CN107122452A - Student's cognitive diagnosis method of sequential - Google Patents
Student's cognitive diagnosis method of sequential Download PDFInfo
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Abstract
The invention discloses a kind of student's cognitive diagnosis method of sequential, including：Obtain the history answering information of multiple students；It is modeled according to the history answering information got using sequential cognitive diagnosis method, capacitation vector and the knowledge-ID correlation matrix after partial order is limited；Knowledge-ID correlation matrix according to Efficiency analysis and after partial order is limited is predicted to the ability value and score of a certain student next period.This method is analyzed and processed continuously, for a long time by test question information and student's answer situation, carrying out, and can accurately analyze whole capability level and acquisition of knowledge degree of the student in different time sections.
Description
Technical field
The present invention relates to a kind of student's cognitive diagnosis method in educational data digging technology field, more particularly to sequential.
Background technology
Cognitive diagnosis be tradition examination test with a kind of improvement for evaluating with it is perfect.The general examination of education is particularly big
The examination of scale, only provides exam score.But by single fraction, student can neither be obtained and specifically grasp or what is not grasped
The conclusion of knowledge, can not obtain student do wrong examination question the reason for, to be remedied；For the student of identical fraction, more can not
Obtain the difference of the state of knowledge and cognitive structure that may be present between them.The information that tradition examination is provided unsuitable
The need for hair tonic exhibition, the main task of cognitive diagnosis is the difference according to student, excavates more Cognitive Processing information.Cognition is examined
It is disconnected pass a test obtain student in test (observable) reaction (test result), so as to deduce the knowledge of student's not observable
State.
Cognitive diagnosis is the state estimation to student's study for a long time, is the important content of intelligent individual character teaching.Accurately
Cognitive diagnosis, student can be helped clearly to learn situation precise knowledge to itself this period, while parent can be aided in
It is that student formulates personalized learning strategy with teacher.Thus how for a long time, exactly diagnose student's acquisition of knowledge degree
Change, be always educational data excavation applications explore a major issue.
At present, the method on cognitive diagnosis mainly has following several：
1) the ability value diagnosis based on the single moment
Answer situation based on item response theory (IRT) only by student on examination question, by topic characteristic function
Computing, to speculate the ability of student.IRT item parametes have：Difficulty, discrimination and conjecture degree.According to the difference of parameter,
Characteristic function can be divided into one-parameter model (difficulty), two-parameter model (difficulty, discrimination) and three parameter model (difficulty, differentiation
Degree, conjecture parameter) etc..Student is modeled as one-dimensional continuous ability value, and represents that student's is comprehensive with this single ability value
Conjunction ability.Under IRT models, student is portrayed as an object with single ability value, except the factor of topic itself (is distinguished
Degree, difficulty, conjecture degree) outside, performance situation of the student on topic is only influenceed by this single integrated ability.
2) acquisition of knowledge diagnosis based on the single moment
After the related competence dimension in knowledge point is introduced, there is scholar to propose knowledge point information (the Q squares associated by examination question
Battle array), and think that the interaction between multiple knowledge points that an examination question is investigated has " internuncial " (to need to be grasped investigation
All knowledge points can just have an opportunity to answer questions examination question) and " compensatory " (one of relevant knowledge that a GPRS is investigated
Point is with regard to energy score).This method belongs to Multidimensional Discrete cognitive diagnosis model (DINA), and the DINA models think the phase between knowledge point
Interaction is " internuncial ".DINA models also introduce examination question knowledge in addition to using answer situation of the student on examination question
Point incidence matrix (Q matrixes), is entered by combining distinctive information in more education sectors to the knowledge point Grasping level of student
Row diagnosis, and student is modeled as an Efficiency analysis on multidimensional knowledge point, wherein every one-dimensional value of vector is 0-1
, represent whether student has grasped the knowledge point in corresponding dimension.DINA models can further combined with the conjecture on examination question and
Error parameter, performance situation of the prediction student on examination question.
3) the ability value diagnosis based on multiple moment.
It is an incremental long process in view of learning as run marathon.The static cognition of single point in time is examined
Disconnected method far can not meet the growing demand of student, have scholar to propose the cognitive diagnosis technology based on sequential,
Such technology can be largely classified into following two models：The IRT models of sequential and BKT models and its related improvement.Sequential
IRT technologies are mainly the factor that the time is added in traditional IRT models, it is believed that the ability value of student, which changes with time, to be met
Specific rule can use linear function fit, so as to according to the answer data of the multiple test of student's history, obtain student and exist
Not ability value in the same time.BKT models are widely applied in intelligent tutoring system, and the data of the application scenarios have a distinctness
Feature, for the examination question with along with or problem, candidate can not answer repeatedly repeatedly in the same time.The model uses hidden Ma Er
The acquisition of knowledge of section's husband's method to student is modeled, while defining four probability：Initial ability value, T moment knowledge points
The transition probability of Grasping level, the probability and the probability of conjecture made a mistake is used to characterize performance and shape of the student at each moment
State.
