1. bookAHEAD OF PRINT
Journal Details
License
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Open Access

The Psychological Memory Forgetting Model Based on the Analysis of Linear Differential Equations

Published Online: 22 Aug 2022
Volume & Issue: AHEAD OF PRINT
Page range: -
Journal Details
License
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Introduction

The development of network technology has brought about the phenomenon of information overload. The collaborative filtering recommendation system proactively provides a list of resources that may interest users based on the historical evaluation information of certain resources. The existing collaborative filtering recommendation system equally treats historical information occurring at different moments and lacks quantitative timeliness analysis [1]. Some research results use linear or nonlinear decay functions to quantify the changes in aging over time. Some scholars use linear functions as the basis for the quantification of timeliness. Some scholars use the kernel function to calculate the decay of aging with time. Some scholars pointed out that the quantification of timeliness is a process of gradual decline over time. Although the above model satisfies the basic characteristics of aging decline over time to a certain extent, it does not consider the variability of the aging decline of resources. In the recommendation system, user interest drift is ubiquitous, which leads to dynamic changes in the degree of time-dependent decline. It shows that popular resources have a slower decay rate. Therefore, how to explore the decay characteristics of timeliness over time according to user interests is the main way to improve the practical value of timeliness quantification in the recommendation system.

Based on this, we propose a collaborative filtering recommendation model (FC-CFRM) using forgetting curves. This model reveals the differentiation law of the forgetting rate of resources from user interest drift. A multi-stage aging quantification method and a time unit mapping function are used to calculate the aging value of historical information with memory characteristics [2]. This provides a basis for rationally quantifying the timeliness of historical information and improving recommendation effects.

Recommended model

FC-CFRM can be defined as 6-tuple < U, O, R, P,δ, f >. Where U = {ui} represents the user collection. O = {oj} represents a collection of resources. R = {ri,j} represents user ui use evaluation information of resource oj. P = {pj (t, k)} represents the time-effect quantification function of different resources. pj (t, k) ∈[0,1], t represents the forgetting rate of the current time step k control aging decays with time. δ is called the lazy update factor. The larger δ is at the same stage, the smaller the adjustment of the forgetting rate by user feedback. f is the utility function [3]. It is used to calculate the user's recommended value for unknown resources. The purpose of f :U × O ×TR. FCCFRM is to find those resources with the highest recommended value to form an evaluation list L. For example: if the resource set evaluated by the user ui is O*, it is the maximum value function. Then the formal expression of the recommendation list L is uiU,L=argmaxoj{OO*}f(ui,Oj,pj) \forall {u_i} \in U,L = \arg \mathop {\max}\limits_{{o_j} \in \left\{{O - {O^*}} \right\}} \,f\left({{u_i},\,{O_j},{p_j}} \right)

The specific implementation process of FC-CFRM is divided into two stages: similarity calculation and evaluation prediction.

Similarity calculation

According to user resource historical evaluation information, similarity calculation is used to determine the degree of similarity between different resources [4]. This provides a basis for subsequent evaluation and prediction. Common similarity calculation methods include the Euclidean distance method, conditional probability method, and Pearson method.

Euclidean distance method

We regard resources as points in the subspace of user evaluation manifolds. The distance between points determines the degree of similarity between resources. The greater the distance between points, the lower the similarity. If let U(a,b) denote the set of users participating in the evaluation of resources oa and ob at the same time, the calculation process of the similar Euclidean distance method is expressed as sim(Oa,Ob)=11+iU(a,b)(riarib)2 sim\left({{O_a},{O_b}} \right) = {1 \over {1 + \sum\limits_{i \in {U_{\left({a,b} \right)}}} {{{\left({{r_{ia}} - {r_{ib}}} \right)}^2}}}}

