View source for CUMULATE parametrized asymptotic knowledge assessment

[[CUMULATE]]'s [[CUMULATE parametrized asymptotic knowledge assessment|parameterized asymptotic knowledge assessment]] algorithm is an attempt to overcome shortcomings of its [[CUMULATE asymptotic knowledge assessment|non-paramterized]] version. Namely, ''one-fits-all'' nature, prohibiting parameter tuning for individual users' abilities and problem complexities.

=Computation=
The formula below is used to update the knowledge levels of concepts (''c'') addressed in a problem (''p''). This formula reflects the following principles (identical to the [[CUMULATE asymptotic knowledge assessment|predecessor]] algorithm).
* there are several domain concepts (knowledge items, rules, productions) involved in solving a problem; the knowledge of each of them is updated proportionally to the others
* knowledge is updated only upon correct user answers, there is no penalty for errors
* solving a problem correctly multiple times will result in diminishing update (growth) of the knowledge level of the concepts as the number of successes grows
in addition:
* the initial level of knowledge, speed of knowledge growth, and penalty for repetitive (correct) solutions to the problem - are now adjustable parameters
[[Image:CUMULATE parameterized asymptotic knowledge assessment.png]], where
* ''Ko'' - is the starting level of knowledge, ''Ko'' &isin; [0, 1]
* ''res'' - result of user action (0 -error, 1 - correct); 
* ''Wc,p'' - is a weight of concept ''c'' in problem ''p''
* &Sigma;''Wc,p'' - is the sum of weights of all concepts in problem ''p''
* ''<sub>succ</sub>att<sub>p</sub>'' - is a number of successful solutions to problem ''p'' prior to current attempt
* ''pV'' - speed of knowledge growth parameter, ''pV'' &isin; [0, 1]
* ''OPP'' - over-practicing parameter, controlling the penalty for repetitively solving one problem (correctly), ''OPP'' &isin; [0, 1]

=Examples=

==Example 1==
Below is a graph of concept's knowledge level growth and the penalty coefficient vs. number of successful attempts to apply it in a problem. The lines denote:
* the blue line denotes a successful use of a concept in a new problem (''pV'' = .5)
* the green line denotes a penalty coefficient -- 1/''(<sub>succ</sub>att<sub>p</sub>''+2)''<sup>OPP</sup>'' -- as if it was the same problem (''OPP'' = .25)
* the red line ''merges'' the prior two graphs

[[Image:CUMULATE parameterized asymptotic knowledge assessment - knowledge growth and penalty.png]]
<br/>([http://chart.apis.google.com/chart?cht=lxy&chs=300x240&chd=t:0,10,20,30,40,50,60,70,80,90,100|00,50,75,88,94,97,98,99,100,100,100|-1|-1,100,84,76,71,67,64,61,59,58,56|-1|00,50,63,66,66,65,63,61,59,58,56&chco=24588E,4C9B46,CF1E2B&chxt=x,y&chxl=1:|0|.2|.4|.6|.8|1|0:|0|1|2|3|4|5|6|7|8|9|10&chm=o,24588E,0,-1,10|o,4C9B46,1,-1,10|o,CF1E2B,2,-1,10&chg=10,20&chdl=knowledge+level|penalty+coefficient|resulting+curve&chdlp=b+&chls=1,1,0|1,1,0|3,6,3&chtt=Knowledge+level+growth+and+change+in+penalty|coefficient+vs.+number+of+successful+attempts&chdlp=b alternatively] via [http://code.google.com/apis/chart/ Google Chart API])

==Example 2==
Below the speed of problem learning is held constant (''pV'' = .5) while the penalty for practicing the same problem changes.

[[Image:CUMULATE parameterized asymptotic knowledge assessment - knowledge growth for diff OPP.png]]
<br/>([http://chart.apis.google.com/chart?cht=lxy&chs=300x240&chd=t:0,10,20,30,40,50,60,70,80,90,100|0,50,75,87,93,97,98,99,100,100,100|-1|0,47,71,83,90,94,97,98,98,99,100|-1|0,42,64,77,85,89,93,95,96,97,98|-1|0,35,54,66,73,79,83,86,88,90,92|-1|0,30,45,55,62,67,70,74,76,78,80|-1|0,25,38,45,51,55,58,61,63,65,66&chco=24588E,4C9B46,F3A030,CF1E2B,78387B,7C807F&chxt=x,y&chxl=1:|0|.2|.4|.6|.8|1|0:|0|1|2|3|4|5|6|7|8|9|10&chm=o,24588E,0,-1,10|o,4C9B46,1,-1,10|o,F3A030,2,-1,10|o,CF1E2B,3,-1,10|o,78387B,4,-1,10|o,7C807F,5,-1,10&chg=10,20&chdl=OPP=.01|OPP=.10|OPP=.25|OPP=.50|OPP=.75|OPP=1.0&chdlp=b&chtt=Knowledge+level+growth+for+different+penalty|parameters+vs.+number+of+successful+attempts alternatively] via [http://code.google.com/apis/chart/ Google Chart API])

==Example 3==
Graph below shows differences the growth of knowledge while speed of learning varies. No penalty is given.

[[Image:CUMULATE parameterized asymptotic knowledge assessment - knowledge growth for diff pV.png]]
<br/>([http://chart.apis.google.com/chart?cht=lxy&chs=300x240&chd=t:0,10,20,30,40,50,60,70,80,90,100|0,1,2,3,4,5,6,7,8,9,10|-1|0,10,19,27,34,41,47,52,57,61,65|-1|0,25,44,58,68,76,82,87,90,92,94|-1|0,50,75,88,94,97,98,99,100,100,100|-1|0,75,94,98,100,100,100,100,100,100,100|-1|0,100,100,100,100,100,100,100,100,100,100&chco=24588E,4C9B46,F3A030,CF1E2B,78387B,7C807F&chxt=x,y&chxl=1:|0|.2|.4|.6|.8|1|0:|0|1|2|3|4|5|6|7|8|9|10&chm=o,24588E,0,-1,10|o,4C9B46,1,-1,10|o,F3A030,2,-1,10|o,CF1E2B,3,-1,10|o,78387B,4,-1,10|o,7C807F,5,-1,10&chg=10,20&chdl=pV=.01|pV=.10|pV=.25|pV=.50|pV=.75|pV=1.0&chdlp=b&chtt=Knowledge+growth+for+different+problem+learnig|speeds+vs.+number+of+successful+attempts alternatively] via [http://code.google.com/apis/chart/ Google Chart API])

=Studies=
Coming soon

= Contacts =
[[User:Myudelson | Michael V. Yudelson]]