In above-mentioned cognitive diagnosis method, although the diagnosis of the ability value based on the single moment and acquisition of knowledge diagnosis can be accurate
Ability value or acquisition of knowledge degree of the student in particular moment really are obtained, but this method is static, is only capable of to one
The data of examination are analyzed, it is impossible to suitable for the dynamic evaluation of student's study for a long time.Ability value based on multiple moment is examined
Disconnected method, although student can be obtained in ability value not in the same time, and Accurate Prediction student is in the score at next moment,
But the ability value does not have practical significance, it can not more explain how the ability value of student changes, the ability value of an one-dimensional
In default of explaining meaning, actual value less, this method can only be diagnosed to be student in ability value lifting not in the same time or
Person have dropped, but can not learn that student regresses in the knowledge of which aspect needs to strengthen, skilled enough in terms of which
Need not extra increase training again.Therefore, using existing cognitive diagnosis method, it is difficult to accurately capture student and learn for a long time
During to the situation of change of the Grasping level of each knowledge point, and have according to explain that what factor result in the change of student
Change, for example, newly learnt some knowledge point, or the oversize forgetting of time interval etc..
The content of the invention
It is an object of the invention to provide a kind of student's cognitive diagnosis method of sequential, by test question information and student
Answer situation, analyze and process continuously, for a long time, can accurately analyze whole capability water of the student in different time sections
Gentle acquisition of knowledge degree.
The purpose of the present invention is achieved through the following technical solutions：
A kind of student cognitive diagnosis method of sequential, including：
Obtain the history answering information of multiple students；
It is modeled according to the history answering information got using sequential cognitive diagnosis method, obtains knowledge point and grasp
Degree and examination question-knowledge point correlation matrix after partial order is limited；
Examination question-knowledge point correlation matrix according to knowledge point Grasping level and after partial order is limited is to a certain student
Score on the knowledge point Grasping level and done examination question of next period is predicted.
The history answering information that the basis is got using sequential cognitive diagnosis method be modeled including：
Initialize Efficiency analysis U, examination question-knowledge point correlation matrix V, and entity difficult parameters b, then student obtain
The posterior probability for being scored at R is expressed as：
Wherein, T is total time hop count, and N is total student's quantity, and M is total examination question number,Tried for student i t-th of period
J answer result is inscribed,For Grasping levels of the student i within t-th of period on each knowledge point, VjIt is examination question j with each
The degree of correlation of knowledge point, bjFor examination question j difficulty, IijRepresent whether student i was examination question j, Iij=1, which represents student i, did
Examination question j, otherwise represents not do；Represent that the score R of student obeys average and is
Variance isGaussian Profile, whereinFormal definition is as follows：
The Q matrixes of given examination question association, | Q |=M × K：
Wherein, K is knowledge point quantity；
It is defined as follows partial ordering relation：For examination question j, if itself and knowledge point q relation are Qjq=1, the pass with knowledge point p
It is for Qjp=0, then it is assumed that for examination question j, knowledge point q is more more relevant than knowledge point p；
Then have：
In above formula,Represent for examination question j, knowledge point q is more more relevant than knowledge point p,Represent for examination question j,
Knowledge point p is more more relevant than knowledge point q；
By given Q matrixes, training data triple is converted to：
Model learning is made to examination question-knowledge point correlation matrix, and defined partial ordering relation can be kept, that is, is maximized
Examination question-knowledge point correlation matrix V is in given partial ordering relationPosterior probability afterwards
In above formula,For examination question-knowledge point correlation matrix V likelihood, p (V) is examination question-knowledge point correlation
The priori of matrix V；Vjq、VjpDegrees of correlation of the examination question j with knowledge point q and p is represented respectively,Represent for examination
Inscribe j, probability more more relevant than knowledge point p knowledge point q；
Posterior probability to examination question-knowledge point correlation matrix V can obtain following form while taking the logarithm：
In above formula, λ is regularization parameter；
Add after partial ordering relation, for certain examination, the one-dimensional knowledge point Grasping level of each student, then base can be obtained
In learning curve the ability value changes of the different examinations of a certain student are caught from forgetting curve；
Assuming that ability value obedience averages of the student i t-th of period isVariance isGaussian Profile：
In above formula,Represent that knowledge point Grasping levels of the student i within t-th of period isProbability,It is the t period students i obtained according to being calculated after the forgetting of hypothesis and the formula of study part
To 1~K of knowledge point Grasping level,For actual Grasping level, I is indicator function, is done for identifying students ' actual situation
Which examination question, if student i has been examination question j, Iij=1.
T period student i are to the calculation formula of knowledge point k Grasping level：
Wherein, αiFor student i personalizing parameters,It is student i in the forgetting part of t-th of period,For student i
In the study part of t-th of period.
Student i is in the forgetting part of t-th of periodThat is student i is from the t-1 period to t-th period
Forget part, be expressed as：
Wherein,Represent the t-1 period student i to knowledge point k Grasping level, e(-Δt/S)Represent by Δ t's
After time, the percentage of residual is remembered, S represents the intensity of memory, and Δ t is the time of the t-1 period to t-th of period
Interval.
Student i is in the study part of t periodsIt is expressed as：
Wherein,Grasping levels of the t-1 period student i to knowledge point k is represented,Represent t-th of period
Exercise number of times of the i to knowledge point k is given birth to,T-th of period is represented, influence of the number of times to learning rate is practised in knowledge point, and r is used
To limit the speed of growth, M' is used for limiting the increasing degree of maximum.
