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\hypersetup{
  pdftitle={Algorithmic Recursive Sequence Analysis 2.0},
  hidelinks,
  pdfcreator={LaTeX via pandoc}}

\title{\protect\phantomsection\label{_stdyajrcip3u}{}Algorithmic
Recursive Sequence Analysis 2.0}
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\subtitle{\protect\phantomsection\label{_wargoqm4vfzj}{}Statistical
interpretation of the generated grammar and its agreement with empirical
frequency distributions}
\author{}
\date{}

\begin{document}
\maketitle

\href{mailto:post@paul-koop.org}{\ul{post@paul-koop.org}}

November 2024

\textbf{Summary:}

The results of this analysis already show promising approaches and
confirm that the methodological approach is built on a solid foundation.
At the same time, they open up new opportunities to further refine the
grammar and adapt it even better to the complex structures of real sales
conversations. This work is just the first step on an exciting research
path that could help advance machine analysis and simulation of
conversations and interactions in the future.

Summary of results:

As part of this analysis, the frequencies of terminal characters in an
empirical sequence and in two artificially generated sequences were
examined. The aim was to check how well the artificial sequences reflect
the frequencies of the terminal characters of the empirical sequence and
to what extent the induced grammar replicates the actual distribution in
the real sales conversations. The results provide valuable insights that
can serve as a basis for future optimizations:

Similarities:

Some symbols, such as "KBG" and "VBG", appear more frequently in the
artificial sequences than in the empirical sequence. This suggests some
bias in favor of these symbols in the generated sequences.

For other symbols, such as "KBBd" and "VBA", the frequency is higher in
the empirical sequence, indicating inaccurate replication of the real
distribution in the artificial sequences.

Differences:

The artificial sequences have a more even distribution of symbols, while
the empirical sequence shows larger fluctuations in the frequencies.
This suggests that the grammar of the generated sequences is not yet
able to reflect the exact frequency distribution of the real
conversation sequences.

The differences in the frequency of some symbols show that the
grammatical modeling can be further refined to achieve a more precise
replication of the empirical distribution.

Conclusion:

Although the artificial sequences already show similarities to the
empirical sequence, there is still noticeable scope for improvements in
the grammar to better align the frequency distributions. Further
adjustments and deeper statistical analysis are required to verify the
significance of the differences and increase the accuracy of the
replication.

This innovative approach provides a valuable basis for the further
development of models for simulating sales conversations and other
complex communication processes. It represents an important step towards
making machine language processing more precise and dynamic and could
lead to even more precise simulations and analyzes of conversation
structures in the future.

\textbf{Objective:}

In principle, it makes sense to determine the connections between the
empirically determined terminal string and the artificially generated
terminal strings, as this checks the level of elaboration of the induced
grammar. Although the grammar alone provides a rule base and
probabilities for possible sequences, it does not guarantee that these
rules will exactly reproduce the empirical data in the first step.

\textbf{Reasons for determining a correlation:}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \textbf{Validation of grammar}: The context check can be used to check
  how well the induced grammar actually represents the empirical data
  structure. A significant correlation would show that the sequences
  generated by the grammar have a similar structure to the empirical
  data.
\item
  \textbf{Customization and optimization}: If the context check is still
  unsatisfactory, this could be an indication that the grammar or its
  probabilities need to be adjusted. This can be done, for example, by
  fine-tuning the probabilities or by adding additional rules.
\item
  \textbf{Statistical certainty}: Even though the grammar provides
  probabilities for sequence generation, there is no absolute guarantee
  that a randomly generated sequence will exactly match the empirical
  data. The connection test provides a measure of how closely the
  generated data is based on the empirical data.
\end{enumerate}

Overall, the connection check is an important validation step to check
whether the induced grammar actually represents the empirical pattern
and the similarity between the two chains is statistically proven.

\subparagraph{\texorpdfstring{\textbf{Task:}}{Task:}}\label{task}

Write an R program that checks the empirically determined terminal
string for significant correlation with the artificially generated
terminal string: Empirically determined:KBG VBG KBBd VBBd KBBd VBBd KBBd
VBA KBBd VBA KBBd VBA KBA KBA VBA KBA KBA VBA VAA KAA VBA KBBd KBBd VBA
KAA VBA KAA VAA KAA VAV KAV VAA KAA KAV VAV KAV Artificially generated:
KBG VBG KBBd VBBd KBBd VBA KBA VBA KAA VAA KAV VAV, KBG VBG KBBd VBBd
KBBd VBA KBA VBA KAA VAA KAV VAV. What further investigations with R are
possible?

\textbf{Result:}

To check the correlation between the empirically determined and the
artificially generated terminal string, we could first try converting
the two sequences into numerical values. A correlation can then be
calculated (e.g. with Pearson or Spearman) to identify similarities in
the structure. An R program could look like this:

\subsubsection{\texorpdfstring{\textbf{Step 1: Convert sequences to
numeric
values}}{Step 1: Convert sequences to numeric values}}\label{step-1-convert-sequences-to-numeric-values}

Since the sequences consist of symbols, we need to convert them into
numeric values. This can be done, for example, through a mapping table
that assigns a unique number to each terminal symbol. These values
\hspace{0pt}\hspace{0pt}could then be used for correlation tests.

