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SuperMemo 2 was great. Its simple algorithm has survived in various mutations to this day in popular apps such as Anki or Mnemosyne. However, the algorithm was dumb in the sense that there was no way of modifying the function of optimum intervals. The findings of 1985 were set in stone. Memory complexity and stability increase were expressed with the same single number: E-factor. It is a bit like using a single lever in a bike to change gears, and the direction of driving.
SuperMemo 2 很棒。它的简单算法已经在流行的应用程序（如Anki或Mnemosyne）中存活至今。然而，该算法是愚蠢的，因为其没有办法修改最优间隔的函数。1985 的发现是板上钉钉的。记忆复杂度和稳定性增加用相同的数字表示：E-factor。这有点像在自行车上使用同一个控制杆来改变档位和方向。
Individual items could adapt the spacing of review by changes to their estimated difficulty. Those changes could compensate for errors in the function of optimum intervals. Even if the algorithm was slow to converge on the optimum, in theory, it was convergent. The main flaw was that, in Algorithm SM-2, new items would not benefit from the experience of old items.
单个的项目可以根据其估计的难度来调整复习的间隔。这些变化可以补偿最优间隔函数中的错误。即使算法收敛速度慢，理论上也是收敛的。主要的缺陷是，在算法 SM-2 中，新项目不会从旧项目的经验中受益。
Algorithm SM-2 is not adaptable. New items do not benefit from the experience of old items
Algorithm SM-4 was the first attempt to arm SuperMemo with universal adaptability. It was completed in February 1989. In the end, adaptability was too slow to show up, but inspiration gathered with Algorithm SM-4 was essential for further progress, esp. in understanding the problem of stability-vs-accuracy in spaced repetition. In short, Algorithm SM-4 was too stable to be accurate. This was quickly remedied in Algorithm SM-5 just 7 months later. Here is an excerpt from my Master's Thesis to explain the details:
算法 SM-4 是第一次尝试用通用的适应性武装 SuperMemo。它于 1989 年 2 月竣工。最后，适应性表现得太慢了，但是用算法 SM-4 收集的灵感对于进一步发展是至关重要的，特别是在理解间隔重复中的稳定性与准确性问题上。简而言之，SM-4 算法太过稳定而不能准确。这在 7 个月后的 SM-5 算法中很快得到了纠正。以下是我的[硕士论文](https://supermemo.guru/wiki/Master's Thesis)的一段摘录。
Archive warning: Why use literal archives?
档案警告:为什么使用文字档案?
This text was part of: "Optimization of learning" by Piotr Wozniak (1990)
本文是 Piotr Wozniak 的《优化学习》(1990)的一部分。
The main fault of Algorithm SM-2 seems to have been the arbitrary shape of the function of optimal intervals. Although very effective in practice and confirmed by years of experimental repetitions, this function could not claim scientifically proved validity, nor could it detect the overall impact of few day variations of optimal intervals on the learning process. Bearing these flaws in mind I decided to employ routine SuperMemo repetitions in the validation of the function of optimal intervals!
算法 SM-2 的主要错误似乎是最优间隔函数的任意形状。虽然在实践中非常有效，并且经过多年的实验重复得到了证实，但是这个函数并不能被科学地证明其有效性，也不能检测出最优间隔的几天变化对学习过程的整体影响。考虑到这些缺陷，我决定使用常规的 SuperMemo 重复来验证最优间隔的作用!
Using optimization procedures like those applied in finding E-Factors I wanted the program to correct the initially proposed function whenever corrections appeared justified.
使用那些在寻找E因子时应用的优化过程，我想让程序在任何时候修正最初提出的函数，只要修正看起来是合理的。
To achieve this goal, I tabulated the function of optimal intervals.
为了达到这个目的，我把最优间隔的函数列成表。
Figure: Matrix of optimal intervals showed up in SuperMemo 4 in 1989, and survived to this day in SuperMemo 17 with few changes. In SuperMemo 4, it was the source of information for optimum spacing. In SuperMemo 17, it is derived from the OF matrix generated by Algorithm SM-15 that is used concurrently with Algorithm SM-17. The picture presents a matrix from SuperMemo 5, and shows a significant departure from the original values of the matrix. In SuperMemo 4, adaptations proceeded at much slower pace
**图片：**最优间隔矩阵出现在1989年的SuperMemo 4中，并在SuperMemo 17中保留至今，几乎没有变化。在SuperMemo 4中，它是优化间距的信息来源。在SuperMemo 17中，它是由算法SM-15生成的矩阵得到的，同时与算法SM-17使用。这幅图展示了一个来自SuperMemo 5的矩阵，并显示了与原始矩阵值的显著偏离。在SuperMemo 4中，改编的速度要慢得多
Particular entries of the matrix of optimal intervals (later called the OI matrix) were initially taken from the formulas used in Algorithm SM-2.
最优间隔矩阵(后来称为OI矩阵)的初始项最初取自算法SM-2中的公式。
SuperMemo 4 (February 1989), in which the new solution was implemented, used the OI matrix to determine values of inter-repetition intervals:
SuperMemo 4(1989年2月)实现了新的解决方案，使用OI矩阵来确定重复间隔的值:
I(n):=OI(n,EF)
where:
- I(n) - the n-th inter-repetition interval of a given item (in days),
- EF - E-Factor of the item,
- OI(n,EF) - the entry of the OI matrix corresponding to the n-th repetition and the E-Factor EF.
上式中：
- I(n) - 给定项目的第n次重复间隔(以天为单位)，
- EF - 项目的E-Factor，
- OI(n,EF) - OI矩阵中对应第 n 次重复和 EF 的项目。
However, the OI matrix was not fixed once for all. In the course of repetitions, particular entries of the matrix were increased or decreased depending on the grades. For example, if the entry indicated the optimal interval to be X and the used interval was X+Y while the grade after this interval was not lower than four, then the new value of the entry would fall between X and X+Y.
然而，OI 矩阵并不是固定的。在重复的过程中，矩阵的具体项随着分数的不同而增加或减少。例如，如果项目指出最优的间隔是 X，使用的间隔是 X+Y，而这个间隔之后的等级不低于 4，那么项目的新值将落在 X 和 X+Y 之间。
Thus the values of the OI entries in the equilibrium state should settle at the point where the stream of poor-retention items balances the stream of good-retention items in its influence on the matrix.
因此，处于平衡状态的 OI 项的值应该稳定在一个点上，在这个点上，低保留项流与高保留项流对矩阵的影响相平衡。
As a consequence, SuperMemo 4 was intended to yield an ultimate definition of the function of optimal intervals.
因此，SuperMemo 4 试图给出最优间隔函数的最终定义。
It did not take me long to realize that the verification-correction cycle in the new algorithm was too long. It was not much different than running the 1985 eperiment on the computer. To determine decade-long intervals, I needed a decade to pass to test the outcomes of the review. This led to Algorithm SM-5 seven months later. Here is the problem with Algorithm SM-4 as described in my Master's Thesis:
我很快就意识到新算法的验证-校正周期太长了。这与在计算机上运行 1985 年实验没有太大的不同。为了确定十年的间隔，我需要十年的时间来测试审查的结果。这导致了七个月后的 算法 SM-5。以下是我在[硕士论文](https://supermemo.guru/wiki/Master's _thesis)中描述的算法 SM-4 的问题:
Archive warning: Why use literal archives?
档案警告:为什么使用文字档案?
This text was part of: "Optimization of learning" by Piotr Wozniak (1990)
本文是 Piotr Wozniak 的《优化学习》(1990)的一部分。
Algorithm SM-4 was implemented in SuperMemo 4 and was used between March 9, 1989 and October 17, 1989. Although the main concept of modifying the function of optimal intervals seemed to be a major step forward, the implementation of the algorithm was a failure. The basic insufficiency of the algorithm appeared to result from formulas applied in modification of the OI matrix.
算法 SM-4 在 SuperMemo 4 中实现，并在 1989 年 3 月9 日至 1989 年 10 月 17 日期间使用。虽然修改最优间隔函数的主要概念似乎是向前迈出的一大步，但是算法的实现是失败的。该算法的基本不足似乎源于在修改OI矩阵时应用的公式。
There were two the most striking flaws:
- modifications were too subtle to rearrange the OI matrix visibly in a reasonably short time,
- for longer inter-repetition intervals, the effect of modification had to wait very long before being steadily fixed, i.e. it took quite a lot of time before the result of a modification of a few-month-long interval could be seen and corrected if necessary
有两个最明显的缺陷:
- 修改太过细微，无法在合理的短时间内明显地重新排列 OI 矩阵，
- 对于较长的间隔重复时间，修改的效果需要很长时间才能稳定固定，即需要很长时间才能看到几个月间隔修改的结果，并在必要时进行修正
After seven months of using Algorithm SM-4, the OI matrices of particular databases did not look much different from their initial states. One could explain this fact by the correctness of my earlier predictions concerning the real values of the optimal inter-repetition intervals, however, as it was later proved by means of Algorithm SM-5, the actual reason of the stability of the matrices was the flaws in the optimization formulae.
在使用 SM-4 算法 7 个月后，特定数据库的 OI 矩阵与它们的初始状态并没有太大的不同。人们可以用我早先关于最优重复间隔真值的预测的正确性来解释这一事实，然而，正如后来用算法SM-5证明的那样，矩阵稳定的实际原因是优化公式中的缺陷。
As far as acquisition rate and retention are concerned, there is no reliable evidence that Algorithm SM-4 brought any progress. Slight improvement could as well be related to general betterment of the software and improvement in item formulation principles
就获取率和保留率而言，没有可靠的证据表明 SM-4 算法带来了任何进展。轻微的改进也可能与软件的总体改进和项目制定原则的改进有关。
Interestingly, you can still see the matrix of optimum intervals in newer versions of SuperMemo. The matrix is not used by the algorithm, however, it is displayed in SuperMemo statistics as it informs the user about the impact of complexity on the prospects of items in the learning process.
