Difference between revisions of "CUMULATE parametrized asymptotic knowledge assessment"

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* the blue line denotes a successful use of a concept in a new problem (''pV'' = .5)
 
* 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 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)
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* the red line ''merges'' the prior two graphs
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* the red line ''merges'' the two graphs above
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[[Image:CUMULATE parameterized asymptotic knowledge assessment - knowledge growth and penalty.png]]
 
[[Image:CUMULATE parameterized asymptotic knowledge assessment - knowledge growth and penalty.png]]
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<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])
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([http://chart.apis.google.com/chart?cht=lxy&chs=300x240&chd=t:0,10,20,30,40,50,60,70,80,90,100%7C00,50,75,88,94,97,98,99,100,100,100%7C-1%7C-1,100,84,76,71,67,64,61,59,58,56%7C-1%7C00,50,63,66,66,65,63,61,59,58,56&chco=24588E,4C9B46,CF1E2B&chxt=x,y&chxl=1:%7C0%7C.2%7C.4%7C.6%7C.8%7C1%7C0:%7C0%7C1%7C2%7C3%7C4%7C5%7C6%7C7%7C8%7C9%7C10&chm=o,24588E,0,-1,10%7Co,4C9B46,1,-1,10%7Co,CF1E2B,2,-1,10&chg=10,20&chdl=knowledge+level%7Cpenalty+coefficient%7Cresulting+curve&chdlp=b+&chls=1,1,0%7C1,1,0%7C3,6,3&chtt=Knowledge+level+growth+and+change+in+penalty%7Ccoefficient+vs.+number+of+successful+attempts&chdlp=b alternatively] via [http://code.google.com/apis/chart/ Google Chart API])
  
 
==Example 2==
 
==Example 2==
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Below the speed of problem learning is held constant (''pV'' = .5) while the penalty for practicing the same problem changes.
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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]]
 
[[Image:CUMULATE parameterized asymptotic knowledge assessment - knowledge growth for diff OPP.png]]
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<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])
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<br/>
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([http://chart.apis.google.com/chart?cht=lxy&chs=300x240&chd=t:0,10,20,30,40,50,60,70,80,90,100%7C0,50,75,87,93,97,98,99,100,100,100%7C-1%7C0,47,71,83,90,94,97,98,98,99,100%7C-1%7C0,42,64,77,85,89,93,95,96,97,98%7C-1%7C0,35,54,66,73,79,83,86,88,90,92%7C-1%7C0,30,45,55,62,67,70,74,76,78,80%7C-1%7C0,25,38,45,51,55,58,61,63,65,66&chco=24588E,4C9B46,F3A030,CF1E2B,78387B,7C807F&chxt=x,y&chxl=1:%7C0%7C.2%7C.4%7C.6%7C.8%7C1%7C0:%7C0%7C1%7C2%7C3%7C4%7C5%7C6%7C7%7C8%7C9%7C10&chm=o,24588E,0,-1,10%7Co,4C9B46,1,-1,10%7Co,F3A030,2,-1,10%7Co,CF1E2B,3,-1,10%7Co,78387B,4,-1,10%7Co,7C807F,5,-1,10&chg=10,20&chdl=OPP=.01%7COPP=.10%7COPP=.25%7COPP=.50%7COPP=.75%7COPP=1.0&chdlp=b&chtt=Knowledge+level+growth+for+different+penalty%7Cparameters+vs.+number+of+successful+attempts alternatively] via [http://code.google.com/apis/chart/ Google Chart API])
  
 
==Example 3==
 
==Example 3==
 
Graph below shows differences the growth of knowledge while speed of learning varies. No penalty is given.
 
Graph below shows differences the growth of knowledge while speed of learning varies. No penalty is given.
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[[Image:CUMULATE parameterized asymptotic knowledge assessment - knowledge growth for diff pV.png]]
 
[[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])
 
<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=
 
=Studies=
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Coming soon
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* [[CUMULATE_user_and_domain_adaptive_user_modeling#Study_1|Study 1]] - comparison of the [[CUMULATE_asymptotic_knowledge_assessment|legacy]] and [[CUMULATE_parametrized_asymptotic_knowledge_assessment|parametrized]] algorithms based on SQL problem-solving data
  
 
= Contacts =
 
= Contacts =
 
[[User:Myudelson | Michael V. Yudelson]]
 
[[User:Myudelson | Michael V. Yudelson]]

Latest revision as of 21:50, 20 January 2010

CUMULATE's parameterized asymptotic knowledge assessment algorithm is an attempt to overcome shortcomings of its 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 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

CUMULATE parameterized asymptotic knowledge assessment.png, where

  • Ko - is the starting level of knowledge, Ko ∈ [0, 1]
  • res - result of user action (0 -error, 1 - correct);
  • Wc,p - is a weight of concept c in problem p
  • ΣWc,p - is the sum of weights of all concepts in problem p
  • succattp - is a number of successful solutions to problem p prior to current attempt
  • pV - speed of knowledge growth parameter, pV ∈ [0, 1]
  • OPP - over-practicing parameter, controlling the penalty for repetitively solving one problem (correctly), OPP ∈ [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/(succattp+2)OPP -- as if it was the same problem (OPP = .25)
  • the red line merges the two graphs above

CUMULATE parameterized asymptotic knowledge assessment - knowledge growth and penalty.png
(alternatively via 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.

CUMULATE parameterized asymptotic knowledge assessment - knowledge growth for diff OPP.png
(alternatively via Google Chart API)

Example 3

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

CUMULATE parameterized asymptotic knowledge assessment - knowledge growth for diff pV.png
(alternatively via Google Chart API)

Studies

Contacts

Michael V. Yudelson

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