This method also includes：
With reference to partial ordering relation, forget, and three parts of study, obtain the final loss function E of model：
Wherein, λUWith λVIt is regularization parameter, IijRepresent whether student i was examination question j, Iij=1, which represents student i, did
Examination question j, otherwise represents not doing the loss function that we can define according to above formula；
The more new algorithm declined using gradient, in each round iteration, is utilized respectively loss function and grasps journey to knowledge point
Degree parameter U, examination question-knowledge point degree of correlation parameter V, each student individuality parameter alpha and examination question difficult parameters b ask inclined
Derivative, and parameter renewal and solution are carried out, until model convergence, the knowledge point palm of the final output student within 1~T period
Degree is held, and obtained examination question-knowledge point correlation matrix is limited by partial order.
Examination question-knowledge point correlation matrix according to knowledge point Grasping level and after partial order is limited is to a certain student
Score on the knowledge point Grasping level and done examination question of subsequent time period be predicted including：
Efficiency analysis contains Grasping levels of the T time section student i to 1~K of knowledge pointThen the T+1 periods are learned
Grasping levels of the raw i to 1~K of knowledge pointCalculation formula be：
Wherein,It is the T+1 period student i to knowledge point k Grasping level, k=1,2 ..., K；Represent
The T period student i is to knowledge point k Grasping level, αiFor student i personalizing parameters,Represent by Δ T+1's
After time, the percentage of memory residual, the time interval of the T period to the T+1 period of Δ T+1,Represent T
+ 1 period student i to knowledge point k exercise number of times,Represent the T+1 period, knowledge point exercise number of times pair
The influence of learning rate；
The T+1 period student i score predictor formula is：
Wherein,For student i examination question j in the T+1 period answer result.
As seen from the above technical solution provided by the invention, for student's history answering information, enter according to the period
Row is divided, with similar item reaction theory IRT form combination pedagogy field priori (Q matrixes) by the energy of student
Force value is corresponded on the every one-dimensional knowledge point investigated a period；For the knowledge point Grasping level between different time sections
Change, this method introduces the big classical theory-forgetting curve in pedagogy field two and learning curve to student's different time sections
Knowledge point dimension is modeled, and then the more accurately change of diagnostic analysis student knowledge point Grasping level in different time sections
Change situation (forget more still improve in study more), compensate for the drawbacks of existing method lacks dynamic and be extremely strong explanatory.
Brief description of the drawings
In order to illustrate the technical solution of the embodiments of the present invention more clearly, being used required in being described below to embodiment
Accompanying drawing be briefly described, it should be apparent that, drawings in the following description are only some embodiments of the present invention, for this
For the those of ordinary skill in field, on the premise of not paying creative work, other can also be obtained according to these accompanying drawings
Accompanying drawing.
Fig. 1 is a kind of flow chart of student's cognitive diagnosis method of sequential provided in an embodiment of the present invention.
Embodiment
With reference to the accompanying drawing in the embodiment of the present invention, the technical scheme in the embodiment of the present invention is carried out clear, complete
Ground is described, it is clear that described embodiment is only a part of embodiment of the invention, rather than whole embodiments.Based on this
The embodiment of invention, the every other implementation that those of ordinary skill in the art are obtained under the premise of creative work is not made
Example, belongs to protection scope of the present invention.
Embodiment
The embodiment of the present invention provides a kind of student's cognitive diagnosis method of sequential, as shown in figure 1, it is mainly including as follows
Step：
Step 11, the history answering information for obtaining multiple students.
The history answering information of each student can include：Knowledge point (Q including Reaction time, involved by answer mesh
Matrix) and each technical ability examination situation (i.e. topic information) and the answer result (right or wrong) of user etc..The topic information with
And Q matrixes mark gained in advance by pedagogy expert, the history answering information of each student can be from on-line study platform intelligence net
Obtain.
The history answering information that step 12, basis are got is modeled using sequential cognitive diagnosis method, is known
Know point Grasping level and examination question-knowledge point correlation matrix after partial order is limited.