有趣的是，你仍然可以在新版本的 SuperMemo 中看到最优间隔矩阵。这个矩阵并没有被算法使用，但是它显示在SuperMemo 统计数据中，因为它告诉用户复杂性对学习过程中项目的前景的影响。
If you compare a matrix produced by SuperMemo 5 in 8 months of use, you will notice significant similarity to the matrix produced in two decades of using Algorithm SM-8:
如果你比较一个使用了 8 个月的 SuperMemo 5 生成的矩阵，你会发现它与使用了 20 年的算法 SM-8 生成的矩阵有显著的相似性：
Figure: Matrix of optimum intervals is no longer used in Algorithm SM-17. However, it can still be generated with procedures of Algorithm SM-15. The columns correspond with easiness of the material expressed as A-Factor. The rows correspond with memory stability expressed as repetition category
**图片：**算法 SM-17 不再使用最优间隔矩阵。但是，仍然可以通过算法 SM-15 的步骤来生成。这些列与以A因子表示的材料的简易度相一致。这些行对应于以重复类别表示的记忆稳定性
Here is the outline of Algorithm SM-4 as described in my Master's Thesis:
以下是我在[硕士论文](https://supermemo.guru/wiki/Master's _thesis)中描述的算法SM-4的概要:：
Archive warning: Why use literal archives?
档案警告:为什么使用文字档案?
This text was part of: "Optimization of learning" by Piotr Wozniak (1990)
本文是 Piotr Wozniak(1990) 的《优化学习》的一部分。
Algorithm SM-4 used in SuperMemo 4.0:
- Split the knowledge into smallest possible items
- With all items associate an E-Factor equal to 2.5
- Tabulate the OI matrix for various repetition numbers and E-Factor categories
- Use the following repetition spacing to obtain the initial OI matrix:
- OI(1,EF)=1
- OI(2,EF)=6
- for n>2 OI(n,EF)=OI(n-1,EF)*EF
- OI(n,EF) - optimal inter-repetition interval after the n-th repetition (in days) for items with E-Factor equal EF,
- Use the OI matrix to determine inter-repetition intervals:
- I(n, EF) = OI(n, EF)
- I(n,EF) - the n-th inter-repetition interval for an item whose E-Factor equals EF (in days),
- OI(n,EF) - the entry of the OI matrix corresponding to the n-th repetition and the E-Factor EF
- After each repetition estimate the quality of the repetition response in the 0-5 grade scale (see Algorithm SM-2).
- After each repetition modify the E-Factor of the recently repeated item according to the formula:
$EF'=EF+(0.1-(5-q)\times(0.08+(5-q)\times0.02))$
- EF' - new value of the E-Factor,
- EF - old value of the E-Factor,
- q - quality of the response in the 0-5 grade scale.
- If EF is less than 1.3 then let EF be 1.3.
- After each repetition modify the relevant entry of the OI matrix.
- An exemplary formula could look as follows (the actual formula used in SuperMemo 4 was more intricate):
$OI'=interval+\cfrac{interval\times (1-\cfrac{1}{EF})}{2}\times(0.25\times q-1)$ $OI''=(1-fraction)\times OI+fraction\times OI'$
- OI - new value of the OI entry,
- OI' - auxiliary value of the OI entry used in calculations,
- OI - old value of the OI entry,
- interval - interval used before the considered repetition (i.e. the last used interval for the given item),
- fraction - any number between 0 and 1 (the greater it is the faster the changes of the OI matrix),
- EF - E-Factor of the repeated item,
- q - quality of the response in the 0-5 grade scale.
- Note that for q=4 the OI does not change and that for q=5 the OI increases 4 times less than it decreases for q=0.
- Note also that the maximum change of the OI equals (I(n)-I(n-1))/2 in terms of the repetition spacing used in Algorithm SM-2 (i.e. (OI-OI/EF)/2).
- If the quality response was lower than 3 then start repetitions for the item from the beginning without changing the E-Factor.
- After each repetition session of a given day repeat again all the items that scored below four in the quality assessment. Continue the repetitions until all of these items score at least four
算法SM-4在SuperMemo4.0中的应用
算法
- 将知识分解为最小的卡片
- 所有卡片的 EF 设置为 2.5
- 列出以重复次数和 EF 为索引的 OI 矩阵
- 使用下面的重复间隔来初始化 OI 矩阵：
- OI(1,EF)=1
- OI(2,EF)=6
- for n>2 OI(n,EF)=OI(n-1,EF)*EF
- OI(n,EF) - 难度为 EF 的卡片在第 n 次复习时的最优间隔
- 使用 OI 矩阵来确定重复间隔：
- I(n, EF) = OI(n, EF)
- I(n, EF) - 难度为 EF 的卡片在第 n 次复习时的间隔
- OI(n, EF) - OI 矩阵对应 n 和 EF 的值
- 每次重复后都用 0-5 的成绩来给回忆质量打分
- 每次重复后都根据下列公式修改 EF：
$EF'=EF+(0.1-(5-q)\times(0.08+(5-q)\times0.02))$
- EF' - EF 的新值
- EF - EF 的旧值
- q - 回忆质量打分
- 如果 EF < 1.3 则令 EF = 1.3
- 每次重复后都修改 OI 矩阵上的相关项
- 一个示例公式如下（SM-4 中实际使用的公式更复杂）：
$OI'=interval+\cfrac{interval\times (1-\cfrac{1}{EF})}{2}\times(0.25\times q-1)$ $OI''=(1-fraction)\times OI+fraction\times OI'$
- OI - OI 对应项的旧值
- OI' - 计算 OI 对应项新值的辅助值
- OI'' - OI 对应项的新值
- interval - 之前的间隔
- fraction - 调和参数
- EF - 重复卡片的 EF
- q - 回忆质量评分
- 注意，当 q = 4 时 OI 不会改变，当 q = 5 时 OI 增长是 q = 0 时减少的 4 分之一
- 如果回忆质量小于 3，卡片重新开始复习，并且不改变其 EF
- 在某一天的每次重复练习之后，再重复那些在回忆质量评分中得分低于4分的卡片。继续重复，直到所有这些卡片得分至少4分。
In addition to slow convergence, Algorithm SM-4 showed that the use of matrix of intervals leads to several problems that could easily be solved by replacing intervals with O-factors. Those additional flaws led to a fast implementation of Algorithm SM-5 yet in 1989. Here is a short analysis that explained the flaws in the use of the matrix of optimum intervals:
SM-4 表明，除了收敛速度慢以外，使用间隔矩阵会导致一些问题，而用 OF 来替换间隔可以很容易解决。这些额外的缺陷使 SM-5 快速实现。以下是使用最优间隔矩阵的缺陷：
Archive warning: Why use literal archives?
This text was part of: "Optimization of learning" by Piotr Wozniak (1990)
档案警告:为什么使用文字档案?
本文是Piotr Wozniak(1990)的《优化学习》的一部分。
- In the course of repetition it may happen that one of the intervals will be calculated as shorter than the preceding one. This is certainly inconsistent with general assumptions leading to the SuperMemo method. Moreover, validity of such an outcome was refuted by the results of application of the Algorithm SM-5. This flaw could be prevented by disallowing intervals to increase or drop beyond certain values, but such an approach would tremendously slow down the optimization process interlinking optimal intervals by superfluous dependencies. The discussed case was indeed observed in one of the databases. The discrepancy was not eliminated by the end of the period in which the Algorithm SM-4 was used despite the fact that the intervals in question were only two weeks long
- E-Factors of particular items are constantly modified thus in the OI matrix an item can pass from one difficulty category to another. If the repetition number for that item is large enough, this will result in serious disturbances of the repetitory process of that item. Note that the higher the repetition number, the greater the difference between optimal intervals in neighboring E-Factor category columns. Thus if the E-Factor increases the optimal interval used for the item can be artificially long, while in the opposite situation the interval can be much too short
- 在重复过程中，可能会发生一个间隔在计算后短与前一个间隔的情况，这与 SuperMemo 的假设不一致。此外，即使可以通过不允许间隔增加或减少超过某值来避免这种情况，但这会极大减慢矩阵的优化过程。
- 特定卡片的 EF 会不断修改，因此在 OI 矩阵中，一个卡片可以从一个难度类别转移到另一个难度类别。如果卡片重复的次数足够大，将会导致卡片重复过程的严重影响。注意，卡片连续重复次数越大，相邻的 EF 列的最优间隔间的差异就越大。英雌，如果 EF 增加，卡片的最优间隔会变得不自然的长，在相反的情况下间隔可能会变得太短。
The Algorithm SM-4 tried to interrelate the length of the optimal interval with the repetition number. This approach seems to be incorrect because the memory is much more likely to be sensitive to the length of the previously applied inter-repetition interval than to the number of the repetition
SM-4 尝试将最优间隔的长度和重复次数相关联。这种方法似乎是不正确的。因为记忆对之前使用的重复间隔长度更敏感，而不是重复次数。
The idea of the matrix of optimum intervals was born on Feb 11, 1989. SuperMemo would keep a collection of optimum intervals for different levels of memory stability and difficulty. It would modify the intervals with the flow of new data. Using this new idea, on Mar 1, 1989, I started using SuperMemo 4 to learn Esperanto. Very early I noticed that the idea needs comprehensive revision. The convergence of the algorithm was excruciatingly slow.
