Manually specifying FFTs

Verbally defining FFTs

Here are some instructions for manually specifying trees:

• Each node must start with the word “If” and should correspond to the form: If <CUE> <DIRECTION> <THRESHOLD>, predict <EXIT>.

• Numeric thresholds should be specified directly (without brackets), like age > 21.

• For categorical variables, factor thresholds must be specified within curly braces, like sex = {male}. For factors with sets of values, categories within a threshold should be separated by commas like eyecolor = {blue,brown}.

• To specify cue directions, standard logical comparisons =, !=, <, >= (etc.) are valid. For numeric cues, only use >, >=, <, or <=. For factors, only use = or !=.

• Positive exits are indicated by True, while negative exits are specified by False.

• The final node of an FFT is always bi-directional (i.e., has both a positive and a negative exit). The description of the final node always mentions its positive (True) exit first. The text Otherwise, predict EXIT that we have included in the example above is actually not necessary (and ignored).

Example

Now, let’s use our verbal description of an FFT (assigned to in_words above) as the my.tree argument of the FFTrees() function. This creates a corresponding FFT and applies it to the heartdisease data:

# Create FFTrees from a verbal FFT description (as my.tree): 
my_fft <- FFTrees(formula = diagnosis ~.,
                  data = heartdisease,
                  main = "My 1st FFT", 
                  my.tree = in_words)

Running FFTrees() with the my.tree argument creates an object my_fft that contains one FFT. A verbal description of this tree can be printed by inwords(my_fft), but we want to print or plot the object to evaluate the tree’s performance on training or testing data. Let’s see how well our manually constructed FFT (my_fft) did on the training data:

# Inspect FFTrees object:
plot(my_fft, data = "train")

Figure 1: An FFT manually constructed using the my.tree argument of FFTrees().

When manually constructing a tree, the resulting FFTrees object only contains a single FFT. Hence, the ROC plot (in the right bottom panel of Figure 1) cannot show a range of FFTs, but locates the constructed FFT in ROC space.

The formal definition of our new FFT is available from the FFTrees object my_fft:

# Get FFT definition(s):
get_fft_df(my_fft)  # my_fft$trees$definitions

#> # A tibble: 1 × 7
#>    tree nodes classes cues         directions thresholds     exits  
#>   <int> <int> <chr>   <chr>        <chr>      <chr>          <chr>  
#> 1     1     3 n;n;c   sex;age;thal =;>=;=     1;45;fd,normal 1;0;0.5

Note that the 2nd node in this FFT (using the age cue) is predicting the noise outcome (i.e., a non-final exit value of 0 or FALSE, shown to the left). As our tree definitions always refer to the signal outcome (i.e., a non-final exit value of 1 or TRUE, shown to the right), the direction symbol of a left exit (i.e., the 2nd node in Figure 1: if age < 45, predict 0 or noise) must be flipped relative to its appearance in the tree definition (if age >= 45, predict 1 or signal). Thus, the plot and the formal definition describe the same FFT.

As it turns out, the performance of our first FFT created from a verbal description is a mixed affair: The tree has a rather high sensitivity (of 91%), but its low specificity (of only 10%) allows for many false alarms. Consequently, its accuracy measures fail to exceed the baseline level.

Creating an alternative FFT

Let’s see if we can come up with a better FFT. The following example uses the cues thal, cp, and ca in the my.tree argument:

# Create a 2nd FFT from an alternative FFT description (as my.tree): 
my_fft_2 <- FFTrees(formula = diagnosis ~.,
                    data = heartdisease, 
                    main = "My 2nd FFT", 
                    my.tree = "If thal = {rd,fd}, predict True.
                               If cp != {a}, predict False. 
                               If ca > 1, predict True. 
                               Otherwise, predict False.")