Conditional probability method

We regard the degree of resource similarity as the probability p(oa | ob) that a certain resource is evaluated by the specified user at the same time that another resource is also evaluated by the user at the same time [5]. If Freq(oa) and Freq(oa, ob) respectively denote the number of users who evaluate resource oa and the number of users who evaluate resources oa and ob at the same time, then the calculation process of the conditional probability method is expressed as sim(oa,ob)=p(oa|ob)Freq(oa)=Freq(oa,ob)×|U|Freq(oa)×Freq(ob)×|U|=Freq(oa,ob)Freq(oa)×Freq(ob) sim\left({{o_a},{o_b}} \right) = {{p\left({{o_a}|{o_b}} \right)} \over {Freq\left({{o_a}} \right)}} = {{Freq\left({{o_a},{o_b}} \right) \times \left| U \right|} \over {Freq\left({{o_a}} \right) \times Freq\left({{o_b}} \right) \times \left| U \right|}} = {{Freq\left({{o_a},{o_b}} \right)} \over {Freq\left({{o_a}} \right) \times Freq\left({{o_b}} \right)}}

Pearson Method

Pearson's method uses the idea of mathematical statistics. We regard resource evaluation as the sampling result of repeated experiments in the same sample space. If r*,a¯ \overline {{r_{*,a}}} and r*,b¯ \overline {{r_{*,b}}} denote the average user ratings of resources oa and ob, respectively, and U(a,b) defines the same Euclidean distance method, the calculation process of Pearson's method is expressed as sim(oa,ob)=iU(a,b)(ri,ar*,a¯)(ri,br*,b¯)iU(a,b)(ri,ar*,a¯)2iU(a,b)(ri,br*,b¯)2 sim\left({{o_a},\,{o_b}} \right) = {{\sum\limits_{i \in {U_{\left({a,b} \right)}}} {\left({{r_{i,a}} - \overline {{r_{*,a}}}} \right)\left({{r_{i,b}} - \overline {{r_{*,b}}}} \right)}} \over {\sqrt {{{\sum\limits_{i \in {U_{\left({a,b} \right)}}} {\left({{r_{i,a}} - \overline {{r_{*,a}}}} \right)}}^2}} \sqrt {\sum\limits_{i \in {U_{\left({a,b} \right)}}} {{{\left({{r_{i,b}} - \overline {{r_{*,b}}}} \right)}^2}}}}}

Evaluation and prediction

According to the similarity calculation result, the evaluation value aggregation operation is performed on all resources similar to the unknown evaluation resource oj to obtain the evaluation prediction value of the designated user ui to oj. We take into account the time-effect characteristics of resource historical evaluation information and introduce the time-effect quantification function into the evaluation prediction method as ri,j=ri,*¯+iO'(ri,ar*,a¯)×Sim(oj,oa)×pa(t,k)iO'Sim(Oj,Oa)×pa(t,k) {r_{i,j}} = \overline {{r_{i,*}}} + {{\sum\limits_{i \in O'} {\left({{r_{i,a}} - \overline {{r_{*,a}}}} \right) \times Sim\left({{o_j},{o_a}} \right) \times {p_a}\left({t,k} \right)}} \over {\sum\limits_{i \in O'} {Sim\left({{O_j},{O_a}} \right)} \times {p_a}\left({t,k} \right)}}

O′ represents a collection of resources that the user ui has evaluated and is similar to oj. ri,*¯ \overline {{r_{i,*}}} represents the average value of the historical evaluation of these | O′ | resources. pa (t, k) Quantifies the timeliness value of historical evaluation of oa according to different time steps t and control parameters k. pa(t, k) is the research focus of the model.

Quantification method of resource timeliness

The German psychologist Ebbinghaus used nonsense syllables as materials and used the method of relearning to reveal the characteristics of memory aging in the forgetting process over time (Figure 1). The evaluation information of the designated resource in the recommendation system reflects the user's subjective recognition of the resource. This degree of subjective acceptance will inevitably decay over time, similar to the forgetting process. Different memory sources have different levels of forgetting attenuation, and memory materials that users are interested in are more likely to be remembered [6]. This is the same as the requirement that different resources in the recommendation system have differences in time decay. Based on this, we use the forgetting curve describing memory characteristics as the basis for measuring the time-effect change of resources.