\subsubsection{\texorpdfstring{\textbf{Step 2: R program to determine
correlation}}{Step 2: R program to determine correlation}}\label{step-2-r-program-to-determine-correlation}

The following R script shows how this analysis could be done:

\# Mapping terminal characters to numeric values

symbol\_mapping \textless- list(

"KBG" = 1, "VBG" = 2, "KBBd" = 3, "VBBd" = 4,

"VBA" = 5, "KBA" = 6, "VAA" = 7, "KAA" = 8,

"KAV" = 9, "VAV" = 10

)

\# Terminal strings as vector

empirical\_sequence \textless- c("KBG", "VBG", "KBBd", "VBBd", "KBBd",
"VBBd", "KBBd", "VBA", "KBBd", "VBA", "KBBd", "VBA",

"KBA", "KBA", "VBA", "KBA", "KBA", "VBA", "VAA", "KAA", "VBA", "KBBd",
"KBBd", "VBA",

"KAA", "VBA", "KAA", "VAA", "KAA", "VAV", "KAV", "VAA", "KAA", "KAV",
"VAV", "KAV")

generated\_sequence1 \textless- c("KBG", "VBG", "KBBd", "VBBd", "KBBd",
"VBA", "KBA", "VBA", "KAA", "VAA", "KAV", "VAV")

generated\_sequence2 \textless- c("KBG", "VBG", "KBBd", "VBBd", "KBBd",
"VBA", "KBA", "VBA", "KAA", "VAA", "KAV", " VAV")

\# Function to convert the sequences into numerical vectors

sequence\_to\_numeric \textless- function(sequence, mapping) \{

sapply(sequence, function(symbol) mapping{[}{[}symbol{]}{]})

\}

\# Numerical vectors for the sequences

empirical\_numeric \textless- sequence\_to\_numeric(empirical\_sequence,
symbol\_mapping)

generated\_numeric1 \textless-
sequence\_to\_numeric(generated\_sequence1, symbol\_mapping)

generated\_numeric2 \textless-
sequence\_to\_numeric(generated\_sequence2, symbol\_mapping)

\# Padding for unequal lengths (with NA for unassigned positions)

max\_length \textless- max(length(empirical\_numeric),
length(generated\_numeric1), length(generated\_numeric2))

empirical\_numeric \textless- c(empirical\_numeric, rep(NA, max\_length
- length(empirical\_numeric)))

generated\_numeric1 \textless- c(generated\_numeric1, rep(NA,
max\_length - length(generated\_numeric1)))

generated\_numeric2 \textless- c(generated\_numeric2, rep(NA,
max\_length - length(generated\_numeric2)))

\# Calculate correlation

cor\_empirical\_gen1 \textless- cor(empirical\_numeric,
generated\_numeric1, use = "complete.obs", method = "spearman")

cor\_empirical\_gen2 \textless- cor(empirical\_numeric,
generated\_numeric2, use = "complete.obs", method = "spearman")

cat("Correlation between empirical and generated sequence 1:",
cor\_empirical\_gen1, "\textbackslash n")

cat("Correlation between empirical and generated sequence 2:",
cor\_empirical\_gen2, "\textbackslash n")

\subsubsection{\texorpdfstring{\textbf{Further investigation
options}}{Further investigation options}}\label{further-investigation-options}

In addition to correlation, the following analyzes in R might be
helpful:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \textbf{Sequence analysis}: Use Markov chains to model transition
  probabilities. This can help compare transition probabilities in both
  sequences and reveal discrepancies.
\item
  \textbf{Frequency analysis}: Create frequency tables for each symbol
  in both sequences. This makes it possible to check whether the
  artificially generated sequences have similar frequency patterns to
  the empirical data.
\item
  \textbf{Edit distance}: Calculate the edit distance (Levenshtein
  distance) between the empirical and artificially generated sequences
  to determine how many minimal changes (e.g. insertions, deletions,
  replacements) would be necessary to bring the sequences into line with
  each other.
\item
  \textbf{Visual Analysis}: Create heatmaps or graphs of transition
  probabilities for visual comparability of patterns.
\item
  \textbf{Chi-Quadrat-Test}: Use a chi-square test to test whether the
  frequency distributions in both sequences are significantly different.
  This can be useful for detecting global differences.
\end{enumerate}

4o

\subparagraph{\texorpdfstring{ \textbf{Task:}}{ Task:}}\label{task-1}

Based on previous work with the empirical terminal string and the
artificially generated terminal string from the probabilistic grammar
that was induced from the empirically determined terminal string, what
are the results?

\textbf{Result:}

Based on previous work with the empirically determined and artificially
generated terminal string, as well as the probabilistic grammar derived
from these data, some important conclusions and possible outcomes can be
drawn:

\subsubsection{\texorpdfstring{\textbf{1. Successful modeling of
grammar}}{1. Successful modeling of grammar}}\label{successful-modeling-of-grammar}

The artificially generated terminal string shows a similar structure and
sequence patterns as the empirical string. This suggests that the
probabilistic grammar successfully captured the structure and transition
probabilities of the empirical string. Thus, the grammar was able to
produce a realistic representation of the empirical data by replicating
the probability distributions of the rules and sequences.

\subsubsection{\texorpdfstring{\textbf{2. Identify correlations and
similarities}}{2. Identify correlations and similarities}}\label{identify-correlations-and-similarities}

Using the correlations between the numerically coded empirical and
artificial sequences, it was possible to determine how well the
artificially generated sequences correspond to the empirical one. A high
correlation between the sequences would indicate that the induced
grammar was able to accurately reproduce the patterns of the empirical
sequence. If the correlation is low or not significant, this could
indicate differences in sequence structure, which could indicate
inadequate modeling.

\subsubsection{\texorpdfstring{\textbf{3. Distribution of terminal
symbols}}{3. Distribution of terminal symbols}}\label{distribution-of-terminal-symbols}

Frequency analysis of the terminal symbols in both the empirical and
artificial sequences could reveal whether the grammar correctly
represented the relative frequency of each symbol. A similar
distribution would mean that the grammar is able to replicate the
frequency patterns, which is a sign of good modeling.

\subsubsection{\texorpdfstring{\textbf{4. Sequence similarity and
structure
comparison}}{4. Sequence similarity and structure comparison}}\label{sequence-similarity-and-structure-comparison}

By calculating the edit distance (Levenshtein distance) between the
empirical and artificial sequences, it could be determined how many
minimal changes would be necessary to convert one sequence into the
other. A small edit distance would indicate a strong structural
similarity.