最优间隔矩阵的概念诞生于 1989 年 2 月 11 日。SuperMemo将为不同水平的记忆稳定性和难度保存一组最优间隔。它将根据新数据流修改间隔。1989 年 3 月 1 日，我开始使用 SuperMemo 4 来学习世界语。我很早就注意到这个想法需要全面修改。算法的收敛速度慢得要命。
By May 5, 1989, I had a new idea in my mind. This was, in essence, the birth of stability increase function, except, in the optimum review of SuperMemo there would be no retrievability dimension. The new algorithm would use the matrix of optimum factors. Instead of remembering optimum intervals, it would remember how much intervals need to increase depending on memory complexity and current memory stability. Slow convergence of Algorithm SM-4 also inspired the need to randomize intervals (May 20, 1989).
1989 年 5 月 5 日，我有了一个新想法。从本质上讲，这就是稳定性增长函数的诞生，除了在 SuperMemo 的最优复习中没有可提取性维度。新算法将使用最优因子矩阵。它不会记住最优的时间间隔，而是记住需要根据记忆复杂度和当前记忆稳定性增加多少时间间隔。算法 SM-4 的缓慢收敛也激发了随机化间隔的需求(1989年5月20日)。
In the meantime, progress of SuperMemo had to be delayed because of school obligations again. With Krzysztof Biedalak, we decided to write a program for developing school tests that could be used with SuperMemo. The project was again used to wriggle out from other obligations in classes with open-minded Dr Katulski who has become a supporter of all-things SuperMemo by now.
与此同时，SuperMemo](https://supermemo.guru/wiki/SuperMemo) 的进展又一次因为学校的义务而被推迟。在 Krzysztof Biedalak 的帮助下，我们决定编写一个程序，用于开发可以与 SuperMemo 一起使用的学校测试。这个项目再次被用来在开明的 Krzysztof 博士的课堂上摆脱其他义务，Krzysztof 现在已经成为 SuperMemo 的支持者。
I spent summer on practical training in the Netherlands when progress was slow due to various obligations (including compulsory excursions!). One of the chief reasons for slowness was extreme dieting that resulted from the need to save money to pay my PC 1512 debts. I also hoped to earn something extra to buy a hard disk for my computer. All my work was possible thanks to the courtesy of Peter Klijn of the University of Eindhoven. He just gave me his PC for my private use for the whole 2-month period of stay. He did not want my good work over SuperMemo to slow down. It was the first time I could keep all my files on a hard disk, and it felt like a move from an old bike to Tesla Model S.
我花了一个夏天在荷兰进行实践训练，由于各种义务(包括强制远足！)进展缓慢。速度慢的主要原因之一是极端节食，这是因为需要存钱来偿还我的PC1512债务。我还希望能多挣些钱给我的电脑买个硬盘。多亏了埃因霍温大学的Peter Klijn的帮助，我的所有工作才得以完成。他只是把他的电脑给了我，让我在整整两个月的时间里私人使用。他不想让我在SuperMemo上出色的工作慢下来。这是我第一次可以把所有的文件都保存在硬盘上，感觉就像是从一辆旧自行车搬到了特斯拉S型车。
Only on Oct 16, 1989, the new Algorithm SM-5 was completed and I started using SuperMemo 5. I remarked in my notes: "A great revolution is in the offing". The progress was tremendous indeed.
直到 1989 年 10 月 16 日，新的算法 SM-5 才完成，我开始使用 SuperMemo 5。我在笔记中写道：“一场伟大的革命即将到来”。确实取得了巨大的进展。
I had a couple of users of SuperMemo 2 that were ready to upgrade to SuperMemo 5. I demanded only one price: they will start with the matrix of optimum factors initialized to a specific value. This was to validate the algorithm and make sure it carries no preconceived prejudice. All preset optimization matrices would converge nicely and fast. This fact was then used to claim universal convergence, and the algorithm was described as such in the first-ever publication about spaced repetition algorithms. Do not read that publication! It has been emasculated by peer review. Read below to understand Algorithm SM-5.
我有几个 SuperMemo 2 的用户准备升级到 SuperMemo 5。我只要求一个条件：他们将从初始化为特定值的最优因子矩阵开始。这是为了验证算法，并确保它没有先入为主的偏见。所有预先设定的优化矩阵将很好地快速收敛。这一事实随后被用来主张普遍收敛，该算法在关于间隔重复算法的第一本出版物中被这样描述。不要读那份出版物！它已经被同行评审阉割了。阅读下面的内容来理解 SM-5 算法。
SuperMemo uses a simple principle: "use, verify, and correct". After a repetition, a new interval is computed with the help of the OF matrix. The "relevant entry" to compute the interval depends on the repetition (category) and item difficulty. After the interval period elapses, SuperMemo calls for the next repetition. The grade is used to tell SuperMemo how well the interval "performed". If the grade is low, we have reasons to believe that the interval is too long and the OF matrix entry is too high. In such cases, we slightly reduce the OF entry. The relevant entry here is the one that was previously used in computing the interval (i.e. before the interval started). In other words, it is the entry that is (1) used to compute the interval (after n-th repetition) and then (2) used to correct the OF matrix (after the n+1 repetition).
SuperMemo 使用了一个简单的原则：“使用、验证和纠正”。在重复之后，在OF 矩阵的帮助下计算一个新的间隔。计算时间间隔的“相关项目”取决于重复(类别)和项目难度。在间隔时间过后， SuperMemo 要求下一次重复。使用成绩来告诉 SuperMemo 这个间隔“表现”得如何。如果成绩较低，我们有理由相信，间隔太长，OF 矩阵 对应的项太大。在这种情况下，我们稍微减少了对应的项。这里的相关项目是之前用于计算间隔(即在间隔开始之前)的项目。换句话说，项目(1)用于计算间隔(n次重复之后)，然后(2)用于纠正OF 矩阵 (n+1次重复之后)。
Here is an outline of Algorithm SM-5 as presented in my Master's Thesis:
以下是我的[硕士论文](https://supermemo.guru/wiki/Master's _thesis)中给出的算法SM-5的概要:
Archive warning: Why use literal archives?
档案警告:为什么使用文字档案?
This text was part of: "Optimization of learning" by Piotr Wozniak (1990)
本文是Piotr Wozniak(1990)的《优化学习》的一部分。
The final formulation of the algorithm used in SuperMemo 5 is presented below (Algorithm SM-5):
- Split the knowledge into smallest possible items
- With all items associate an E-Factor equal to 2.5
- Tabulate the OF matrix for various repetition numbers and E-Factor categories. Use the following formula:
- OF(1,EF):=4
- for n>1 OF(n,EF):=EF
- where:
- OF(n,EF) - optimal factor corresponding to the n-th repetition and the E-Factor EF
- Use the OF matrix to determine inter-repetition intervals:
- I(n,EF)=OF(n,EF)*I(n-1,EF)
- I(1,EF)=OF(1,EF)
- where:
- I(n,EF) - the n-th inter-repetition interval for an item of a given E-Factor EF (in days)
- OF(n,EF) - the entry of the OF matrix corresponding to the n-th repetition and the E-Factor EF
- After each repetition assess the quality of repetition responses in the 0-5 grade scale (cf. Algorithm SM-2)
- After each repetition modify the E-Factor of the recently repeated item according to the formula:
- EF':=EF+(0.1-(5-q)*(0.08+(5-q)*0.02))
- where:
- EF' - new value of the E-Factor
- EF - old value of the E-Factor
- q - quality of the response in the 0-5 grade scale
- If EF is less than 1.3 then set EF to 1.3
- After each repetition modify the relevant entry of the OF matrix. Exemplary formulas constructed arbitrarily and used in the modification could look like this:
- OF':=OF*(0.72+q*0.07)
- OF*:=(1-fraction)*OF+fraction*OF'*
- where:
- OF - new value of the OF entry
- OF' - auxiliary value of the OF entry used in calculations
- OF - old value of the OF entry
- fraction - any number between 0 and 1 (the greater it is the faster the changes of the OF matrix)
- q - quality of the response in the 0-5 grade scale
- Note that for q=4 the OF does not change. It increases for q>4 and decreases for q<4.
- If the quality response was lower than 3 then start repetitions for the item from the beginning without changing the E-Factor
- After each repetition session of a given day repeat again all items that scored below four in the quality assessment. Continue the repetitions until all of these items score at least four
SuperMemo 5算法的最终公式如下(算法SM-5):
- 将知识分成尽可能小的项目
- 所有项目的 E-Factor 等于 2.5
- 将不同重复次数和 EF 类别的矩阵制表。使用以下公式：
- OF(1,EF):=4
- for n>1 OF(n,EF):=EF
- 其中：
- OF(n,EF) - 对应给定简易度和连续复习次数 n 的最优因子
- 使用 OF 矩阵确定重复间隔：
- I(n,EF)=OF(n,EF)*I(n-1,EF)
- I(1,EF)=OF(1,EF)
- 其中：
- I(n,EF) - 给定 EF 值的卡片的连续第 n 个间隔
- OF(n,EF) - OF 矩阵对应给定简易度和连续复习次数 n 的 OF 值
- 在每次重复之后，在 0-5 级量表中评估成绩 (cf.算法SM-2)
- 每次重复后，根据公式修改最近重复项的e因子：
- EF':=EF+(0.1-(5-q)*(0.08+(5-q)*0.02))
- 其中：
- EF' - 简易度的新值
- EF - 简易度的旧值
- q - quality of the response in the 0-5 grade scale
- 如果 EF 小于 1.3，则将 EF 设置为 1.3
- 每次重复后修改矩阵的相关项目。任意构造并在修改中使用的示例性公式可以是这样的：
- OF':=OF*(0.72+q*0.07)
- OF*:=(1-fraction)*OF+fraction*OF'*
- 其中：
- OF *- OF 的新值
- OF' - 计算 OF 新值的辅助值
- OF - OF 的旧值
- fraction - 0~1之间的一个值，越大 OF 改变越大
- q - 回忆质量打分 0 - 5
- 请注意，对于 q=4 ，OF 不变。q>4 时增大，q<4 时减小。
- 如果 q 小于 3，则不改变 EF 的值，项目从头开始复习。
- 在某一天的每一次重复之后，重复所有在回忆质量打分中得分低于 4 分的项目。继续重复，直到所有这些项目得分至少 4 分
In accordance with previous observations, the entries of the OF matrix were not allowed to drop below 1.2. In Algorithm SM-5, by definition, intervals cannot get shorter in successive repetitions. Intervals are at least 1.2 times as long as their predecessors. Changing the E-Factor category increases the next applied interval only as many times as it is required by the corresponding entry of the OF matrix.