As FFTrees aims to interpret the my.tree argument to the best of its abilities, there is some flexibility in entering a verbal description of an FFT. For instance, we also could have described our desired FFT in more flowery terms:

# Create a 2nd FFT from an alternative FFT description (as my.tree): 
my_fft_2 <- FFTrees(formula = diagnosis ~.,
                    data = heartdisease, 
                    main = "My 2nd FFT", 
                    my.tree = "If thal equals {rd,fd}, we shall say True.  
                               When Cp differs from {a}, let's predict False. 
                               Whenever CA happens to exceed 1, we will insist on True.
                               Else, we give up and go away.") 

However, as the vocabulary of FFTrees is limited, it is safer to enter cue directions in their symbolic form (i.e., using =, <, <=, >, >=, or !=).¹ To verify that FFTrees interpreted our my.tree description as intended, let’s check whether the FFT of inwords(my_fft_2) yields a description that corresponds to what we wanted:

inwords(my_fft_2)

#> [1] "If thal = {rd,fd}, decide True."                 
#> [2] "If cp != {a}, decide False."                     
#> [3] "If ca > 1, decide True, otherwise, decide False."

As this seems (a more prosaic version of) what we wanted, let’s visualize this tree (to evaluate its performance) and briefly inspect its tree definition:

# Visualize FFT:
plot(my_fft_2)

# FFT definition:
get_fft_df(my_fft_2)  # my_fft_2$trees$definitions
#> # A tibble: 1 × 7
#>    tree nodes classes cues       directions thresholds exits  
#>   <int> <int> <chr>   <chr>      <chr>      <chr>      <chr>  
#> 1     1     3 c;c;n   thal;cp;ca =;=;>      rd,fd;a;1  1;0;0.5
# Note the flipped direction value for 2nd cue (exit = '0'):
# 'if (cp  = a), predict 1' in the tree definition corresponds to 
# 'if (cp != a), predict 0' in the my.tree description and plot.  

Figure 2: Another FFT manually constructed using the my.tree argument of FFTrees().

This alternative FFT is nicely balancing sensitivity and specificity and performs much better overall. Nevertheless, it is still far from perfect — so check out whether you can create even better ones!

Overview: Retrieving, manipulating, and using FFT definitions

Figure 3: Overview of 4 tree definition and conversion functions (in yellow)
and 6 tree trimming functions (in blue).

Figure 3 provides an overview of the functions provided for converting and manipulating FFT definitions. We can distinguish between three cycles:

• Cycle 1: Obtaining and collecting sets of FFT definitions. The main FFTrees() function creates an FFTrees object x with and from corresponding tree definitions. The following functions get, collect, and evaluate FFT definitions:
  1. get_fft_df() extracts the set of current FFT definitions (from x$trees$definitions).
     
  2. add_fft_df() adds one or more FFT definitions to an existing set of FFT definitions. Thus, the function allows collecting sets of FFT definitions.
     
  3. the tree.definitions argument of the main FFTrees() function allows evaluating sets of FFT definitions for some data.
• Cycle 2: Converting between two tree definition formats (both stored as data frames):
  1. read_fft_df() selects a single FFT from a set of FFT definitions and converts it into a tidy data format (as a data frame in which each row represents a node).
  2. write_fft_df() converts a single FFT from the tidy data frame (with 1-row-per-node) into the 1-row-per FFT format used by add_fft_df() and the tree.definitions argument of FFTrees().
• Cycle 3: Manipulating an individual tree definition by a set of tree editing and trimming functions:
  • add_nodes() allows adding new nodes to an FFT;
  • drop_nodes() allows deleting nodes of an FFT;
  • edit_nodes() allows changing existing nodes of an FFT;
  • flip_exits() allows changing the exit direction of existing nodes;
  • reorder_nodes() allows changing the order of FFT nodes;
  • select_nodes() allows filtering existing nodes of an FFT.

The following example illustrates the interplay of these functions in a typical workflow. When looking at Figure 3, we first move down on the right side (from retrieving sets of FFT definitions to manipulating individual FFTs) and then move up on the left side (to re-convert individual FFTs into sets of FFT definitions).

Example

We illustrate a typical workflow by redefining some FFTs that were built in the Tutorial: FFTs for heart disease and evaluating them on the (full) heartdisease data.