First have to initialization knowledge point Grasping level parameter U, examination question-knowledge point correlation matrix V, and examination question difficulty
Parameter b, then after known parameters U, V, b, the probability that student is scored at R can be expressed as：
Wherein,Score R probability is obtained for student, T is total time hop count, and here by one
Cutting in individual month is a period, and N is total student's quantity, and M is total examination question number,Tried by student i within t-th of period
J answer result is inscribed,For Grasping levels of the student i within t-th of period on each knowledge point, VjIt is examination question j with each
The degree of correlation of knowledge point, bjFor examination question j difficulty, IijRepresent whether student i was examination question j, Iij=1, which represents student i, did
Examination question j, otherwise represents not do,Represent that the score R of student obeys average and is
Variance isGaussian Profile, whereinFormal definition is as follows：
In above formula, the Efficiency analysis U of student is no practical significance, in order to realize U pairs of the Efficiency analysis of student
The Q matrixes of examination question association should be given on corresponding knowledge point, | Q |=M × K：
Wherein, K is knowledge point quantity；
It is defined as follows partial ordering relation：For examination question j, if itself and knowledge point q relation are Qjq=1, the pass with knowledge point p
It is for Qjp=0, then it is assumed that for examination question j, knowledge point q is more more relevant than knowledge point p；
Then have：
In above formula,Represent for examination question j, knowledge point q is more more relevant than knowledge point p,Represent for examination question j,
Knowledge point p is more more relevant than knowledge point q；Similar,
By given Q matrixes, training data triple is converted to：DT:(M×K×K)I
Wish that model acquires examination question-knowledge point correlation matrix V, the partial ordering relation in initial Q matrix can be kept, (i.e.:For examination
J is inscribed, if the Q of expert's markjq=0and Qjp=1, it is intended that the V that model is acquiredjq＞ Vjp)
This is equivalent to maximization examination question-knowledge point correlation matrix V in given partial ordering relationPosterior probability afterwards：
In above formula,Examination question-knowledge point correlation matrix V posterior probability is represented, can be with according to Bayesian formula
Conversion is obtained, and examination question-knowledge point correlation matrix V priori of the posterior probability equal to examination question-knowledge point correlation matrix V is multiplied by
Examination question-knowledge point correlation matrix V likelihood score,It is examination question-knowledge point correlation matrix V likelihood, p (V) is
Examination question-knowledge point correlation matrix V priori, Vjq、VjpDegrees of correlation of the examination question j with knowledge point q and p is represented respectively,Represent for examination question j, probability more more relevant than knowledge point p knowledge point q；
Posterior probability to examination question-knowledge point correlation matrix V can obtain following form while taking the logarithm：
WhereinRepresent the logarithmic form of examination question-knowledge point correlation matrix V posterior probability, both members
Need while being converted into logarithmic form, λ is regularization parameter.
Add after partial ordering relation, for certain examination, the one-dimensional knowledge point Grasping level of each student can be obtained, be
Realize the modeling of sequential, the two big classical theories in KPT model reference pedagogy field：Learning curve and forgetting curve are caught
Catch the ability value changes of a certain student's difference examination；It is specific as follows：
Assuming that knowledge point Grasping level obedience averages of the student i within t-th of period isVariance isGauss point
Cloth：
WhereinRepresent that knowledge point Grasping levels of the student i within t-th of period isProbability,Be according to
We assume that forgetting and learn part formula after calculate obtained Grasping level,For actual Grasping level, I refers to
Show function, for identifying which examination question students ' actual situation did, if student i has been examination question j, Iij=1.
Influenceed by two aspect factors：The forgetting for the knowledge point that student has learnt to the last period and student are at t-th
Between study in section to each knowledge point；Represent that student i is living through forgetting in t-th of period
With after study to 1~K of knowledge point Grasping level (being calculated according to the following equation).
Within t-th of period, student i is to the calculation formula of knowledge point k Grasping level：
Wherein, αiFor student i personalizing parameters,The forgetting part for being student i within t-th of period,To learn
Study parts of the raw i within t-th of period.
For forgeing part, Chinese mugwort this great forgetting curve of guest thinks that man memory power can be gradually with the postponement forgetting amplitude of time
Reduce, use for reference this thought, student i is in the forgetting part of t-th of periodI.e. student i is from the t-1 period to t
The forgetting part of individual period, is expressed as：
Wherein,When representing t-1 period, student i is to knowledge point k Grasping level, e(-Δt/S)Δ is passed through in expression
After t time, the percentage of residual is remembered, S represents the intensity of memory, and Δ t is the t-1 period to t-th of period
Time interval.
And the grasp of knowledge or technical ability can gradually tend towards stability again with the first rapid rise of the increase of exercise number of times, specifically
, student i is in the study part of t-th of periodIt is expressed as：
Wherein,Represent the t-1 period, student i to knowledge point k Grasping level,Represent t-th of time
In section, student i does topic number of times on the k of knowledge point,Represent in t-th of period, number of times is practised to study in knowledge point
The influence of rate, r is used for limiting the speed of growth, and M' is used for limiting the increasing degree of maximum.
With reference to partial ordering relation, forget, and three parts of study, we can obtain the final loss function E of model such as
Shown in lower formula：
Wherein, λUWith λVIt is regularization parameter, IijRepresent whether student i was examination question j, Iij=1, which represents student i, did
Examination question j, otherwise represents not do.The loss function that we can define according to above formula, the more new algorithm declined using gradient,
In each round iteration, knowledge point Grasping level parameter U, examination question-knowledge point degree of correlation of the loss function to student are utilized respectively
Parameter V, each student individuality parameter alpha and examination question difficult parameters b seek partial derivative, and carry out parameter renewal and solution, directly
To model convergence, knowledge point Grasping level of the student within 1~T period can be exported, and limit what is obtained by partial order
Examination question-knowledge point correlation matrix.Efficiency analysis is that, in the long-term dynamics diagnostic result of this T period, can help to student
Student does detailed summary for the learning state of this period in past, while formulating the specific aim learning strategy of next stage.
Step 13, the examination question according to knowledge point Grasping level and after partial order is limited-knowledge point correlation matrix pair
Score on the knowledge point Grasping level and done examination question of a certain student's subsequent time period is predicted.
