256 D. Pelusi
parameters (see [32]) of the corresponding training semesters. For each semesters pair,
we compute the GS coefficient through formula 3 and define the following quantity:
fGS=nGS
np, (6)
wherenpis the number of semester pairs with positive profit andnGSis the number
of profitable semesters with similarity index GS that lies between two extrema. In
particular, we consider the GS ranges: 0.0–0.1, 0.1–0.2,..., until 0.9–1.0. So, the
quantityfGSrepresents a measure of the frequency of global similarity values based
on their memberships at the ranges named above.
In order for the algorithm results to have a statistical sense, we need to apply our
technique to many samples that have a trend similar to the exchange rates of the year
considered. To solve this problem, we use a technique described by Efron [12], called
the bootstrap method [17, 34]. The key idea is to resample from the original data,
either directly or via a fitted model,^2 to create various replicate data sets, from which
the variability of the quantities of interest can be assessed without long-winded and
error-prone analytical calculations. In this way, we create some artificial exchange
rate series, each of which is of the same length as the original series.
3 Experimental results
We apply our algorithm to hourly Euro-Dollar exchange rates and consider 2006 as
the year for trading. To create samples with trends similar to the Euro-Dollar exchange
rate 2006, we use parametric bootstrap methods [3].
In the parametric bootstrap setting, we consider an unknown distributionFto
be a member of some prescribed parametric family and obtain a discrete empirical
distributionFn∗by estimating the family parameters from the data. By generating
an iid random sequence from the distributionFn∗, we can arrive at new estimates of
various parameters of the original distributionF.
The parametric methods used are based on assuming a specific model for the data.
After estimating the model by a consistent method, the residuals are bootstrapped. In
this way, we obtain sample sets with the same length of exchange rates as 2006.
Table 1 shows the results with 10, 100, 200, 300, 400 and 500 samples. On the
rows, we have the samples number and on the columns we have the global similarity
frequency defined in formula (6). We can note that there are no results for the ranges
0.5–0.6, 0.6–0.7, until 0.9–1.0 because they give null contribution, that is there are
no global similarity values belonging to the above-named ranges. Moreover, we can
observe that for 10 samples, the range with highest frequency is 0.0–0.1, that is, it
is more likely that with a similarity coefficient between 0 and 0.1 we have a positive
profit than for the other ranges.
The statistics results for 100 samples show that the range with the greatest fre-
quency is 0.1–0.2. For 200, 300, 400 and 500 samples we obtain about the same value.
(^2) We use a GARCH model (see [4]).