This paper addresses the problem of forecasting categorical time series, which have wide application in all areas of activity where judgement and expert judgement are used. One of the main problems in forecasting such series is to account for the linguistic uncertainty that arises in expert judgement, moving from qualitative to quantitative assessment and vice versa. This paper proposes a categorical time series prediction model based on computing with words. A codebook of 32, 15, 11, 9, 8, 7, 6 or 4 words is suggested for comparing categorical time series attributes. The number of words depends on the level of detail of the categorical attributes of the series. Words from codebooks are described using discrete interval type-2 fuzzy sets, which allows for the linguistic uncertainty of the categorical attributes of the series. Based on the proposed model, a fuzzy algorithm for categorical time series forecasting is developed, consisting of five steps: word model definition, fuzzy relationship definition, fuzzy relationship grouping, fuzzy forecasting, result interpretation. The quality of the proposed model is confirmed by three estimated characteristics: mean absolute prediction error; mean square prediction error; mean relative prediction error. Time series, categorical data, discrete interval type-2 fuzzy set, uncertainty, computing with ORCID: 0000-0002-9046-8015 (O. Tymchuk); 0000-0002-6343-4469 (A. Pylypenko); 0000-0001-9021-3680 (M. Iepik))
Categorical Times Series (CTS) is a time series in which observations at each point in time have categorical values (nominal or ordinal). Obviously, such series are quite common in practice, although in the literature they are paid much less attention than series with numerical continuous variables. The source of CTS can be both devices and expert judgments. For example, IP addresses, web addresses, area codes are obtained from the web server; diagnoses are recorded as a result of medical examination; sequences of letters and words are processed during speech recognition, etc.
During the last twenty years of the last century a number of different approaches to CTS modelling
have been proposed. Mostly these models were based on Markov chain model and discrete ARMA
model. Particular attention was paid to the theory of generalized linear models, the authors of which are
P. McCullagh and J. Nelder (1989) [
Parametric methods of time series analysis assume that the data are based on a stationary process. CTS is considered stationary if the marginal distribution of the data is constant over the period of time for which they were collected. The correlation between consecutive values is a function only of their distance from each other, not of their position in the series. However, there are many examples of categorical series that do not meet this definition of stationarity. A thorough analysis of non-stationarity
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of CTS is given in Heinz Kaufmann [
In the early 2000s, the wavelet method [
In recent publications, the authors continue to study the problem of stationarity and likelihood
maximization of CTS [
The purpose of this paper is to develop a fuzzy time series model that will allow to perform calculations with words. Based on the developed model, a fuzzy algorithm for forecasting categorical time series is proposed and the quality of forecasting is evaluated.
In this paper the theory of interval type-2 fuzzy sets (IT2 FS) is applied. It is accepted that the degree
of belonging of an element of a universal set to a fuzzy subset is not defined for each element of a
universal set uniquely, but there is a certain blur. That is to say, the degree of membership is fuzzy and
takes values in the interval [
( ) = ( − 1)° ( − ), ( ) ∈ , ( − 1) ∈ , ∀ , ∈ {1, … , }, where ( ) – is the predicted value given as a granulated term, ( − 1) – last value in the T2 FTS, – the codebook, – number of granulated terms in the codebook, ( − ),
= ̅1̅,̅̅̅ – is a relation that describes the fuzzy relationship between ( ) and ( − ),
– number of time series values, ° – Zadeh's Compositional Rule of Inference (MAX MIN).
A granulated term is described by a word and a discrete interval type-2 fuzzy set (DIT2FS): = 〈 〉,
∈ {1, … , },
= 〈 , ̃ 〉, where – granulated term, – word,
̃ – DIT2FS, which describes the word.
The proposed model (1) allowed us to develop a fuzzy algorithm for forecasting time series, which
is based on the basic principles of the theory of type-2 fuzzy sets [
Step 3: Grouping fuzzy logic relationships. This paper uses Chen's [
Inference (MAX MIN)
Step 4: Fuzzy forecasting. Fuzzy forecasting is defined using the Zadeh's Compositional Rule of
̃ ( ) = {
(
( −1)), if −1 in
,
the approach to forecasting fuzzy time series [
Step 1: Identifying the word model. Generally, when interpreting numbers and predicting values,
people tend to move to categories such as "a little", "a bit", "reasonably", "good", "a lot", etc. Therefore,
each categorical feature in the time series is matched with a word from the codebook. This paper
proposes the use of codebooks provided by J.M. Mendel. Codebooks of 32, 15, 11, 9, 8, 7, 6 and 4
words (granulated terms) are allowed. The choice of codebook depends on the level of categorical detail
of the time series. Typically, granulated terms are represented as a pair - word and DIT2FS, which is
defined on primary variable [
Step 2: Identifying fuzzy relationships. After each categorical feature is matched with a codebook word, a fuzzy logic relationship matrix FR is constructed +1 – i+1-th time series value (next value).
where – i-th time series value (current state), where ̃ ( ) – predicted value in the form of DIT2FS, −1 – last value of the time series, represented as a granulated term, ̃ ( − 1) – last value of the time series, represented as DIT2FS.
