# Theory

## Contents

## Available tools for memory and sleep analysis

To study your memory and sleep, use options available in SuperMemo in the following places:

- Workload
- Learning statistics
- Element statistics
- Analysis
- Memory graphs
- Repetition history
- Sleep Chart

## Examples

### Repetition calendar

You can use **Toolkit : Workload** (*Ctrl+W*) to inspect the daily and the monthly calendar of repetitions. You can view the number of repetitions scheduled, number of scheduled topics, number of scheduled items, as well as the record of past repetitions, and past retention, or how many new items were memorized.

#### Daily workload

#### Monthly workload

### 4D memory graphs

With **Toolkit : Memory : 4D graphs** you can visualize your memory in 4D:

Stability increaseFigure:A 3D graph of SInc[] matrix made regular by checkingAverageand rotated around theaxis (vertical). The graph is based on 272 repetition cases (all data points included) for items with difficulty=0.05. TheAveragecheckbox helps you extract best data using data-rich neighboring entries and theoretical SInc[] predictions where no data is available. This is particularly useful for scarce data in new collections (as the one used in the picture)

### Collection analysis

You can use **Toolkit : Statistics : Analysis** (*Shift+Alt+A*) to inspect your forgetting curves, daily changes to your measured forgetting index, superiority of the latest Algorithm SM-17 over the previous generation spaced repetition algorithm, your learning overload, and many more:

#### Forgetting curves

##### Typical exponential forgetting curve

Figure:Toolkit : Statistics : Analysis : Forgetting (UF)for 20 repetition number categories multiplied by 20 A-Factor categories. In the picture, blue circles represent data collected during repetitions. The larger the circle, the greater the number of repetitions recorded. The red curve corresponds with the best-fit forgetting curve obtained by exponential regression. For ill-structured material the forgetting curve is crooked, i.e. not exactly exponential. The horizontal aqua line corresponds with the requested forgetting index, while the vertical green line shows the moment in time in which the approximated forgetting curve intersects with the requested forgetting index line. This moment in time determines the value of the relevant R-Factor, and indirectly, the value of the optimum interval. For the first repetition, R-Factor corresponds with the first optimum interval. The values of O-Factor and R-Factor are displayed at the top of the graph. They are followed by the number of repetition cases used to plot the graph (i.e. 21,303). At the beginning of the learning process, there is no repetition history and no repetition data to compute R-Factors. It will take some time before your first forgetting curves are plotted. For that reason, the initial value of the RF matrix is taken from the model of a less-than-average student. The model of average student is not used because the convergence from poorer student parameters upwards is faster than the convergence in the opposite direction. TheDeviationparameter displayed at the top tells you how well the negatively exponential curve fits the data. The lesser the deviation, the better the fit. The deviation is computed as a square root of the average of squared differences (as used in the method of least squares).

##### First review forgetting curve

Figure:The first forgetting curve for newly learned knowledge collected with SuperMemo. Power approximation is used in this case due to the heterogeneity of the learning material freshly introduced in the learning process. Lack of separation by memory complexity results in superposition of exponential forgetting with different decay constants. On a semi-log graph, the power regression curve is logarithmic (in yellow), and appearing almost straight. The curve shows that in the presented case recall drops merely to 58% in four years, which can be explained by a high reuse of memorized knowledge in real life. The first optimum interval for review at retrievability of 90% is 3.96 days. The forgetting curve can be described with the formula R=0.9906*power(interval,-0.07), where 0.9906 is the recall after one day, while -0.07 is the decay constant. In this is case, the formula yields 90% recall after 4 days. 80,399 repetition cases were used to plot the presented graph. Steeper drop in recall will occur if the material contains a higher proportion of difficult knowledge (esp. poorly formulated knowledge), or in new students with lesser mnemonic skills. Curve irregularity at intervals 15-20 comes from a smaller sample of repetitions (later interval categories on a log scale encompass a wider range of intervals).