Figure 1

Forgetting process forgetting curve

Single-stage aging method

We regard the current aging body as an information storage room. The initial timeliness is used as the input of the room, and the retention of timeliness based on the historical evaluation information of the forgetting feature resources is a function pa (t, k) of the time step t. Where k is used to control the forgetting rate of aging [7]. We convert the dynamic process into a differential equation form, which is expressed as dpa(t,k)/dt=kpa(t,k) {dp}_a\left({t,k} \right)/dt = - {kp}_a\left({t,k} \right)

Solve the original function pa (t, k) of equation (6), which is expressed as pa(t,k)=p0ekt {p_a}\left({t,k} \right) = {p_0}{e^{- kt}}

The forgetting curve indicated by t ∈ (0, ∞) should take the first quadrant of the coordinate system. p0 is the intercept of the curve on the y axis, the initial value of the immediate effect. Since the initial memory aging values of the tested materials begin to decline from 100%, p0 = 1 is taken. The single-stage aging quantification method of resource oa is based on equation (8) and obtains the aging value of resource historical evaluation information according to the current time step t and the forgetting rate k. pa(t,k)=ekt {p_a}\left({t,\,k} \right) = {e^{- kt}}

The forgetting rate k is the main parameter reflecting the attenuation difference of the resource forgetting curve [8]. We will establish a time-effective multi-stage quantification method based on user interest drift.

Multi-stage aging method

In this section, the method of determining control parameters is understood as the adjustment process of the forgetting curve in multiple stages based on the phenomenon of memory enhancement in memory psychology.

Memory enhancement refers to the memory step (recovery of complete memory) after learning and a new forgetting process with a lower forgetting rate. If let t1,t2 and t3 respectively denote the time step at which 3 relearning occurs, the forgetting curve adjustment process is shown in Figure 2.

Figure 2

Multi-stage forgetting curve adjustment

Figure 2 can be regarded as a dynamic adjustment process of forgetting rate caused by changes in user interest. If tn−1 and tn are any adjacent time steps where the memory enhancement phenomenon occurs on the forgetting curve, and kn−1 is the forgetting rate of the forgetting curve from tn−1 to tn, then the resource tn represents the time-dependent quantitative function of the forgetting stage. pa(t,kn1)=ekn1(ttn1),t(tn1,tn) {p_a}\left({t,{k_{n - 1}}} \right) = {e^{- {k_{n - 1}}\left({t - {t_{n - 1}}} \right)}},t\, \in \left({{t_{n - 1}},{t_n}} \right)

We need to further determine kn−1 learning strategy. We shift the new forgetting curve after adjusting the forgetting rate to the x axis in reverse to make it have a common starting point with the original forgetting curve. We let the forgetting rates before and after the adjustment be kn−1 and kn respectively, and the above operation process is shown in Figure 3.

Figure 3

Analysis of aging forgetting rate adjustment

The rate of forgetting on the curve is adjusted based on user feedback. If β is the decay difference between the old and the new forgetting curve at the time step tn, then according to equation (9), the value of β is expressed as β=ekn(tntn1)ekn1(tntn1) \beta = {e^{- {k_n}\left({{t_n} - {t_{n - 1}}} \right)}} - {e^{- {k_{n - 1}}\left({{t_n} - {t_{n - 1}}} \right)}}

The numerical relationship between kn−1 and kn is established by β, which is expressed as kn=ln(ekn1(tntn1))(tntn1) {k_n} = {{\ln \left({{e^{- {k_{n - 1}}\left({{t_n} - {t_{n - 1}}} \right)}}} \right)} \over {- \left({{t_n} - {t_{n - 1}}} \right)}}

