![]() |
![]() | Data Menu | ![]() | ![]() | Distribution Menu |
Visualize Menu
The visualize menu contains entries for nonparametric
estimation procedures. The options differ in the univariate and
multivariate modes.
The following options are provided by the Visualize Menu:
Univariate Mode
Kernel Density (univariate mode)
This option plots a kernel density with a given bandwidth for the active
data set. Kernel densities can be applied to data sets of type
Xtremes Univariate Data and Xtremes Censored Data. The dialog box
provides the following options:
Histogram
This option plots a histogram of the active data set and can be
applied to Xtremes Grouped Data and Xtremes Discrete Data.
Scatterplot (2D)
For Xtremes Time Series and Xtremes Multivariate Data, a scatterplot
of points ( i, y[i] ) or ( x[i], y[i] ) becomes available in a
special plot window which is a plain plot window equipped with
the additional facility to plot and to handle a scatterplot.
Different data sets are represented in different scatterplot
windows. First select two components of the active data set
in the case of Xtremes Multivariate Data.
A related facility is Clusterplot in
the Visualize Menue.
Time series facilities (scatterplot)
Plot a
Least Squares Polynomial
in the scatterplot window (based on the active data) or
execute Moving Average, Lowess to obtain
moving averages and residuals - where the latter are still
represented by a scatterplot - in separate scatterplot windows.
Another option
is Seasonal
Component.
Cutting facilities (scatterplot)
Points below a threshold can be made inactive by using
the Threshold
option in the local menu
of the scatterplot window (rightclick in the window) or by using
the (scissors) point selection mouse mode. The
inactive points are marked in a green color. The active points can
be saved to a file by executing
the Save Actual Points
option.
Conversion of a scatterplot into a curve
The full plot
options (option mouse mode) become applicable
by copying the scatterplot into the clipboard using
the moving mouse mode. After this operation,
the scatterplot - with connected points - is dealt with as a
curve. Afterwards, drag or copy the "scatterplot curve" to another
plot or scatterplot window according to your choice.
Boxplot
A boxplot
for Xtremes Univariate Data x[1],...,x[n] consists of
The boxplot option can be applied as well to Xtremes Multivariate
Data. The boxplots for the single univariate data sets (in the
different columns) are plotted at the positions 1,2,3, ....
An exception is made if there are exclusively real numbers in
the column headers. In that case, these real numbers determine
the plotting positions.
We remark that each boxplot is dealt with in the same manner
as a single function in a plotting window.
Boxplots are not dealt with in Statistical Analysis because
they are adjusted to normal data. The data plotted outside
of the interval I are called outliers (indicating that the normal
modeling may not be correct).
Clusterplot
Let the active data set be of type Xtremes Time Series. Denote by k the
number of exceedances over a threshold u and denote by m(k) the
number of clusters. The sample mean cluster size (relative
to u or k) is given by
mcsize(k) := k/m(k), k=1,...,n.
One may execute Visualize ... Clusterplot to obtain
a scatterplot of (1/n, x[1]), ..., (1,
x[n]) and, additionally, plots of (k, mcsize(k)) and (k,
1/mcsize(k)) for k=1,...,n in the graphics
windows Sample Mean Cluster Size and Reciprocal Sample
Mean Cluster Size.
Let u and, thus, k be
fixed. Denote by |m| the cluster size of a cluster m of exceedance
times. Then
P[k]({x}) := |{m : |m| = x}| / m(k), x = 1,...,k
defines the sample cluster size distribution P[k]. Notice
that mcsize(k) is the mean of the distribution P[k]. Creating
a threshold in the Clusterplot window (SHIFT+leftclick)
also opens a
window Sample Cluster Size Distribution displaying P[k]({x}) for x=1,...,k by
means of a histogram.
For more information, see Cluster
Options.
Sample QF
This option plots the sample qf for the active data set which must be
of type Xtremes Univariate Data, Xtremes Censored Data or Xtremes
Grouped Data. The empirical qf for grouped data is the qf pertaining to
the histogram.