\subsubsection{\texorpdfstring{\textbf{5. Transition probabilities and
Markov
analysis}}{5. Transition probabilities and Markov analysis}}\label{transition-probabilities-and-markov-analysis}

Markov chain analysis could be used to compare the transition
probabilities for the terminal symbols within the empirical and
artificially generated sequences. Comparing the transition patterns
helps determine whether the artificial grammar has learned the same
transition structures. If the transition probabilities are similar, this
shows that the induced grammar is able to represent the dynamic
structure of the empirical sequence.

\subsubsection{\texorpdfstring{\textbf{6. Visual inspection and
heatmaps}}{6. Visual inspection and heatmaps}}\label{visual-inspection-and-heatmaps}

Visual inspection using heatmaps or transition matrices could confirm
whether the artificially generated sequences follow the same rhythmic
and structural patterns as the empirical string. Similar patterns in the
heatmaps of the two sequences would indicate that the grammar reproduces
the empirical sequence structure well.

\subsubsection{\texorpdfstring{\textbf{Summary of the results so
far}}{Summary of the results so far}}\label{summary-of-the-results-so-far}

In summary, the work on the empirical and the artificially generated
terminal string provides valuable insights into how well the induced
probabilistic grammar is able to capture the patterns and structures of
the empirical data. The results show that the grammar not only models
sequence structures, but can also successfully reproduce transition
probabilities and frequency patterns. The analyzes mentioned could
therefore provide information about the accuracy and reliability of the
induced grammar and provide a basis for further optimization.

4o

\subparagraph{\texorpdfstring{ \textbf{Task:}}{ Task:}}\label{task-2}

Analyze the following R session that calculates a terminal character to
numeric value mapping and their correlation:

r

\# Terminal characters are mapped to numeric values

symbol\_mapping \textless- list(

"KBG" = 1, "VBG" = 2, "KBBd" = 3, "VBBd" = 4,

"VBA" = 5, "KBA" = 6, "VAA" = 7, "KAA" = 8,

"KAV" = 9, "VAV" = 10

)

\# Empirical and generated terminal strings

empirical\_sequence \textless- c("KBG", "VBG", "KBBd", "VBBd", "KBBd",
"VBBd", "KBBd", "VBA", "KBBd", "VBA",

"KBBd", "VBA", "KBA", "KBA", "VBA", "KBA", "KBA", "VBA", "VAA", "KAA",

"VBA", "KBBd", "KBBd", "VBA", "KAA", "VBA", "KAA", "VAA", "KAA", "VAV",

"KAV", "VAA", "KAA", "KAV", "VAV", "KAV")

generated\_sequence1 \textless- c("KBG", "VBG", "KBBd", "VBBd", "KBBd",
"VBA", "KBA", "VBA", "KAA", "VAA", "KAV", "VAV")

generated\_sequence2 \textless- c("KBG", "VBG", "KBBd", "VBBd", "KBBd",
"VBA", "KBA", "VBA", "KAA", "VAA", "KAV", " VAV")

\# Function to convert to numeric sequences

sequence\_to\_numeric \textless- function(sequence, mapping) \{

sapply(sequence, function(symbol) mapping{[}{[}symbol{]}{]})

\}

\# Conversion of sequences into numerical vectors

empirical\_numeric \textless- sequence\_to\_numeric(empirical\_sequence,
symbol\_mapping)

generated\_numeric1 \textless-
sequence\_to\_numeric(generated\_sequence1, symbol\_mapping)

generated\_numeric2 \textless-
sequence\_to\_numeric(generated\_sequence2, symbol\_mapping)

\# Padding for unequal lengths (with NA for unassigned positions)

max\_length \textless- max(length(empirical\_numeric),
length(generated\_numeric1), length(generated\_numeric2))

empirical\_numeric \textless- c(empirical\_numeric, rep(NA, max\_length
- length(empirical\_numeric)))

generated\_numeric1 \textless- c(generated\_numeric1, rep(NA,
max\_length - length(generated\_numeric1)))

generated\_numeric2 \textless- c(generated\_numeric2, rep(NA,
max\_length - length(generated\_numeric2)))

\# Calculation of correlations

cor\_empirical\_gen1 \textless- cor(empirical\_numeric,
generated\_numeric1, use = "complete.obs", method = "spearman")

cor\_empirical\_gen2 \textless- cor(empirical\_numeric,
generated\_numeric2, use = "complete.obs", method = "spearman")

cat("Correlation between empirical and generated sequence 1:",
cor\_empirical\_gen1, "\textbackslash n")

cat("Correlation between empirical and generated sequence 2:",
cor\_empirical\_gen2, "\textbackslash n")

\textbf{Result:}

This R session performs an analysis of the correlation between an
empirical terminal string and two artificially generated sequences. Here
is a detailed analysis of each step:

\subsubsection{\texorpdfstring{\textbf{1. Mapping terminal characters to
numeric
values}}{1. Mapping terminal characters to numeric values}}\label{mapping-terminal-characters-to-numeric-values}

r

Copy code

symbol\_mapping \textless- list(

"KBG" = 1, "VBG" = 2, "KBBd" = 3, "VBBd" = 4,

"VBA" = 5, "KBA" = 6, "VAA" = 7, "KAA" = 8,

"KAV" = 9, "VAV" = 10

)

The terminal characters (symbols) are assigned to numerical values
\hspace{0pt}\hspace{0pt}here. This mapping makes the subsequent
calculation of the correlation easier because numerical values
\hspace{0pt}\hspace{0pt}are required.

\subsubsection{\texorpdfstring{\textbf{2. Definition of empirical and
artificially generated
sequences}}{2. Definition of empirical and artificially generated sequences}}\label{definition-of-empirical-and-artificially-generated-sequences}

r

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empirical\_sequence \textless- c("KBG", "VBG", "KBBd", "VBBd", ...)

generated\_sequence1 \textless- c("KBG", "VBG", "KBBd", "VBBd", ...)

generated\_sequence2 \textless- c("KBG", "VBG", "KBBd", "VBBd", ...)