(1 – pa (tn, kn−1)) is the upper limit of the value of β taken by the forgetting curve at the time step tn. We divide (1 – pa (tn, kn−1)) into δ line segments. The quantitative relationship between the degree of adjustment β and the degree of maintenance of timeliness is expressed as β=1pa(tn,kn1)δ \beta = {{1 - {p_a}\,\left({{t_n},{k_{n - 1}}} \right)} \over \delta}

In the recommendation system, δ can be set to a positive integer greater than 1. We substitute equation (12) into equation (10) and eliminate β to obtain the calculation formula of kn : kn=ln(1+(δ1)ekn1(tntn1)lnδtn1tn {k_n} = {{\ln \left({1 + \left({\delta - 1} \right){e^{- {k_{n - 1}}\,\left({{t_n} - {t_{n - 1}}} \right)}} - \ln \delta} \right.} \over {{t_{n - 1}} - {t_n}}}

The adjustment process of the forgetting rate presents a recursive relationship. If the initial forgetting rate is k0, combining equations (9) and (13) can determine the time-effect quantification function of the resource history evaluation information at any time in the multi-stage forgetting process. This satisfies the relevant requirements in the evaluation and forecasting process.

Time unit mapping function

When pa (t, k) ∈ (0,1], f needs to meet the one-to-one mapping from system time T :[0, ∞) to time step t :[0, ∞), which is expressed as T:[0,)ft:[0,) T:\left[ {0,\,\infty} \right)\mathop {ft}\limits_ \to :\,\left[ {0,\,\infty} \right)

1) When pa (tϑ, k) = 0.1, it indicates that the evaluation record at the current time step has lost its recommendation effect. If the timeliness continues to decline, there is no practical significance to the process of quantifying the timeliness of the record. Based on this, the critical time step t is introduced in formula (3), which is expressed as pa(t,k)={ekt0t<tϑ;0.1,tϑt {p_a}\left({t,\,k} \right) = \left\{{\matrix{{{e^{- kt}}} & {0 \le t < {t_\vartheta};} \cr {0.1,} & {{t_\vartheta} \le t} \cr}} \right.

If Tϑ is used to represent the critical system time corresponding to tϑ, then combined with equation (15), equation (14) can be further expressed as f:{[0,Tϑ)[0,tϑ),pa(t,k)[0.1,1);[Tϑ,)tϑ,pa(t,k)=0.1. f:\,\,\left\{{\matrix{{\left[ {0,\,\,{T_\vartheta}} \right) \to \,\left[ {0,\,\,{t_\vartheta}} \right),} & {{p_a}\left({t,k} \right) \in \left[ {0.1,\,1} \right);} \cr {\left[ {\,{T_\vartheta},\,\infty \,} \right) \to \,{t_\vartheta},} & {{p_a}\left({t,k} \right) = 0.1.} \cr}} \right.

According to formula (16), the time unit mapping function of the single-stage aging quantification method can be expressed as f:t={ln(10)kTϑ0T<Tϑ;ln(10)kTϑT f:t = \left\{{\matrix{{{{\ln \left({10} \right)} \over {k{T_\vartheta}}}} & {0 \le T < {T_\vartheta};} \cr {{{\ln \left({10} \right)} \over k}} & {{T_\vartheta} \le T} \cr}} \right.

2) We substitute the forgetting rate km into Eq. (15) to find tϑm t_\vartheta ^m . We use the determined function form of equation (17) to find the Tϑm T_\vartheta ^m corresponding to tϑm t_\vartheta ^m under the forgetting rate km : Tϑm=k0Tϑ0(km)1 T_\vartheta ^m = {k_0}T_\vartheta ^0\,{\left({{k_m}} \right)^{- 1}}