Efficiency analysis was contained within the T period, Grasping levels of the student i to 1~K of knowledge pointThen T+1
Individual period, Grasping levels of the student i to 1~K of knowledge pointCalculation formula be：
Wherein,For the T+1 period, student i is to knowledge point k Grasping level, k=1,2 ..., K；Represent
The T period student i is to knowledge point k Grasping level, αiFor student i personalizing parameters,Represent by Δ T+1
Time after, remember the percentage of residual, Δ T+1 is the time interval of the T period to the T+1 period,Represent
The exercise number of times that student i is done on the k of knowledge point in the T+1 period,Represent in the T+1 period, practise
Influence of the number of times to learning rate；
The T+1 periods, score predictor formulas of the student i on done examination question is：
Wherein,The answer result on examination question j is done in the T+1 period by student i.
Such scheme of the present invention, for student's history answering information, is divided according to the period, anti-with similar item
Answer the priori (Q matrixes) in theory IRT form combination pedagogy field that the ability value of student is corresponded into an examination institute
On the every one-dimensional knowledge point investigated；For the change of the knowledge point Grasping level between different time sections, this method introduces education
Big classical theory-the forgetting curve in field two and learning curve are modeled to the knowledge point dimension in student's different time sections,
And then more accurately diagnostic analysis student knowledge point Grasping level in different time sections situation of change (forget it is more still
Improve in study more), it compensate for the drawbacks of existing method lacks dynamic and be extremely strong explanatory.
Through the above description of the embodiments, those skilled in the art can be understood that above-described embodiment can
To be realized by software, the mode of necessary general hardware platform can also be added to realize by software.Understood based on such,
The technical scheme of above-described embodiment can be embodied in the form of software product, the software product can be stored in one it is non-easily
The property lost storage medium (can be CD-ROM, USB flash disk, mobile hard disk etc.) in, including some instructions are to cause a computer to set
Standby (can be personal computer, server, or network equipment etc.) performs the method described in each embodiment of the invention.
The foregoing is only a preferred embodiment of the present invention, but protection scope of the present invention be not limited thereto,
Any one skilled in the art is in the technical scope of present disclosure, the change or replacement that can be readily occurred in,
It should all be included within the scope of the present invention.Therefore, protection scope of the present invention should be with the protection model of claims
Enclose and be defined.
Claims (7)
1. a kind of student's cognitive diagnosis method of sequential, it is characterised in that including：
Obtain the history answering information of multiple students；
It is modeled according to the history answering information got using sequential cognitive diagnosis method, obtains knowledge point Grasping level
And the examination question after partial order is limited-knowledge point correlation matrix；
Examination question-knowledge point correlation matrix according to knowledge point Grasping level and after partial order is limited is next to a certain student
Score on the knowledge point Grasping level and done examination question of individual period is predicted.
2. student's cognitive diagnosis method of a kind of sequential according to claim 1, it is characterised in that described according to acquisition
To history answering information using sequential cognitive diagnosis method be modeled including：
Initialize Efficiency analysis U, examination question-knowledge point correlation matrix V, and entity difficult parameters b, then student obtain score
It is expressed as R posterior probability：
Wherein, T is total time hop count, and N is total student's quantity, and M is total examination question number,It is student i in t-th of period examination question j
Answer result,For Grasping levels of the student i within t-th of period on each knowledge point, VjIt is examination question j with each knowledge
The degree of correlation of point, bjFor examination question j difficulty, IijRepresent whether student i was examination question j, Iij=1, which represents student i, did examination question
J, otherwise represents not do；Represent that the score R of student obeys average and isVariance
ForGaussian Profile, whereinFormal definition is as follows：
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The Q matrixes of given examination question association, | Q |=M × K：
Wherein, K is knowledge point quantity；
It is defined as follows partial ordering relation：For examination question j, if itself and knowledge point q relation are Qjq=1, the relation with knowledge point p is
Qjp=0, then it is assumed that for examination question j, knowledge point q is more more relevant than knowledge point p；
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<mi>j</mi>
<mi>q</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>1</mn>
<mi>a</mi>
<mi>n</mi>
<mi>d</mi>
<mi> </mi>
<msub>
<mi>Q</mi>
<mrow>
<mi>j</mi>
<mi>p</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>q</mi>
<msubsup>
<mo>></mo>
<mi>j</mi>
<mo>-</mo>
</msubsup>
<mi>p</mi>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>i</mi>
<mi>f</mi>
<mi> </mi>
<msub>
<mi>Q</mi>
<mrow>
<mi>j</mi>
<mi>q</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
<mi>a</mi>
<mi>n</mi>
<mi>d</mi>
<mi> </mi>
<msub>
<mi>Q</mi>
<mrow>
<mi>j</mi>
<mi>p</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>;</mo>
</mrow>
In above formula,Represent for examination question j, knowledge point q is more more relevant than knowledge point p,Represent for examination question j, knowledge
Point p is more more relevant than knowledge point q；
