STUDY ON TRENDS OF SOCIAL MOOD DYNAMICS WITH WAVELET ANALYSIS METHODS

STUDY ON TRENDS OF SOCIAL MOOD DYNAMICS WITH WAVELET ANALYSIS METHODS

Natalia Semenchuk

docent, candidate of physico- mathematical sciences, Yanka Kupala State University of Grodno,

Belarus, Grodno  

Elena Banyukevich

senior lecturer, Yanka Kupala State University of Grodno,

Belarus, Grodno

ABSTRACT

The development of a general concept and implementation of a data storage This paper presents the study on trends and features in social mood of an urban population based on wavelet analysis methods. The study dataset is survey data on life satisfaction of an urban population obtained in a format of discrete time series. The method used is wavelet transformation of the dataset with Coifman scaling function. The motivation of the paper is development of algorithms for estimaing a smoothed component (a trend in data) and defining periodical components (patterns of data functioning) in case of their availability. In the beginning basic concepts and definitions of multiple-scale analysis, scaling, wavelet methods and discrete wavelet transformation are given. Properties of Coifman scaling function are described and its applicability to analyzing social mood dynamics is argued. Then study methodology is described which combines an algorithm for constructing a smoothed component and an algorithm for evaluating a periodic component based on the Coifman scaling function. The application of the algorithms is illustrated with tests on model examples. Eventually the proposed algorithms are applied to various sociological data and social mood dymanics is shown. We also discuss effectiveness of the proposed algorithms in processing large volumes of data from social networks.

Keywords: time series; smoothed component; discrete wavelet transformation; periodic scaling functions; spectral density.

1. Introduction

This paper is focused on the problem of analyzing data dynamics over time – defining ‘behaviour’ of time series data, expressed in its changing characteristics, such as trend, mean, periodic fluctuations, dispersion, and covariance. Of special interest aremethods capable of analyzing dynamics in data on several time series of considerable lenghts easily accessed though such channels as online portals, social networks, etc. For example, data collection of customer behaviour in an online store can be done automatically every day. This paper presents wavelet analysis methods as an approach to identify trends and patterns of functioning (hidden periodicities) in such data.

2. Basic concepts of wavelet analysis

It is known [1], multi-dimensional analysis (MDA) in space  is a sequence of nested subspaces V_j , jÎ Z of space , responding to the following properties:

1.   for any ,

2. ,

3.  ,

4. ,

5. there exists  (scaling function), and ,  include the orthonormal basis of space .

In this case, the functions

                                         (1)

include the orthonormal basis of space , . Function , where ,  is the orthonormal basis of space  ,  - orthogonal complement of space  and  called wavelet. At the same time functions of type

,                                       (2)

form basis of space , which is the orthogonal complement from space  to space , . The idea of multi-dimensional analysis belongs to S.G. Mallat [2] and Y. Meyer [3]. Any function  can be represented as a convergent series:

                         (3)

where

,                                         (4)

,                                         (5)

, , .

Amounts in (3) are usually finite because of limitations of scaling functions carriers and wavelets. For wavelet transformation of , given by the formula (3), coefficients ,  need to be calculated and are defined by ratios (4) and (5), . The problem of calculating a large number of integrals with a required accuracy appears to be at stake. The solution to this problem is rapid wavelet transformation, proposed by S.G. Mallat [2]. Mallat’s algorithm allows calculating wavelet expansion coefficients without integration by applying algebraic operations based on convolution. Another problem is the choice of initial coefficients. Direct calculation in the form of integrals (4) and (5) is a time-consuming operation. Additionally it does not provide the required accuracy if . In [4, p. 122] several ways to solve this problem are presented. For example, a set of values of the function : , where x_k pass a changing area of an x variable with a constant step,  , can be used as initial decomposition coefficients. Such a method for selecting the initial coefficients is the most applicable in wavelet analysis of discrete time series.

Coiflets, as well as Daubechies wavelets and simmlets, are related to wavelet functions with a compact carrier. The method of constructing such functions was proposed by Ingrid Daubechies; Daubechies wavelets are used in [5]. Coiflets are a special case of Daubechies wavelets with zero moments of scaling function [6]. Figure 1 shows the graph of Coifman scaling function (coiflet [7]) of 4th order used further.

Figure 1. Scaling function

3. Algorithm for constructing a smoothed time series

Let  be T consecutive observations of a random process, at regular intervals, .

Approximation coefficients are defined as

,                                 (6)

where  is the scaling function calculated as in (1).

We propose a following algorithm to construct the smoothed time series.

Step 1. Select the type of the scaling function. It depends on the length of the implementation.

Step 2. Calculate coefficients of the discrete wavelet transformation ,  as in (6).

Step 3. Construct a scalogram  using the formula

, ,

where  − wavelet is defined in (2).

Step 4. Analyze the graphic of the scalogram. If the peak exists at a high level , then a high-frequency component is present in the studied series. If the peak exists at a low level , it means that there is a low-frequency component in the studied series.

Step 5. Construct a smoothed time series using the following formula:

,                                        (7)

where  is the scaling function calculated as in (1),  are approximation coefficients calculated at Step 2, .

4. Applying proposed algorithm for constructing a smoothing component

Here we use the following function for initial set of values

where  with spacing 1/T, where .

The noisy signal was obtained by adding the values of the normal distribution function to the initial signal

,

where  with spacing 1/T, where ,  − is a mean square value and m=0 is an expectation.

Figure 2 shows a graph of the original function (in black) and noisy function with the normal distribution (in gray).

Figure 2. Initial function and noisy function

Fig. 3 shows the graphs of the smoothed component initial function (in black) and noise function (in gray) on level 3.

Figure3. Initial and noise functions

During construction of smoothed components at all levels j=1,..,9 of scaling function the best smoothed component of the noisy signal was obtained at the 3d level. If the level increases, the smoothed component gives the worst results and approaches 0 value. The best results for a pure signal were obtained at the first three levels.

5. Study on trends and features of social mood dynamics with wavelet analysis methods

Figure 4 illustrates the time series collected from 2013 to 2019 by month from various opinion polls.

Figure 4. Sociological survey data

Dashed line shows general satisfaction with life (Does your life suit you?) (% of the positive answers).

Black line shows satisfaction with housing services (Are you satisfied with housing and communal services?) (% of the positive answers).

Gray line shows customer loyalty (Are you satisfied with the shop?) (% of the positive answers).

The smoothed components constructed for each row based algorithm described in paragraph 2 are presented at Figure 5.

Figure 5. Smoothed components

It is obvious there is a definite periodic component in the black and dashed rows.

Conclusion

The model examples considered in the paper and the results of the real data analysis show that proposed algorithms allow to process data from various sociological surveys in automated way. The advantage of these algorithms lies in their simplicity, rapidness and wavelets ability to adapt to nonstandard data behavior. This enables the researchers to identify hidden trends in the dynamics of time series data and can be applied for the studies of big digitally born user-generated data.

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