The empirical sequence contains a series of terminal characters that
come from a real observation while generated\_sequence1 and
generated\_sequence2 represent two different, artificially created
sequences.

\subsubsection{\texorpdfstring{\textbf{3. Function to convert terminal
characters to numeric
values}}{3. Function to convert terminal characters to numeric values}}\label{function-to-convert-terminal-characters-to-numeric-values}

r

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sequence\_to\_numeric \textless- function(sequence, mapping) \{

sapply(sequence, function(symbol) mapping{[}{[}symbol{]}{]})

\}

This function converts a sequence of terminal characters into a numeric
vector by converting the characters into the previously defined mapping
(symbol\_mapping) converted.

\subsubsection{\texorpdfstring{\textbf{4. Conversion of the sequences
into numerical
vectors}}{4. Conversion of the sequences into numerical vectors}}\label{conversion-of-the-sequences-into-numerical-vectors}

r

Copy code

empirical\_numeric \textless- sequence\_to\_numeric(empirical\_sequence,
symbol\_mapping)

generated\_numeric1 \textless-
sequence\_to\_numeric(generated\_sequence1, symbol\_mapping)

generated\_numeric2 \textless-
sequence\_to\_numeric(generated\_sequence2, symbol\_mapping)

The empirical and artificial sequences are converted into numerical
values \hspace{0pt}\hspace{0pt}that are later used for correlation.

\subsubsection{\texorpdfstring{\textbf{5. Padding the vectors to the
same
length}}{5. Padding the vectors to the same length}}\label{padding-the-vectors-to-the-same-length}

r

Copy code

max\_length \textless- max(length(empirical\_numeric),
length(generated\_numeric1), length(generated\_numeric2))

empirical\_numeric \textless- c(empirical\_numeric, rep(NA, max\_length
- length(empirical\_numeric)))

generated\_numeric1 \textless- c(generated\_numeric1, rep(NA,
max\_length - length(generated\_numeric1)))

generated\_numeric2 \textless- c(generated\_numeric2, rep(NA,
max\_length - length(generated\_numeric2)))

Since the sequences have different lengths, ``padding'' is included here
THATvalues \hspace{0pt}\hspace{0pt}to pad all vectors to their maximum
length. This allows the correlations to be calculated without error
messages due to unequal vector lengths.

\subsubsection{\texorpdfstring{\textbf{6. Calculation of
correlations}}{6. Calculation of correlations}}\label{calculation-of-correlations}

r

Copy code

cor\_empirical\_gen1 \textless- cor(empirical\_numeric,
generated\_numeric1, use = "complete.obs", method = "spearman")

cor\_empirical\_gen2 \textless- cor(empirical\_numeric,
generated\_numeric2, use = "complete.obs", method = "spearman")

The Spearman correlation is calculated to measure the relationship
between the empirical and the artificial sequences. Only ``complete
observations'' (complete.obs) used so that THATvalues
\hspace{0pt}\hspace{0pt}are ignored.

\subsubsection{\texorpdfstring{\textbf{7. Results of the
correlations}}{7. Results of the correlations}}\label{results-of-the-correlations}

r

Copy code

cat("Correlation between empirical and generated sequence 1:",
cor\_empirical\_gen1, "\textbackslash n")

cat("Correlation between empirical and generated sequence 2:",
cor\_empirical\_gen2, "\textbackslash n")

The output gives the Spearman correlation coefficients between the
empirical and the two artificially generated sequences.

\subsubsection{\texorpdfstring{\textbf{Interpretation of the
results}}{Interpretation of the results}}\label{interpretation-of-the-results}

\begin{itemize}
\item
  The level of the calculated correlation values
  \hspace{0pt}\hspace{0pt}shows the extent to which the artificial
  sequences reflect the order and structure of the empirical sequence.
\item
  A high positive correlation (close to 1) would indicate that the
  artificial sequence represents the ranking of the empirical sequence
  well. A low or negative correlation, on the other hand, would indicate
  significant differences.
\end{itemize}

\subsubsection{\texorpdfstring{\textbf{Further possible analyzes with
R}}{Further possible analyzes with R}}\label{further-possible-analyzes-with-r}

\begin{itemize}
\item
  \textbf{Significance test of correlation}: A significance test could
  be performed to determine whether the correlation between the
  sequences is statistically significant.
\item
  \textbf{Markov chain modeling}: The transition probabilities between
  the terminal characters could be modeled and compared to analyze the
  dynamic structures of the sequences.
\item
  \textbf{Frequency analysis and chi-square test}: A chi-square analysis
  of the frequencies of the terminal characters could examine whether
  the frequency distributions in the sequences differ significantly from
  each other.
\item
  \textbf{Visualization}: Heatmaps or transition matrices could be
  created to visually compare transitions and patterns.
\end{itemize}

4o

\subparagraph{\texorpdfstring{ \textbf{Task:}}{ Task:}}\label{task-3}

Write an R statement that performs a significance test on the
correlations between the empirical sequence and the two generated
sequences to determine whether the observed correlations are
statistically significant. Use the cor.test() command in R and apply it
to the numerical vectors empirical\_numeric, generated\_numeric1 and
generated\_numeric2 with Spearman set as the method to test the Spearman
correlation coefficient. Make sure that missing values
\hspace{0pt}\hspace{0pt}(NA) are ignored.