根据之前的观察，OF 矩阵的项不允许低于 1.2。在算法 SM-5 中，根据定义，间隔不能在连续的重复中变得更短。时间间隔至少是其前身的 1.2 倍。更改 E-Factor 类别仅增加下一个应用的间隔，增加的倍数与 OF 矩阵相应项所需的倍数相同。
Anki manual includes a passage that is unexpectedly critical of Algorithm SM-5 (Apr 2018). The words are particularly surprising as Algorithm SM-5 has never been published in full (the version above is just a rough outline). Despite the fact that the words of criticism were clearly uttered in good will, they hint at the possibility that if Algorithm SM-2 was superior over Algorithm SM-5, perhaps it is also superior over Algorithm SM-17. If that was the case, I would have wasted the last 30 years of research and programming. To this day, Wikipedia "criticises" "SM3+". "SM3+" is a label first used in Anki manual that has been used at dozens of sites on the web (esp. those that prefer to stick with the older algorithm for its simplicity). A comparison between Algorithm SM-2 and Algorithm SM-17 is presented here. Hopefully it leaves no doubt.
Anki 手册中包含了一段对 SM-5 算法出乎意料的批评(2018 年 4 月)。这些话特别令人惊讶，因为算法 SM-5 从来没有完整地发表过(上面的版本只是一个粗略的大纲)。尽管批评的话语显然是善意的，但它们暗示了一种可能性，即如果算法 SM-2是优于算法 SM-5，或许它也优于算法 SM-17。如果是这样的话，我就浪费了过去 30 年的研究和编程。时至今日，维基百科还在“批评” “SM3+”。“SM3+” 是最初在Anki手册中使用的一个标签，已经在网络上的几十个站点使用(特别是那些因其简单性而倾向于坚持使用较旧算法的人)。这里比较了算法 SM-2 和算法 SM-17 。希望这不会有任何疑问。
My Master's Thesis, published in excerpts in 1998 at supermemo.com, included only a rough description of the Algorithm SM-5. For the sake of clarity, dozens of minor procedures were not published. Those procedures would require a lot of tinkering to ensure good convergence, stability, and accuracy. This type of tinkering requires months of learning combined with analysis. There has never been a ready out-of-the box version of Algorithm SM-5.
我的[硕士论文](https://supermemo.guru/wiki/Master's _thesis)于 1998 年在 supermemo.com 上摘录发表，其中只包括对算法 SM-5 的粗略描述。为了清楚起见，数十个次要程序没有公布。这些程序需要进行大量修改，以确保良好的收敛性、稳定性和准确性。这种修修补补需要几个月的学习和分析相结合。算法 SM-5 从来没有现成的开箱即用版本。
The source code of Algorithm SM-5 has never been published or opened, and the original algorithm could only be tested by users of SuperMemo 5, in MS DOS. SuperMemo 5 became freeware in 1993. It is important to notice that random dispersal of intervals around the optimum value was essential for establishing convergence. Without the dispersal, the progression of the algorithm would be agonizingly slow. Similarly, matrix smoothing was necessary for consistent behavior independent of the scarcity of data collected for different levels of stability and item complexity.
算法SM-5的源代码从未发表或公开过，最初的算法只能在 MS DOS 下由 Supermemo 5 的用户进行测试。SuperMemo 5 在 1993 年成为免费软件。需要注意的是，围绕最优值的随机分散间隔是建立收敛性的关键。如果没有[分散](https://supermemo.guru/wiki/History_of_spaced_repetition_(print)# random_dispersal_of_interval)，算法的进展将会慢得令人痛苦。同样，[矩阵平滑](https://supermemo.guru/wiki/History_of_spaced_repetition_(print)# Matrix_smoothing)对于一致的行为是必要的，独立于为不同级别的稳定性和项目复杂性收集的数据的稀缺性。
Multiple evaluations done in 1989, and since, have pointed to an unquestionable superiority of Algorithm SM-5 and later algorithms over Algorithm SM-2 in any metric studied. Algorithm SM-17 might actually be used to measure the efficiency of Algorithm SM-5 if we had volunteers to re-implement that ancient code for use with our universal metric. We have done comparisons for Algorithm SM-2 thus far. The implementation cost was simply insignificant. Needless to say, Algorithm SM-2 lags well behind in its predictive powers, esp. for suboptimum levels of retrievability (i.e. situations in which the user does not religiously stick to the prescribed schedule).
在 1989 年和之后进行的多次评估表明，在任何研究的度量中，算法 SM-5 和之后的算法毫无疑问地优于算法 SM-2。算法 SM-17 实际上可以用来衡量算法 SM-5 的效率，如果我们有志愿者用我们的通用度量重新实现古老的代码。到目前为止，我们已经对 SM-2 算法进行了比较。实现的成本微不足道。不用说，SM-2 算法在预测能力方面远远落后，特别是在可提取性的次优水平(在用户没有严格遵守规定时间表的情况下)。
Even a basic understanding of the underlying model should make it clear that a good implementation would yield dramatic benefits. SuperMemo 5 would adapt its function of intervals to the user's memory. SuperMemo 2 was set in stone. I am very proud that wild guesses made in 1985 and 1987 stood the test of time, but no algorithm should trust the judgment of a humble student with 2 years experience in the implementation of spaced repetition algorithms. Instead, SuperMemo 4 and all successive implementations made fewer and fewer guesses, and provided better and faster adaptability. Of all those implementations, only SuperMemo 4 was slow to adapt and was replaced in 7 months by a superior solution.
即使对底层模型有基本的了解，也应该清楚一个好的实现将产生巨大的好处。SuperMemo 5 将根据用户的记忆来调整它的间隔函数。SuperMemo 2 是固定不变的。我非常自豪的是，1985年和1987年的大胆猜测经受住了时间的考验，但任何算法都不应该相信一个拥有 2 年间隔重复算法实现经验的卑微学生的判断。相反，Supermemo 4 和所有后续实现进行的猜测越来越少，并且提供了更好、更快的适应性。在所有这些实现中，只有 Supermemo 4 适应速度慢，并在 7 个月内被更好的解决方案取代。
There is no ill-will in Anki criticism but I would not be surprised if the author pushed for an implementation and speedy move to self-learning rather than spending time on tinkering with procedures that did not seem to work as he expected. In contrast, back in 1989, I knew Algorithm SM-2 was flawed, I knew Algorithm SM-5 was superior, and I would spare no time and effort in making sure the new concept was perfected to its maximum theoretical potential.
Anki 的批评没有恶意，但如果作者推动实施并迅速转向自学，而不是花时间修补那些似乎不像他预期的那样有效的程序，我不会感到惊讶。相比之下，在1989年，我知道算法SM-2是有缺陷的，我知道算法SM-5是更好的，我将不遗余力地确保新概念被完善到其最大的理论潜力。
Excerpt from the Anki manual (April 2018):
Anki手册摘录(2018年4月):
Anki was originally based on the SuperMemo SM5 algorithm. However, Anki’s default behaviour of revealing the next interval before answering a card revealed some fundamental problems with the SM5 algorithm. The key difference between SM2 and later revisions of the algorithm is this:
- SM2 uses your performance on a card to determine the next time to schedule that card
- SM3+ use your performance on a card to determine the next time to schedule that card, and similar cards
Anki最初是基于 SuperMemo 的 SM5 算法。然而，Anki 在回答卡片之前显示下一个间隔的默认行为揭示了 SM5 算法的一些基本问题。SM2 于后来修改的算法之间的关键区别是：
- SM2 根据您的一张卡片上的表现来安排该卡片下一次的时间
- SM3+ 根据您的一张卡片上的表现来安排该卡片，以及类似卡片下一次的时间
The latter approach promises to choose more accurate intervals by factoring in not just a single card’s performance, but the performance as a group. If you are very consistent in your studies and all cards are of a very similar difficulty, this approach can work quite well. However, once inconsistencies are introduced into the equation (cards of varying difficulty, not studying at the same time every day), SM3+ is more prone to incorrect guesses at the next interval - resulting in cards being scheduled too often or too far in the future.
后一种方法不仅考虑了单个卡片的表现，还考虑了作为一个组的表现，从而保证了选择更准确的间隔时间。如果你在学习上非常一致，并且所有的牌都有一个非常相似的难度，这种方法可以很好地工作。然而，一旦在公式中引入了不一致性(不同难度的卡片，不是每天在同一时间学习)，SM3+ 更容易在下一个间隔中出现不正确的猜测——导致牌被安排得太频繁或太遥远。
Furthermore, as SM3+ dynamically adjusts the "optimum factors" table, a situation can often arise where answering "hard" on a card can result in a longer interval than answering "easy" would give. The next times are hidden from you in SuperMemo so the user is never aware of this.