Step 5: Interpretation of the result. The prediction result can be both a numerical value and a word
from the codebook. To obtain the result in numerical form, you need to reduce the type ̃ ( ). Type
reduction can be performed using the ECM algorithm for DIT2FS [
A classic example of CTS is the data set on sleep in newborns [
When comparing categorical assessments obtained, for example, by experts, with integer numerical
values, linguistic uncertainty arises, which needs to be eliminated. This problem is considered in the
theory of fuzzy sets, where the degree of membership of an element of a universal set to a fuzzy subset
(3)
(4)
can be any real number from the interval [
After each infant’s sleep state is matched with a word from the codebook, a fuzzy logical relationship matrix is constructed → +1 (Table 2), where is the infant's sleep state at time i (current state), and +1 is the infant's sleep state at time i + 1 (next value).
The relationship groups derived from Table 2 are shown in Table 3.
Forecasting the next infant's sleep state is based on operations on the FOUs of the current state and the corresponding group from Table 3.
To match the forecast value to a categorical attribute, we use Jaccard's similarity index: select the attribute that will have the greatest similarity between the FOU that describes that attribute and the resulting DITFS.
Software was developed based on the developed categorical time series forecasting algorithm. The programming language Python and the libraries numpy and matplotlib were used to implement the software. Description of the modules of the developed software:
1. words_model.py – module that defines 32-, 15-, 11, -9-, 8-, 7-, 6- and 4-word codebooks. Each codebook is presented as a dictionary. An example of a 4-word codebook presentation is shown in Figure 3.
2. std_mfs.py – module in which standard membership functions (trapmf, trimf, gaussmf, smf, zmf, sigmf) are defined.
3. dit2fou.py, dit2ft.py and dit2lv.py – modules which contain a class describing the FOU DIT2FS, a term and a linguistic variable respectively. The class structure of the modules is shown in the class diagram (Figure 4). 4. similarity_measure.py – module in which functions are defined to calculate the similarity relationship between FOUs of two DITFSs A and B on primary variable X. The following functions are defined in the module: Jaccard Similarity Measure; Zhao, Xiao, Li and Deng's Similarity Measure, Yang and Lin's Similarity Measure. 5. forecasting_categorical.py – module containing a basic function for predicting a categorical time series. The function signature forecasting: forecasting(*, data=None, categories=None, model_words='words_32', sm_method='jaccard', steps=1) data – dataset, categories – a list of categorical features from the dataset to be associated with codebook words, , → +1 91, al → al 92, al → al 93, al → al 94, al → al 95, al → al 96, al → al 97, al → ah 98, ah → ah 99, ah → ah 100, ah → ah 101, ah → ah 102, ah → ah 103, ah → ah 104, ah → ah 105, ah → ah 106, ah → tr model_words – codebook type, default is 32-word codebook sm_method – method for determining the similarity between FOUs DIT2FSs, the default is Jaccard Similarity Measure, steps – number of foreseeable values, default is 1.
6. main.py – entry point to the application.
A diagram of how the modules interact is shown in Figure 5. To use the developed software, run the script from the main.py module, specifying the necessary prediction parameters. An example of the Python code from the main.py module for the test case described in the next section is shown in Listing main.py (Figure 6).
To evaluate the obtained forecasting result, we again resort to replacing categorical values with their numerical equivalents. Figure 3 shows three rows of data: Real, Prediction and Error (Error = Real – Prediction, mean(Error) = –0,315, std(Error) = 0.912). Д To test the algorithm, a test set of 77 observations was taken. The graph of the fuzzy time series simulation result is shown in Figure 7. 2) root mean squared error (RMSE): mean absolute percentage error (MAPE): 1 =1 = ∑| | = 0,605,
∑ = √ =1
2 − 1
= 0,959, = 1 ∑ =1 | |
× 100% = 18,179%, – the actual (true) value, – number of observations. where – error, difference between predicted and true values in t period, These indicators can be further used to compare forecasting models and choose the best one. (5) (6) (7)