The tabbed dialog available from **Tools : Analysis** in SuperMemo provides matrices and graphs that illustrate the current state of the learning process in the currently opened knowledge system. 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 (see Optimization algorithm used in spacing repetitions in SuperMemo for a description of most general concepts related to Algorithm SM-8).

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

**Distributions**

**Interval distribution** - distribution of inter-repetition intervals in a given knowledge system

**A-Factor distribution** - distribution of A-Factors in a given knowledge system (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 knowledge system (only memorized items are considered, i.e. there is no zero-repetitions category)

**Lapses distribution** - distribution of the number of times particular items of the knowledge system have been forgotten (only memorized items 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 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 Wozniak’s 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**

**G-FI 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.

**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 **notebook dialog.

**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)

**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.