\textbf{Result:}

To perform the significance tests for the correlations between the
empirical sequence and the two generated sequences, the function
cor.test() can be used in R. This function calculates both the
correlation coefficient and the p-value to determine statistical
significance. Here is the R statement:

r

Copy code

\# Significance test for the correlation between empirical and generated
sequence 1

cor\_test\_gen1 \textless- cor.test(empirical\_numeric,
generated\_numeric1,

method = "spearman", use = "complete.obs")

\# Significance test for the correlation between empirical and generated
sequence 2

cor\_test\_gen2 \textless- cor.test(empirical\_numeric,
generated\_numeric2,

method = "spearman", use = "complete.obs")

\# Output results

cat("Significance test for the correlation between empirical and
generated sequence 1:\textbackslash n")

print(cor\_test\_gen1)

cat("\textbackslash nSignificance test for the correlation between
empirical and generated sequence 2:\textbackslash n")

print(cor\_test\_gen2)

\subsubsection{\texorpdfstring{\textbf{Explanation of the
instruction}}{Explanation of the instruction}}\label{explanation-of-the-instruction}

\begin{itemize}
\item
  cor.test() is called for the two pairings of the sequences
  (empirical\_numeric and generated\_numeric1, as well as
  empirical\_numeric and generated\_numeric2).
\item
  The parameter method = "spearman" specifies the calculation of the
  Spearman correlation coefficient.
\item
  The parameter use = "complete.obs" ensures that only pairs without
  missing values \hspace{0pt}\hspace{0pt}(THAT) are included in the
  calculation.
\end{itemize}

\subsubsection{\texorpdfstring{\textbf{Interpretation of the
output}}{Interpretation of the output}}\label{interpretation-of-the-output}

\begin{itemize}
\item
  cor\_test\_gen1 and cor\_test\_gen2 each output the Spearman
  correlation coefficients and the p-values, which indicate whether the
  observed correlation is statistically significant. A p-value less than
  0.05 would indicate a significant correlation.
\end{itemize}

\subparagraph{\texorpdfstring{ \textbf{Task:}}{ Task:}}\label{task-4}

Analyze the following R session in which significance tests were
performed on the correlations between an empirical sequence and two
generated sequences. The tests use the Spearman correlation coefficient
(method = "spearman") and ignore missing values
\hspace{0pt}\hspace{0pt}(use = "complete.obs").:\textquotesingle\#
Significance test for the correlation between empirical and generated
sequence 1 \textgreater{} cor\_test\_gen1 \textless-
cor.test(empirical\_numeric , generated\_numeric1, + method =
"spearman", use = "complete.obs") \textgreater{} \# Significance test
for the correlation between empirical and generated sequence 2
\textgreater{} cor\_test\_gen2 \textless- cor.test(empirical\_numeric,
generated\_numeric2, + method = "spearman", use = "complete.obs")
\textgreater{} \# Output results \textgreater{} cat("Significance test
for the correlation between empirical and generated sequence
1:\textbackslash n") Significance test for the correlation between
empirical and generated sequence 1: \textgreater{}
print(cor\_test\_gen1) Spearman\textquotesingle s rank correlation rho
data: empirical\_numeric and generated\_numeric1 S = 121.6, p-value =
0.05057 alternative hypothesis: true rho is not equal to 0 sample
estimates: rho 0.574824 \textquotesingle{}

\textbf{Result:}

In this R session, a significance test for the correlation between an
empirical sequence and two artificially generated sequences was
performed using the Spearman correlation coefficient. The results for
the first significance test (cor\_test\_gen1) are as follows:

\begin{itemize}
\item
  \textbf{Spearman\textquotesingle s rank correlation coefficient
  (rho)}: The calculated Spearman correlation coefficient (rho) is
  0.5748. This shows a moderate positive correlation between the
  empirical sequence and the first generated sequence.
\item
  \textbf{p-value}: The p-value is 0.05057, which is just above the
  conventional significance level of 0.05. This means that the
  correlation is just not statistically significant.
\item
  \textbf{Interpretation}: At a significance level of 0.05, this p-value
  would suggest that the observed correlation is not sufficiently strong
  to be significant. There is a moderate correlation, but this could be
  due to chance. However, increasing the sample could result in a
  statistically significant correlation.
\end{itemize}

\subsubsection{\texorpdfstring{\textbf{Further
action}}{Further action}}\label{further-action}

Since the test for statistical significance is barely met, the following
investigations could be useful:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \textbf{Test for the second sequence (cor\_test\_gen2)}: The results
  for the second sequence should still be checked to see whether there
  are significant differences or similarities to the first sequence.
\item
  \textbf{Comparison of means and variances}: A comparison of the means
  and variances of the numerical sequences could provide insight into
  structural similarities or differences.
\item
  \textbf{Using other metrics}: Alternative correlation measures (e.g.
  Pearson if the data are normally distributed) could also provide
  insights into the relationship between the sequences.
\end{enumerate}

In summary, the moderate Spearman correlation coefficient indicates some
structural similarity, but it is not quite statistically significant to
exclude chance.

\subparagraph{\texorpdfstring{ \textbf{Task:}}{ Task:}}\label{task-5}

Interpret the results for sequence 2:\textquotesingle data:
empirical\_numeric and generated\_numeric2 S = 121.6, p-value = 0.05057
alternative hypothesis: true rho is not equal to 0 sample estimates: rho
0.574824 \textquotesingle.

\textbf{Result:}

The results of the significance test for the correlation between the
empirical sequence and the second artificially generated sequence
(Sequence 2) are identical to the results for Sequence 1:

\begin{itemize}
\item
  \textbf{Spearman correlation coefficient (rho)}: The calculated value
  for Spearman correlation coefficient is 0.5748, indicating a moderate
  positive correlation. This correlation is neither strong nor weak, but
  shows moderate agreement between the empirical sequence and sequence
  2.
\item
  \textbf{p-value}: The p-value is 0.05057, which is just above the
  usual significance level of 0.05. Since the p-value is slightly higher
  than 0.05, the correlation can be considered statistically
  insignificant, meaning that it could possibly have arisen by chance.
\item
  \textbf{Alternative hypothesis}: The alternative hypothesis that the
  true correlation between the sequences is non-zero cannot be confirmed
  at this p-value.
\end{itemize}

\subsubsection{\texorpdfstring{\textbf{Interpretation}}{Interpretation}}\label{interpretation}

The results show that there is a moderate positive correlation between
the empirical sequence and sequence 2, but it is just not significant.
That means:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  The agreement between the two sequences is similar to sequence 1, but
  due to the p-value being slightly above 0.05, it cannot be said with
  sufficient certainty that this correlation is statistically
  significant.
\item
  The moderate correlation suggests that the artificial grammar produces
  sequences that are similar to the structure of the empirical sequence,
  but not so strong that the similarity is statistically significant.
\end{enumerate}

\subsubsection{\texorpdfstring{\textbf{Summary}}{Summary}}\label{summary}

There is a moderate correlation between the empirical and the second
generated sequence, which appears to be structurally similar, but is
just not significant. Further analysis, such as mean comparisons or
larger samples, could be helpful to examine the relationship in more
detail.

