A short tutorial on Fuzzy Time Series

But some key features distinguish the Fuzzy Time Series e turn it on a attractive option:ReadabilityManageabilitySimplicityScalabilityHereafter I going to assume that you don’t have a machine learning (with focus on fuzzy systems) and time series background and I will present the key concepts of these fields..Then the Fuzzy Time Series methods will be introduced with the help of the pyFTS library..Let’s go?What are Fuzzy Sets?SourceIf you already know about Fuzzy Logic and Fuzzy Sets, you can go ahead to the next section..But classic / rigid sets do not give us this flexibility.The Fuzzy Logic, proposed by Zadeh (1965), state a duality instead of this dichotomy: a certain element may belong and simultaneously do not belong to the same set at certain levels, such that the membership is a value in the interval [0, 1]..The fuzzy sets have no strict boundaries and they are usually overlapping, so — using the previous example — I can be medium high and medium, or 90% medium and 10% high for example.Given X, a Numerical Variable, such that X ∈ ℝ — for instance an height measure — its Universe of Discourse, abbreviated to U, is the is the range of values that this variable can assume, such that U = [ min(X), max(X) ].A linguistic variable A is the transformation of the values of the numerical variable X into a set of words / linguistic terms (what we call fuzzification)..Each word/linguistic ter is a fuzzy set ã ∈ Ã, and each fuzzy set ã is associated to a function μ (mu greek letter), such that μã: X →[0,1] (this mean that μã receives an input value from X and return an output value on interval [0,1]).Let’s get back to the numerical variable height, measured in centimeters..If x is smaller than a, or greater than c, we say that x is completely outside of the fuzzy set (int other words: its membership grade is equal to 0).So, what would our sets look like for the linguistic variable Ã?Suppose a person is 163cm tall..If you want to know more about Fuzzy Logic and Fuzzy Systems….But where are here to talk about time series, right?.So let’s talk about time!What are Time Series?If you already know about time series, you can go to the next section, this one is very introductory!Time series are sets of data representing the behavior of one (or more) random variable over time, and its main characteristic is that the successive records of this variable are not independent of each other and their analysis must take into account the order in that were collected.According to Ehlers (2009), “the neighboring observations are dependent and we are interested in analyzing and modeling this dependence”..That to predict future values of the time series I use past / lagged values of the same series.A simple example is the autoregressive AR(p) model, where p indicates the number of lagged variables used on forecasting..Where do fuzzy sets come into this story?What are Fuzzy Time Series?SourceThe use of fuzzy sets for modeling and predicting time series arises almost intuitively, first based on the ability of fuzzy models to approximate functions, but also on the readability of rules using linguistic variables that make them more accessible to experts and non-experts analysis.The pioneer work on fuzzy time series is Song and Chisson(1993) but here we present the evolution published by Chen(1996)..The idea is to divide the Universe of Discourse from time series in intervals/partitions (the fuzzy sets), and learn how each area behaves (extracting rules through the time series patterns)..In other words: let’s create a linguistic variable to represent the numerical time series, and these areas will be the linguistic terms of our variable.When we create a linguistic variable to represent the universe of discourse, we create a “vocabulary,” and then the fuzzyfied series is composed of words in that vocabulary.. More details

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