Sample DF
This option plots the sample df for the active data set which must be
of type Xtremes Univariate Data, Xtremes Censored Data or Xtremes
Grouped Data.
The empirical df for grouped data is the df pertaining to the histogram.
Sample Mean Excess
This option plots the empirical mean excess function for the active data set which must be of
type Xtremes Univariate Data or Xtremes Grouped Data.
Options of dialog box:
Sample Median Excess
This option plots the sample median excess function for the active data
set which must be of
type Xtremes Univariate Data or Xtremes Grouped Data. If X
has the df F, the median excess function is the median of the
conditional distribution F( · |u) of X - u given X > u,
m(u,F) = F**(-1)(1/2,u) ,
with F**(-1) being the qf of the df F.
We obtain the sample median excess function as m( · ,F[n]). For
our computations we evaluate
m( x[n-k+1:n], F[n] ), 6 <= k <= n,
and take a linear interpolation thereof. The sample median excess
function for grouped data is the excess function pertaining to the
histogram.
Sample Hazard Function
For Xtremes Univariate Data, take the kernel hazard function as an
estimator of the hazard function h[F], where f[n,b] is the kernel
density for the Epanechnikov kernel. Xtremes asks you to enter a
bandwidth beta in the dialog box Enter bandwidth. The default
value
is 1. For Xtremes Grouped Data, again with frequencies n[j] in
cells (t[j], t[j+1]), the sample hazard function is defined via the
histogram.
Sample Autocovariance
It is assumed that the data come from a weakly stationary
time series. Display the sample autocovariance function
r(h) = (1/n) Sigma (x[i]- mu) (x[i+h] -
mu) ,
with mu denoting the sample mean.
It is advisable to detrend and deseasonalize the data using
the options in the local menu of
the scatterplot window.
Sample Autocorrelation
Display the sample autocorrelation function
rho(h) = r(h) / r(0)
with r(h) being
the sample autocovariance function.
Sample Path
Plot the sample path of the active univariate data set, i.e. the sample df
multiplied by the sample size.
Multivariate Mode
Scatterplot (3D)
Plot data points (x , y, z) in a three-dimensional coordinate system.
For plotting, one must select three variables of the multivariate
data set. This option is applicable to Xtremes Multivariate Data.
Kernel Density (multivariate)
Let k be a univariate kernel (see univariate kernel
density). Then, u(x) = k(x[1]) ... k(x[d]) defines
a d-variate kernel.
For visualization, only bivariate Gaussian densities are utilized.
Bandwidth and direction of the kernel may be adjusted by using the
parameter varying mouse mode.
Sample DF (multivariate)
Plot the sample df for the given multivariate data set. One must select
two components first.
Sample SF (multivariate)
Plot the sample survivor function (sf) for the given
multivariate data set. One must select two components
first.
Sample Canonical Dependence Function
Plot the sample canonical dependence function for the given
multivariate data set. One must select two components
first.
For details see Statistical Analysis, (10.17) and
the comments in Chapters 9 and 10.
Surface Plot
The Surface Plot option performs
a surface plot of data stored in a multivariate data
set, which must follow the structure described below.
Given a multivariate data set with points
x[i] = x0 + (i-1)(n-1)(x1-x0),
y[j] = y0 + (j-1)(m-1)(y1-y0),
z[i,j] = f( x[i], y[i] ),
1 <= i <= n, 1 <= j <= m, one can display a surface plot of
the function obtained by an interpolation of the points (x[i], y[j],
z[i,j]).
Xtremes detects if a lattice is defined by (x[i], y[j]),
yet one must store the points
ordered according to their x- and y-coordinates.
Example:
The following data set contains supporting points of the function
f(x,y) = xy. Only the actual data are listed:
"x" "y" "z"
1 1 1
1 2 2
1 3 3
2 1 2
2 2 4
2 3 6
3 1 3
3 2 6
3 3 9
...
![]() |