此外，当SM3+动态调整“最优因子”表时，经常会出现这样的情况：在一张卡片上回答“困难”比回答“容易”的间隔时间更长。下一次时间在 SuperMemo 中是隐藏的，所以用户永远不会知道这个。
After evaluating the alternatives, the Anki author decided that near-optimum intervals yielded by an SM2 derivative are better than trying to obtain optimum intervals at the risk of incorrect guesses. An SM2 approach is predictable and intuitive to end users, whereas an SM3+ approach hides the details from the user and requires users to trust the system (even when the system may make mistakes in the scheduling)
在对备选方案进行评估之后，Anki 的作者认为，通过模仿 SM2 产生的接近最优间隔要比冒着错误猜测的风险获得最优间隔更好。SM2 方法对用户来说是可预测的和直观的，而 SM3+ 方法对用户隐藏了细节，并要求用户信任系统(即使系统可能在调度中出错)
Of all the above claims, only one is true. SuperMemo 2 is indeed more intuitive. This problem has plagued SuperMemo for years. Each version is more complex, and it is hard to hide some of that complexity from users. We will keep trying though.
在所有这些说法中，只有一个是正确的。SuperMemo 2确实更直观。多年来，这个问题一直困扰着 SuperMemo。每个版本都更复杂，很难向用户隐藏其中的某些复杂性。我们将继续努力。
This is why Anki criticism is not justified:
这就是为什么 Anki 批评是不合理的原因：
- the fact that SuperMemo 5 used past performance of all items to maximize performance on new items is an advantage, not a "problem". Even more, that is the key to the power of adaptability
- 事实上，SuperMemo 5 利用所有项目的过去表现来最大化新项目的表现，这是一个优势，而不是一个“问题”。更重要的是，这是适应性的关键
- inconsistent grading has been a problem for all algorithms. On average, adaptability helps find out the average effect of mis-grading, esp. if inconsistencies are themselves consistent (i.e. the user keeps committing similar offences in similar circumstances)
- 不一致的评分一直是所有算法的一个问题。平均而言，适应性有助于发现错误评分的平均影响，特别是如果不一致本身是一致的(即用户在相似的情况下不断犯类似的错误)
- mixed difficulties are handled by SuperMemo 5 much better. While both difficulty and stability increase are coded by E-factor in SuperMemo 2, in SuperMemo 5 and later, those two properties of memory are separated
- 混合的困难被 SuperMemo 5 处理得更好。在 SuperMemo 2 中难度和稳定性增长都是用 EF 编码的，而在 SuperMemo 5 及以后的版本中，这两种记忆特性是分开的
- interval predictions have been proven superior in SuperMemo 5, and the claim "more prone to incorrect guesses" can only be explained by errors in implementation
- 间隔预测在 SuperMemo 5 中被证明更优越，而“更容易出错”的说法只能用实现中的错误来解释
- lower grades could bring longer intervals if matrix smoothing is not implemented. That part of the algorithm has only been described verbally in my thesis
- 如果没有实现矩阵平滑，较低的评分可能带来更长的间隔。这部分算法在我的论文中只做了口头描述
- intervals and repetition dates have always been prominently displayed in SuperMemo, even in simpler implementations (e.g. for handheld devices, smartphones, etc.). Nothing is hidden from the user. Most of all, forgetting index and burden statistics are in full view, and make it possible to see if SuperMemo keeps its retention promise, and at what cost to workload
- 即使在更简单的实现中(例如，对于手持设备、智能手机等)，间隔和重复日期也总是显著地显示在 SuperMemo 中。没有向用户隐藏任何东西。最重要的是，遗忘指数和工作量的统计数据是完全可见的，并且有可能看到 SuperMemo 是否履行了它承诺的保留，以及在怎样的工作量代价下。
- the algorithm and all its details were in full display in SuperMemo 5. In particular, changes to the OF matrix would show at each repetition. For anyone with a basic understanding of the algorithm, its operations were fully transparent. Comprehensive tracking could also be implemented in Anki, or others applications, but might require lots of tinkering. It would be costly. In the end, Anki's choice might indeed be good on account of simplicity (in terms of implementation, operations, tracking, debugging, maintenance, user's intuitions, etc.)
- 算法及其所有细节在 SuperMemo 5 中得到了充分的展示。特别是，在每次重复时将显示对矩阵的更改。对于任何对算法有基本了解的人来说，它的操作都是完全透明的。全面的跟踪也可以在 Anki 或其他应用程序中实现，但可能需要大量的修改。这将是昂贵的。最后，Anki 的选择可能确实很好，因为它很简单(从实现、操作、跟踪、调试、维护、用户直觉等方面来说)。
- the complaint of "hiding details" is the polar opposite to the claim that neural networks are superior for spacing repetitions. Unlike neural networks implementation, the operations of Algorithm SM-5 are relatively easy to analyze
- “隐藏细节”的抱怨与神经网络在间隔重复方面更优越的主张是截然相反的。与神经网络实现不同，算法SM-5的操作比较容易分析
Our official response published at supermemopedia.com in 2011 seems pretty accurate today:
我们2011年在 supermemopedia.com 上发表的官方回应今天看来相当准确:
Archive warning: Why use literal archives?
档案警告:为什么使用文字档案?
It is great that Anki introduces its own innovations while still giving due credit to SuperMemo. It is true that SuperMemo's Algorithm SM-2 works great as compared with, for example, Leitner system, or SuperMemo on paper. However, the superiority of Algorithm SM-5 over SM-2 is unquestionable. Both in practice and in theory. It is Algorithm SM-2 that has the intervals hard-wired and dependent only on item's difficulty, which is approximated with a heuristic formula (i.e. a formula based on a guess derived from limited pre-1987 experience). It is true that you cannot "spoil" Algorithm SM-2 by feeding it with false data. It is so only because it is non-adaptable. You probably prefer your word processor with customizable fonts even though you can mess up the text by applying Wingdings.
很棒的是，Anki 介绍了自己的创新，同时仍然给予 SuperMemo 应有的赞誉。的确，与莱特纳系统或纸面上的 Supermemo 相比，SuperMemo 的算法 SM-2 工作得很好。然而，SM-5 算法相对于 SM-2 算法的优越性是毋庸置疑的。无论是在实践中还是在理论上都是如此。算法 SM-2 的时间间隔是固定的，并且只依赖于项目的难度，该难度可以用一个启发式公式来近似(即一个基于 1987 年前有限经验得出的猜测的公式)。确实，您不能通过向算法 SM-2 提供错误数据来“破坏”它。之所以如此，只是因为它不具有适应性。您可能更喜欢使用可自定义字体的文字处理程序，尽管您可能会通过应用修饰把文本弄乱。
Algorithm SM-2 simply crudely multiplies intervals by a so-called E-Factor, which is a way of expressing item difficulty. In contrast, Algorithm SM-5 collects data about user's performance and modifies the function of optimum intervals accordingly. In other words, it adapts to the student's performance. Algorithm SM-6 goes even further and modifies the function of optimum intervals so that to achieve a desired level of knowledge retention. The superiority of those newer algorithms has been verified in more ways than one, for example, by measuring the decline in workload over time in fixed-size databases. In cases studied (small sample), the decline of workload with newer algorithms was nearly twice as fast as compared with older databases processed with Algorithm SM-2 (same type of material: English vocabulary).
SM-2算法简单地用一个所谓的 E-Factor 来粗略地乘以间隔，这是一种表示项目难度的方法。相反，算法 SM-5 收集用户的表现数据，并相应地修改最优间隔的函数。换句话说，它适应学生的表现。算法 SM-6 甚至更进一步，修改了最优间隔的函数，以达到期望的知识保留水平。这些新算法的优越性已经通过多种方式得到了验证，例如，通过测量固定大小数据库中工作负载随时间的下降情况。在研究的案例中(小样本)，与使用算法 SM-2 处理的旧数据库(相同类型的材料：英语词汇)相比，使用新算法处理的工作量下降速度几乎是后者的两倍。
All SuperMemo algorithms group items into difficulty categories. Algorithm SM-2 gives each category a rigid set of intervals. Algorithm SM-5 gives each category the same set of intervals too, however, these are adapted on the basis of user's performance, i.e. not set in stone.
所有的 SuperMemo 算法都将项目分成难度类别。算法 SM-2 给每个类别一个严格的间隔集。算法 SM-5 也给了每个类别一组间隔，但是这些间隔是根据用户的表现来调整的，也就是说不是固定不变的。
"Bad grading" is indeed more important in Algorithm SM-5 than it is in Algorithm SM-2 as false data will result in "false adaptation". However, it is always bad to give untrue/cheat grades in SuperMemo, whichever algorithm you use.
“糟糕的评分”在算法 SM-5 中确实比在算法 SM-2 中更严重，因为错误的数据会导致“错误的适应”。然而，在 SuperMemo 中给出不真实/作弊的分数总是不好的，无论你使用哪种算法。
With incomplete knowledge of memory, adaptability is always superior to rigid models. This is why it is still better to adapt to an imprecise average (as in Algorithm SM-5) than to base the intervals on an imprecise guess (as in Algorithm SM-2). Needless to say, the last word goes to Algorithm SM-8 and later, as it adapts to the measured average
在有关记忆的知识不完整的情况下，适应性总是优于僵化的模型。这就是为什么适应一个不精确的平均值(如算法SM-5)要比根据一个不精确的猜测(如算法 SM-2)来确定间隔要好。 毋庸置疑，最后一个词进入算法 SM-8 之后又变了，因为它适应了测得的平均值
SuperMemo 5 was so obviously superior that I did not collect much data to prove my point. I made only a few comparisons for my Master's Thesis and they left no doubt.
SuperMemo 5的优势如此明显，以至于我没有收集太多数据来证明我的观点。我只拿我的硕士论文做了几次比较，结果毫无疑问。
Archive warning: Why use literal archives?
档案警告:为什么使用文字档案?
