Statistical properties of fluctuations of time series representing appearances of words in nationwide blog data and their applications: An example of modeling fluctuation scalings of nonstationary time series

To elucidate the nontrivial empirical statistical properties of fluctuations of a typical nonsteady time series representing the appearance of words in blogs, we investigated approximately $3\times {10}^9$ Japanese blog articles over a period of six years and analyze some corresponding mathematical models. First, we introduce a solvable nonsteady extension of the random diffusion model, which can be deduced by modeling the behavior of heterogeneous random bloggers. Next, we deduce theoretical expressions for both the temporal and ensemble fluctuation scalings of this model, and demonstrate that these expressions can reproduce all empirical scalings over eight orders of magnitude. Furthermore, we show that the model can reproduce other statistical properties of time series representing the appearance of words in blogs, such as functional forms of the probability density and correlations in the total number of blogs. As an application, we quantify the abnormality of special nationwide events by measuring the fluctuation scalings of 1771 basic adjectives.

Figure 1

(a) An example of daily time series of raw word appearances for the “yowai (weak)” $F_j\left(t\right)$. (b) The daily time series of the normalized total number of blogs $m\left(t\right)$ (see Appendix pp8). (c) The daily time series of word appearances scaled by the normalized total number of blogs, for “yowai” ${\tilde{F}}_j\left(t\right)=F_j\left(t\right)/m\left(t\right)$ [defined by Eq. (23)]. (d) The differential time series of word appearances scaled by the normalized total number of blogs, for “yowai” $\delta {\tilde{F}}_i\left(t\right)\equiv F\left(t\right)/m\left(t\right)-F(t-1)/m(t-1)$ [defined by Eq. (30)]. From these figures, we can confirm that the time variation of raw word appearances $F_j\left(t\right)$ shown in the panel (a) is almost the same as that of the total number of blogs $m\left(t\right)$ shown in the panel (b).

Figure 2

(a) TFS of raw time series $V\left[F_j\right]$. The data shown are empirical results of 1771 adjectives (black triangles), corresponding to the theoretical curve given by Eq. (19) or (20) (red dashed line). The blue dashed-dotted line indicates $y=x^{0.5}$, which corresponds to the Poisson distribution. (b) The correlations of raw time series between individual time series $\text{cor}[F_j,m]$ and the scaled total number of blogs $m\left(t\right)$ (i.e., the shared Poisson parameter). The data shown are empirical results of 1771 adjectives (black plus triangles), the theoretical curve given by Eq. (56) (red dashed line), and the theoretical upper bound, which is considered for the finite number of observables given in Eq. (C9) (gray dashed-dotted line). From these panels, we can confirm that the empirical data are in good agreement with the theoretical curve.

Figure 3

(a) TFS of normalized time series $V{\left[{\tilde{F}}_j\right]}^{0.5}$ [${\tilde{F}}_j\left(t\right)$ is defined by Eq. (23)]. The data shown are empirical results of 1771 adjectives (black triangles), and the corresponding results of theoretical curves given by Eq. (27) or (28) (red dashed line). The green dashed-dotted line indicates the correction of the theoretical lower bound, which is considered in the nonstationary $V\left[r_j\right]$ given by Eq. (29). In addition, the thin blue dashed-dotted line indicates $\propto x^{0.5}$, and thin gray dashed line is the theoretical curve of the raw time series given by Eq. (20). (b) The TFS of the differential of the time series of word appearances $V\left[\delta {\tilde{F}}_j\right]$ [$\delta {\tilde{F}}_j\left(t\right)$ is defined by Eq. (30)]. The black triangles indicate the actual data, the red dashed line indicates the theoretical lower bound given by Eq. (32) or (33), and the blue dashed-dotted line indicates $\propto x^{0.5}$. We can confirm that the theoretical curve is in accordance with the empirical data from the panel (b).

