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URFinder allows you to simultaneously investigate relationships between the nodes of a system and also between the different states the nodes can be in. URFinder achieves this by allowing you to search for relationship parameters U and R (on the node repertoire and set of nodes of the system respectively) that minimise the expected float entropy (efe) of the system. The software also allows you to collect observations of efe, under choosing U and R uniformly at random, in order to plot efe-histograms. You can copy and paste the results tables into Microsoft Excel. URFinder is freeware and the author is happy to distribute the source code. However, the author asks that if users want to release modified versions then they should indicate their modifications in the help pages. URFinder was written during research that resulted in the paper “Quasi-Conscious Multivariate Systems”.

Inputting data and resulting outputs

*These instructions include an example for you to try.

Stage 1

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Stage 2

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Stage 3

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Stage 4

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  1. The first mode calculates the weighted relation R(U_op1,S_i), on the set of nodes S, given by U_op1 (the first option in the Stage 3 table of options for the weighted relation on the node repertoire) and S_i (the ith data element in the Stage 2 table). Note that for nodes a,b in S we have R(U_op1,S_i)(a,b):= U_op1(f_i(a),f_i(b)), where f_i is the function that maps a node to its state as defined by the ith data element in the Stage 2 table. To specify which S_i to use you have to enter the integer i in the field titled “From T use data element ...”.
  2. The second mode calculates the weighted relation R_T on the set of nodes S, where for a,b in S, R_T(a,b) is the mean of the values R(U_op1,S_i)(a,b) over all of the data elements S_i. This mode is seldom used.
  3. The third mode calculates and displays the float entropy fe(R_op1,U_op1,S_i) of data element S_i with respect to the first option given for R, the first option given for U and the metric d_n used. Note that we have (for simplified notation, R and U in place of R_op1 and U_op1) 
    fe(R,U,S_i):=log_2(#{S_j in Omega_S,V : d_n(R,R(U,S_j))<=d_n(R,R(U,S_i))}),
    where Omega_S,V is the set of all possible data elements that a system, with set of nodes S and node repertoire V, can have; e.g. for #S=5, #V=4 we have #Omega_S,V=4^5=1024. The float entropy of a data element is a measure of the amount of information required (in addition to that given by R and U) in order to specify that data element; the unit of measure is bits. When using this mode the metric d_n needs to be specified in order to give a distance between the weighted relations as matrices; by default n=1 but other allowed values are 2,3,4, and "infinity". The value 1 specifies the Manhattan metric (giving the L_1 distance), 2 specifies the Euclidian metric, 3 and 4 give the L_3 and L_4 distances respectively, and infinity gives the Supremum metric.
  4. The forth mode calculates and displays the expected float entropy efe(R_op1,U_op1,T) over the elements of T with respect to the first option given for R, the first option given for U and the metric d_n used. The expected float entropy efe(R_op1,U_op1,T) is the mean of the float entropies given by the elements of T. Therefore, efe(R_op1,U_op1,T) is the expectation for T but if T is merely sample data of some system then efe(R_op1,U_op1,T) is only an approximation of the expected float entropy for that system. In this case, by the law of large numbers T needs to be suitably large to give a good approximation.
  5. The fifth mode calculates and displays the distance d_n(R_op1,R(U_op1,S_i)) with respect to the first option given for R, the first option given for U and the metric d_n used. This is a distance between the weighted relations as matrices. The definition of R(U_op1,S_i) is given in the help details for the first mode.
  6. The sixth mode calculates and displays d_n(R_op1,U_op1,T) which is the mean of d_n(R_op1,R(U_op1,S_i)) over all S_i in T. For the definition of d_n(R_op1,R(U_op1,S_i)) see the help details for the fifth mode.
  7. The seventh mode is one of the two most useful modes (the other being mode eight). Recall that we can enter multiple options for the entries in the Stage 3 and Stage 4 tables (options for U and R respectively) by separating the options with a colon (e.g. the example above, for R, includes the entry 0.625:0.875). Mode seven searches over the options given for U and R and returns the tables that minimise the expected float entropy. Whilst uniqueness is not always guaranteed, for given T, the aim is to find U and R for which efe(R,U,T) is minimised. How the options are searched over can be set for U and R independently. For each there are three ways to search. The first tries every combination of the options given in the table (this is the most comprehensive search). The second way respects the order in which the options appear in the cells of the table. That is the first entry in the cells are tried together, then the second entry in the cells are tried together and so on, but, for instance, the first entry in one cell is not tried with the second entry of another cell. The third way to search has been ineffective and shouldn’t be used. It finds the option that gives the smallest expected float entropy one cell at a time, fixes the option for that cell accordingly and then moves onto the next cell. Whichever of the search options is used, if the result of the search isn’t unique then the table that is returned has multiple entries in the cells separated with colons. The first entry in the cells is one result; the second entry in the cells is another, and so on.
  8. The eighth mode calculates and displays observations of expected float entropy as a random variable. The random variable is given by efe(R_random,U_random,T), where the entries in R_random and U_random are chosen at random. Either the entries are independently chosen uniformly at random from a finite number of equally spaced values between 0 and 1 (in which case the user enters the number of equally spaced values to use) or the user enters “infinity” as the number of values to use (in which case the software chooses the entries independently and uniformly at random using double-precision). The number of expected float entropy observations to be calculated and displayed must also be entered. Subsequently the data generated can be copied and pasted into other software (Excel for example) and used to produce an efe-histogram. If the elements of T come from a source that is far from random (such as localised sampling of photographs for example) then the efe-histogram will have a long left tail. It turns out that, for non-random systems, certain choices of R and U are isolated from the rest in the sense that they give much lower efe values and, hence, the system defines relationships. In addition to the efe data generated, the software also displays an example of U_random and R_random so that the user has an example of what the software generates.
  9. The ninth mode is similar to the third mode except it calculates the float entropy, fe(R_op1,U_op1,S_i), for every data element S_i in T with respect to the first option given for R, the first option given for U and the metric d_n used. The results are displayed in an output table. The mode can be used when wanting to fill in missing node values for an incomplete system state when solutions for R and U, that minimise the expected float entropy of the system, are already known.

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Help-V1-2015
J. W. Mason