To obtain a set of existing tree definitions, we use our default algorithms to create an FFTrees object x:

# Create an FFTrees object x:
x <- FFTrees(formula = diagnosis ~ .,           # criterion and (all) predictors
             data = heart.train,                # training data
             data.test = heart.test,            # testing data
             main = "Heart Disease 1",          # initial label
             decision.labels = c("low risk", "high risk"),  # exit labels
             quiet = TRUE)                      # hide user feedback

As we have seen in the Tutorial, evaluating this expression yields a set of 7 FFTs. Rather than evaluating them individually (e.g., by using print(x) or plot(x) commands to inspect specific trees), we now extract the tree definitions to select and modify individual FFTs. As the definitions are stored as part of the FFTrees object x, this could be achieved by evaluating x$trees$definitions or by summary(x)$definitions. Alternatively, we can use the get_fft_df() utility function on x to obtain the set of generated tree definitions:

# Get tree definitions of x:
(tree_dfs <- get_fft_df(x))

#> # A tibble: 7 × 7
#>    tree nodes classes cues             directions thresholds          exits    
#>   <int> <int> <chr>   <chr>            <chr>      <chr>               <chr>    
#> 1     1     3 c;c;n   thal;cp;ca       =;=;>      rd,fd;a;0           1;0;0.5  
#> 2     2     4 c;c;n;c thal;cp;ca;slope =;=;>;=    rd,fd;a;0;flat,down 1;0;1;0.5
#> 3     3     3 c;c;n   thal;cp;ca       =;=;>      rd,fd;a;0           0;1;0.5  
#> 4     4     4 c;c;n;c thal;cp;ca;slope =;=;>;=    rd,fd;a;0;flat,down 1;1;0;0.5
#> 5     5     3 c;c;n   thal;cp;ca       =;=;>      rd,fd;a;0           0;0;0.5  
#> 6     6     4 c;c;n;c thal;cp;ca;slope =;=;>;=    rd,fd;a;0;flat,down 0;0;0;0.5
#> 7     7     4 c;c;n;c thal;cp;ca;slope =;=;>;=    rd,fd;a;0;flat,down 1;1;1;0.5

The resulting R object tree_dfs is a data frame with 7 variables. Each of its 7 rows defines an FFT in the context of our current FFTrees object x (see the vignette on Creating FFTs with FFTrees() for help on interpreting tree definitions). As the “ifan” algorithm responsible for creating these trees yields a family of highly similar FFTs (which vary only by their exits, and may truncate some cues), we may want to explore alternative versions of these trees.

All 7 FFTs defined in tree_dfs share the three cues thal, cp, and ca, in this order. Let’s assume we had reasons to explore the cue order thal, ca, and cp.

Obtaining individual tree definitions

Before we can apply our tree editing functions, we first select an initial FFT that we want to manipulate further. This is the job of read_fft_df() (see Cycle 2 of Figure 3):

• read_fft_df() selects an FFT definition and converts it into a tidy data frame.

In our present case, we select and convert tree 1:

(fft_1 <- read_fft_df(ffts_df = tree_dfs, tree = 1))

#>   class  cue direction threshold exit
#> 1     c thal         =     rd,fd    1
#> 2     c   cp         =         a    0
#> 3     n   ca         >         0  0.5

The resulting R object fft_1 selected the definition of tree 1 and converted it into a tidy data frame. Each of its 3 rows defines a node of this FFT. This format can be understood and manipulated more easily than the more compact format used in row 1 of tree_dfs (but both formats can be converted into each other by the complementary functions read_fft_df() and write_fft_df()).

Manipulating individual tree definitions

Having selected one FFT and converted it into a tidy data frame fft_1, we can use various tree trimming functions to edit or manipulate the definition of an individual tree (see Cycle 3 of Figure 3).

Reordering nodes
In our example, our selected FFT uses the cues thal, cp, and ca, but we wanted to explore the alternative cue order thal, ca, and cp. In other words, we need to swap the order of the 2nd and 3rd cues of fft_1. This is the purpose of the reorder_nodes() function:

• reorder_nodes() allows changing the order of FFT nodes.