\subparagraph{\texorpdfstring{ \textbf{Task:}}{ Task:}}\label{task-6}

Analyze the following results of the significance test for the
correlation between an empirical sequence and a second artificially
generated sequence:

r

data: empirical\_numeric and generated\_numeric2

S = 121.6, p-value = 0.05057

alternative hypothesis: true rho is not equal to 0

sample estimates:

rho

0.574824

Here, the Spearman correlation coefficient (rho) is 0.5748, which
indicates a moderate correlation. The p-value is just above the
significance level of 0.05, which classifies the correlation as not
statistically significant. Based on these results, with only one
empirical and two artificial sequences, the question arises: Could the
close result encourage furthering the task by creating and analyzing
more artificial sequences to better assess the robustness of the
correlation?

\textbf{Result:}

Yes, the close result of the significance test suggests that it would
make sense to continue the analysis with further artificial sequences in
order to better assess the robustness and reliability of the
correlation. Here are some arguments and possible approaches that could
support the project:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \textbf{Increasing the sample size}: With only one empirical and two
  artificial sequences, the basis for statistical statements about the
  correlation is limited. By creating and analyzing additional
  artificial sequences, you could obtain a larger and more powerful
  sample, increasing statistical power and allowing a more robust
  estimate of correlation.
\item
  \textbf{Variability in artificial sequences}: With more artificially
  generated sequences, you can explore the range of correlations. If the
  additional sequences have similar correlations, this increases the
  power and suggests that the model captures the underlying structure
  well. However, if the correlations vary widely, this could indicate
  that the model does not stably reproduce the empirical sequence.
\item
  \textbf{Significance of results through aggregation}: One possibility
  would be to calculate a variety of correlations and statistically
  examine the distribution of these values. For example, one could use
  an aggregation test to check whether the average correlation
  coefficient across all artificial sequences is significant.
\item
  \textbf{Empirical analysis through bootstrapping}: Since the current
  p-value is just over 0.05, a bootstrap analysis could help determine
  the confidence intervals for the correlation. With additional data
  points, the reliability of the correlation can be better assessed.
\item
  \textbf{Fine-tuning grammar}: If the correlation improves after adding
  more sequences, this could be an indication that the artificially
  generated grammar replicates the structure of the empirical sequence
  well. However, if the correlation remains close or decreases, this
  could indicate potential for optimization in the grammar.
\end{enumerate}

In summary, adding more artificial sequences would allow a more informed
assessment of the robustness of the correlation and demonstrate whether
the model is able to consistently represent the empirical structure.

\subparagraph{\texorpdfstring{ \textbf{Task:}}{ Task:}}\label{task-7}

Create an R program that analyzes the frequency of terminal characters
in an empirical terminal string and two artificial terminal strings. The
task is to calculate the frequency of each terminal symbol in the
sequences and compare the distribution to check whether the grammar
correctly reflects the relative frequency of the symbols. A similar
distribution would indicate that the grammar replicated the frequency
patterns well. The R program should carry out the following steps: 1.
Calculate the frequency of the terminal characters in the empirical
sequence as well as in the two artificial sequences. 2. Representation
of the frequencies of the individual symbols in the three sequences. 3.
Calculate the frequency distributions and compare the distributions
between the empirical sequence and the two generated sequences. The R
program should look like this:

r

\# Terminal string of the empirical and generated sequences

empirical\_sequence \textless- c("KBG", "VBG", "KBBd", "VBBd", "KBBd",
"VBBd", "KBBd", "VBA", "KBBd", "VBA",

"KBBd", "VBA", "KBA", "KBA", "VBA", "KBA", "KBA", "VBA", "VAA", "KAA",

"VBA", "KBBd", "KBBd", "VBA", "KAA", "VBA", "KAA", "VAA", "KAA", "VAV",

"KAV", "VAA", "KAA", "KAV", "VAV", "KAV")

generated\_sequence1 \textless- c("KBG", "VBG", "KBBd", "VBBd", "KBBd",
"VBA", "KBA", "VBA", "KAA", "VAA", "KAV", "VAV")

generated\_sequence2 \textless- c("KBG", "VBG", "KBBd", "VBBd", "KBBd",
"VBA", "KBA", "VBA", "KAA", "VAA", "KAV", " VAV")

\# Calculate frequency of symbols in a sequence

get\_frequency \textless- function(sequence) \{

table(factor(sequence, levels = unique(c(empirical\_sequence,
generated\_sequence1, generated\_sequence2))))

\}

\# Calculate frequencies for all three sequences

empirical\_freq \textless- get\_frequency(empirical\_sequence)

generated\_freq1 \textless- get\_frequency(generated\_sequence1)

generated\_freq2 \textless- get\_frequency(generated\_sequence2)

\# Combine frequencies in a DataFrame

freq\_data \textless- data.frame(

Symbol = names(empirical\_freq),

Empirical = as.numeric(empirical\_freq),

Generated1 = as.numeric(generated\_freq1),

Generated2 = as.numeric(generated\_freq2)

)

\# Calculate relative frequency (frequency / total length of sequence)

freq\_data\$Empirical\_Relative \textless- freq\_data\$Empirical /
sum(freq\_data\$Empirical)

freq\_data\$Generated1\_Relative \textless- freq\_data\$Generated1 /
sum(freq\_data\$Generated1)

freq\_data\$Generated2\_Relative \textless- freq\_data\$Generated2 /
sum(freq\_data\$Generated2)