#### Daily changes to the measured forgetting index

Figure:Exemplary graph enabling a more meaningful analysis of the forgetting index. Changes to forgetting index inAnalysisuse the daily measured forgetting index (previously: less informative cumulative measured forgetting index value was taken for the entire period since the last use ofToolkit : Statistics : Reset parameters : Forgetting index record). Note that the priority queue may distort the actual retention in your collection as measured values are primarily taken from top-priority material. Thus measured forgetting index should be understood as "forgetting index measured at repetitions", not as "overall measured forgetting index".

#### R-Metric

Figure:R-Metricgraph demonstrates superiority of Algorithm SM-18 over the old Algorithm SM-15 for the presented collection used in the testing period of full 4 years dating back to Apr 2, 2015. It was plotted using 24,104 data points (i.e. repetition cases with data from both algorithms), and smoothed up to show the trends. Multiple spots beneath the line of 0 at the vertical axis (`R-metric<0`

) have been smoothed out (they correspond with days when the previous version of the algorithm appeared superior in a smaller sample of repetitions). Some positive and negative trends correspond with changes in the algorithm as data were collected in the new algorithm's testing period. A gradual increase in the metric in the months Feb-May 2016, might be a statistical aberration, or it might be the result of new interval values and a bigger R-metric for intervals departing from the optimum used in earlier SuperMemos. The latter interpretation might suggest that the benefits of Algorithm SM-18 can gradually increase over time

#### Priorities missed due to overload

### Collection statistics

You can use **Toolkit : Statistics : Statistics** to supervise and understand your learning process:

### Element data

You can use **Toolkit : Statistics : Element data** to inspect learning statistics of a given element:

### Element repetition history

You can use **Learning : Statistics : Repetition history** (*Shift+Ctrl+H*) (on the element menu) to inspect the history of repetitions for individual elements:

Figure:Repetition historydialog box displaying the history of repetitions for the current element related to the Malpighian body and the renal tubules. In this example, the item has been repeated 10 times thus far. It was forgotten only once at the 3rd repetition on Apr 20, 1999 (after 97 days since the previous repetition). Since then it has been recalled successfully every time. It was last repeated on Apr 01, 2019 at 12:08:57. The hour data is present only for the last repetition due to the fact that SuperMemo registers the repetition hour only as of SuperMemo 13 onwards (hours are used in correlating retention with sleep data available fromSleepChart). The item is scheduled for repetition in roughly 4.5 years (on Sep 18, 2023)

### Element memory status

You can use **Learning : Statistics : Memory status** (on the element menu) to see how history of repetitions translates into the changes in two variables of long-term memory (see also: SuperMemo Algorithm):

Figure:Changes in memory status over time for an exemplary item. The horizontal axis represents time spanning the entire repetition history. The top panel shows retrievability (tenth power, R^10, for easier analysis). Retrievability grid in gray is labelled by R=99%, R=98%, etc. The middle panel displays optimum intervals in navy. Repetition dates are marked by blue vertical lines and labelled in aqua. The end of the optimum interval where R crosses 90% line is marked by red vertical lines (only if intervals are longer than optimum intervals). The bottom panel visualizes stability (presented as`ln(S)/ln(days)`

for easier analysis). The graph shows that retrievability drops fast (exponentially) after early repetitions when stability is low, however, it only drops from 100% to 94% in long 10 years after the 7th review. All values are derived from an actual repetition history and the DSR model.

### Repetitions graph

Repetitions graph for any subsets of elements can be shown in the browser with **Tools : Repetitions graph** on the browser menu:

Figure: The horizontal axis corresponds with the repetition number and the vertical axis represents intervals (logarithmic scale). Despite a popular belief, the semi-log scale does not produce a linear graph here. Clearly the increase in the length of intervals slows down with successive repetitions. Moreover, the graph corresponding with zero lapses (red curve), results from the superposition of items with lower and faster increase in intervals (determined by difficulty). The bell-shaped curve is determined by all contributing items (below repetition number 10) and then only by difficult items or items with low forgetting index for which the increase in the length of intervals is significantly slower (above repetition 10). To see the above graph in your own collection, useTools : Repetitions graphon the browser menu

### Sleep Chart

You can use **Toolkit : Sleep Chart** (*F12*) to optimize the timing of your sleep as well as the timing of your learning:

#### Sleep and repetitions timeline

Inspect the timeline of repetitions and sleep:

Figure:Sleep and learning timeline. Sleep blocks are marked in blue. Learning blocks in red. Total learning time on individual days is displayed on the right. The currently selected sleep block turns yellow. Its length is displayed at the bottom. A sleep block whose color bleeds from blue into pink denotes interrupted sleep (e.g. with an alarm clock). A sleep block with the color changing from black to blue marks a delayed sleep episode (e.g. caused by watching a late tennis match).