In contrast to the read_fft_df() function above, the tree editing functions of Cycle 3 all require an individual FFT-definition (as a tidy data frame) as their first argument fft. In the case of reorder_nodes(), the second argument order specifies a numeric vector for the desired node positions (i.e., the integers from 1 to nrow(fft) in any order). Thus, swapping the order of the 2nd and 3rd nodes of fft_1 is achieved as follows:

(my_fft_1 <- reorder_nodes(fft = fft_1, order = c(1, 3, 2)))

#> reorder_nodes: Former exit node now is node 2

#>   class  cue direction threshold exit
#> 1     c thal         =     rd,fd    1
#> 2     n   ca         >         0    1
#> 3     c   cp         =         a  0.5

Swapping the final two nodes implies that the final node of fft_1 (using the numerical cue ca) becomes a non-final cue, whereas the previous middle node (using a categorical cue cp) becomes a final cue (see the corresponding feedback message). By default, the previous exit cue becomes a signal exit (here: exit 1, i.e., predicting a TRUE value for the diagnosis criterion). Saving the resulting tree definition (stored in the same tidy data frame format as fft_1) as an R object my_fft_1 will allow us to collect our modified FFTs later.

Flipping exits As we had no specific exit structure in mind when deciding to explore the cue order thal, ca, and cp, we will explore all possible exit directions. As every non-final node allows for two alternative exits, a tree containing \(n\) nodes has \(2^{(n-1)}\) exit structures (i.e., \(2^2 = 4\) for 3 cues). The \(3\) exit structures in addition to my_fft_1 can be obtained by flipping the exit directions of the first, second, or both non-final nodes. Each alternative can be achieved by applying the flip_exits() function to my_fft_1:

• flip_exits() allows changing the exit direction of existing nodes.

For instance, the tree definition with a signal exit at the first node of my_fft_1 can be reversed into a non-signal node as follows:

(my_fft_2 <- flip_exits(my_fft_1, nodes = 1))

#>   class  cue direction threshold exit
#> 1     c thal         =     rd,fd    0
#> 2     n   ca         >         0    1
#> 3     c   cp         =         a  0.5

Using magrittr pipes to combine steps

The tree conversion and editing functions do not need to be used separately. As their first argument represents the data object to be manipulated, they can be viewed as verbs in a tree manipulation language and chained into longer command sequences by using the magrittr pipe operator. For instance, reversing the exit of the 2nd cue of my_fft_1 could be achieved as follows:

library(magrittr)

(my_fft_3 <- my_fft_1 %>% 
  flip_exits(nodes = 2))

#>   class  cue direction threshold exit
#> 1     c thal         =     rd,fd    1
#> 2     n   ca         >         0    0
#> 3     c   cp         =         a  0.5

Similarly, the five steps from FFTrees object x to the tree definition with the alternative exit structure of my_fft_4 could be chained into a single pipe as follows:

(my_fft_4 <- x %>% 
  get_fft_df() %>%
  read_fft_df(tree = 1) %>%
  reorder_nodes(order = c(1, 3, 2)) %>%
  flip_exits(nodes = c(1, 2)))

#> reorder_nodes: Former exit node now is node 2

#>   class  cue direction threshold exit
#> 1     c thal         =     rd,fd    0
#> 2     n   ca         >         0    0
#> 3     c   cp         =         a  0.5

To evaluate our four alternative tree definitions, we need to re-convert them into the 1-line-per-FFT format (using write_fft_df()) and gather them in a single data frame (using add_fft_df()). In Figure 3, this corresponds to moving upwards on the left (from individual FFT definitions in a tidy data frame to combining sets of FFT definitions in a single data frame).

