introfdim                package:fdim                R Documentation

_I_n_t_r_o_d_u_c_t_i_o_n _t_o _t_h_e _c_a_l_c_u_l_a_t_i_o_n _o_f _t_h_e _d_i_m_e_n_s_i_o_n _f_r_a_c_t_a_l.

_D_e_s_c_r_i_p_t_i_o_n:

     Shows the theoretical basis regarding the fractal dimension
     measurement.

_D_e_t_a_i_l_s:

     Fractals burst into the open in early 1970s. Their breathtaking
     beauty  captivated many a layman and a professional alike. 
     Striking fractal images can often be obtained with very elementary
     means.  However, the definition of fractals is far from being
     trivial and depends  on a formal definition of dimension. It takes
     a few chapters of an Advanced Analysis book to rigorously define a
     notion of dimension.  The important thing is that the notion is
     not unique and even more importantly, for a given set, various
     definitions may  lead to different numerical results. When the
     results differ the set is called fractal.  Or in the words of
     Benoit Mandelbrot, the father of fractals: A fractal is by
     definition a set for which the Hausdorff-Besicovitch dimension
     strictly exceeds the topological dimension.

     The topological dimension of a smooth curve is, as one would
     expect, one and that of a sphere is two which may  seem very
     intuitive. However, the formal definition was only given in 1913
     by the Dutch mathematician  L. Brouwer (1881-1966). A (solid) cube
     has the topological dimension of three because in any
     decomposition of  the cube into smaller bricks there always are
     points that belong to at least four (3+1) bricks. The Brouwer
     dimension is obviously an integer.  The Hausdorff-Besicovitch
     dimension, on the other hand, may be a fraction.  Formal
     definition of this quantity requires a good deal of the Measure
     Theory.  But fortunately for a class of sets Hausdorff-Besicovitch
     dimension can be easily evaluated.  This sets are known as the
     self-similar fractals and, because of that ease, the property of
     self-similarity is often  considered to be germane to fractals in
     general. The applet below illustrates the idea of self-similarity.

     The similarity dimension of the snowflake curve is finite. This is
     arrived at in the following manner.  If it were a straight line we
     could split it into to smaller segments each half the length of
     the "parent" line.  The length of the line would be the sum of the
     two smaller segments. If we were talking about areas, then taking
     a  square and splitting it into 4 smaller squares with areas 1/4
     of the "parent" square.  We would observe that the four smaller
     areas sum up to the original size. Notice that when the side of a
     square is halved,  its area decreases by the factor of 4 which is
     (1/2)^2. For a cube, acting similarly, decreasing its size by a
     factor of 2,  results in smaller cubes each with the volume
     (1/8)=(1/2)^3 of the "parent" cube. We can detect a commonality in
     these  three examples. Given a shape of size S. It's split into N
     similar smaller shapes each with the size S/N so  that N*(S/N)=S.
     In each of the three cases we used a different function S. If a is
     a linear dimension of the shape we  have S(a)=a for a line segment
     and S(a)=a^2 and S(a)=a^3 for the square and cube, respectively.
     Thus, N*(S/N)=S can  be rewritten as N*(a/M)^D=a^D where a is the
     "linear size" of the shape, M is the number of linear parts, and N
     is the  total number of the resulting smaller shapes. This gives
     NM^(-D)=1 or N=M^(-D). In all three cases we took M=2 and D  was
     successively 1, 2, and 3. We see that D=log(N)/log(M) is what we
     would call the dimension in all three cases.

     This quantity D is known as the similarity dimension. It applies
     to shapes that are composed of a few copies of  themselves whose
     "linear" size is smaller than that of the "parent" shape by a
     factor of M. Returning to the snowflake,  we have N=4 and M=3 In
     this case D=log(4)/log(3) is somewhere between 1 and 2. 

     The Koch's snowflake has no self-intersection and is obtained from
     a line segment as an image of a continuous function.  By one of
     Brouwer's theorems this function preserves the topological
     dimension of the segment (which is, of course 1).  Finally, the
     curve has topological dimension 1 whereas its
     Hausdorff-Besicovitch dimension is log(4)/log(3). 

     The main funcions are fdim and slopeopt, the first one for 
     calculating the object with pairs (size of elements, number of
     elements with points inside). In order to fulfill the objective of
     the function it evaluate the slope of the linear regression model
     by using a first stimation of confidence. After the first
     stimation and in order to make afordable to check other confidence
     paramenters we provide the second function. By this way the user
     only evaluate the sequence of points once. After that he can
     change the confidence parameters and calculate the slope.

     The other functions are complementary, in order to provide sets of
     points  (plane, line, sphere, and so on).

_A_u_t_h_o_r(_s):

     Francisco Javier Martinez de Pison.
     francisco.martinez@dim.unirioja.es
     Joaquin Ordieres Mere.             
     joaquin.ordieres@dim.unirioja.es
     Manuel Castejon Limas.             
     manuel.castejon@dim.unirioja.es
     Fco. Javier de Cos Juez.           
     francisco-javier.de-cos@dim.unirioja.es

_R_e_f_e_r_e_n_c_e_s:

     Halsey C.T., Mogens H.J., Kandanoff L.P., Procaccia I., Shraiman
     B.I. "Fractal Measures and their singularities: The
     caracterization of strange sets". Physical Review  vol 33, n 2.
     1986

     Roberts J.A., Cronin A. "Unbiased estimation of multi-fractal
     dimensions of finite data sets" <URL:
     http://www.sci.usq.edu.au/pub/MC/staff/robertsa/multif.htm> . July
     1996.

     David M. Alexander, Phil Sheridan, Paul D. Bourke, Otto
     Konstandatos. "Global and local similarity of the primary visual
     cortex: mechanisms of orientation preference". HELNET -
     International Workshop on Neural Networks, September 1997

     Geoffrey B. West, James H. Brown, Brian J. Enquist. "The Fourth
     Dimension of Life: Fractal Geometry and Allometric Scaling of
     Organisms"., Santa Fe Institute of Research. 1999

     Christo Faloutsos, Volker Gaede. "Analisys of n-dimensional
     Quadtrees Using the Hausdorff Fractal Dimension". Mumbai (Bombay),
     Proceedings of the 22nd  VLDB Conference, India, 1996.

     Alberto Belussi, Christo Faloutsos. "Estimating the Selectivity of
     Spatial Queries Using the 'Correlation' Fractal Dimension".,
     Zurich, Switzerland, Proceedings of the 21st VLDB Conference,
     1995.

     Menndez Fernndez C.; Ordieres Mer J.; Ortega Fernndez F.
     "Importance of information pre-processing importance in the
     improvement of neural networks results.". International Journal on
     Expert System and Neural Networks, Vol. 13, No. 2, pp. 95-103. May
     1996.


_S_e_e _A_l_s_o:

     `fdim', `slopeopt', `makefract'

