Temporal fluctuation scaling in nonstationary time series using the path integral formalism

We model the time evolution of the mean and the variance of nonstationary time series using the path integral formalism with the purpose to obtain the temporal fluctuation scaling presents in complex systems. To this end, we first show how the probability of change between two times of a stochastic variable can be written in terms of a Feynman kernel, where the cumulant generating function of statistical moments is identified as the Hamiltonian of the system. Thus, by including the effects of a stochastic drift and a temporal logarithmic term in the cumulant generating function, we find analytical expressions describing the temporal evolutions of the mean and the variance in terms of cumulants. Starting from these expressions, we obtain the temporal fluctuation scaling written as a general analytical relation between the variance and the mean, in such a way that this relation satisfies a power law, with the exponent being a function on time. Additionally, we study several financial time series associated with changes of prices for some stock indexes and currencies. For this financial time series, we find that the temporal evolution of the mean and the variance, the temporal fluctuation scaling, and the temporal evolution of the exponent which are obtained from this path integral approach are in agreement with those obtained using the empirical data.

Figure 1

Evolution of the mean of the time series of the Nikkei 225 stock index measured daily from January $4,1984$ to August $24,2020$ taking the average of the total accumulated data each window size $WS$. Left: Log-return time series ($WS=138$ data). Center: Volatility time series ($WS=159$ data). Right: Log-return absolute time series ($WS=159$ data).

Figure 2

Evolution of the mean of the time series of the S&P 500 stock index measured daily from January $3,1950$ to August $24,2020$ taking the average of the total accumulated data each window size $WS$. Left: Log-return time series ($WS=171$ data). Center: Volatility time series ($WS=102$ data). Right: Log-return absolute time series ($WS=102$ data).

Figure 3

Evolution of the mean of the time series of the FTSE stock index measured daily from October $20,1997$ to August $24,2020$ taking the average of the total accumulated data each window size $WS$. Left: Log-return time series ($WS=146$ data). Center: Volatility time series ($WS=117$ data). Right: Log-return absolute time series ($WS=172$ data).

Figure 4

Evolution of the mean of the time series of the Nikkei 225 stock index measured daily from January $4,1984$ to August $24,2020$ by scaling the data weekly (every five data) and taking the average of the total accumulated data each window size $WS$. Left: Log-return time series ($WS=186$ data). Center: Volatility time series ($WS=159$ data). Right: Log-return absolute time series ($WS=125$ data).

Figure 5

Evolution of the mean of the time series of the S&P 500 stock index measured daily from January $3,1950$ to August $24,2020$ by scaling the data weekly (every five data) and taking the average of the total accumulated data each window size $WS$. Left: Log-return time series ($WS=95$ data). Center: Volatility time series ($WS=60$ data). Right: Log-return absolute time series ($WS=60$ data).

Figure 6

Evolution of the mean of the time series of the Dow Jones stock index measured every 2 sec from October $10,2014$ at 09:30:03 to October $31,2014$ at 16:20:00 by scaling the data every 40 sec (every 20 data) and taking the average of the total accumulated data each window size $WS$. Left: Log-return time series ($WS=113$ data). Center: Volatility time series ($WS=173$ data). Right: Log-return absolute time series ($WS=173$ data).

Figure 7

Evolution of the mean of the time series of the FTSE stock index measured daily from October $20,1997$ to August $24,2020$ by scaling the data weekly (every five data) and taking the average of the total accumulated data each window size $WS$. Left: Log-return time series ($WS=50$ data). Center: Volatility time series ($WS=138$ data). Right: Log-return absolute time series ($WS=194$ data).

Figure 8

Evolution of the mean of the time series of the Nikkei 225 stock index measured daily from January $4,1984$ to August $24,2020$ moving the time series in 3000 data (12 and a half years) and taking the average of the total accumulated data each window size $WS$. Left: Log-return time series ($WS=84$ data). Center: Volatility time series ($WS=17$ data). Right: Log-return absolute time series ($WS=14$ data).