Re-converting and collecting sets of tree definitions

Having created four modified tree definitions (i.e., my_fft_1 to my_fft_4), we now want to re-convert them from the tidy data frame (with each row representing an FFT node) to the (non-tidy) data frame format (in which each row represents and entire FFT). The more compact format allows us to gather several FFT definitions in one data frame, so that we can evaluate them together for some data in a call to the main FFTrees() function. In Figure 3, the corresponding steps are shown on the left sides of Cycle 2 and Cycle 1.

To re-convert between the two tree definition formats (both stored as data frames), we use the write_fft_df() function:

• write_fft_df() converts a single FFT from the tidy data frame (with 1-row-per-node) into the 1-row-per FFT format used by add_fft_df() and the tree.definitions argument of FFTrees().

In addition to the FFT definition to convert (e.g., my_fft_1), the tree argument allows providing a numeric ID which will later allow us to identify the FFT within a set:

(my_tree_dfs <- write_fft_df(my_fft_1, tree = 1))

#>   tree nodes classes       cues directions thresholds   exits
#> 1    1     3   c;n;c thal;ca;cp      =;>;=  rd,fd;0;a 1;1;0.5

This creates an R object my_tree_dfs as a (non-tidy) data frame in which the definition of my_fft_1 occupies only one row.

Next, we add the other FFT definitions to the my_tree_dfs data frame. This can be achieved by the add_fft_df() function:

• add_fft_df() adds one or more FFT definitions to an existing set of FFT definitions. Thus, the function allows collecting sets of FFT definitions.

However, as this first requires converting each of the tidy FFT definitions into the more compact 1-row-per FFT format, we use a series of command pipes. Each pipe starts from a tidy FFT definition and applies write_fft_df() (with an ID) and add_fft_df (to my_tree_dfs), before re-assigning the modified data frame as my_tree_dfs again:

my_tree_dfs <- my_fft_2 %>%
  write_fft_df(tree = 2) %>%
  add_fft_df(my_tree_dfs)

my_tree_dfs <- my_fft_3 %>%
  write_fft_df(tree = 3) %>%
  add_fft_df(my_tree_dfs)

my_tree_dfs <- my_fft_4 %>%
  write_fft_df(tree = 4) %>%
  add_fft_df(my_tree_dfs)

my_tree_dfs

#>   tree nodes classes       cues directions thresholds   exits
#> 1    1     3   c;n;c thal;ca;cp      =;>;=  rd,fd;0;a 1;1;0.5
#> 2    2     3   c;n;c thal;ca;cp      =;>;=  rd,fd;0;a 0;1;0.5
#> 3    3     3   c;n;c thal;ca;cp      =;>;=  rd,fd;0;a 1;0;0.5
#> 4    4     3   c;n;c thal;ca;cp      =;>;=  rd,fd;0;a 0;0;0.5

The resulting data frame my_tree_dfs contains 4 FFT definitions in its 4 rows. Per our design (above), they only differ in the values of their tree ID and the structures of their exits.

Applying sets of tree definitions to data

Having edited and collected a set of tree definitions, we now can evaluate the corresponding FFTs on some data. We can do this by providing our set of tree definitions (stored in the my_tree_dfs data frame) to the tree.definitions argument of the main FFTrees() function:

• The tree.definitions argument of the main FFTrees() function allows evaluating sets of FFT definitions for some data.

When using the main FFTrees() function with a set of tree.definitions (as a data frame in which each line contains the definition of one FFT), we can provide an existing FFTrees object (e.g., x from above). Importantly, however, the input of tree.definitions prevents the generation of new FFTs (via the “ifan” or “dfan” algorithms) and instead evaluates the FFT definitions provided on the data specified:²

# Evaluate tree.definitions for an existing FFTrees object y:
y <- FFTrees(object = x,                      # an existing FFTrees object
             tree.definitions = my_tree_dfs,  # new set of FFT definitions
             main = "Heart Disease 2"         # new label
             )

As we can see from the feedback messages, providing an FFTrees object (here x) re-uses its datasets, parameters, and labels, unless they are overwritten by new values. The resulting FFTrees object y contains the summary statistics resulting from applying the 4 FFT definitions of my_tree_dfs to the datasets that were used to create x (i.e., data = heart.train and data.test = heart.test).