Tϑ0 T_\vartheta ^0 is the critical system time corresponding to the initial forgetting rate. Use Tϑ0 T_\vartheta ^0 as the segmentation condition to obtain the time unit mapping function suitable for the multi-stage aging quantification method: f:t={ln(10)kmTϑmΔT,0ΔT<(km)1Tϑ0k0;ln(10)km,(km)1Tϑ0k0ΔT. f:t = \left\{{\matrix{{{{\ln \left({10} \right)} \over {{k_m}T_\vartheta ^m}}\Delta T,\,0 \le \Delta T < {{\left({{k_m}} \right)}^{- 1}}T_\vartheta ^0{k_0};} \hfill \cr {{{\ln \left({10} \right)} \over {{k_m}}},{{\left({{k_m}} \right)}^{- 1}}T_\vartheta ^0{k_0} \le \Delta T.} \hfill \cr}} \right.

k0 represents the initial forgetting rate. ΔT represents the system time increment between the current system time and the last user feedback.

Simulation analysis
Data preparation

We recommend the classic data set MovieLens in the system for simulation analysis. MovieLens comprises more than 10,000,000 evaluations made by 71567 users on 10681 different movies on the entire network. Each evaluation record includes user identification, movie identification, evaluation value, and time stamp. We take the first 10,000 time-adjacent resource historical evaluation information as the initial simulation data set in the original data set. The initial experimental data set is divided into training and test sets by cross-validation, and the percentage is divided. The training set is used to obtain the forgetting rate of resource historical evaluation information. The test set will be evaluated and predicted based on the time-effect analysis results obtained from the training.

Obtain the time-sensitive forgetting rate of the resource according to the resource identifier in the test record. Calculate the aging value based on the system time difference to make recommendations. After the recommendation is over, the forgetting rate is adjusted for the resource of the record feedback. At the same time, the test records after evaluation and prediction are moved to the end of the training set. This provides a basis for calculating the similarity between resources in the subsequent recommendation process.

The average absolute error (MAE) is used to measure the recommended effect of FC-CFPM in different experiments, and its calculation method is shown in equation (20). b(n) represents the predicted value calculated by the recommendation system. bp(n) represents the true evaluation value obtained based on user feedback. MAE reflects the average difference between the recommended predicted value and the actual user feedback value. Obviously, the smaller the value of MAE, the better the evaluation and prediction effect of the recommendation system. MAE=n=1N|b(n)bp(n)|N MAE = {{\sum\limits_{n = 1}^N {\left| {b\left(n \right) - {b^p}\left(n \right)} \right|}} \over N}

Parameter setting

The specific setting method of the initial critical system time Tϑ0 T_\vartheta ^0 of the resource and the corresponding initial forgetting rate k0 is shown in Table 1.

Parameter settings

Parameter name Parameter meaning Value
Tϑ0 T_\vartheta ^0 Initial critical system time 6E(ΔT)
k0 Initial forgetting rate 1
Value analysis of inertia factor δ

The laziness factor δ is used to control the degree of forgetting rate adjustment. The larger δ is, the less obvious the update effect of the time-sensitive forgetting rate of the user evaluation feedback on the resource. The value of the laziness factor δ cannot be the same as other system parameters. It is obtained by calculating the relevant statistical characteristics on the training set and running FC-CFRM based on the laziness factor δ taking 10, 20, 50, and 100. δ reasonable value determined according to the results of MAE. The simulation result is shown in Figure 4.

Figure 4

The value of δ effects the recommendation effect

Each curve in Figure 4 represents the 10-time segmented MAE sampling of FC-CFRM under different δ values. The segment length is 200. The MAE value of δ=10 is comparable to other experimental results on the 1200 to 1400 test records (MAE=0.84817). All other data segments have higher MAE values. It is manifested as a worse recommendation effect.

When δ = 20, 50, 100, the maximum difference of MAE under different data segments is 0.0148. The minimum value on the 1600 to 1800 records located at δ = 20 and = 100 is 0.09. The average value on the 800th to 1000th records of the curve with 20 and 50 is 0.000312. The MAE curve of δ=20, 50, 100 adopts a linear fitting method for similarity analysis, and the fitted straight lines are consistent.