By given Q matrixes, training data triple is converted to：
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>D</mi>
<mi>T</mi>
</msub>
<mo>:</mo>
<mrow>
<mo>(</mo>
<mi>M</mi>
<mo>&times;</mo>
<mi>K</mi>
<mo>&times;</mo>
<mi>K</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>D</mi>
<mi>T</mi>
</msub>
<mo>:</mo>
<mo>=</mo>
<mo>{</mo>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>,</mo>
<mi>q</mi>
<mo>,</mo>
<mi>p</mi>
<mo>)</mo>
</mrow>
<mo>|</mo>
<mi>q</mi>
<msubsup>
<mo>></mo>
<mi>j</mi>
<mo>+</mo>
</msubsup>
<mi>p</mi>
<mo>}</mo>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>;</mo>
</mrow>
Model learning is made to examination question-knowledge point correlation matrix, and defined partial ordering relation can be kept, i.e. maximization examination
Topic-knowledge point correlation matrix V is in given partial ordering relationPosterior probability afterwards
<mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>V</mi>
<mo>|</mo>
<msubsup>
<mo>></mo>
<mi>j</mi>
<mo>+</mo>
</msubsup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mo>></mo>
<mi>j</mi>
<mo>+</mo>
</msubsup>
<mo>|</mo>
<mi>V</mi>
<mo>)</mo>
</mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>V</mi>
<mo>)</mo>
</mrow>
</mrow>
1
<mrow>
<munder>
<mo>&Pi;</mo>
<mrow>
<mi>j</mi>
<mo>&Element;</mo>
<mi>M</mi>
</mrow>
</munder>
<mi>p</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mo>></mo>
<mi>j</mi>
<mo>+</mo>
</msubsup>
<mo>|</mo>
<mi>V</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munder>
<mo>&Pi;</mo>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>,</mo>
<mi>q</mi>
<mo>,</mo>
<mi>p</mi>
<mo>)</mo>
<mo>&Element;</mo>
<msub>
<mi>D</mi>
<mi>T</mi>
</msub>
</mrow>
</munder>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>q</mi>
<msubsup>
<mo>></mo>
<mi>j</mi>
<mo>+</mo>
</msubsup>
<mi>p</mi>
<mo>|</mo>
<mi>V</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>q</mi>
<msubsup>
<mo>></mo>
<mi>j</mi>
<mo>+</mo>
</msubsup>
<mi>p</mi>
<mo>|</mo>
<mi>V</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow>
<mo>-</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>V</mi>
<mrow>
<mi>j</mi>
<mi>q</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>V</mi>
<mrow>
<mi>j</mi>
<mi>p</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
In above formula,For examination question-knowledge point correlation matrix V likelihood, p (V) is examination question-knowledge point correlation matrix V
Priori；Vjq、VjpDegrees of correlation of the examination question j with knowledge point q and p is represented respectively,Represent for examination question j, know
Know the point q probability more more relevant than knowledge point p；
Posterior probability to examination question-knowledge point correlation matrix V can obtain following form while taking the logarithm：
<mfenced open = "" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>I</mi>
<mi>n</mi>
<mi> </mi>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>V</mi>
<mo>|</mo>
<msubsup>
<mo>></mo>
<mi>j</mi>
<mo>+</mo>
</msubsup>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>I</mi>
<mi>n</mi>
<mi> </mi>
<mi>p</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mo>></mo>
<mi>j</mi>
<mo>+</mo>
</msubsup>
<mo>|</mo>
<mi>V</mi>
<mo>)</mo>
</mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>V</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<mi>I</mi>
<mi>n</mi>
<munder>
<mo>&Pi;</mo>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>,</mo>
<mi>q</mi>
<mo>,</mo>
<mi>p</mi>
<mo>)</mo>
<mo>&Element;</mo>
<msub>
<mi>D</mi>
<mi>T</mi>
</msub>
</mrow>
</munder>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow>
<mo>-</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>V</mi>
<mrow>
<mi>j</mi>
<mi>q</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>V</mi>
<mrow>
<mi>j</mi>
<mi>p</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
</mfrac>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>V</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<munder>
<mo>&Sigma;</mo>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>,</mo>
<mi>q</mi>
<mo>,</mo>
<mi>p</mi>
<mo>)</mo>
<mo>&Element;</mo>
<msub>
<mi>D</mi>
<mi>T</mi>
</msub>
</mrow>
</munder>
<mi>I</mi>
<mi>n</mi>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow>
<mo>-</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>V</mi>
<mrow>
<mi>j</mi>
<mi>q</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>V</mi>
<mrow>
<mi>j</mi>
<mi>p</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
</mfrac>
<mo>+</mo>
<mi>I</mi>
<mi>n</mi>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>V</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<munder>
<mo>&Sigma;</mo>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>,</mo>
<mi>q</mi>
<mo>,</mo>
<mi>p</mi>
<mo>)</mo>
<mo>&Element;</mo>
<msub>
<mi>D</mi>
<mi>T</mi>
</msub>
</mrow>
</munder>
<mi>I</mi>
<mi>n</mi>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow>
<mo>-</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>V</mi>
<mrow>
<mi>j</mi>
<mi>q</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>V</mi>
<mrow>
<mi>j</mi>
<mi>p</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
</mfrac>
<mo>-</mo>
<mi>&lambda;</mi>
<mo>|</mo>
<mo>|</mo>
<mi>V</mi>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
In above formula, λ is regularization parameter；
Add after partial ordering relation, for certain examination, the one-dimensional knowledge point Grasping level of each student can be obtained, then based on
Curve is practised from forgetting curve to catch the ability value changes of the different examinations of a certain student；
Assuming that ability value obedience averages of the student i t-th of period isVariance isGaussian Profile：
In above formula,Represent that knowledge point Grasping levels of the student i within t-th of period isProbability,It is the t period students i obtained according to being calculated after the forgetting of hypothesis and the formula of study part
To 1~K of knowledge point Grasping level,For actual Grasping level, I is indicator function, is done for identifying students ' actual situation
Which examination question, if student i has been examination question j, Iij=1.
