Fig 1.
Statistically symmetric process curves in the parameter space of binary Markov process.
Independence transformation symmetry (green) and odd reversion symmetry (orange); the uniform process is the only intersection between independent processes and reversible processes.
Fig 2.
Time series of mid-quote of the pair U.S. dollar (USD) and Japanese yen (JPY) in the foreign exchange market.
(a-1) Time series of mid-quote of USD/JPY from 2011, October, 30th to 2011, November, 5th in original sampling time of 0.1s (clock time). (a-2) Corresponding time series of volume of orders. (b-1) Zoomed graph—gray box in (a-1)—shows when the Japanese government intervened in the market on October, 31st; the action of the Japanese government can be divided in two phases, first causing a quick appreciation of the U.S. dollar (from GMT 1h25 to 2h40—red box) followed by an almost constant price period (from GMT 2h40 to 5h55—blue box). (b-2) Corresponding time series of volume of orders, with a high number of orders during the period under the effect of the intervention.
Fig 3.
Local analysis of the independence statistical symmetry for the currency pair USD/JPY in the week from 2011, October, 30th to 2011, November, 5th.
(a-1) Mid-quote time series in tick time with intervals evaluated as independence asymmetric shown in green (for window size w = 1000). (a-2) Color plot showing window size effect on the hypothesis test; color code refers to the decimal logarithm of the probability and points are colored only if
is below the threshold 10−7, i.e., if they are classified as independence asymmetric. (b-1) and (b-2) details the Japanese government intervention; red box indicates the quick increase in the mid-quote and the blue box the period of almost constant mid-quote. The period corresponding to the increase in the mid-quote is classified as independence symmetric and the almost constant period, independence asymmetric.
Fig 4.
Local analysis of the time reversion statistical symmetry for the currency pair USD/JPY in the week from 2011, October, 30th to 2011, November, 5th.
(a-1) Mid-quote time series in tick time with intervals evaluated as time reversion asymmetric shown in orange (for window size w = 1000). (a-2) Color plot showing window size effect on the hypothesis test; color code refers to the decimal logarithm of the probability and points are colored only if
is below the threshold 10−7, i.e., if they are classified as time reversion asymmetric. (b-1) and (b-2) details the Japanese government intervention; red box indicates the quick increase in the mid-quote and the blue box the period of almost constant mid-quote. Time reversion asymmetric intervals only occur during the intervention and a short period right after it.
Fig 5.
Details of local analysis of the time reversion statistical symmetry for price time series and corresponding price difference integrated sign time series for week from 2011, October, 30th to 2011, November, 5th.
(a) Zoomed USD/JPY mid-quote time series in tick time with intervals evaluated as time reversion asymmetric shown in orange (for window size w = 1000). (b) Corresponding time series for the integrated sign of the price difference in the USD/JPY market. During the intervention period, local tendencies due to persistent movement of the price in one direction—and not due to changes of large magnitude—are opposite to the mid-quote general tendency.
Fig 6.
Mid-quote time series in tick time and local analysis of the independence statistical symmetry for the currency pair USD/JPY in nine weeks of October and November of 2011.
(a) from October, 2nd to October, 8th, (b) from October, 9th to October, 15th, (c) from October, 16th to October, 22nd, (d) from October, 23rd to October, 29th, (e) from October, 30th to November, 5th, (f) from November, 6th to November, 12th, (g) from November, 13th to November, 19th, (h) from November, 20th to November, 26th and (i) from November, 27th to December, 3rd.
Fig 7.
Mid-quote time series in tick time and local analysis of the time reversion statistical symmetry for the currency pair USD/JPY in nine weeks of October and November of 2011.
(a) from October, 2nd to October, 8th, (b) from October, 9th to October, 15th, (c) from October, 16th to October, 22nd, (d) from October, 23rd to October, 29th, (e) from October, 30th to November, 5th, (f) from November, 6th to November, 12th, (g) from November, 13th to November, 19th, (h) from November, 20th to November, 26th and (i) from November, 27th to December, 3rd. All weeks are characterized mainly as time reversion symmetric, except during special intervals where significant deviations from the symmetric hypothesis for several window sizes appear (indicated by red arrows), in particular four occasions: In weeks (a) and (b), the symmetry breaking occurs in a small interval in the beginning of week, when the foreign exchange market opens; in week (e), there are several asymmetric intervals during the Japanese government intervention; and in week (i) there is one short asymmetric period corresponding to the announcement of global central bank intervention.
Fig 8.
Details of local analysis of the time reversion statistical symmetry for price time series and corresponding price difference integrated sign time series for weeks from 2011, October, 2nd to October, 8th, from October, 9th to October, 15th and from November, 27th to December, 3rd.
Zoomed USD/JPY mid-quote time series in tick time with intervals evaluated as time reversion asymmetric shown in orange (for window size w = 1000) for week (a-1) from October, 2nd to October, 8th, (b-1) from October, 9th to October, 15th and (c-1) from November, 27th to December, 3rd. Corresponding time series for the integrated sign of the price difference in the USD/JPY market for week (a-2) from October, 2nd to October, 8th, (b-2) from October, 9th to October, 15th and (c-2) from November, 27th to December, 3rd. The first two asymmetric periods occur in the beginning of the respective weeks and present similar trend with zigzag structures, while the last asymmetric interval shows local tendencies with prices moving in a preferred direction that agrees with the mid-quote general tendency.