Elsevier

Fuzzy Sets and Systems

Volume 361, 15 April 2019, Pages 33-55
Fuzzy Sets and Systems

Interpretation-aware Cognitive Map construction for time series modeling

https://doi.org/10.1016/j.fss.2018.05.013Get rights and content

Abstract

We raise a few issues regarding time series modeling using Cognitive Maps, which is an example of a qualitative rather than a purely quantitative approach. Methods that operate at the level of concepts instead of numerical values are a worthy alternative for time series processing thanks to certain desirable properties such as abstraction and ease of interpretation. Those features are unavailable in classical time series modeling algorithms, where the main concern is numerical accuracy of a prediction expressed as a series of numerical values. In this work, we argue that it is desirable for Cognitive Map-based models to be simplified, ideally to the point where their descriptiveness and numerical accuracy is not compromised. In particular, we show experimentally that both the size of the map and the size of the set of possible weight values can be minimized without increasing the error rate.

Introduction

Time series modeling is one of the main interests of mathematical and information sciences. This is mostly caused by the omnipresence of real-world phenomena, which can be described using numerical values that form such series. Multiple efforts in the area of time series modeling were strengthened when collecting time series data became a habit and a compelling source of information. Time series analysis allows us not only to learn about the modeled phenomenon itself, but also to gain knowledge about its predicted state in the future – importance of those premises contributes towards the development of a variety of methods. Nowadays, awareness of the need for data mining became apparent as the use of instruments for collecting time series data became a part of our daily routines. Generalizing this into other domains of human activity, such as manufacturing, finances, and natural phenomena prediction, we can say without hesitation that time series modeling became a key problem of data science.

Throughout the years, a wide set of well-known tools has been created for this task. Dominant methods are numerical and rooted in statistical approaches, and their theoretical maturity has reached a very high level. Unfortunately, the choice of such methods is not always optimal. The main issue is the fact that they operate solely at the level of numerical values. Popular tools include, i.a., the ARIMA model family, exponential smoothing methods, frequency domain methods and forecasting techniques based on regression [1], [4], [5], and they compete with one another mostly at the level of numerical accuracy, while being oblivious to the importance of the ease of human interpretation. It is often difficult or even impossible to understand the data or the created model without having the necessary theoretical background, and it can also lead to information overload. In addition, many standard information processing methods do not consider the possibility of model simplification. Most importantly, these models allow us to form predictions exclusively at the level of single, consecutive values, and the results often do not generalize.

It must be emphasized that the area of time series modeling was developing rapidly in order to accommodate a growing volume of time series data. Appearance of new trends in data science, such as big data, heterogeneous data sources or non-numeric data proved that there is a need for a shift in time series modeling paradigms. These problems are especially apparent when data are obtained from fields that have been only recently recognized as a source of information. Examples of such unconventional information channels include retail service recommendations and sentiments expressed in social networks. They consist of information (ordered with respect to time) about preferences, hobbies, and other important aspects of the functioning of humans and their environment. A specific nature of such information streams requires special attention. Let us reiterate that the aforementioned popular numerical techniques are often incapable of distinguishing important aspects of such data – mostly owing to substantial uncertainty (and thus unpredictability) of the human behavior. In addition, standard methods for time series visualization for non-numeric data may lead to a biased problem representation. As a consequence, a human expert might construct a misleading model.

Let us support this discussion with an example from the domain of distributed systems security, in which there is an urgent need for alternative approaches to non-numeric time series modeling. Let us imagine that we are monitoring and recording system events and parameters. Due to spatial dispersion it can happen that events occur at the same time. Also, events do not occur within an equally spaced time interval. In addition, events are not comparable and we do not want to introduce an evaluation saying that, for instance, denial of some service is more serious than some unexpectedly opened socket. Firstly, it is not desirable to represent such data as a value in time plot, as we usually do for regular, numeric time series. Secondly, relationships between events have to be taken into account. The research community steps forward and provides solutions to deal with such problems. Let us mention a paper in which authors propose to mine for structured behaviors [35].

Effective processing of data that is available in non-standard information channels is a highly desired task which can lead to a deeper understanding of dependencies in such data. These issues can be partly mitigated by another approach to time series modeling with state-based data structures, which include Artificial Neural Networks (ANN) [9] and Hidden Markov Models (HMM) [16]. Referenced methods were found to be useful in a wide spectrum of important applications, including speech recognition and synthesis, biochemistry, sequence modeling and general time series prediction. Still, in that case, the prediction takes place at the level of numerical values, i.e. without any abstraction added by the model.

In this paper, we argue that it is crucial to investigate the potential of alternative knowledge processing methods for time series modeling, and we concentrate on Cognitive Maps (CMs) [2]. At this point, we would like to acknowledge the fact that certain issues connected with modeling at the level of concepts have already been raised in other works on this topic [25], [28]. Nonetheless, there is still a significant lack of proper understanding of the motivation and associated issues such as the aforementioned human perception and understanding of the model. Most works concentrate on the technical aspects of presented methods, and for this reason an intuitive, illustrated explanation of the relation between modeling at the level of concepts and modeling at the level of numerical values seems to be lacking. The primary contribution of this paper is that it offers a self-contained study which should be sufficient to understand the advantages and disadvantages of time series modeling with CMs at the conceptual (and not only at the technical) level. The reader can also find an experimental evaluation of custom algorithms that can be used in order to take advantage of certain features of CMs which are relevant to human interpretation.

