^{i}$ endswapped is trueendendif swapped is falsebreak loopendend |

Listing 9: Algorithm of the extraction operation. |

Input: aggregate $A$, number of set $k$ and position of element $m$Output: aggregate $R$for $i=1:N$ dofor ${k}_{i}=1:\left(m-1\right)$ do ${r}_{{k}_{i}}={a}_{{k}_{i}}^{i}$for ${k}_{i}=m:\left({n}_{i}-1\right)$ do ${r}_{{k}_{i}}={a}_{{k}_{i}+1}^{i}$end |

Listing 10: Algorithm of the insert operation. |

Input: aggregate $A$, number of set $k$ and position of element $m$Output: aggregate $R$for $i=1:N$ dofor ${k}_{i}=1:\left(m-1\right)$ do${r}_{{k}_{i}}={a}_{{k}_{i}}^{i}$ endif $k=i$ then${r}_{m}={a}_{m}^{k}$ else${r}_{m}=\varnothing $ endfor ${k}_{i}=m:{n}_{i}$ do${r}_{{k}_{i}+1}={a}_{{k}_{i}}^{i}$ endend |

Listing 11: Algorithm of the singling operation. |

Input: aggregate $A$Output: aggregate $R$m = 2 while not all found dofor ${k}_{1}=m:{n}_{1}$ doif ${a}_{{k}_{1}}^{1}={a}_{{k}_{1}-1}^{1}$ then$m={k}_{1}$ break loopendall foundendfor $i=1:N$ doextract ${a}_{m}^{i}$endend |

## 6. Multi-Image Notion and Formation Algorithm

**Definition**

**6.**

- 1.
- Normalization of partial multi-images (a normalized multi-image (normalized tuple) is a multi-image (tuple) to the elements of which dummy elements are added; a dummy element is a value that is absent in the tuple before its normalization; an empty element ∅ can be used as a dummy element; at other steps of the multi-image processing, dummy elements are ignored):$$\begin{array}{c}{\widehat{I}}_{j}={I}_{j}\u22ca\left({\langle \varnothing \rangle}_{k=1}^{{E}_{i}^{j}}\succ {a}_{{n}_{j}}^{j}\right)=\u27e6T,{M}_{j}\mid \langle {t}_{i}\rangle {}_{i=1}^{{\tau}_{j}},\langle \langle {a}_{{i}_{j}}^{j}\rangle {}_{{i}_{j}=1}^{{n}_{j}},{\langle \varnothing \rangle}_{l=1}^{{E}_{i}^{j}}\rangle \u27e7=\\ =\u27e6T,{M}_{j}\mid \langle {t}_{i}\rangle {}_{i=1}^{{\tau}_{j}},\langle {\widehat{a}}_{{i}_{j}}^{j}\rangle {}_{{i}_{j}=1}^{{n}_{j}+{E}_{i}^{j}}\u27e7\end{array}$$

- 2.
- Uniting the normalized partial multi-images:$${I}_{U}=\underset{j=1}{\overset{N}{{\displaystyle \mathsf{U}}}}{\widehat{I}}_{j}=\u27e6T,{M}_{1},\dots {M}_{N}\mid \langle \langle {t}_{i}\rangle {{}_{i=1}^{{\tau}_{j}}\rangle}_{j=1}^{N},\langle {\widehat{a}}_{{i}_{1}}\rangle {}_{{i}_{1}=1}^{{n}_{1}+{E}_{i}^{1}},\dots \langle {\widehat{a}}_{{i}_{N}}\rangle {}_{{i}_{N}=1}^{{n}_{N}+{E}_{i}^{N}}\u27e7$$

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 9.**An example of applying the exclusive intersection operation to two quasi-compatible aggregates.

**Figure 14.**An example of applying the symmetric difference operation to two quasi-compatible aggregates.

**Figure 15.**An example of applying the symmetric difference operation to two incompatible aggregates.

Features | TTS | TS | TMS | ASA |
---|---|---|---|---|

Ordering | yes | no | no | yes |

Logical operations | no | yes | yes | yes |

Fuzziness | partly | partly | partly | yes |

Data aggregation | no | no | partly | yes |

Temporality | yes | no | no | yes |

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**MDPI and ACS Style**

Pester, A.; Sulema, Y.; Dychka, I.; Sulema, O. Temporal Multimodal Data-Processing Algorithms Based on Algebraic System of Aggregates. *Algorithms* **2023**, *16*, 186.
https://doi.org/10.3390/a16040186

**AMA Style**

Pester A, Sulema Y, Dychka I, Sulema O. Temporal Multimodal Data-Processing Algorithms Based on Algebraic System of Aggregates. *Algorithms*. 2023; 16(4):186.
https://doi.org/10.3390/a16040186

**Chicago/Turabian Style**

Pester, Andreas, Yevgeniya Sulema, Ivan Dychka, and Olga Sulema. 2023. "Temporal Multimodal Data-Processing Algorithms Based on Algebraic System of Aggregates" *Algorithms* 16, no. 4: 186.
https://doi.org/10.3390/a16040186