Comparative analysis of recommendation effects

The comparative analysis of the recommendation effect measures the recommendation effect of different models on the test set. The classic collaborative filtering recommendation model that lacks consideration of the timeliness of resource historical evaluation information is expressed as (N-CFRM). The collaborative filtering recommendation model with simple index k=1 and time-effect quantification is expressed as (E-CFRM). The collaborative filtering recommendation model based on the quantification of the mechanical forgetting curve is expressed as (F-CFRM). The collaborative filtering recommendation model based on the forgetting curve is expressed as (FC-CFRM). Recommendation effect analysis uses the formula (21) to calculate the degree of MAE difference between other recommendation models *-CFRM and N-CFRM. If the value is negative, the recommendation effect is worse than N-CFRM. On the contrary, if the value is positive and larger, the recommendation effect is better. The experimental results are shown in Figure 5. Div=(MAENCFRMMAE*CFRM)/MAE*CFRM Div = \left({MA{E_{N - CFRM}} - MA{E_{* - CFRM}}} \right)/MA{E_{* - CFRM}}

Figure 5

Comparative analysis of recommendation effects

Each segment in Figure 5 contains the degree of MAE difference between the three recommended models and N-CFRM. The segment length is 200. In general, the recommendation error of all recommendation models shows a fluctuating state on different segmented data sets. Among them, E-CFRM has 7 positive segments and 3 negative segments. F-CFRM has 8 positive segments and 2 negative segments. FC-CFRM are all positive segments. Table 2 shows the statistical characteristics of MAE for each recommendation model, indicating its specific recommendation effect.

MAE statistical analysis

Recommended method Max Minimum variance average value
N-CFRM 0.88285 0.6601 0.005915 0.784458
F-CFRM 0.89202 0.64958 0.006362 0.773788
FC-CFRM 0.85071 0.62898 0.005661 0.754041
E-CFRM 0.89084 0.6601 0.005927 0.785858

The average recommendation effect ranking of each recommended method is E-CFRM<N-CFRM<F-CFRM<FC-CFRM. The order of stability is F-CFRM < E-CFRM < N-CFRM < FC-CFRM

Since E-CFRM adopts an artificial method to set the forgetting rate of the exponential function, it ignores the influence of user interest on the time-dependent quantification of different resources. Therefore, its performance is worse than N-CFRM in terms of recommendation effect and recommendation stability.

From the experimental results of F-CFRM, it can be seen that adopting the forgetting curve can better fit the change of resource aging with time. Its performance is that the average recommendation effect is higher than that of N-CFRM. Still, it ignores the characteristics of the multi-stage forgetting process in the quantification of resource timeliness. The recommendation effect of F-CFRM will fluctuate greatly when user interests change. FC-CFRM can adopt a useful historical information time-sensitive quantitative model to track changes in user interests.

Conclusion

We propose a collaborative filtering recommendation model using forgetting curves based on the temporal characteristics of historical evaluation. The experimental results show that the model reasonably reveals the changing law of forgetting rate with user interest and provides stable and high-quality recommendation effects.

Figure 1

Forgetting process forgetting curve
Forgetting process forgetting curve

Figure 2

Multi-stage forgetting curve adjustment
Multi-stage forgetting curve adjustment

Figure 3

Analysis of aging forgetting rate adjustment
Analysis of aging forgetting rate adjustment

Figure 4

The value of δ effects the recommendation effect
The value of δ effects the recommendation effect

Figure 5

Comparative analysis of recommendation effects
Comparative analysis of recommendation effects

Parameter settings

Parameter name Parameter meaning Value
Tϑ0 T_\vartheta ^0 Initial critical system time 6E(ΔT)
k0 Initial forgetting rate 1