We now can assess and visualize the performance of the FFTs in the usual ways, e.g., by the summary(), print() and plot() functions for y or its trees:

summary(y)

#> Heart Disease 2
#> 
#> FFTrees 
#> - Trees: 4 fast-and-frugal trees predicting diagnosis
#> - Parameters: algorithm = 'ifan', max.levels = 4,
#>               stopping.rule = 'exemplars', stopping.par = 0.1,
#>               goal = 'bacc', goal.chase = 'bacc', goal.threshold = 'bacc'.
#> 
#> 
#> Table: Tree definitions.
#> 
#> | tree| nodes|classes |cues       |directions |thresholds |exits   |
#> |----:|-----:|:-------|:----------|:----------|:----------|:-------|
#> |    1|     3|c;n;c   |thal;ca;cp |=;>;=      |rd,fd;0;a  |1;0;0.5 |
#> |    2|     3|c;n;c   |thal;ca;cp |=;>;=      |rd,fd;0;a  |0;1;0.5 |
#> |    3|     3|c;n;c   |thal;ca;cp |=;>;=      |rd,fd;0;a  |1;1;0.5 |
#> |    4|     3|c;n;c   |thal;ca;cp |=;>;=      |rd,fd;0;a  |0;0;0.5 |
#> 
#> 
#> Table: Tree statistics on training data [p(True) = 44%].
#> 
#> | tree|   n| hi| fa| mi| cr| sens| spec|  far|  ppv|  npv| dprime|  acc| bacc| wacc| cost_dec| cost_cue| cost|  pci|  mcu|
#> |----:|---:|--:|--:|--:|--:|----:|----:|----:|----:|----:|------:|----:|----:|----:|--------:|--------:|----:|----:|----:|
#> |    1| 150| 54| 18| 12| 66| 0.82| 0.79| 0.21| 0.75| 0.85|   1.69| 0.80| 0.80| 0.80|     0.20|        0| 0.20| 0.86| 1.76|
#> |    2| 150| 44|  7| 22| 77| 0.67| 0.92| 0.08| 0.86| 0.78|   1.79| 0.81| 0.79| 0.79|     0.19|        0| 0.19| 0.88| 1.58|
#> |    3| 150| 63| 42|  3| 42| 0.95| 0.50| 0.50| 0.60| 0.93|   1.66| 0.70| 0.73| 0.73|     0.30|        0| 0.30| 0.85| 1.98|
#> |    4| 150| 28|  2| 38| 82| 0.42| 0.98| 0.02| 0.93| 0.68|   1.74| 0.73| 0.70| 0.70|     0.27|        0| 0.27| 0.87| 1.68|
#> 
#> 
#> Table: Tree statistics on test data [p(True) = 48%].
#> 
#> | tree|   n| hi| fa| mi| cr| sens| spec|  far|  ppv|  npv| dprime|  acc| bacc| wacc| cost_dec| cost_cue| cost|  pci|  mcu|
#> |----:|---:|--:|--:|--:|--:|----:|----:|----:|----:|----:|------:|----:|----:|----:|--------:|--------:|----:|----:|----:|
#> |    1| 153| 64| 19|  9| 61| 0.88| 0.76| 0.24| 0.77| 0.87|   1.86| 0.82| 0.82| 0.82|     0.18|        0| 0.18| 0.87| 1.69|
#> |    2| 153| 49|  8| 24| 72| 0.67| 0.90| 0.10| 0.86| 0.75|   1.71| 0.79| 0.79| 0.79|     0.21|        0| 0.21| 0.87| 1.70|
#> |    3| 153| 70| 45|  3| 35| 0.96| 0.44| 0.56| 0.61| 0.92|   1.55| 0.69| 0.70| 0.70|     0.31|        0| 0.31| 0.85| 1.90|
#> |    4| 153| 28|  0| 45| 80| 0.38| 1.00| 0.00| 1.00| 0.64|   2.44| 0.71| 0.69| 0.69|     0.29|        0| 0.29| 0.87| 1.71|

# Visualize individual FFTs:
# plot(y, tree = 1)

Comparing the accuracy statistics of our new FFTs (in object y) to our original FFTs (in object x) shows that swapping the 2nd and 3rd cue had hardly an effect. Upon reflection, this is not surprising: Most people are still classified into the same categories as before. However, if we were to evaluate the costs of classification (e.g., by considering the pci and mcu measures or the cost measures for cue usage), we could still detect differences between FFTs that show the same accuracy.