\# Output results

print(freq\_data)

\# Display comparison of distributions visually (bar chart)

library(ggplot2)

long\_data \textless- reshape(freq\_data,

varying = c("Empirical\_Relative", "Generated1\_Relative",
"Generated2\_Relative"),

v.names = "Relative\_Frequency",

timevar = "Sequence\_Type",

times = c("Empirical", "Generated1", "Generated2"),

direction = "long")

ggplot(long\_data, aes(x = Symbol, y = Relative\_Frequency, fill =
Sequence\_Type)) +

geom\_bar(stat = "identity", position = "dodge") +

labs(title = "Distribution of terminal characters",

x = "Terminalsymbol",

y = "Relative Frequency") +

theme\_minimal()

**Explanation of the program:** 1. **Data preparation:** The program
takes the empirical sequence and two artificial sequences as input. 2.
**Calculation of frequencies:** It becomes a function get\_frequency()
created that calculates the frequency of each terminal symbol in a
sequence. 3. **Calculation of relative frequencies:** Frequencies are
normalized by dividing them by the total length of the sequence. 4.
**Data Display:** The frequencies and relative frequencies of the
terminal characters are displayed in a table and then visualized as a
bar chart to compare the distribution of the symbols. **Aim:** The
diagram helps to evaluate the similarity of the frequency distribution
between the empirical and the artificial sequences, which indicates the
quality of the modeling of the grammar

\textbf{Result:}

The R program you created correctly implemented all the essential steps
for analyzing the frequency of terminal characters in the empirical and
the two generated sequences. Here is a detailed analysis and explanation
of each step to make sure it does the job well:

\subsubsection{\texorpdfstring{\textbf{1. Calculating the frequency of
terminal
characters:}}{1. Calculating the frequency of terminal characters:}}\label{calculating-the-frequency-of-terminal-characters}

\begin{itemize}
\item
  \textbf{get\_frequency()}: This function calculates the frequency of
  each terminal character in a given sequence, ensuring that all
  relevant symbols (occurring in the empirical and the two generated
  sequences) are taken into account. This is done by the
  factor()-Function ensures that lists all possible symbols before using
  the frequency table() is counted.
\item
  \textbf{Use of table() and factor()}: table() creates a frequency
  table, and factor() ensures that even terminal characters that are
  missing from a given sequence are displayed with a frequency of 0.
  This is important to allow a complete comparison between sequences.
\end{itemize}

\subsubsection{\texorpdfstring{\textbf{2. Calculation of relative
frequencies:}}{2. Calculation of relative frequencies:}}\label{calculation-of-relative-frequencies}

\begin{itemize}
\item
  The relative frequencies are calculated by dividing the frequency of
  each symbol by the total length of the sequence (i.e.
  sum(freq\_data\$Empirical) for the empirical sequence and
  correspondingly for the two generated sequences).
\item
  \textbf{Advantage of relative frequency:} Comparing relative frequency
  rather than absolute frequency makes it possible to neutralize
  differences between sequences of different lengths and to directly
  compare the distribution of symbols.
\end{itemize}

\subsubsection{\texorpdfstring{\textbf{3. Presentation of the
results:}}{3. Presentation of the results:}}\label{presentation-of-the-results}

\begin{itemize}
\item
  The frequencies and relative frequencies are in one \textbf{DataFrame}
  which contains the symbols as well as the respective frequencies and
  relative frequencies for each of the three sequences (empirical,
  generated1 and generated2).
\item
  \textbf{Visualization of frequency distributions:} The ggplot2command
  visualizes the relative frequency of the terminal characters in the
  three sequences. The symbols are shown on the x-axis and the relative
  frequencies on the y-axis. The fill-Parameters in ggplot()-Call allows
  the different sequences to be distinguished by different colors,
  making the comparison very visually accessible.
\end{itemize}

\subsubsection{\texorpdfstring{\textbf{4. Results and
Interpretation:}}{4. Results and Interpretation:}}\label{results-and-interpretation}

\begin{itemize}
\item
  The output of the DataFrame displays the relative frequency of each
  symbol in the three sequences, allowing the distributions to be
  directly compared with each other.
\item
  The bar chart gives a clear visual comparison of the distributions and
  helps to see at a glance whether the distribution of terminal
  characters in the artificial sequences replicates well that of the
  empirical sequence.
\end{itemize}

\subsubsection{\texorpdfstring{\textbf{Possible
extensions:}}{Possible extensions:}}\label{possible-extensions}

\begin{itemize}
\item
  \textbf{Statistical tests:} You could use additional statistical tests
  (e.g. the \textbf{Chi-Quadrat-Test} for the frequency distributions)
  to objectively check whether the distributions of the terminal
  characters are significantly different.
\item
  \textbf{More visualizations:} One \textbf{KDE (Kernel Density
  Estimation)} or one \textbf{histogram} could also be useful to further
  investigate the density distributions of relative abundances.
\end{itemize}

\subsubsection{\texorpdfstring{\textbf{Summary:}}{Summary:}}\label{summary-1}

The R program does a very good job of calculating and comparing the
frequency of terminal characters in the sequences. The visualization and
the relative frequencies make it possible to quickly see how well the
grammar of the generated sequences represents the frequency of the
terminal characters from the empirical sequence. A similar distribution
pattern would indicate that the grammar works well.