Figure 9

Evolution of the mean of the time series of the S&P 500 stock index measured daily from January $3,1950$ to August $24,2020$ moving the time series in 12 000 data (50 yr) and taking the average of the total accumulated data each window size $WS$. Left: Log-return time series ($WS=82$ data). Center: Volatility time series ($WS=186$ data). Right: Log-return absolute time series ($WS=186$ data).

Figure 10

Evolution of the mean of the time series of the Dow Jones stock index measured every 2 sec from October $10,2014$ at 09:30:03 to October $31,2014$ at 16:20:00 moving the time series in 220 000 data (8 days, 6 h, 26 min, and 40 sec) and taking the average of the total accumulated data each window size $WS$. Left: Log-return time series ($WS=74$ data). Center: Volatility time series ($WS=200$ data). Right: Log-return absolute time series ($WS=200$ data).

Figure 11

Evolution of the mean of the time series of the FTSE stock index measured daily from October $20,1997$ to August $24,2020$ moving the time series in 500 data (2 yr) and taking the average of the total accumulated data each window size $WS$. Left: Log-return time series ($WS=20$ data). Center: Volatility time series ($WS=198$ data). Right: Log-return absolute time series ($WS=200$ data).

Figure 12

Evolution of the mean of the time series of the COP-USD foreign exchange measured daily from July $16,2010$ to February $11,2020$ taking the average of the total accumulated data each window size $WS$. Left: Log-return time series ($WS=20$ data). Center: Volatility time series ($WS=30$ data). Right: Log-return absolute time series ($WS=30$ data).

Figure 13

Evolution of the variance of the time series of the COP-USD foreign exchange measured daily from July $16,2010$ to February $11,2020$ taking the variance of the total accumulated data each window size ($WS=132$ data) for all time series. Left: Log-return time series. Center: Volatility time series. Right: Log-return absolute time series.

Figure 14

Evolution of the mean of the time series of the COP-USD foreign exchange measured daily from July $16,2010$ to March $13,2020$ taking the mean of the total accumulated data keeping fixed the optimal window size found for each time series and taking the error in each data as the weight of the variance [found with the regression of Eq. (33)] when dividing by the total number of means $N/WS$. Left: Log-return time series ($WS=20$ data). Center: Volatility time series ($WS=30$ data). Right: Log-return absolute time series ($WS=30$ data). The subcharts in each series represent an enlargement of the tail of the series used to observe the quality of the fit.

Figure 15

Evolution of the mean of the time series of the COP-USD foreign exchange measured daily from July $16,2010$ to August $25,2020$ taking the mean of the total accumulated data keeping fixed the optimal window size found for each time series and taking the error in each data as the weight of the variance [found with the regression of Eq. (33)]. Left: Log-return time series ($WS=20$ data). Center: Volatility time series ($WS=30$ data). Right: Log-return absolute time series ($WS=30$ data). The subcharts in each series represent an enlargement of the tail of the series used to observe the quality of the fit.

Figure 16

Evolution of the variance of the time series as a function of the mean of the time series measured daily until August 24, 2020 taking the mean and variance of the total accumulated data, keeping the optimal window size fixed $WS$. (Left) Absolute log-return of COP-USD measured from July 16, 2010 with regression parameters $K=0.1350\pm 0.0052$, $\alpha =0.9465\pm 0.0082$ and GBE of $5.77\%$ ($WS=30$ data). (Right) Volatility of FTSE measured from October 20, 1997 with regression parameters $K=0.2081\pm 0.0054$, $\alpha =1.8901\pm 0.0101$, and GBE of $3.72\%$ ($WS=117$ data).

Figure 17

Evolution of the TFS exponent as a function of the time measured daily until August 24, 2020 taking the mean and variance of the total accumulated data, keeping the optimal window size fixed $WS$. (Left) Absolute log-return of COP-USD measured from July 16, 2010 with regression parameters $K=0.1350\pm 0.0052$, $\alpha =0.9465\pm 0.0082$ and GBE of $5.77\%$ ($WS=30$ data). (Right) Volatility of FTSE measured from October 20, 1997 with regression parameters $K=0.2081\pm 0.0054$, $\alpha =1.8901\pm 0.0101$, and GBE of $3.72\%$ ($WS=117$ data).