In this paper, we propose a new methodology for time series classification. It employs two techniques: the fuzzy transform (F-transform) and the well-known decision tree classifier. A combination of these two tools appears to result in a new classification method that shows good statistical properties and could be a noteworthy alternative for considered as best for time series 1NN classifier.
Time series data form an increasing proportion of the world’s data supply. The omnipresence of
time series and the exponentially growing size of databases resulted in an explosion of interest
in Data Mining methods adapted to the specificity of time series. In typical time series mining
tasks, such as indexing, grouping, classification, forecasting, segmentation, summarization, and
anomaly detection, the analysis of the similarity between the series plays an important role
(see, e.g. [
The so-called fuzzy transform (or F-transform, for short), introduced by Perfilieva [
The main goal of this contribution is to compare the best distance measures in the most popular 1NN method with the new classification method – random forest based on F-transform. We want to test the usefulness of the F-transforms in time series classification.
The paper is organized as follows: in Section 2 we recall basic information on the F-transform. Next, in Section 3, we make a short introduction to time series classification and propose how to apply the F-transform for this task. Then, in Section 4, we describe the investigation conducted to examine the proposed classification method and discuss some of the experimental results.
In this section, we provide only the definition and basic concepts related to the F-transform.
For more details we refer, e.g., to [
Let us consider a continuous real function : [, ] → R. The F-transform is defined concerning the so-called fuzzy partition of the domain [, ] by a finite number of fuzzy sets satisfying some axioms specified in the following definition.
Definition 1. Let 1 < . . . < denote fixed nodes within [, ], such that 1 = and = and ⩾ 3. We say that fuzzy sets 1, . . . , form a fuzzy partition of [, ] if they satisfy the following conditions for = 1, . . . , : 1. () = 1; 2. () = 0 for ∈/ (− 1, +1), where for uniformity of notation, we put 0 = 1 = and +1 = = ; 3. is continuous; 4. is strictly increasing in [− 1, ] and strictly decreasing in [, +1]; 5. ∑︀
=1 () = 1 for each ∈ [, ].
The last axiom is known as orthogonality or the Ruspini condition. The membership functions
of fuzzy sets 1, . . . , forming a fuzzy partition are called basic functions. It is worth noting
that the shapes of basic functions are not predetermined and can be selected to meet some
additional specific properties. A fuzzy partition is called uniform if the nodes are equidistant
(i.e. = − 1 + ℎ for = 2, . . . , and some fixed ℎ) and fuzzy sets 2, . . . , − 1 are shifted
copies of symmetrized 1 (or , for details see [
Definition 2. Let 1, . . . , denote a fuzzy partition of [, ] and let : [, ] → R be a continuous function. The -tuple (1, . . . , ) of real numbers given by
∫︀ () ∫︀ = ()() , = 1, . . . , , (1) is a (direct) fuzzy transform (F-transform) of with respect to the given fuzzy partition.
In practical applications, is usually not given analytically. Instead, we are provided with some data points obtained from observations or measurements. Thus Def. 2 can be modified by replacing integrals in (1) by finite sums.
To be more strict, let 1, . . . , denote a fuzzy partition of [, ] and let the function : [, ] → R be given at fixed points 1, . . . , ∈ [, ], where > . We say that the set of points {1, . . . , } is suficiently dense with respect to the fuzzy partition 1, . . . , if for every ∈ {1, . . . , } there exist ∈ {1, . . . , } such that () > 0. Now we are able to define the so-called discrete F-transform.
Definition 3. Let 1, . . . , denote a fuzzy partition of [, ] and let : [, ] → R be a function known at points 1, . . . , ∈ [, ]. Moreover, let us assume that the set {1, . . . , } is suficiently dense with respect to the fuzzy partition 1, . . . , . Then the -tuple (1, . . . , ) of real numbers given by (2) (3) = ∑︀∑=︀1 ()() , = 1, . . . , ,
=1 () is a (direct) discrete fuzzy transform (discrete F-transform) of with respect to the given fuzzy partition.
In applications we usually refer to the F-transform without specifying explicitly whether it is for a continuous or discrete problem, i.e. according to Def. 2 or Def. 3, assuming that the reader can recognize which is the actual underlying concept from the context.
The F-transform (as well as the discrete F-transform) of will be denoted by [ ] = (1, . . . , ), where the number is called the -th component of the F-transform. One can realize that the components of the F-transform are just weighted mean values of the original function , where the weights are determined by the basic functions 1, . . . , .
The F-transform itself would not be interesting enough without its inverse formula which allows reconstructing from [ ].
Definition 4. Let [ ] = (1, . . . , ) be the direct F-transform of with respect to a fuzzy partition 1, . . . , of [, ]. Then the function , : [, ] → R given by
,() = ∑︁ · (), ∈ [, ],
=1 is called the inverse F-transform of .
It is seen that the inverse F-transform is a continuous function on [, ]. However, what’s more,
it can be shown that the sequence of the inverse F-transform {,}∞=3 converges uniformly
to the initial function as → ∞. Moreover, this result is valid both when a fuzzy partition is
uniform [
Thus, to conclude, while the direct F-transform [ ] may serve as a discrete approximate representation of a function : [, ] → R, the inverse F-transform , is a suitable continuous approximation of .
Both of these statements make the F-transform an extremely useful tool in various fields
and applications. One of them is time series analysis. Until now, it was used there mainly and
successfully for prediction (see, e.g., [
Time series classification is an important problem in data analysis. Although many algorithms
have been proposed the nearest neighbor (NN) classifier still seems to be the most appreciated
one. It is so mostly because of its simplicity and noticeably good performance in many situations.