This text was part of: "Optimization of learning" by Piotr Wozniak (1990)
本文是Piotr Wozniak(1990)的《优化学习》的一部分。
3.8. Evaluation of the Algorithm SM-5
3.8.算法 SM-5 的评估
The Algorithm SM-5 has been in use since October 17, 1989, and has surpassed all expectations in providing an efficient method of determining the desired function of optimal intervals, and in consequence, improving the acquisition rate (15,000 items learnt within 9 months). Fig. 3.5 indicates that the acquisition rate was at least twice as high as that indicated by combined application of the SM-2 and SM-4 algorithms!
算法 SM-5 自 1989 年 10 月 17 日开始使用，它提供了一种确定理想的最优间隔函数的有效方法，超出了所有人的预期，从而提高了习得率(在 9 个月内学习了 15,000 个项目)。图 3.5 显示习得速率至少是组合应用 SM-2 和 SM-4 算法的两倍！
Figure: Changes of the work burden in databases supervised by SM-2 and SM-5 algorithms.
**图片：**在 SM-2 和 SM-5 算法的监督下，数据库中工作负担的变化
The knowledge retention increased to about 96% for 10-month-old databases. Below, some knowledge retention data in selected databases are listed to show the comparison between the SM-2 and SM-5 algorithms:
对于 10 个月的数据库，知识保留率提高到 96% 左右。下面列出了选定数据库中的一些知识保留率数据，以显示 SM-2 和 SM-5 算法之间的比较：
- Date - date of the measurement,
- 日期 - 测量日期，
- Database - name of the database; ALL means all databases averaged
- 数据库 - 数据库的名称；ALL 表示所有数据库的平均值
- Interval - average current interval used by items in the database
- 间隔 - 数据库中项目使用的平均当前间隔
- Retention - knowledge retention in the database
- 保留率 - 数据库中的知识保留率
- Version - version of the algorithm applied to the database
- 版本 - 数据库所应用的算法版本
| Date | Database | Interval | Retention | Version |
|---|---|---|---|---|
| Dec 88 | EVD | 17 days | 81% | SM-2 |
| Dec 89 | EVG | 19 days | 82% | SM-5 |
| Dec 88 | EVC | 41 days | 95% | SM-2 |
| Dec 89 | EVF | 47 days | 95% | SM-5 |
| Dec 88 | all | 86 days | 89% | SM-2 |
| Dec 89 | all | 190 days | 92% | SM-2, SM-4 and SM-5 |
In the process of repetition the following distribution of grades was recorded:
在复习过程中，记录了下列成绩分布情况:
| Quality | Fraction |
|---|---|
| 0 | 0% |
| 1 | 0% |
| 2 | 11% |
| 3 | 18% |
| 4 | 26% |
| 5 | 45% |
This distribution, in accordance to the assumptions underlying the Algorithm SM-5, yields the average response quality equal to 4. The forgetting index equals to 11% (items with quality lower than 3 are regarded as forgotten). Note, that the retention data indicate that only 4% of items in a database are not remembered. Therefore forgetting index exceeds the percentage of forgotten items 2.7 times.
根据算法 SM-5 的假设，该分布产生的平均回忆质量等于 4。遗忘指数等于 11%(成绩低于 3 的项目被认为是被遗忘的)。请注意，保留率数据表明数据库中只有 4% 的项目没有被记住。因此，遗忘指数超过遗忘项目百分比的 2.7 倍。
In a 7-month old database, it was found that 70% of items had not been forgotten even once in the course of repetitions preceding the measurement, while only 2% of items had been forgotten more than 3 times
在一个 7 个月前的数据库中，发现 70% 的项目在测量之前的重复过程中甚至没有忘记一次，而只有 2% 的项目遗忘次数超过 3 次
Anki's criticism of SuperMemo 5 calls for a simple proof in the light of modern spaced repetition theory. We can show that today's model of memory can be mapped onto the models underlying both algorithms: Algorithm SM-2 and Algorithm SM-5, and the key difference between the two is the missing adaptability of the function of optimal intervals (represented in Algorithm SM-5 by matrix of optimum factors).
Anki 对 SuperMemo 5 的批评需要根据现代的间隔重复理论来做一个简单的证明。我们可以表明，今天的记忆模型可以映射到两种算法基础上的模型：算法 SM-2 和算法 SM-5，两者之间的关键区别是最优间隔函数的适应性缺失(在算法SM-5中由最优因子矩阵表示)。
Let SInc=f(C,S,R) be a stability increase function that takes complexity C, stability S, and retrievability R as arguments. This function determines the progressive increase in review intervals in optimum learning.
SInc = f (C、S、R) 是一个稳定性增长函数，复杂性 C、稳定 S、可恢复性 R 作为参数。这个函数决定了最优学习中复习间隔的递增。
Both Algorithms, SM-2 and SM-5 ignore the retrievability dimension. In theory, if both algorithms worked perfectly, we could assume they aim at R=0.9. As it can be measured in SuperMemo, both algorithms fail at that effort for they do not know relevant forgetting curves. They simply do not collect forgetting curve data. This data-collecting possibility was introduced only in Algorithm SM-6 in 1991.
两种算法，SM-2 和 SM-5 都忽略了可提取性维度。理论上，如果两种算法都能完美运行，我们可以假设它们的目标是 R=0.9。正如可以在 SuperMemo 中测量的那样，这两种算法都失败了，因为它们不知道相关的遗忘曲线。他们只是不收集遗忘曲线数据。这种数据收集的可能性是在 1991 年才在算法 SM-6 中引入的。
However, if we assume that 1985 and 1987 heuristics were perfect guesses, in theory, the algorithm could use SInc=F(C,S) with constant R of 90%.
然而，如果我们假设 1985 和 1987 启发式是完美的猜测，在理论上，该算法可以使用SInc=F(C,S)，常数R为90%。
Due to the fact that SM-2 uses the same number, EF for stability increase and for item complexity, for SM-2 we have SInc=f(C,S) equation represented by EF=f'(EF,interval), where it can be shown easily with data that f<>f'. Amazingly, the heuristic used in SM-2 made this function work by decoupling the actual link between the EF and item complexity. As data shows that SInc keeps decreasing with an increase in S, in Algorithm SM-2, by definition, all items would need to gain on complexity with each review if EF was to represent item complexity. In practical terms, Algorithm SM-2 uses EF=f'(EF,interval), which translates to SInc(n)=f(SInc(n-1),interval).
由于 SM-2 使用相同的数字 EF 用于稳定性增加和项目复杂性，对于 SM-2，我们有以 EF=f'(EF,interval) 的形式表示的 SInc=f(C,S) 方程，其中的数据可以很容易地显示 f<>f' 。令人惊讶的是，SM-2 中使用的启发式通过解耦 EF 和项目复杂性之间的实际联系，使这个函数发挥作用。由于数据显示 SInc 随着 S 的增加而不断减少，在算法 SM-2 中，根据定义，如果要用 EF 来表示项的复杂度，那么所有项都需要在每次复习时获得复杂度。在实际应用中，算法 SM-2 使用 EF=f'(EF,interval)，即 SInc(n)=f(SInc(n-1),interval)。
Let us assume that the EF=f(EF,interval) heuristic was an excellent guess as claimed by proponents of Algorithm SM-2. Let SInc be represented by O-factor in Algorithm SM-5. We might then represent SInc=f(C,S) as OF=f(EF,interval).
让我们假设 EF=f(EF,interval) 启发式正如 SM-2 算法的支持者所声称的那样，是一个很好的猜想。令 SInc 在算法 SM-5 中由 O-factor 表示。然后我们可以将 SInc=f(C,S) 表示为 OF=f(EF,interval)。
For Algorithm SM-2, OF would be constant and equal to EF, in Algorithm SM-5, OF is adaptable and can be modified depending on algorithm's performance. It seems pretty obvious that penalizing the algorithm for bad performance by a drop to OF matrix entries, and rewarding it by an increase in OF entries is superior to keeping OF constant.
对于算法 SM-2, OF 是常数，等于 EF，在算法 SM-5 中，OF是可适应的，可以根据算法的表现进行修改。很明显，对表现差的算法进行惩罚，降低OF 矩阵对应的项，并通过增加对应的项来奖励它，这要优于保持不变。
On a funny twist, as much as supporters of Algorithm SM-2 claim it performs great, supporters of neural network SuperMemo kept accusing algebraic algorithms of: lack of adaptability. In reality, the adaptability of Algorithm SM-17 is best to-date as it is based on the most accurate model of memory.
有趣的是，尽管 SM-2 算法的支持者声称它表现得很好，但神经网络 SuperMemo 算法的支持者却不断指责代数算法：缺乏适应性。实际上，算法 SM-17 的适应性是最好的，因为它是基于最精确的记忆模型。
It is conceivable that heuristics used in SM-2 were so accurate that the original guess on OF=f(EF,interval) needed no modification. However, as it has been shown in practical application, the OF matrix quickly evolves, and converges onto values described in this publication (Wozniak, Gorzelanczyk 1994). They differ substantively from the assumption wired into Algorithm SM-2.
可以想象，在 SM-2 中使用的启发法是如此精确，以致于原来对 OF=f(EF,interval) 的猜测不需要修改。然而，正如在实际应用中所显示的那样，OF 矩阵迅速发展，并收敛到这篇出版物(Wozniak, Gorzelanczyk 1994)中描述的值。它们与算法 SM-2 中的假设有本质上的不同。
Summary:
总结：
- sm17 (2016): SInc=f(C,S,R), 3 variables, f is adaptable
- sm17 (2016): SInc=f(C,S,R)， 3 个变量，f 是有适应性的
- sm5 (1989): SInc=f(C,S,0.9), 2 variables, f is adaptable
- sm5 (1989): SInc=f(C,S,0.9)， 2 个变量，f 是有适应性的
- sm2 (1987): SInc=f(SInc,S,0.9) - 1 variable, f is fixed
- sm2 (1987): SInc=f(SInc,S,0.9) - 1 个变量，f 是固定的
Algorithm SM-5 showed fast convergence, which was quickly demonstrated by users who began with univalent OF matrices. It was quite a contrast with Algorithm SM-4.
