The tabbed dialog available from **Tools
: Statistics : Analysis **in SuperMemo provides matrices and graphs that illustrate the
current state of the learning process in the currently opened collection. Some of these
graph can be understood without understanding Algorithm
SM-8; however, most of them require general understanding of how SuperMemo computes
the optimum spacing of repetitions.

The following tabs and subtabs are available in the **Analysis** dialog:

**Distributions**

Interval distribution- distribution of inter-repetition intervals in a given collection

A-Factor distribution- distribution of A-Factors in a given collection (note, that the distribution itself is not used in Algorithm SM-8, and merely results from it)

Repetitions distribution- distribution of the number of repetition in a given collection (only memorized elements are considered, i.e. there is no zero-repetitions category)

Lapses distribution- distribution of the number of times particular elements of the collection have been forgotten (only memorized elements are considered)

**Curves **- four hundred
forgetting curves are independently plotted in order to compute the RF matrix. These
correspond to twenty repetition number categories multiplied by twenty A-Factor categories (note that for
data representation convenience, the
columns of the RF matrix for the first repetition are indexed by the number of memory lapses
rather than A-Factor). By choosing a proper combination of tab at the bottom of the graph,
you can select a forgetting curve of interest. Horizontal axis represents time expressed
as: (1) U-Factor, i.e. the ratio of subsequent inter-repetition intervals, or (2) days
(only for the first repetition). Vertical axis represents knowledge retention in
percent:

Blue circles in the picture represent data collected during repetitions. The greater the circle, the greater the number of repetitions recorded. The red curve corresponds with the best-fit forgetting curve obtained by exponential regression (note that for ill-structured material the forgetting curve is flattened, i.e. not exactly exponential). The horizontal green 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 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. At the beginning of the learning process, there is no repetition history and no repetition data that could be used to compute R-Factors. 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)

**Graphs**

FI-G graph- G-FI graph correlates the expected forgetting index with the grade obtained at repetitions. You can imagine that the forgetting curve graph might use average grade instead of retention on its vertical axis. If you correlated this grade with the forgetting index (which is 100% minus retention), you arrive at the G-FI graph

G-AF graph- G-AF graph correlates the first grade obtained by an item with the ultimate estimation of its A-Factor value. At each repetition, the current element's old A-Factor estimation is removed from the graph and the new estimation is added. This graph is used by the Algorithm SM-8 to quickly estimate the first value of A-Factor at the moment when all we know about an element is the first grade it has scored in its first repetition

DF-AF graph- DF-AF graph shows decay constants of power approximation of R-Factors along columns of the RF matrix. The horizontal axis represents A-Factor, while the vertical axis represents D-Factor (i.e. Decay Factor). D-Factor is a decay constant of power approximation of curves that can be inspected with theApproximationstab of theAnalysisdialog box

First interval graph- the length of the first interval after the first repetition depends on the number of times a given item has been forgotten. Note that the first repetition here means the first repetition after forgetting (i.e. not the first repetition ever). In other words, a twice repeated item will have the repetition number equal to one after it has been forgotten (i.e. the repetition number will not equal three). The first interval graph shows exponential regression curve that approximates the length of the first interval for different numbers of memory lapses (including the zero-lapses category that corresponds with newly memorized items). In the graph below, blue circles correspond to data collected in the learning process (the greater the circle, the more repetitions have been recorded). In the presented case, newly memorized items are optimally repeated after seven days; however, items that have been forgotten 10 times (which is rare in SuperMemo) will require the interval of two days (due to logarithmic scaling, the size of the circle is not linearly proportional to the data sample and the number of repetition cases for lapses=0 is by far larger than for lapses=10 as can be seen inDistributions :Lapses distribution). The graph below includes data from over 90,000 repetitions

**Matrices**

O-Factor matrix- matrix of optimal factors indexed by the repetition number and A-Factor (only for the first repetition, A-Factor is replaced with memory lapses)

R-Factor matrix- matrix of retention factors

Cases matrix- matrix of repetition cases used to compute the corresponding entries of the RF matrix (double click an entry to view the relevant forgetting curve). This matrix can be edited manually

Optimal intervals- matrix of optimum intervals derived from the OF matrix

D-Factor vector- vector of D-Factor values for different A-Factor values (also repetition cases used in computing particular D-Factors)

**3-D Graphs** - 3-D graphs that visually illustrate the
changes to OF, RF and Cases matrices

Approximations- twenty power approximation curves that show the decline of R-Factors along columns of the RF matrix. For each A-Factor, with increasing values of the repetition number, the value of R-Factor decreases (at least theoretically it should decrease). Power regression is used to illustrate the degree of this decline that is best reflected by the decay constant called here D-Factor. By choosing the A-Factor tab at the bottom of the graph, you can view a corresponding R-Factor approximation curve. The horizontal axis represents the repetition number, while the vertical axis represents R-Factor. The value of D-Factor is shown at the top of the graph. The blue polyline shows R-Factors as derived from repetition data. The red curve shows the fixed-point power approximation of R-Factor (fixed-point approach is used as for the repetition number equal two, R-Factor equals A-Factor). The green curve shows the fixed-point power approximation of R-Factor taken from the OF matrix. This is equivalent to substituting the D-Factor obtained by fixed-point power approximation of R-Factors with D-Factor obtained from DF-AF linear regression.

**Frequently Asked Questions**

*(Steven
Trezise, USA, Apr 20, 1999)*

**Question:**

In my collection, I have items for which I have done between 1 and 8 repetitions.
However, when I look at the Cases matrix, there are no entries beyond repetition 3

**Answer:
**The algorithm used by
SuperMemo updates all optimization matrices using repetition categories, not the
actual repetition number (you can view the optimization matrices with

**Tools : Statistics : Analysis : Matrices**). A repetition category is an expected number of repetitions needed to reach the currently used interval. Once the matrices change, the estimation of repetition category may change too. If, for example, you score well in repetitions and your intervals become longer, it will take fewer repetitions to get to a given interval. In such a case, you might be at 8-th repetition while your repetition category will be 3. All matrices such as OF matrix, RF matrix, etc. will be updated in the third row (not in the 8-th row)

*(Zoran
Maximovic, Serbia,
Sep 25, 2000)*

**Question:**

In approximation graphs in **Tools ****
: Statistics : Analysis**, some of
the curves "jump out" of the graph area. What is wrong?

**Answer:
**
This was a harmless bug
in the algorithm in SuperMemo 98/99. The assumption is that intervals cannot
grow beyond the value of A-Factor. For that reason, the maximum R-Factor should
equal the relevant A-Factor. However, in plotting the forgetting curves, higher
values of U-Factors are used as repetitions may be delayed (e.g. with

**Mercy**, user procrastination, etc.). The algorithm puts a cap on the maximum R-Factor value (along the theoretical assumption that R-Factors cannot be greater than corresponding A-Factors). However, the implementation used the maximum U-Factor value as the cap (the one used in plotting the forgetting curve). Consequently, R-Factors could grow larger than A-Factors and the curve would "jump out" of the graph, which displays the correct cap.

This bug should have little effect on the learning process. The higher cap does not invalidate the correctness of R-Factors. It just does not prevent very long intervals in case of very good repetition results.

This bug has been fixed in SuperMemo 2000