Table 1.
A survey of the EWS literatures between 2014 and 2017, showing the titles and first authors of the papers, corresponding reference numbers, their publication dates, whether the data used was real or simulated, the types of complex systems, the common EWIs, and customized EWIs (if any).
Table 2.
The two foreign exchange pairs: AUD-JPY and CHF-JPY, and the periods their time series data were available over.
Fig 1.
(a) A part of the T0 = 15 s AUDJPY exchange rate (red) on 6th Oct, 2008, and the Gaussian smoothed time series (blue) that tracks it very closely. (b) The residue time series, obtained by subtracting the Gaussian smoothed time series from the exchange rate. In these plots, the bandwidth of the Gaussian kernel used is 100T0.
Fig 2.
(a) The exchange rate time series and (b) the indicator time series. The pair of blue dashed vertical lines represents the time window over the exchange rate time series in (a) was used to compute one indicator value (the open blue circle in (b)). We slide this window along to obtain the pair of red dashed vertical lines, within which we obtain a second indicator value (the open red open circle in (b)). Repeating this we obtained the indicator time series in (b), which is then Gaussian smoothed (blue dashed curve). For the T2 rolling window indicated by the pair of blue solid vertical lines, its endpoint is the last value on the Gaussian-smoothed indicator time series (the solid blue dot in (b)). The endpoint value of the next T2 rolling window (the pair of red solid vertical lines) is shown in (b) as the solid red dot on the Gaussian-smoothed curve.
Fig 3.
The histogram of endpoints of T2 rolling windows over the entire AUD-JPY time series from 2005 to 2010.
Table 3.
Parameters and their test ranges to determine the optimal combination for EWSs, as well as to test the effects of data frequency.
Fig 4.
Venn diagram illustration for maximum spreads, in terms of percentile and statistical significance.
In this illustration, A represents the set of maximum spreads {yms} over the entire time series using a given combination of parameters. B1 and B2 represent the elements in A with yms values above 90th percentile and 95th percentile. C represents the set of the maximum spreads corresponding to statistically significant EWSs.
Fig 5.
(a) The histogram of maximum spreads yms in Set A. (b) The histogram of maximums spreads yms in Set A with signs (positive for boom and negative for bust). The blue and red solid vertical lines indicate the 90th and 95th percentiles yms10 and yms5 respectively. The maximum spread is defined to be positive, but in this histogram, we restore their signs. We can think of a large positive maximum spread as a boom, and a large negative maximum spread as a bust. The blue and red solid vertical lines in (b) correspond to the signed yms10 and yms5 values respectively.
Fig 6.
Statistically significant EWSs for the AUD-JPY exchange rate between 1996 and 2004, obtained from the (a) lag-1 autocorrelation, (b) variance, (c) low-frequency power spectrum, (d) concurrent signals with historical p value for endpoint < = 0.025, and (e) concurrent signals with historical p value for endpoint < = 0.06. For subplot (a), (b), and (c), we used a historical p value for the endpoint of 0.025. In (d), we show the concurrent EWSs for the same historical p value. More statistically significant concurrent EWSs can be included in (e) by increasing the historical p value of the endpoint to p ≤ 0.06. In this figure, a statistically significant EWS begins at a solid blue vertical line and ends at a solid red vertical line.
Fig 7.
Statistically significant EWSs for the AUD-JPY exchange rate between 2005 and 2010, obtained from the (a) lag-1 autocorrelation, (b) variance, (c) low-frequency power spectrum, (d) concurrent signals with historical p value for endpoint < = 0.025, and (e) concurrent signals with historical p value for endpoint < = 0.06. For subplot (a), (b), and (c), we used a historical p value for the endpoint of 0.025. In (d), we show the concurrent EWSs for the same historical p value. More statistically significant concurrent EWSs can be included in (e) by increasing the historical p value of the endpoint to p ≤ 0.06. In this figure, a statistically significant EWS begins at a solid blue vertical line and ends at a solid red vertical line.
Fig 8.
Statistically significant EWSs for the CHF-JPY exchange rate between 2008 and 2009, obtained from the (a) lag-1 autocorrelation, (b) variance, (c) low-frequency power spectrum, (d) concurrent signals with historical p value for endpoint < = 0.025, and (e) concurrent signals with historical p value for endpoint < = 0.06. For subplot (a), (b), and (c), we used a historical p value for the endpoint of 0.025. In (d), we show the concurrent EWSs for the same historical p value. More statistically significant concurrent EWSs can be included in (e) by increasing the historical p value of the endpoint to p ≤ 0.06. In this figure, a statistically significant EWS begins at a solid blue vertical line and ends at a solid red vertical line.
Table 4.
Parameter combinations with T0 increasing from the optimal value up to 6 hr, to test whether the EWSs can be discovered at longer time intervals.
Fig 9.
Concurrent EWSs (short green bands) for various time intervals (left axis) compared with the EWSs for 30-s time interval (long red bands) for AUD-JPY from 1996 to 2004.
The historical endpoint requirement is p ≤ 0.2. The exchange rate is plotted (black curve) is plotted in the background with axis to the right.
Fig 10.
Concurrent EWSs (short green bands) for various time intervals (left axis) compared with the EWSs for 15-s time interval (long red bands) for AUD-JPY from 2005 to 2010.
The historical endpoint requirement is p ≤ 0.2. The exchange rate is plotted (black curve) is plotted in the background with axis to the right.
Fig 11.
Concurrent EWSs (short green bands) for various time intervals (left axis) compared with the EWSs for 30-s time interval (long red bands) for CHF-JPY from 2009 to 2009.
The historical endpoint requirement is p ≤ 0.2. The exchange rate is plotted (black curve) is plotted in the background with axis to the right.
Fig 12.
The histograms of the ratios ((a), (c), and (e)) and precisions (P1) ((b), (d), and (f)) of the 100,000 samples with 250 days trial period, for the indicators AC(1), Var, and LFPS respectively for the data set AUD-JPY from 1996 to 2004. The red vertical lines mark
, and in the legends we give the proportion of samples with
in the 100,000 samples as rate of exceeding 1. The pool values of
and P1 are marked by black vertical lines.
Fig 13.
The histograms of the ratios ((a), (c), and (e)) and precisions (P1) ((b), (d), and (f)) of the 100,000 samples with 250 days trial period, for the indicators AC(1), Var, and LFPS respectively for the data set AUD-JPY from 2005 to 2010. The red vertical lines mark
, and in the legends we give the proportion of samples with
in the 100,000 samples as rate of exceeding 1. The pool values of
and P1 are marked by black vertical lines.
Fig 14.
The histograms of the ratios ((a), (c), and (e)) and precisions (P1) ((b), (d), and (f)) of the 100,000 samples with 250 days trial period, for the indicators AC(1), Var, and LFPS respectively for the data set CHF-JPY. The red vertical lines mark
, and in the legends we give the proportion of samples with
in the 100,000 samples as rate of exceeding 1. The pool values of
and P1 are marked by black vertical lines.
Fig 15.
Rates of exceeding 1 for the ratios with increasing trial time period, for the indicators AC(1), Var, and LFPS and data sets (a) AUD-JPY (1996–2004), (b) AUD-JPY (2005–2010), and (c) CHF-JPY (2008–2009). Each data point is computed with 100,000 samples.