In comparison to other methods for concept-based and linguistic time series forecasting discussed in the literature, the Cognitive Map-based method presented in this paper distinguishes itself through:

  • Graph-based formalism. A model is constructed in a form of a weighted directed graph, describing relationships between relevant concepts. This representation provides an immediate summary of the entire modeled system. We can see straightaway not only direct relationships, but also indirect relationships (we see paths between all vertices, as one network). In contrast, other approaches to linguistic or concept-based time series forecasting apply conditional clauses of natural language of the form IF X is A THEN Y is B or some other predefined syntax for expressing a forecast. It can be argued that they are descriptive and intuitive, but not as transparent and not as illustrative as a model based on a graph.

  • Higher level of abstraction. The method is built on an initial assumption that the system is modeled using a finite collection of related concepts. After relevant concepts are identified, there is a range of information representation models that can be considered for implementation (crisp, fuzzy, interval-based, intuitionistic, rough, etc., consult Section 3). In other words, there is a spectrum of choices that enables capturing uncertainty in a flexible manner.

    In this paper, we analyze crisp and fuzzy models. Notwithstanding the choice of the information representation scheme, the model always remains expressible using a chosen collection of concepts. When the model is furnished with a conceptually advanced information representation scheme, the concepts-based representation of the phenomena is reinforced rather than replaced. Consistency and coherence of various families of Cognitive Maps (Fuzzy Cognitive Maps, Intuitionistic Cognitive Maps, etc.) is evident when we consider the methodology of model construction, namely, the fact that it is analogous for all kinds of maps. A capability of Cognitive Maps to provide predictions based on a predefined syntax and a collection of linguistic expressions is not a goal in itself: this is one of modeling outcomes, existing in parallel with the graph-based model and quantified predictions. In contrast, adjusting some of the other methods for fuzzy linguistic time series forecasting would require a substantial shift in the methodology, as they are typically paired with a well-defined fuzzy transform.

Section snippets

Paper motivation and objectives

Having in mind the shortcomings of classical approaches to time series modeling, we take under investigation qualitative aspects of models based on Cognitive Maps. As mentioned before, the main motivation for this discussion is the fact that the majority of works concentrate on specific elements of the modeling procedure while ignoring the core problem, namely the interpretation of the model and its qualitative (interpretation-based) analysis. The evident need for non-standard approaches to

Related work

The formalism and applications of first Cognitive Maps (CMs) were conceived on the grounds of biology, psychology, and sociology, where the intuitiveness and practical potential of such a model was first observed [2]. At the beginning, information sciences showed little interest in this method, which in consequence caused further applications and development of CMs to gradually diminish. The revival of this formalism is attributed to Kosko [15] and his works on an extension of Cognitive Maps to

Time series modeling using Cognitive Maps

In this section, we describe more formally and in more detail methods for time series modeling using Cognitive Maps, which were introduced in previous paragraphs. Let us first reiterate that CMs represent the model using a weighted directed graph whose vertices describe the phenomena (concepts) and whose edges determine the dependencies between these concepts. In a graphical representation, the direction of an edge indicates the direction of a dependency, and a value from the set {1,1}

Interpretation-aware construction

When it comes to interpreting the model by its recipients (end users, that is, human beings), we argue that simplicity is the key. This is justified, because certain desirable aspects of Cognitive Maps such as knowledge structuralization that can be easily represented in a compact graphical form, which we have discussed in the preceding sections, can be lost if the model becomes too complex. This simplicity can be related to two main properties of CMs, namely a) the number of nodes, and b)

Experimental results

We present a practical evaluation of algorithms which relate to interpretation-aware Cognitive Map construction. The results were obtained on the machine equipped with the Intel i7-2600 processor running at 3.4 GHz.

Data sets which were used for the empirical study are described below:

Limitations of the Cognitive Map model

In general, the Cognitive Map model is able to successfully model all cyclic dependencies. This ability is essentially inherent, since the model is instantiated with a graph that can have cycles. However, the limitations of the Cognitive Map model are imposed by the dimensionality of a designed modeling space. We have discussed only the two-dimensional case for the sake of clarity. Let us recall that in this case the first dimension represents a current value (i) and the second dimension a

Conclusions

We believe that human interpretation is an important factor when it comes to time series processing and that it has been mostly neglected up to this point. Operating at the level of concepts rather than numerical values is beneficial for the potential recipients of the created models (i.e. humans). We took the experience of the recipients further into account and raised the problem of model simplification (in the intuitive sense of this word) in order to allow for an easy interpretation of both

References (37)

  • A.K. Tsadiras

    Comparing the inference capabilities of binary, trivalent and sigmoid fuzzy cognitive maps

    Inf. Sci.

    (2008)
  • R.R. Yager

    A new approach to the summarization of data

    Inf. Sci.

    (1982)
  • L.A. Zadeh

    Fuzzy sets

    Inf. Control

    (1965)
  • J.S. Armstrong

    Principles of ForecastingA Handbook for Researchers and Practitioners, vol. 30

    (2001)
  • R. Axelrod

    Structure of DecisionThe Cognitive Maps of Political Elites

    (1976)
  • G.E.P. Box et al.

    Time Series AnalysisForecasting and Control

    (2015)
  • R.G. Brown

    Exponential Smoothing for Predicting Demand

    (1956)
  • Z. Chunying et al.

    Research of rough cognitive map model

  • Cited by (5)

    View full text