Archive warning: Why use literal archives?
This text was part of: "Optimization of learning" by Piotr Wozniak (1990)
算法SM-5表现出了快速的收敛性，这很快被以单价[矩阵]开始的用户所证明(https://supermemo.guru/wiki/OF_matrix)。这与SM-4算法形成了鲜明的对比。
档案警告:为什么使用文字档案?
本文是[Piotr Wozniak]的《优化学习》(https://supermemo.guru/wiki/Piotr_Wozniak)(1990)的一部分。
3.6. Improving the predetermined matrix of optimal factors
3.6.改进最优因子的预定矩阵
The optimization procedures applied in transformations of the OF matrix appeared to be satisfactorily efficient resulting in fast convergence of the OF entries to their final values.
However, in the period considered (Oct 17, 1989 - May 23, 1990) only those optimal factors which were characterized by short modification-verification cycles (less than 3-4 months) seem to have reached their equilibrial values.
It will take a few further years before more sound conclusions can be drawn regarding the ultimate shape of the OF matrix. The most interesting fact apparent after analyzing 7-month-old OF matrices is that the first inter-repetition interval should be as long as 5 days for E-Factor equal 2.5 and even 8 days for E-Factor equal 1.3! For the second interval the corresponding values were about 3 and 2 weeks respectively.
在矩阵的转换中应用的优化过程似乎是令人满意的有效的，从而使项目快速收敛到它们的最终值。
然而，在考虑的时期(1989年10月17日- 1990年5月23日)，只有那些以修改-验证周期短(不到3-4个月)为特征的最优因素似乎达到了平衡值。
关于矩阵的最终形状，还需要几年的时间才能得出更合理的结论。在分析了7个月大的矩阵后，最有趣的事实是，对于e因子等于2.5的矩阵，第一次重复间隔应该长达5天，对于e因子等于1.3的矩阵，甚至应该长达8天!第二个时间间隔对应的值分别为3周和2周左右。
The newly-obtained function of optimal intervals could be formulated as follows:
新得到的[最优间隔]函数(https://supermemo.guru/wiki/Optimal_interval)表达式如下:
I(1)=8-(EF-1.3)/(2.5-1.3)*3
I(2)=13+(EF-1.3)/(2.5-1.3)*8
for i>2 I(i)=I(i-1)*(EF-0.1)
where:
- I(i) - interval after the i-th repetition (in days)
- EF - E-Factor of the considered item.
上式中：
- I(I) -第I次重复后的间隔(以天为单位)
- EF - E-Factor。
To accelerate the optimization process, this new function should be used to determine the initial state of the OF matrix (Step 3 of the SM-5 algorithm). Except for the first interval, this new function does not differ significantly from the one employed in Algorithms SM-0 through SM-5. One could attribute this fact to inefficiencies of the optimization procedures which, after all, are prejudiced by the fact of applying a predetermined OF matrix. To make sure that it is not the fact, I asked three of my colleagues to use experimental versions of the SuperMemo 5.3 in which univalent OF matrices were used (all entries equal to 1.5 in two experiments and 2.0 in the remaining experiment).
为了加速优化过程，应该使用这个新函数来确定矩阵的初始状态(SM-5算法的第3步)。除了第一个间隔外，这个新函数与SM-0到SM-5算法中使用的函数没有显著的区别。人们可以把这一事实归因于优化过程的低效，毕竟，这是由应用预先确定的矩阵的事实所造成的。为了确保这不是事实，我让我的三个同事使用SuperMemo 5.3的实验版本，其中使用了一价矩阵(在两个实验中所有的项目都等于1.5，在剩下的实验中等于2.0)。
Although the experimental databases have been in use for only 2-4 months, the OF matrices seem to slowly converge to the form obtained with the use of the predetermined OF matrix. However, the predetermined OF matrices inherit the artificial correlation between E-Factors and the values of OF entries in the relevant E-Factor category (i.e. for n>3 the value OF(n,EF) is close to EF). This phenomenon does not appear in univalent matrices which tend to adjust the OF matrices more closely to requirements posed by such arbitrarily chosen elements of the algorithm as initial value of E-Factors (always 2.5), function modifying E-Factors after repetitions etc.
虽然实验数据库的使用时间只有2-4个月，但矩阵的收敛速度似乎缓慢地趋近于使用预先确定的矩阵所得到的形式。然而，矩阵的预定继承了E-Factor与相关E-Factor类元素值之间的人工相关性(即对于n>3， (n,EF)的值接近于EF)。这种现象在单叶矩阵中不存在，单叶矩阵倾向于根据算法中任意选择的元素如e因子初值(始终为2.5)、函数重复后修改e因子等要求对矩阵进行更紧密的调整。
Some of the criticism from the author of Anki might have been a result of not employing matrix smoothing that was an important component of the algorithm and is still used in Algorithm SM-17.
Archive warning: Why use literal archives?
This text was part of: "Optimization of learning" by Piotr Wozniak (1990)
一些[Anki的作者批评](https://supermemo.guru/wiki/History_of_spaced_repetition_(印刷)# Criticism_of_Algorithm_SM-5)可能是由于没有采用矩阵平滑算法的一个重要组成部分,至今仍用于算法SM-17 (https://supermemo.guru/wiki/Algorithm_SM-17)。
档案警告:为什么使用文字档案?
本文是[Piotr Wozniak]的《优化学习》(https://supermemo.guru/wiki/Piotr_Wozniak)(1990)的一部分。
3.7. Propagation of changes across the matrix of optimal factors
3.7.变化在最优因子矩阵上的传播
Having noticed the earlier mentioned regularities in relationships between entries of the OF matrix I decided to accelerate the optimization process by propagation of modifications across the matrix. If an optimal factor increases or decreases then we could conclude that the OF factor that corresponds to the higher repetition number should also increase.
This follows from the relationship OF(i,EF)=OF(i+1,EF), which is roughly valid for all E-Factors and i>2. Similarly, we can consider desirable changes of factors if we remember that for i>2 we have OF(i,EF')=OF(i,EF*)*EF'/EF* (esp. if EF' and EF are close enough). I used the propagation of changes only across the OF matrix that had not yet been modified by repetition feed-back. This proved particularly successful in case of univalent OF matrices applied in the experimental versions of SuperMemo mentioned in the previous paragraph.
注意到前面提到的矩阵中各元素之间关系的规律性，我决定通过在矩阵中传播修改来加速优化过程。如果一个最优因子增加或减少，那么我们可以得出结论，与较高的重复次数相对应的因子也应该增加。
这是从(i,EF)= (i+1,EF)的关系得出的，它对所有e - factor和i>2大致有效。类似地，如果我们记得对于i>2，我们有of (i,EF')= of (i,EF*)*EF'/EF*(特别是如果EF'和EF*足够接近的话)，我们可以考虑理想的因子变化。我只在还没有被重复反馈修改过的矩阵上使用变化的传播。这证明了特别成功的情况下，一价矩阵适用于SuperMemo的实验版本在前一段提到
The proposed propagation scheme can be summarized as this:
所提出的传播方案可以概括为:
-
After executing Step 7 of the Algorithm SM-5 locate all neighboring entries of the OF matrix that has not yet been modified in the course of repetitions, i.e. entries that did not enter the modification-verification cycle. Neighboring entries are understood here as those that correspond to the repetition number +/- 1 and the E-Factor category +/- 1 (i.e. E-Factor +/- 0.1)
-
Modify the neighboring entries whenever one of the following relations does not hold:
- for i>2 OF(i,EF)=OF(i+1,EF) for all EFs
- for i>2 OF(i,EF')=OF(i,EF*)*EF'/EF*
- for i=1 OF(i,EF')=OF(i,EF*)*
-
For all the entries modified in Step 2 repeat the whole procedure locating their yet unmodified neighbors.
-
执行算法SM-5的第7步后，定位在重复过程中尚未被修改的of矩阵的所有相邻的项，即未进入修改-验证循环的项。相邻项在这里理解为对应重复次数+/- 1和e因素类别+/- 1(即e因素+/- 0.1)的项。
-
当下列关系之一不成立时，修改相邻的项:
-
for i>2 (i,EF)=OF(i+1,EF) for all EFs
-
for i>2 OF(i,EF')=OF(i,EF*)*EF'/EF*
-
因为i=1 OF(i,EF')=OF(i,EF*)*
3.对于第2步中修改的所有项目，重复查找它们尚未修改的邻居的整个过程。
Propagation of changes seems to be inevitable if one remembers that the function of optimal intervals depends on such parameters as:
- student's capacity
- student's self-assessment habits (the response quality is given according to the student's subjective opinion)
- character of the memorized knowledge etc.
Therefore it is impossible to provide an ideal, predetermined OF matrix that would dispense with the use of the modification-verification process and, to a lesser degree, propagation schemes.
变化的传播似乎是不可避免的，如果你记得最优间隔的函数取决于这样的参数:
- 学生的能力
- 学生的自我评价习惯(响应质量根据学生的主观意见给出)
- 记忆知识的性质等。
因此，不可能提供一个理想的、预先确定的矩阵，而不使用修改-验证过程和(在较小程度上)传播方案。
时间间隔的随机散布
One of the key improvements in Algorithm SM-5 was random dispersal of intervals. On one hand, it dramatically accelerated the optimization process, on the other, it caused a great deal of confusion in users of nearly all future versions of SuperMemo: "why do the same items with the same grade use a different interval at each try?". Minor deviations are precious. This was laid bare when "naked" Algorithm SM-17 was released in early SuperMemo 17. It could be seen that users who keep a lot of leeches in their collections would easily hit "local minima" from which they could never get out. The random dispersion was restored with some delay. The period of "nakedness" was needed for accurate observations of the algorithm, esp. in multi-decade learning process like my own.