Look for the best time for learning or sleep (see Sleep Chart for details):

Figure:Sleep and repetitions timeline displaying repetitions blocks of the current collection (in red) and sleep blocks (in blue) with recomputed circadian approximations on the current data. Blue and red continuous lines are predictions of optimum sleep time using the SleepChart model (based on sleep statistics). Yellow continuous line shows the prediction of the maximum of circadian sleepiness (circadian middle-of-the-night peak) using a phase response curve model. Note that, theoretically, the yellow line should roughly fall into the middle between blue and red lines. However, when a disruption of the sleep pattern is severe, those lines might diverge testifying to the fact that it is very hard to build models that fully match the chaotic behavior of the sleep control system subjected to a major perturbation. Aqua dots point to the predicted daytime dip in alertness (i.e. the time when a nap might be most productive).

#### Alertness

See how your brain gradually loses its power during the day:

Figure:Toolkit : Sleep Chart : Alertness (H)graph makes it possible for you to visually inspect how recall (and grades) decrease during the waking day. It also shows the impact of circadian factors with grades slightly lower immediately after waking and slightly higher after the mid-day dip in the 9th hour. The blue dots are recall data illustrating decline in performance during a waking day from 87.5% at waking to 81.5% at midday nadir (size of the circles corresponds with the number of repetitions collected, where minimum 50 repetitions are needed to paint a circle in this particular graph). The yellow line is the estimated homeostatic alertness derived from the sleep log data. The aqua line represents the circadian alertness estimate derived from the same sleep log data. Recall is a resultant of the impact of both sleep-drive processes: homeostatic (yellow) and circadian (aqua). 0.1 (hours) (at the bottom) is the minimum sleep block length taken into consideration. 101,230 repetitions cases were taken to plot the graph. TheDeviationparameter displayed at the top tells you how well the chosen approximation curve fits the data (in the picture: negatively exponential recall curve). The lesser the deviation, the better the fit. The deviation is computed as a square root of the average of squared differences (as used in the method of least squares). DepressedUse Rwill use grade-R correlations for an average student to estimate recall (R derived from the DSR model is not used here, as on theAlertness (C)tab, due to the slowness of the computation). Those correlated R figures may differ from actual recall

#### Two-component sleep model

Look for the time of the day that should give you maximum learning power (*Shift+click* a day in the sleep timeline). See when your learning is not likely to be effective and when you should rather go to sleep:

Figure:The predictions of the two-component sleep model about the homeostatic and circadian status of your alertness. The horizontal axis represents time. Blue blocks show the actual sleep episodes. Aqua line shows the 24h circadian sleep drive with a mid-day siesta hump. Green line is an inverse of the homeostatic sleep drive and can be interpreted as homeostatic alertness, which declines exponentially during wakefulness and is quickly restored by slow-wave sleep (for simplicity, as in Borbely model, the entire sleep block is assumed to have a contribution proportional to its length, as the SleepChart model does not account for sleep stages). Yellow vertical lines show the prediction of the circadian bathyphase (circadian middle-of-the-night peak). Bathyphase computations are done with the help of a phase response curve model (as opposed to a statistical model used in earlier versions of SleepChart). Red line shows the resultant alertness (peaks are best for learning, valleys are best for sleep). For example, Alertness on Oct 1, 2008 at 7:43 was predicted to be at 59% of the maximum but would keep increasing fast in the first 2 hours of wakefulness (a typical symptom of a night sleep that is terminated too early). The picture shows two peaks in alertness on Oct 1, 2008, at 9 am and at 7 pm. Those periods would likely be best suited for learning on that day.