Generalized Dimensions dialog box
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Generalized Dimensions dialog box |
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Use this dialog box to calculate the generalized dimensions of selected series.
Refer to Generalized Dimensions, Grassberger & Procaccia (1983a), Grassberger & Procaccia (1983b), Hausdorff and Capacity Dimension, Renyi (1971), Prichard & Price (1992), Theiler (1986), Theiler (1990), Theiler (1990a) and Theiler & Lookman (1993) for additional information on this subject.
 | This algorithm is O(N2). |
Input
Specifies the worksheet location of the input data for this operation.
Provides a reference to the extent of the data available in the currently active worksheet. Use this range as a guide for typing valid extents in the Input Range box.
Type in this box the range of cells containing the data you want to use as input for this command.
Check this box to indicate that the data specified by the Input Range box is organized in rows as opposed to columns (the default).
Single Series
Select this option to specify that the data specified by the Input Range box represent individual series.
Multivariate
Select this option to specify that the data specified by the Input Range box represent one multidimensional series.
Refer to Specifying Input Arguments for additional information on this subject.
Output
Specifies the worksheet location for placing the results of this operation.
Select this option to place the results of this operation in a new document workbook.
Select this option to place the results of this operation in a new worksheet.
Select this option to place the results of this operation starting at the worksheet location specified in the box to the right of this option.
Check this box to plot the results of this operation.
Refer to Specifying Output Locations for additional information on this subject.
Order and Embedding
Specifies the minimum q-order at which to start calculating the generalized dimensions. For practical purposes we have restricted this value to a minimum of -32.
Specifies the maximum q-order at which to end calculating the generalized dimensions. For practical purposes we have restricted this value to a maximum of 32.
Specifies the method that will be used to calculate the information dimension.
L'Hopital
Select this option to calculate the information dimension using L’Hôpital rule.
As 1.1
Select this option to calculate the information dimension for q=1.1.
As 0.9
Select this option to calculate the information dimension for q=0.9.
Specifies the minimum embedding dimension at which to calculate the specified generalized dimensions for the given data series.
If you want to calculate generalized dimensions for more than one dimensional embedding, type in this box an increment for the dimensional component for the embeddings.
If you want to calculate generalized dimensions for more than one dimensional embedding, type in this box the number of dimensional steps for the embeddings. For example. specifying 3 as the Min. Dimensions, 2 as the Dimension Incr. and 3 as Dimension Steps, will compute the specified generalized dimensions for embeddings of 3, 5 and 7 dimensions.
Specifies the time delay used for the specified dimensional embeddings.
Distance and Metrics tab
Specifies the minimum epsilon at which to calculate the Cq integrals.
Specifies the maximum epsilon at which to calculate the Cq integrals.
The number of epsilon steps between Minimum Epsilon and Maximum Epsilon at which the Cq integrals are calculated.
Epsilon
Select this option if you want the epsilon points selected equidistantly in the interval [Minimum Epsilon, Maximum Epsilon].
Log Epsilon
Select this option if you want the epsilon points selected equidistantly in the interval [Log10(Minimum Epsilon), Log10(Maximum Epsilon)].
Specifies the method by which distances are computed.
Loo - Max
Select this option if you want to use the Minkowski Max, 
, distance metric for measuring distances.
L1 - Manhattan
Select this option if you want to use the Minkowski Manhattan, L1, distance metric for measuring distances.
L2 - Euclidean
Select this option if you want to use the Minkowski Euclidean, L2, distance metric for measuring distances.
Reference Points tab
Specifies the number of points used for calculating the specified generalized dimensions. Typing 0.1 in this box, for example, directs the algorithm to use 10% of the data points for calculating the specified generalized dimensions.
| Particularly useful to decrease computing time for long data series since this algorithm is O(N2). |
Specifies the temporal window for excluding nearest neighbors. Typing 10 in this box, for example, excludes all nearest neighbors which are at the most 10 time steps away from the points of interest.
Refer to Theiler (1986) for additional information on temporal correlations and exclusion windows.
Dimension Estimation tab
Specifies the method used for estimating the generalized dimensions Dq.
Click this option to select Ellner's maximum likelihood estimator. This estimator is given by 
, which we generalize for all q as 
, and where R1 and R2 are the upper and lower epsilon values, respectively. The r represent different epsilon values in the interval. The listed integral is approximated by one of the integration methods available in the Numerical Integration Options dialog box (refer to Integration Options later in this topic for additional information.)
Click this option to use an ordinary least squares (OLS) regression to approximate with a first degree polynomial the set of points, {Log(e), Log(Cq)}. In this case, the slope of the approximating polynomial is reported as Dq.
Click this option to use numerical differentiation to estimate Dq. In this case, the set of points, {Log(e), Log(Cq)}, is differentiated at each (Log(e), Log(Cq)) point and the mean of these derivatives is reported as Dq. The method used for differentiation can be selected from the Numerical DIfferentiation dialog box (refer to Differentiation Options later in this topic for additional information.)
Click this option to use derivatives from a Savitzky-Golay approximation to estimate Dq. In this method, the set of points, {Log(e), Log(Cq)}, is approximated at each (Log(e), Log(Cq)) point by piecewise fixed-degree polynomials and the mean of these derivatives is reported as Dq. Preferences for the Savitzky-Golay operation may be specified in the Savitzky-Golay Options dialog box (refer to Savitzky-Golay Options later in this topic for additional information.)
| Note that the main difference between the Numerical Differentiation and the Savitzky-Golay methods is that the first interpolates each of the points with a single function, while the second approximates sequential points with a polynomial of predetermined degree. In this last case, set of Cq’s is intrinsically smoothed prior to differentiation by obvious reasons. |
Click this option to select the Takens-Theiler estimator. This estimator is given by 
, which we generalize for all q as 
, where R1 is the upper epsilon value. The r represent different epsilon values in the interval. The listed integral is approximated by one of the integration methods available in the Numerical Integration Options dialog box (refer to Integration Options later in this topic for additional information.)
Click this button to display the Numerical Integration Options dialog box, where you can specify preferences for the integration required as a consequence of the selected estimation method.
Click this button to display the Numerical DIfferentiation dialog box, where you can specify preferences for the differentiation required as a consequence of the selected estimation method.
Click this button to display the Savitzky-Golay Options dialog box, where you can specify preferences for differentiating using Savitzky-Golay approximating polynomials.
Smoothing tab
Allows you to specify smoothing options for computing the generalized dimensions Dq.
Check this box if you want to smooth the Cq integral before estimating the generalized dimensions, Dq.
Check this box if you want to smooth the Dq estimates.
Click this option to select smoothing using exponential smoothing.
Refer to the Exponential Smoothing dialog box for additional information on this subject.
Click this button to display the Exponential Smoothing Options dialog box, where you can specify preferences for this operation.
Click this option to select smoothing by a moving average filter.
Refer to the Moving Average dialog box for additional information on this subject.
Click this button to display the Moving Average Options dialog box, where you can specify preferences for this operation.
Click this option to select smoothing by Savitzky-Golay approximation.
Refer to the Savitzky-Golay Options dialog box for additional information on this subject.
Click this button to display the Savitzky-Golay Options dialog box, where you can specify preferences for this operation.
Closes the dialog box and carries out this operation.
Closes the dialog box without carrying out this operation.