Figure 4

The box ensemble scaling of the differential of the time series of word appearances $E_c^{\zeta }\left[\delta F\left(t\right)\right]$, where $\delta F_j\left(t\right)\equiv F_j\left(t\right)-F_j(t-1)$ [defined by Eq. (45)]. (a) The box EFSs: 06.07.2007 (black triangles) for a typical date; 26.06.2010 (red circles) in the case of a special nationwide event, the FIFA World Cup. The corresponding theoretical lower bounds given by Eq. (47) or (48): the black solid line for 06.07.2007, the red dashed line for 26.06.2010. The thick purple dashed line indicates the theoretical curve in the case of the box size ${\zeta }_j=0$, as given by Eq. (49) or (50) and the blue thin dashed-dotted line is $y=x^{0.5}$. (b) The corresponding figure for the great east Japan earthquake on 11.03.2011 (black triangles), 16.03.2011 (red circles), and 29.03.2011 (green circles); the inserted figure shows the deviations from theory, with $l\left(t\right)$ as defined by Eq. (52), where vertical lines correspond to the dates of the main figure. (c) The time series of deviations from the theoretical lower bound $l\left(t\right)$ defined by Eq. (52) [i.e., the ensemble abnormality $l\left(t\right)$], is shown by the black line. The vertical lines indicate the dates shown in panels (a) and (b): 06.07.2007 (the red dashed line), 26.06.2010 (the green dashed-dotted line), 11.03.2011, 16.03.2011, and 29.03.2011 (the blue dashed-double-dotted line). The horizontal lines are the 50th percentile of the set $\left\{l\right(t\left)\right\}$ (the red dashed line) and the 90th percentile (the green dashed-dotted line). (d) The relation between the deviation $l$ and the EFSs. The median of the EFS regarding the dates on which $l$ is in the lower 50th percentile $y_{<50;50}\left(\mu \right)$, defined by Eq. (53), where we use dates under the 50th percentile as shown by the horizontal red dashed line in panel (c). The error bars, corresponding to the 25th percentile and the 75th percentile, $y_{<50;25}\left(\mu \right)$ and $y_{<50;75}\left(\mu \right)$. The green lines indicate the corresponding statistics of the upper 90th percentile, $y_{>90;50}\left(\mu \right)$, for which we use the dates above the 90th percentile shown by the horizontal green dashed-dotted line in the panel (c). The thick purple dashed line indicates the theoretical curve given by Eq. (49), and the blue thin dashed-dotted line indicates $y=x^{0.5}$.

Figure 5

The temporal probability density function of word counts for fixed words. (a)–(d) The probability density function of word counts $F_j\left(t\right)$ for words whose mean word counts ${\check{c}}_j$ are small: (a) “jyamakusai,” which means “bother” (${\check{c}}_j=5.67$), (b) “komuzukashii,” which means “difficult” (${\check{c}}_j=1.94$), (c) “monoui,” which means “languid” $[{\check{c}}_j\left(t\right)=0.24]$, (d) “yuyusii,” which means “momentous” (${\check{c}}_j=0.016$). The black triangles indicate the empirical data and red circles indicate the Poisson distribution whose Poisson parameter corresponds to ${\check{c}}_j$. We can confirm that the empirical distribution obeys the Poisson distribution. (e) The probability density functions of the differential of the scaled time series of word counts $v_j\left(t\right)=\delta (F_j\left(t\right)/m\left(t\right))/{\sigma }_v(c_j\left(t\right),m\left(t\right),{\mathrm{\Delta }}_0\left(t\right))$, in the case that the means of the word counts ${\check{c}}_j$ are very large: “ii,” which means “good” (${\check{c}}_j=110080$) is indicated by the black solid line; “kurai,” which means “dark” (${\check{c}}_j=45090$) is indicated by the red dashed line; and “ooi,” which means “many” (${\check{c}}_j=22535$) is indicated by the green dashed-dotted line. The peach colored thick dashed line indicates the scaled $t$ distribution whose degree of freedom takes a value of 2.64, and the gray thin dotted line is the normal distribution that is in agreement with the central part of the empirical distributions, namely, the normal distribution whose mean is 0 and standard deviation is 0.6. Here, ${\sigma }_v(c_j\left(t\right),m\left(t\right),{\mathrm{\Delta }}_0\left(t\right))=\sqrt{2\overline{c_j\left(t\right)/m\left(t\right)}+2\overline{c_j{\left(t\right)}^2{\mathrm{\Delta }}_0{\left(t\right)}^2}}$. The inserted figure displays the corresponding cumulative distribution functions in a log-log scale. We can confirm that all distributions depict almost the same curve, as predicted by our theory, and also confirm that this common distribution can be approximated by a $t$ distribution.