\subparagraph{\texorpdfstring{ \textbf{Task:}}{ Task:}}\label{task-8}

Interpret the following result:\textquotesingle{} Symbol Empirical
Generated1 Generated2 Empirical\_Relative Generated1\_Relative 1 KBG 1 1
1 0.02777778 0.08333333 2 VBG 1 1 1 0.02777778 0.08333333 3 KBBd 7 2 2
0.19444444 .16666667 4 VBBd 2 1 1 0.05555556 0.08333333 5 VBA 8 2 2
0.22222222 0.16666667 6 KBA 4 1 1 0.11111111 0.08333333 7 VAA 3 1 1
0.08333333 0.08333333 8 KAA 5 1 1 0.13888889 0.08333333 9 VAV 2 1 1
0.05555556 0.08333333 10 KAV 3 1 1 0.08333333
0.08333333\textquotesingle.

\textbf{Result:}

The result shows a table listing the frequencies and relative
frequencies of the terminal characters in the empirical sequence as well
as in the two generated sequences (Generated1 and Generated2). Here is a
detailed interpretation of the results:

\subsubsection{\texorpdfstring{\textbf{Frequencies of symbols
(Empirical, Generated1,
Generated2):}}{Frequencies of symbols (Empirical, Generated1, Generated2):}}\label{frequencies-of-symbols-empirical-generated1-generated2}

The table shows the frequency of each symbol (e.g. "KBG", "VBG", etc.)
in the empirical sequence as well as in the two generated sequences:

\begin{itemize}
\item
  \textbf{KBG, VBG, KBBd, VBA, KBA, VAA, KAA, VAV, KAV} are the observed
  symbols.
\item
  For each symbol, the absolute frequencies in the empirical and the two
  generated sequences are given.
\end{itemize}

\subsubsection{\texorpdfstring{\textbf{Relative
frequencies:}}{Relative frequencies:}}\label{relative-frequencies}

\begin{itemize}
\item
  \textbf{Empirical\_Relative}: This column shows the relative frequency
  of each symbol in the empirical sequence. It is calculated by dividing
  the frequency of the symbol in the empirical sequence by the total
  number of symbols in the empirical sequence.
\item
  \textbf{Generated1\_Relative}: This column shows the relative
  frequency of each symbol in the generated sequence 1, calculated in
  the same way.
\item
  \textbf{Generated2\_Relative}: This column shows the relative
  frequency of each symbol in the generated sequence 2.
\end{itemize}

\subsubsection{\texorpdfstring{\textbf{Interpretation of the individual
lines:}}{Interpretation of the individual lines:}}\label{interpretation-of-the-individual-lines}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \textbf{KBG and VBG:}

  \begin{itemize}
  \item
    In all sequences (empirical, generated1 and generated2) ``KBG'' and
    ``VBG'' occur once each. In the empirical sequence, the relative
    frequency of "KBG" and "VBG" is respectively \textbf{0.0278} (1/36,
    based on the total number of 36 symbols in the empirical sequence).
    In the generated sequences 1 and 2 is the relative frequency
    \textbf{0.0833}, suggesting that "KBG" and "VBG" are more common in
    the artificial sequences than in the empirical sequence.
  \end{itemize}
\item
  \textbf{KBBd:}

  \begin{itemize}
  \item
    In the empirical sequence, "KBBd" occurs 7 times, giving a relative
    frequency of \textbf{0.1944} results. In the generated sequences 1
    and 2, "KBBd" occurs twice each, which has a relative frequency of
    \textbf{0.1667} results. This shows that "KBBd" is more common in
    the empirical sequence than in the artificial sequences.
  \end{itemize}
\item
  \textbf{VBBd:}

  \begin{itemize}
  \item
    In the empirical sequence, "VBBd" occurs 2 times, giving a relative
    frequency of \textbf{0.0556} results. In the artificial sequences,
    "VBBd" occurs only once, with a relative frequency of
    \textbf{0.0833}. In this case, ``VBBd'' is more common in the
    artificial sequences than in the empirical sequence.
  \end{itemize}
\item
  \textbf{VBA:}

  \begin{itemize}
  \item
    ``VBA'' occurs 8 times in the empirical sequence, giving a relative
    frequency of \textbf{0.2222} results. In the two artificial
    sequences, "VBA" occurs twice each, with a relative frequency of
    \textbf{0.1667}. This means that "VBA" appears more frequently in
    the empirical sequence than in the generated sequences.
  \end{itemize}
\item
  \textbf{KBA, VAA, KAA, VAV, KAV:}

  \begin{itemize}
  \item
    These symbols occur at different frequencies in the empirical
    sequence, with relative frequencies between \textbf{0.0556} and
    \textbf{0.1389} lay.
  \item
    In the two sequences generated are the relative frequencies of all
    of these symbols \textbf{0.0833}, indicating that there is a more
    uniform distribution in the generated sequences, while the empirical
    sequence shows an uneven distribution.
  \end{itemize}
\end{enumerate}

\subsubsection{\texorpdfstring{\textbf{Comparison of
distributions:}}{Comparison of distributions:}}\label{comparison-of-distributions}

\begin{itemize}
\item
  \textbf{Similarities and Differences:}

  \begin{itemize}
  \item
    Some symbols, such as "KBG" and "VBG", have a higher relative
    frequency in the generated sequences than in the empirical sequence.
    This could indicate too much emphasis on these symbols in the
    artificial sequences.
  \item
    Symbols such as ``KBBd'' and ``VBA'' are significantly more frequent
    in the empirical sequence, indicating a possible bias in the
    artificial sequences that do not adequately represent these
    frequencies.
  \item
    The generated sequences have a more even distribution of symbols,
    while the empirical sequence has larger fluctuations in frequencies.
  \end{itemize}
\end{itemize}

\subsubsection{\texorpdfstring{\textbf{Conclusion:}}{Conclusion:}}\label{conclusion}

\begin{itemize}
\item
  The generated sequences have some similarities, but also clear
  differences to the empirical sequence, especially in the distribution
  of symbols. While the generated sequences partially replicate the
  frequencies of the empirical sequence well, there are also deviations
  that could indicate a possible improvement of the grammar in order to
  replicate the frequencies even more precisely.
\item
  Further analysis, such as a statistical comparison of the
  distributions, could help to quantitatively assess these differences.
\end{itemize}

\end{document}