3. student's cognitive diagnosis method of a kind of sequential according to claim 2, it is characterised in that the t periods are learned
Give birth to i is to the calculation formula of knowledge point k Grasping level：
<mrow>
<mover>
<msubsup>
<mi>U</mi>
<mrow>
<mi>i</mi>
<mi>k</mi>
</mrow>
<mi>t</mi>
</msubsup>
<mo>&OverBar;</mo>
</mover>
<mo>=</mo>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>-</mo>
<msub>
<mi>&alpha;</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<msubsup>
<mi>F</mi>
<mrow>
<mi>i</mi>
<mi>k</mi>
</mrow>
<mi>t</mi>
</msubsup>
<mo>+</mo>
<msub>
<mi>&alpha;</mi>
<mi>i</mi>
</msub>
<msubsup>
<mi>L</mi>
<mrow>
<mi>i</mi>
<mi>k</mi>
</mrow>
<mi>t</mi>
</msubsup>
<mo>;</mo>
</mrow>
Wherein, αiFor student i personalizing parameters,It is student i in the forgetting part of t-th of period,It is student i
The study part of t period.
4. student's cognitive diagnosis method of a kind of sequential according to claim 3, it is characterised in that student i is at t-th
The forgetting part of periodThat is student i, to the forgetting part of t-th of period, is expressed as from the t-1 period：
<mrow>
<msubsup>
<mi>F</mi>
<mrow>
<mi>i</mi>
<mi>k</mi>
</mrow>
<mi>t</mi>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>U</mi>
<mrow>
<mi>i</mi>
<mi>k</mi>
</mrow>
<mrow>
<mi>t</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>&CenterDot;</mo>
<msup>
<mi>e</mi>
<mrow>
<mo>(</mo>
<mo>-</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>/</mo>
<mi>S</mi>
<mo>)</mo>
</mrow>
</msup>
<mo>;</mo>
</mrow>
Wherein,Represent the t-1 period student i to knowledge point k Grasping level, e(-Δt/S)Represent the time by Δ t
Afterwards, the percentage of memory residual, S represents the intensity of memory, and Δ t is between the t-1 period to the time of t-th of period
Every.
5. student's cognitive diagnosis method of a kind of sequential according to claim 3, it is characterised in that student i is in t
Between section study partIt is expressed as：
<mrow>
<msubsup>
<mi>L</mi>
<mrow>
<mi>i</mi>
<mi>k</mi>
</mrow>
<mi>t</mi>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>U</mi>
<mrow>
<mi>i</mi>
<mi>k</mi>
</mrow>
<mrow>
<mi>t</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>&CenterDot;</mo>
<mfrac>
<mrow>
<msup>
<mi>M</mi>
<mo>&prime;</mo>
</msup>
<msubsup>
<mi>f</mi>
<mi>k</mi>
<mi>t</mi>
</msubsup>
</mrow>
<mrow>
<msubsup>
<mi>f</mi>
<mi>k</mi>
<mi>t</mi>
</msubsup>
<mo>+</mo>
<mi>r</mi>
</mrow>
</mfrac>
<mo>;</mo>
</mrow>
Wherein,Grasping levels of the t-1 period student i to knowledge point k is represented,Represent t-th of period student i
To knowledge point k exercise number of times,T-th of period is represented, influence of the number of times to learning rate is practised in knowledge point, and r is used for
The speed increased is limited, M' is used for limiting the increasing degree of maximum.
