  ## Analysis of the learning process

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 for the sake of computing the RF matrix. These correspond to twenty repetition number categories multiplied by twenty A-Factor categories (note that for the first repetition, the columns of the RF matrix 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 in the case of the first repetition). Vertical axis represents knowledge retention in percent: Blue circles in the picture represent repetitions (the greater the circle, the greater the number of repetitions). Red curve corresponds with the best-fit forgetting curve obtained by exponential regression.
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. 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.
Note that 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 Wozniaks model of memory, and they correspond with the parameters of memory that characterize 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 the Approximations tab of the Analysis dialog 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 may also mean the first repetition after forgetting. 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).

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.

(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

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?