MAE statistical analysis

Recommended method Max Minimum variance average value
N-CFRM 0.88285 0.6601 0.005915 0.784458
F-CFRM 0.89202 0.64958 0.006362 0.773788
FC-CFRM 0.85071 0.62898 0.005661 0.754041
E-CFRM 0.89084 0.6601 0.005927 0.785858

Yang, F., Liu, B., Zhao, L., & Peng, X. Recognition of the Purchasing Intentions of WeChat Users Based on Forgetting Curve. Rev. d’Intelligence Artif., 2019;33(1):61–65 YangF. LiuB. ZhaoL. PengX. Recognition of the Purchasing Intentions of WeChat Users Based on Forgetting Curve Rev. d’Intelligence Artif. 2019 33 1 61 65 10.18280/ria.330111 Search in Google Scholar

Ahmedov, S. A., & Yuldashev, H. D. ON ABATTLE OVER THE ‘EBBINGHAUS FORGETTING CURVE’USING CONTROL CHARTS. Scientific Bulletin. Physical and Mathematical Research., 2020; 2(1):100–103 AhmedovS. A. YuldashevH. D. ON ABATTLE OVER THE ‘EBBINGHAUS FORGETTING CURVE’USING CONTROL CHARTS Scientific Bulletin. Physical and Mathematical Research. 2020 2 1 100 103 Search in Google Scholar

Touchent, K., Hammouch, Z. & Mekkaoui, T. A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives. Applied Mathematics and Nonlinear Sciences., 2020; 5(2): 35–48 TouchentK. HammouchZ. MekkaouiT. A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives Applied Mathematics and Nonlinear Sciences 2020 5 2 35 48 10.2478/amns.2020.2.00012 Search in Google Scholar

Yokuş, A. & Gülbahar, S. Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation. Applied Mathematics and Nonlinear Sciences., 2019; 4(1): 35–42 YokuşA. GülbaharS. Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation Applied Mathematics and Nonlinear Sciences 2019 4 1 35 42 10.2478/AMNS.2019.1.00004 Search in Google Scholar

Ferreira, J. Â., Kreling, J. P. D., & Ozório, A. K. Learning e Forgetting Curve Theories, aplicadas ao Planejamento e Programação da Produção. Brazilian Journal of Development., 2020; 6(12):94914–94928 FerreiraJ. Â. KrelingJ. P. D. OzórioA. K. Learning e Forgetting Curve Theories, aplicadas ao Planejamento e Programação da Produção Brazilian Journal of Development 2020 6 12 94914 94928 10.34117/bjdv6n12-105 Search in Google Scholar

Zhao, L., Chen, L., Liu, Q., Zhang, M., & Copland, H. Artificial intelligence-based platform for online teaching management systems. Journal of Intelligent & Fuzzy Systems., 2019;37(1):45–51 ZhaoL. ChenL. LiuQ. ZhangM. CoplandH. Artificial intelligence-based platform for online teaching management systems Journal of Intelligent & Fuzzy Systems 2019 37 1 45 51 10.3233/JIFS-179062 Search in Google Scholar

Guo, Y., Ji, J., Ji, J., Gong, D., Cheng, J., & Shen, X. Firework-based software project scheduling method considering the learning and forgetting effect. Soft computing., 2019; 23(13):5019–5034 GuoY. JiJ. JiJ. GongD. ChengJ. ShenX. Firework-based software project scheduling method considering the learning and forgetting effect Soft computing 2019 23 13 5019 5034 10.1007/s00500-018-3165-2 Search in Google Scholar

Boutis, K., Pecaric, M., Carrière, B., Stimec, J., Willan, A., Chan, J., & Pusic, M. The effect of testing and feedback on the forgetting curves for radiograph interpretation skills. Medical teacher., 2019; 41(7):756–764 BoutisK. PecaricM. CarrièreB. StimecJ. WillanA. ChanJ. PusicM. The effect of testing and feedback on the forgetting curves for radiograph interpretation skills Medical teacher 2019 41 7 756 764 10.1080/0142159X.2019.157009831046500 Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo