Geoscience Reference
In-Depth Information
With the bootstrap method, the time series is resampled randomly many times
to build up a body of statistics for the likelihood of the sample mean occurring by
chance. In practice this is done using a standard random number generator to pro-
duce a new artificial time series from the members of the original time series (not
necessarily all, since one sample may reappear in the randomization many time),
then calculating the mean of the artificial series. Done many times, this produces
a new random variable,
y
, describing the means of the artificial series, from which
a probability distribution function (pdf) may be estimated from the histogram of
y
values. In the present case, the expected value (true mean) of
y
is the
w
T
co-
variance.Toestablish,e.g.,upperandlower95%confidencelimits, abscissa values
of the cumulative distribution function (cdf, integral of the pdf) were evaluated for
ordinates 0.025 and 0.975, respectively.Figure 3.5 illustrates the procedurefor the
four15-minturbulencerealizations.Ineachcase,200artificialtimeseriesoflength
7,200weresynthesizedbyusingarandomnumbergeneratortoassignindicesfrom
the original
w
T
data
time series (the upper right panel corresponds to the series
showninFig.3.4).Meansofeachartificial
wT
serieswerethenassignedtotheran-
domvariable
y
.EachpanelinFig.3.5shows30-binhistogramsoftheyvariablefor
eachoftherealizationsin thehour-longdataset, wheretheabscissa is
×
y
−
y
z
=
σ
y
w,T btstrp: 222.5000
w,T btstrp: 222.5104
0.8
0.8
σ
=2.02e−007
σ
=2.46e−007
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−
4
−
2
0
2
4
−
4
−
2
0
2
4
w,T btstrp: 222.5208
σ
=2.73e−007
w,T btstrp: 222.5312
σ
=2.31e−007
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−
4
−
2
0
2
4
−
4
−
2
0
2
4
(
x
)
Fig. 3.5
Histograms of the artificial mean values
for 200 artificial time series from a random
sampling of the original 15-min time series of the
w
T
products (length: 7200). Abscissa is
y
=
(
is the sample std. deviation. Dashed curves are a normal distribution for zero
mean and unit standard deviation
x
−
x
)
/
σ
were
σ