Details

We just created a new FFTrees object y by using FFTrees object x for a set of customized FFTs defined by the tree.definitions argument. This circumvented the FFT building algorithms and used the provided FFT definitions instead. Thus, the ordinary distinction between training and test data no longer applies in this context: As no model is being fitted here, both sets were used to evaluate the FFTs in tree.definitions on these data. Acknowledging this, we may use the full dataset of heartdisease, rather than splitting it into two distinct subsets:

# Create a new FFTrees object z:
z <- FFTrees(formula = diagnosis ~ .,
             data = heartdisease,             # using full dataset
             tree.definitions = my_tree_dfs,  # new set of FFT definitions
             main = "Heart Disease 3"         # new label
             )

As before, we can evaluate the performance of our FFTs by obtaining a summary, or printing or plotting individual FFTs from the FFTrees object z:

# Summarize results: 
summary(z)

# Visualize an FFT:
plot(z, tree = 1)

Finally, if we mostly care about comparing our new FFTs with the automatically created ones, we could have added our new set my_tree_dfs to the old set (tree_dfs, obtained from x above).

(all_fft_dfs <- add_fft_df(my_tree_dfs, tree_dfs))

#> # A tibble: 11 × 7
#>     tree nodes classes cues             directions thresholds          exits    
#>    <int> <int> <chr>   <chr>            <chr>      <chr>               <chr>    
#>  1     1     3 c;c;n   thal;cp;ca       =;=;>      rd,fd;a;0           1;0;0.5  
#>  2     2     4 c;c;n;c thal;cp;ca;slope =;=;>;=    rd,fd;a;0;flat,down 1;0;1;0.5
#>  3     3     3 c;c;n   thal;cp;ca       =;=;>      rd,fd;a;0           0;1;0.5  
#>  4     4     4 c;c;n;c thal;cp;ca;slope =;=;>;=    rd,fd;a;0;flat,down 1;1;0;0.5
#>  5     5     3 c;c;n   thal;cp;ca       =;=;>      rd,fd;a;0           0;0;0.5  
#>  6     6     4 c;c;n;c thal;cp;ca;slope =;=;>;=    rd,fd;a;0;flat,down 0;0;0;0.5
#>  7     7     4 c;c;n;c thal;cp;ca;slope =;=;>;=    rd,fd;a;0;flat,down 1;1;1;0.5
#>  8     8     3 c;n;c   thal;ca;cp       =;>;=      rd,fd;0;a           1;1;0.5  
#>  9     9     3 c;n;c   thal;ca;cp       =;>;=      rd,fd;0;a           0;1;0.5  
#> 10    10     3 c;n;c   thal;ca;cp       =;>;=      rd,fd;0;a           1;0;0.5  
#> 11    11     3 c;n;c   thal;ca;cp       =;>;=      rd,fd;0;a           0;0;0.5

Doing so (e.g., by using the add_fft_df() function) creates a superset of 11 tree definitions, which can be evaluated together on the heartdisease data:

# Create a new FFTrees object a:
all <- FFTrees(formula = diagnosis ~ .,       
               data = heartdisease,             # using full dataset
               tree.definitions = all_fft_dfs,  # new set of FFT definitions
               main = "Heart Disease 4",        # new label
)

# Summarize results: 
summary(all)

Evaluating the performance of the corresponding FFTs (e.g., by summary(all)) shows that reversing the final two cues had little to no effects on accuracy (but note minor differences in costs, e.g., mcu).³