Figure 6

The probability density function of the differential of word counts scaled into the standard deviation $v_j\left(t\right)\equiv \delta (F_j\left(t\right)/m\left(t\right))/{\sigma }_v(c_j\left(t\right),m\left(t\right),{\mathrm{\Delta }}_0\left(t\right))$, given by Eq. (57). The red line shows the theoretical distribution given by Eq. (G4) for empirical parameters, the green line shows the theoretical distribution of the steady time series with the same ${\check{c}}_j$ as given by Eq. (G4), for $c_j\left(t\right)={\check{c}}_j$ and $m\left(t\right)=1$. The blue line indicates the scaled differential of the Poisson distribution with the Poisson parameter $c_j$, whose standard deviation takes a value of 1; and the peach colored line indicates the differential of the $t$ distribution whose degree of freedom takes a value of 2.64 and has a standard deviation of 1. The data are shown for (a) “ii,” which means “good” (${\check{c}}_j=110080$), (b) “sukunai,” which means “a few” (${\check{c}}_j=7098$), (c) “hosoi,” which means “slight” (${\check{c}}_j=1137$), (d) “tegowai,” which means “awkward” (${\check{c}}_j=87.00$), (e) “iyarashi,” which means “odious” (${\check{c}}_j=13.42$), (f) “komuzukashii,” which means “difficult” (${\check{c}}_j=1.97$), (g) “monoui,” which means “languid” (${\check{c}}_j=0.24$), and (h) “yuyusii,” which means “momentous” (${\check{c}}_j=0.016$).

Figure 7

Quantification of impacts of nationwide events as measured by the ensemble abnormality $l\left(t\right)$ given by Eq. (52), which is quantified by the deviation of empirical ensemble scalings from the theoretical lower bound using 1771 adjectives. The points indicate comparisons of the same event (or date) between years for the great east Japan earthquake on 11.3 (brown filled circle); the date of the lowest mean of deviation on 14.08 (grass green filled triangle); Halloween on 31.10 (emerald green filled diamond); Christmas Eve on 24.12 (aqua unfilled triangle triangle); New Year's Eve on 31.12 (blue plus); Mother's Day on 13.05.2007, 11.05.2008, 10.05.2009, 09.05.2010, 08.05.2014, 12.05.2013, and 11.05.2014 (purple cross); and the dates of National elections on 29.07.2007, 30.08.2009, 11.07.2010, 16.11.2012, and 21.07.2013 (peach colored unfilled diamond). From this figure, we can compare the impacts of various nationwide events by the same standard.

Figure 8

Temporal dependency of the probabilistic abnormality of the word “Lekue,” which mainly refers to a steamer in a microwave oven (in Japan), and was a hit product in 2010 in Japan. The top figure indicates the normalized word count of Lekue ${\check{F}}_j\left(t\right)=F_j\left(t\right)/m\left(t\right)$, and the bottom figure indicates the corresponding probabilistic abnormality $a_j\left(t\right)$ given by Eq. (61) (the green thin line). The red thick line is the abnormality smoothed by the moving weighted average ${\overline{a}}_j\left(t\right)={\sum }_{T=-15}^{15}{a}_j\left(t\right){\varphi }_n(t-T;\text{mean}=0,sd=6)$, where ${\varphi }_n\left(x\right)$ is the probability density function of the normal distribution. The peach dashed vertical lines indicate dates with large deviations, on which $a_j\left(t\right)\ge 4$, and the bottom blue vertical solid lines show the dates on which Lekue was broadcasted on television in the Tokyo area. From this figure, we can confirm that the probabilistic abnormality was high from the middle of 2010 until the middle of 2011 (the red thick line), and in the same period there were many television broadcasts regarding Lekue, which are one of cause of abnormality.