study_AAconvergence: Function which studies the convergence of Archetypal Analysis...

In archetypal: Finds the Archetypal Analysis of a Data Frame

Function which studies the convergence of Archetypal Analysis when using the PCHA algorithm

Description

First it finds an AA solution under given arguments while storing all iteration history (save_history = TRUE). Then it computes the LOWESS [1] of SSE and its relevant UIK point [2]. Study is performed for iterations after that point. The list of B-matrices and archetypes that were found are stored. The archetypes are being aligned, while the B-matrices are used for computing the used rows-weights, leading rows-weights and maybe percentage of used rows on Convex Hull. The Aitken SSE extrapolation plus the relevant error are computed. The order and rate of convergence are estimated. Finally a multi-plot panel is being created if asked.

Usage

study_AAconvergence(df, kappas, method = "projected_convexhull", 
                    rseed = NULL, chvertices = NULL, plot = FALSE, ...)

Arguments

df	The data frame with dimensions n x d
kappas	The number of archetypes
method	The method that will be used for computing initial approximation: projected_convexhull, see find_outmost_projected_convexhull_points convexhull, see find_outmost_convexhull_points partitioned_convexhull, see find_outmost_partitioned_convexhull_points furthestsum, see find_furthestsum_points outmost, see find_outmost_points random, a random set of kappas points will be used
rseed	The random seed that will be used for setting initial A matrix. Useful for reproducible results.
chvertices	The vector of rows which represents the vertices for Convex Hull (if available)
plot	If it is TRUE, then a panel of useful plots is created
...	Other arguments to be passed to function archetypal, except save_history which must always be TRUE

Details

If we take natural logarithms at the next approximate equation

\epsilon_{n+1} = c\epsilon_{n}^p

for n = 1, 2, 3, \ldots, then we'll find

\log(\epsilon_{n+1}) = \log(c)+p\log(\epsilon_{n})

Thus a reasonable strategy for estimating order p and rate c is to perform a linear regression on above errors, after a selected iteration. That is the output of order_estimation and rate_estimation.

Value

A list with members:

1. SSE, a vector of all SSE from all AA iterations

2. SSE_lowess, a vector of LOWESS values for SSE

3. UIK_lowess, the UIK point [2] of SSE_lowess

4. aitken, a data frame of Aitken [3] extrapolation and error for SSE after UIK_lowess iteration

5. order_estimation, the last term in estimating order of convergence, page 56 of [4], by using SSE after UIK_lowess iteration

6. rate_estimation, the last term in estimating rate of convergence, page 56 of [4], by using SSE after UIK_lowess iteration

7. significance_estimations, a data frame with standard errors and statistical significance for estimations

8. used_on_convexhull, the % of used rows which lie on Convex Hull (if given), as a sequence for iterations after UIK_lowess one

9. aligned_archetypes, the archetypes after UIK_lowess iteration are being aligned by using align_archetypes_from_list. The history of archetypes creation.

10. solution_used, the AA output that has been used. Some times useful, especially for big data.

References

[1] Cleveland, W. S. (1979) Robust locally weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc. 74, 829–836.

[2] 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: http://dx.doi.org/10.2139/ssrn.3043076

[3] Aitken, A. "On Bernoulli's numerical solution of algebraic equations", Proceedings of the Royal Society of Edinburgh (1926) 46 pp. 289-305.

[4] Atkinson, K. E.,An Introduction to Numerical Analysis, Wiley & Sons,1989

See Also

check_Bmatrix

Examples

{
# Load data "wd2"
data(wd2)
ch = chull(wd2)
sa = study_AAconvergence(df = wd2, kappas = 3, rseed = 20191119,
                         verbose = FALSE, chvertices = ch)
	names(sa)
	# [1] "SSE"                      "SSE_lowess"               "UIK_lowess"              
	# [4] "aitken"                   "order_estimation"         "rate_estimation"         
	# [7] "significance_estimations" "used_on_convexhull"       "aligned_archetypes"      
	# [10] "solution_used"        
	# sse=sa$SSE
	# ssel=sa$SSE_lowess
	sa$UIK_lowess
	# [1] 36
	# sa$aitken
	sa$order_estimation
	# [1] 1.007674
	sa$rate_estimation
	# [1] 0.8277613
	sa$significance_estimations
	#        estimation   std.error   t.value      p.value
	# log(c) -0.1890305 0.014658947 -12.89523 5.189172e-12
	# p       1.0076743 0.001616482 623.37475 3.951042e-50
	# sa$used_on_convexhull
	# sa$aligned_archetypes
	data.frame(sa$solution_used[c("SSE","varexpl","iterations","time")])
	#        SSE   varexpl iterations time
	# 1 1.717538 0.9993186         62 8.39
	# Plot class "study_AAconvergence"
	plot(sa)

}

archetypal documentation built on May 29, 2024, 8:46 a.m.