Performs archetypal analysis by using Convex Hull approximation under a full control of all algorithmic parameters. It contains functions useful for finding the proper initial approximation, the optimal number of archetypes and function for applying main algorithm.
The main function is archetypal which is a modification of PCHA algorithm, see [1], [2],
suitable for R language. It provides control to the entrire set of involved parameters and has two main options:
initialrows=NULL, then a method from 'projected_convexhull', 'convexhull', 'partitioned_convexhull', 'furthestsum', 'outmost', 'random' is used
initialrows=a vector of kappas rows, then given rows form the initial solution for AA
This is the main function of the package, but extensive trials has shown that:
AA may be very difficult to run if a random initial solution has beeen chosen
for the same data set the final Sum of Squared Errors (SSE) may be much smaller if initial solution is close to the final one
even the quality of AA done is affected from the starting point
This is the reason why we have developed a whole set of methods for choosing initial solutionfor main PCHA algorithm.
There are three functions that work with the Convex Hull (CH) of data set.
find_outmost_convexhull_points computes the CH of all points
find_outmost_projected_convexhull_points computes the CH for
all possible combinations of variables taken by n (default=2)
find_outmost_partitioned_convexhull_points makes partitions
of data frame, then computes CH for each partition and finally gives the CH of overall union
The most simple method for estimating an initial solution is find_outmost_points
where we just compute the "outmost points", ie those that are the most frequent outmost for all
available points.
The default method "FurthestSum" (FS) of PCHA (see [1], [2]) is used by find_furthestsum_points which applies
FS for many times (default=10) and then finds the most frequent points.
Of course "random" method is available for comparison reasons and that gives a random set of kappas points as initial solution.
All methods give the number of rows for the input data frame.
For that task find_optimal_kappas is available which
runs for each kappas from 1 to maxKappas (default=15) ntrials (desault=10) times AA,
stores SSE, VarianceExplained from each run and then computes knee point by nusing UIK method, see [3].
By using function check_Bmatrix we can evaluate the overall quality of
applied method and algorithm. Quality can be considered high:
if every archetype is being created by a small number of points
if relevant weights are not numerically insignificant
Of course we must take into account the SSE and VarianceExplained, but if we have to compare two solutions with similar termination status, then we must choose that of the simplest B matrix form.
[1] M Morup and LK Hansen, "Archetypal analysis for machine learning and data mining", Neurocomputing (Elsevier, 2012). https://doi.org/10.1016/j.neucom.2011.06.033.
[2] Source: http://www.mortenmorup.dk/index_files/Page327.htm , last accessed 2019-06-07
[3] Christopoulos, Demetris T., Introducing Unit Invariant Knee (UIK) As an Objective Choice for Elbow Point in Multivariate Data Analysis Techniques (March 1, 2016). Available at SSRN: https://ssrn.com/abstract=3043076 or http://dx.doi.org/10.2139/ssrn.3043076