6. student's cognitive diagnosis method of a kind of sequential according to claim 3, it is characterised in that this method is also wrapped
Include：
With reference to partial ordering relation, forget, and three parts of study, obtain the final loss function E of model：
<mfenced open = "" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>E</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>t</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>T</mi>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>N</mi>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>M</mi>
</munderover>
<msub>
<mi>I</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<msup>
<mrow>
<mo>(</mo>
<mi>f</mi>
<mo>(</mo>
<msubsup>
<mi>U</mi>
<mi>i</mi>
<mi>t</mi>
</msubsup>
<mo>,</mo>
<msub>
<mi>V</mi>
<mi>j</mi>
</msub>
<mo>,</mo>
<msub>
<mi>b</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
<mo>-</mo>
<msubsup>
<mi>R</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
<mi>t</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>-</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>t</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>T</mi>
</munderover>
<munder>
<mo>&Sigma;</mo>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>,</mo>
<mi>q</mi>
<mo>,</mo>
<mi>p</mi>
<mo>)</mo>
<mo>&Element;</mo>
<msub>
<mi>D</mi>
<mi>T</mi>
</msub>
</mrow>
</munder>
<mi>I</mi>
<mi>n</mi>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow>
<mo>-</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>V</mi>
<mrow>
<mi>j</mi>
<mi>q</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>V</mi>
<mrow>
<mi>j</mi>
<mi>p</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>+</mo>
<mfrac>
<msub>
<mi>&lambda;</mi>
<mi>U</mi>
</msub>
<mn>2</mn>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>t</mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mi>T</mi>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>N</mi>
</munderover>
<mo>|</mo>
<mo>|</mo>
<mover>
<msubsup>
<mi>U</mi>
<mi>i</mi>
<mi>t</mi>
</msubsup>
<mo>&OverBar;</mo>
</mover>
<mo>-</mo>
<msubsup>
<mi>U</mi>
<mi>i</mi>
<mi>t</mi>
</msubsup>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
<mo>+</mo>
<mfrac>
<msub>
<mi>&lambda;</mi>
<mi>U</mi>
</msub>
<mn>2</mn>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>N</mi>
</munderover>
<mo>|</mo>
<mo>|</mo>
<msubsup>
<mi>U</mi>
<mi>i</mi>
<mn>1</mn>
</msubsup>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
<mo>+</mo>
<mfrac>
<msub>
<mi>&lambda;</mi>
<mi>V</mi>
</msub>
<mn>2</mn>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>M</mi>
</munderover>
<mo>|</mo>
<mo>|</mo>
<msub>
<mi>V</mi>
<mi>j</mi>
</msub>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
Wherein, λUWith λVIt is regularization parameter, IijRepresent whether student i was examination question j, Iij=1, which represents student i, did examination question
J, otherwise represents not doing the loss function that we can define according to above formula；
The more new algorithm declined using gradient, in each round iteration, is utilized respectively loss function and knowledge point Grasping level is joined
Number U, examination question-knowledge point degree of correlation parameter V, each student individuality parameter alpha and examination question difficult parameters b seek partial derivative,
And parameter renewal and solution are carried out, until model convergence, final output student grasps journey in the knowledge point within 1~T period
Degree, and limit obtained examination question-knowledge point correlation matrix by partial order.
7. student's cognitive diagnosis method of a kind of sequential according to claim 1 or 6, it is characterised in that according to knowledge
The knowledge of point Grasping level and examination question-knowledge point correlation matrix after partial order is limited to a certain student's subsequent time period
Point Grasping level and done examination question on score be predicted including：
Efficiency analysis contains Grasping levels of the T time section student i to 1~K of knowledge pointThen i couples of T+1 periods student
1~K of knowledge point Grasping levelCalculation formula be：
<mrow>
<msubsup>
<mi>U</mi>
<mi>i</mi>
<mrow>
<mi>T</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>=</mo>
<mo>{</mo>
<msubsup>
<mi>U</mi>
<mrow>
<mi>i</mi>
<mn>1</mn>
</mrow>
<mrow>
<mi>T</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>,</mo>
<msubsup>
<mi>U</mi>
<mrow>
<mi>i</mi>
<mn>2</mn>
</mrow>
<mrow>
<mi>T</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>,</mo>
<mo>...</mo>
<msubsup>
<mi>U</mi>
<mrow>
<mi>i</mi>
<mi>K</mi>
</mrow>
<mrow>
<mi>T</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>}</mo>
<mo>;</mo>
</mrow>
<mrow>
<msubsup>
<mi>U</mi>
<mrow>
<mi>i</mi>
<mi>k</mi>
</mrow>
<mrow>
<mi>T</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>&ap;</mo>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>-</mo>
<msub>
<mi>&alpha;</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<msubsup>
<mi>U</mi>
<mrow>
<mi>i</mi>
<mi>k</mi>
</mrow>
<mi>T</mi>
</msubsup>
<msup>
<mi>e</mi>
<mrow>
<mo>-</mo>
<mfrac>
<mrow>
<mi>&Delta;</mi>
<mi>T</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>S</mi>
</mfrac>
</mrow>
</msup>
<mo>+</mo>
<msub>
<mi>&alpha;</mi>
<mi>i</mi>
</msub>
<msubsup>
<mi>U</mi>
<mrow>
<mi>i</mi>
<mi>k</mi>
</mrow>
<mi>T</mi>
</msubsup>
<mfrac>
<mrow>
<msubsup>
<mi>Mf</mi>
<mi>k</mi>
<mrow>
<mi>T</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msubsup>
</mrow>
<mrow>
<msubsup>
<mi>f</mi>
<mi>k</mi>
<mrow>
<mi>T</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>+</mo>
<mi>r</mi>
</mrow>
</mfrac>
<mo>;</mo>
</mrow>
Wherein,It is the T+1 period student i to knowledge point k Grasping level, k=1,2 ..., K；Represent T
Period, student i was to knowledge point k Grasping level, αiFor student i personalizing parameters,Represent the time by Δ T+1
Afterwards, the percentage of memory residual, the time interval of the T period to the T+1 period of Δ T+1,Represent T+1
Period student i to knowledge point k exercise number of times,The T+1 period is represented, number of times is practised to study in knowledge point
The influence of rate；
The T+1 period student i score predictor formula is：
<mrow>
<msubsup>
<mi>R</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
<mrow>
<mi>T</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>&ap;</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>exp</mi>
<mo>&lsqb;</mo>
<mo>-</mo>
<msub>
<mi>V</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<msubsup>
<mi>U</mi>
<mi>i</mi>
<mi>T</mi>
</msubsup>
<mo>-</mo>
<msub>
<mi>b</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
</mrow>
</mfrac>
<mo>;</mo>
</mrow>
Wherein,For student i examination question j in the T+1 period answer result.
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