The 1NN classifiers are mainly used for time series due to the dimensionality of the data. Their
high performance, especially with dynamic time warping (DTW) and its modified versions used
as the distance measure has been confirmed by many experiments (see, e.g., [
In our study, we set ourselves the goal to make use of renowned classification methods, such as the random forest classifier, but in the field of time series. The reason we use the F-transform is to reduce the high dimensionality of time-series data. Indeed, using the F-transform we can compress time series to significantly smaller vectors and then perform the classification task on them.
Let = { : ∈ } be a given time series. Obviously, might be viewed as a function () = defined on a fixed time interval which is not given analytically but instead some measurements at points ∈ are available. Assuming { : ∈ } is suficiently dense with respect to the fuzzy partition 1, . . . , and following (2) the F-transform of is given by [] = (1, . . . , ), where
∑︀ () = ∈ ∑︀ () ∈ , = 1, . . . , .
(4) Looking at the time series representation given by (4) it becomes clear that the use of the F-transform makes it possible to reduce significantly the dimensionality of the data.
Further on we will conduct tests to check whether our approach combining decision trees and the F-transform can compete with other widespread classification tools.
To investigate how the proposed F-transform-based random forest classifier behaves and to
compare it with classifiers utilizing various distances and similarity measures we conducted
an extensive experimental study. It was performed on benchmark datasets from the UCR
Time Series Classification Archive [
In our experiment, we analyzed 29 datasets included in the UCR repository (i.e. PowerCons,
Cofee, BME, SmoothSubspace, Wafer, Plane, Strawberry, ItalyPowerDemand, Meat, GunPoint,
CBF, UMD, BeetleFly, Symbols, MoteStrain, ECG200, Trace, SwedishLeaf, FaceFour, Yoga, Beef,
Wine, Fish, Fungi, ShapesAll, Ham, FiftyWords, ElectricDevices, BirdChicken). They have been
specially selected to deal with a wide range of possible problems that can be encountered when
analyzing time series, since they difer in the number of observations, the sizes of the training
and test sets, the shape of their trajectories, etc. For instance, PowerCons contains data on
the energy consumption of French households over two seasons - heating and summer. The
collected trajectories contain clear observations of outliers related to the increased energy
consumption. Cofee dataset contains spectrograms of two diferent types of cofee - Robusta
and Arabica. The BME set is artificially generated data that represents three types of trajectories:
containing a local maximum at the beginning of the series, containing a local maximum at the
end of the series, and having no shots. Wafer contains trajectories corresponding to the records
from several sensors monitoring the production process, the so-called silicon wafers, where the
observation labels say whether the record describes a normal or a disturbing process. The Plane
set contains the outlines of seven diferent airplane models converted into a one-dimensional
time series. The thing that characterizes the contour data is often the high variability over
a small period, which corresponds to small indentations in the shape. Strawberry contains
spectrograms of fruit mousses made from strawberries or from strawberries with the addition
of other fruits. For a more detailed description of the datasets, we refer to [
The suggested F-transform-based random forest classifier was compared with the most
efective variants of the 1NN algorithm equipped with 55 metrics and similarity measures described
in [
Figure 1 presents a comparison of the considered classification methods. Box-plots given
there shows the distribution of their accuracy obtained for all 29 benchmarks. As to be expected,
there is no definite winner but the top eficient classifiers are based on distances related to
DTW and its modifications. This result is in line with the conclusions of the research conducted
by Górecki and Piasecki [
Good properties of the method utilizing the F-transform are confirmed by the drawing in Figure 2 where each considered method is placed on a map describing two features: the mean error and standard deviation (to be more strict, the position of the method corresponds to the upper left corner of the rectangle representing the given method). As it is easily seen the proposed classification method based on the F-transform reveals the smallest standard deviation of the error and one of the smallest mean errors.
Additionally, all considered methods were grouped according to clusters, where the mean errors and standard deviations were used as features. Figure 2 shows the results of k-means clustering, where the optimal number of clusters turned out to be equal to 3. Hence we have obtained 3 disjoint groups: weak and unstable (yellow), moderate (green), and highly eficient and stable (blue). Our classification method based on the F-transform belongs to the “blue” cluster but we marked it in red to make it easier to identify in the drawing.
Our research confirmed that none of the classification methods neither any of the distance measures nor our F-transform classifier is the best for all available datasets. However, there is a group of methods performing significantly better than the others. This group includes the proposed classifier based on the F-transform. The limited space does not allow the presentation and discussion of all the results obtained, but - as we believe - the reader may be convinced of the high quality of the adsorbed method based on what is presented in this contribution. Anyway, classifiers obtained with the F-transform challenge the methods considered to be the best at present, i.e. the 1NN algorithm with an appropriate metric.
Moreover, the F-transform enables use in a classification other than distance-based methods (e.g. decision trees, logistic regression) as it significantly reduces the dimensionality of the data. Due to the F-transform, we can jump to a completely new level of classification and use completely diferent methods, which - as it turns out - work well. We hope that our study will also contribute to increasing the interest in the F-transform, showing its usefulness not only in forecasting (as demonstrated earlier), but also in other time series analysis tasks.
Many questions and problems are still open. In particular, we want to examine if there is a significant relationship between the fuzzy partition selection and the resulting classification. We have observed that the F-transform works efectively for the classification of spectrographic data. Hence, we want to indicate a class of time series for which the F-transform method reveals the best properties as a classifier. Finally, in the nearest future, we plan to examine how to apply the F-transform in other time series analysis problems such as cluster analysis or anomaly detection.