算法SM-5的关键改进之一是间隔的随机散布。一方面，它极大地加快了优化过程，另一方面，它给几乎所有未来版本的SuperMemo的用户造成了极大的困惑:“为什么相同等级的相同项目每次使用不同的间隔时间?”。微小的偏差是宝贵的。当“naked”算法SM-17在早期的SuperMemo 17中发布时，这个问题就暴露出来了。可以看出，那些收藏了大量水蛭的用户很容易就会遇到“局部极小值”，从此他们再也出不来了。随机离散度在一定的延迟下恢复。“赤裸”这段时间对算法的准确观察是需要的，尤其是在我这样的几十年的学习过程中。
Archive warning: Why use literal archives?
This text was part of: "Optimization of learning" by Piotr Wozniak (1990)
档案警告:为什么使用literal archives?
这篇文章是[Piotr Wozniak]的一部分:“改进学习”(https://supermemo.guru/wiki/Piotr_Wozniak) (1990)
3.5. Random dispersal of optimal intervals
3.5.最优间隔的随机散布
To improve the optimization process further, a mechanism was introduced that may seem to contradict the principle of optimal repetition spacing. Let us reconsider a significant fault of the Algorithm SM-5: A modification of an optimal factor can be verified for its correctness only after the following conditions are met:
- the modified factor is used in calculation of an inter-repetition interval
- the calculated interval elapses and a repetition is done yielding the response quality which possibly indicates the need to increase or decrease the optimal factor
为了进一步改进优化过程，引入了一种可能与最优重复间隔原则相矛盾的机制。让我们重新考虑一下算法SM-5的一个重大错误:只有在满足以下条件时，才能对一个最优因子的修改进行正确性验证:
- 修正因子用于计算重复间隔
- 计算的时间间隔过去了，重复的时间间隔产生的响应质量可能表明需要增加或减少最优因子
This means that even a great number of instances used in modification of an optimal factor will not change it significantly until the newly calculated value is used in determination of new intervals and verified after their elapse.
The process of verification of modified optimal factors after the period necessary to apply them in repetitions will later be called the modification-verification cycle. The greater the repetition number the longer the modification-verification cycle and the greater the slow-down in the optimization process.
这意味着即使大量的实例用于修改一个最优因子，也不会显著地改变它，直到新计算的值被用于确定新的间隔，并在它们过去后进行验证。
在重复使用一段时间后对修改过的最优因素进行验证的过程以后将称为修改-验证周期。重复次数越大，修改-验证周期越长，优化过程的慢化程度越大。
To illustrate the problem of modification constraint let us consider calculations from Fig. 3.4.
One can easily conclude that for the variable INTERVAL_USED greater than 20 the value of MOD5 will be equal 1.05 if the QUALITY equals 5. As the QUALITY=5, the MODIFIER will equal MOD5, i.e. 1.05. Hence the newly proposed value of the optimal factor (NEW_OF) can only be 5% greater than the previous one (NEW_OF:=USED_OF*MODIFIER). Therefore the modified optimal factor will never reach beyond the 5% limit unless the USED_OF increases, which is equivalent to applying the modified optimal factor in calculation of inter-repetition intervals.
Bearing these facts in mind I decided to let inter-repetition intervals differ from the optimal ones in certain cases to circumvent the constraint imposed by a modification-verification cycle.
为了说明修改约束的问题，让我们考虑图[3.4]中的计算(http://super-memory.com/english/ol/sm5.htm)。
我们可以很容易地得出结论，对于变量INTERVAL_USED大于20，如果质量等于5，那么MOD5的值将等于1.05。当质量=5时，修改器将等于MOD5，即1.05。因此，新提出的最优因子(NEW_OF)只能比前一个(NEW_OF:=USED_OF*MODIFIER)大5%。因此，修改后的最优因子永远不会超过5%的极限，除非USED_OF增加，这相当于将修改后的最优因子应用于计算重复间隔。
考虑到这些事实，我决定在某些情况下让重复间隔与最优间隔不同，以绕过修改-验证周期的约束。
I will call the process of random modification of optimal intervals dispersal.
If a little fraction of intervals is allowed to be shorter or longer than it should follow from the OF matrix then these deviant intervals can accelerate the changes of optimal factors by letting them drop or increase beyond the limits of the mechanism presented in Fig. 3.4. In other words, when the value of an optimal factor is much different from the desired one then its accidental change caused by deviant intervals shall not be leveled by the stream of standard repetitions because the response qualities will rather promote the change than act against it.
Another advantage of using intervals distributed round the optimal ones is elimination of a problem which often was a matter of complaints voiced by SuperMemo users - the lumpiness of repetition schedule. By the lumpiness of repetition schedule I mean accumulation of repetitory work in certain days while neighboring days remain relatively unburdened. This is caused by the fact that students often memorize a great number of items in a single session and these items tend to stick together in the following months being separated only on the base of their E-Factors.
我把这个过程称为最优间隔的随机修正散布。
如果允许一小部分时间间隔更短或更长时间比它应该遵循从矩阵的这些不正常的间隔可以加速最优因素的变化,让他们减少或增加超越极限的机制提出了图(3.4)(http://super-memory.com/english/ol/sm5.htm)。换句话说，当一个最优因子的值与期望的值有很大的差异时，由于响应特性更倾向于促进变化，而不是与之相反，因此由偏差间隔引起的偶然变化不会被标准重复流拉平。
使用最优间隔的另一个好处是消除了SuperMemo用户经常抱怨的一个问题——重复调度的混乱。我所说的重复计划的块块性，是指重复工作在某天的积累，而相邻的日子则相对没有负担。这是由于学生们在一次学习中往往记住了大量的项目，而这些项目往往在接下来的几个月里粘在一起，仅根据他们的电子因素分开。
Dispersal of intervals round the optimal ones eliminates the problem of lumpiness. Let us now consider formulas that were applied by the latest SuperMemo software in dispersal of intervals in proximity of the optimal value. Inter-repetition intervals that are slightly different from those which are considered optimal (according to the OF matrix) will be called near-optimal intervals. The near-optimal intervals will be calculated according to the following formula:
在最优间隔周围散布间隔消除了块状问题。现在让我们考虑一下最新的SuperMemo软件在散布接近最优值的间隔时所使用的公式。与被认为是最优的间隔(根据矩阵的形式)稍有不同的重复间隔称为接近最优的间隔。近似最优间隔按下式计算:
NOI=PI+(OI-PI)*(1+m)
where:
- NOI - near-optimal interval
- PI - previous interval used
- OI - optimal interval calculated from the OF matrix (cf. Algorithm SM-5)
- m - a number belonging to the range <-0.5,0.5> (see below)
or using the OF value:
NOI=PI*(1+(OF-1)*(1+m))
NOI=PI+(OI-PI)*(1+m)
上式中：
- NOI-接近最优间隔
- PI-使用以前的间隔
- OI-由矩阵计算的最优间隔(cf.算法SM-5)
- m-属于<-0.5、0.5>范围的数字(见下文)
The modifier m will determine the degree of deviation from the optimal interval (maximum deviation for m=-0.5 or m=0.5 values and no deviation at all for m=0).
In order to find a compromise between accelerated optimization and elimination of lumpiness on one hand (both require strongly dispersed repetition spacing) and the high retention on the other (strict application of optimal intervals required) the modifier m should have a near-zero value in most cases.
修改器m将确定偏离最优间隔的程度(m =-0.5或m=0.5值的最大偏差，m=0时完全没有偏差)。
为了在加速优化和消除块度(两者都需要很强的分散重复间隔)和高保留(严格应用最优间隔)之间找到一个折衷点，m在大多数情况下应该有一个接近于零的值。
The following formulas were used to determine the distribution function of the modifier m:
- the probability of choosing a modifier in the range <0,0.5> should equal 0.5:
- the probability of choosing a modifier m=0 was assumed to be hundred times greater than the probability of choosing m=0.5:
- the probability density function was assumed to have a negative exponential form with parameters a and b to be found on the base of the two previous equations:
利用以下公式确定修饰语m的分布函数:
- -在<0,0.5>范围内选择一个修饰语的概率应该等于0.5:
- 假设选择一个修饰语m=0的概率是选择m=0.5的概率的100倍:
- 假设概率密度函数为负指数形式，参数a和b在前两个方程的基础上:
The above formulas yield values a=0.04652 and b=0.09210 for m expressed in percent.
From the distribution function
上述公式对于以百分比表示的m，则得a=0.04652, b=0.09210。
从分布函数
integral from -m to m of aexp(-babs(x))dx = P (P denotes probability)
从-m到m的积分aexp(-bab (x))dx = P (P表示概率)
we can obtain the value of the modifier m (for m>=0):
我们可以得到修饰符m的值(对于m>=0):
m=-ln(1-b/a*P)/b
Thus the final procedure to calculate the near-optimal interval looks like this:
因此，计算接近最优间隔的最终步骤如下:
a:=0.047;
b:=0.092;
p:=random-0.5;
m:=-1/bln(1-b/aabs(p));
m:=m*sgn(p);
NOI:=PI*(1+(OF-1)*(100+m)/100);
where:
- random - function yielding values from the range <0,1) with a uniform distribution of probability
- NOI - near-optimal interval
- PI - previously used interval
- OF - pertinent entry of the OF matrix
a:=0.047;
b:=0.092;
p:=random-0.5;
m:=-1/bln(1-b/aabs(p));
m:=m*sgn(p);
NOI:=PI*(1+(OF-1)*(100+m)/100);
- random -函数的取值范围<0,1)，具有均匀分布的概率
- NOI -接近最优间隔
- PI -以前使用的间隔
- 矩阵的相关项