Calculate topology characteristics for different IQR factors

Description

This function analyzes how different IQR (Interquartile Range) factors affect the topology's characteristics. It helps users determine the optimal factor for their specific data by showing how the factor choice impacts the base size and set sizes in the resulting topology.

Usage

analyze_topology_factors(data, factors = NULL, plot = TRUE)

Arguments

Details

The function works by:

1. Calculating different thresholds using IQR/factor

2. Creating a subbase using these thresholds

3. Generating the base from intersections of subbase elements

4. Analyzing the resulting topology's characteristics

A larger factor results in a smaller threshold, which typically leads to a finer topology with more distinct sets but smaller set sizes.

Value

A data.frame with the following columns:

factor

Numeric. The IQR factor used for threshold calculation.

threshold

Numeric. The calculated threshold value (IQR/factor).

base_size

Integer. Number of sets in the topological base.

max_set_size

Integer. Size of the largest set in the base.

min_set_size

Integer. Size of the smallest set in the base.

If plot = TRUE, also generates a line plot showing the relationship between IQR factors and base sizes as a side effect.

Examples

# Generate sample data
data <- rnorm(100)

# Analyze topology with default factors
results <- analyze_topology_factors(data)
print(results)

# Use custom factors
custom_results <- analyze_topology_factors(data, factors = c(2, 4, 6))
print(custom_results)

Calculate multiple threshold methods for topology analysis

Description

Calculate multiple threshold methods for topology analysis

Usage

calculate_thresholds(data)

Arguments

Value

A named list containing five threshold values:

mean_diff

Numeric. Threshold based on mean of absolute differences between adjacent sorted values.

median_diff

Numeric. Threshold based on median of absolute differences between adjacent sorted values.

sd

Numeric. Threshold based on standard deviation of the data.

iqr

Numeric. Threshold based on IQR divided by 4.

dbscan

Numeric. Threshold based on DBSCAN-like density estimation using k-th nearest neighbor distance.

Calculate topology base size for a given threshold

Description

Calculate topology base size for a given threshold

Usage

calculate_topology(data, threshold)

Arguments

Value

integer scalar. The number of sets in the topological base. This value represents the complexity of the topology - larger values indicate more complex topological structure.

Create a complete topology from data points with neighborhood structure

Description

Create a complete topology from data points with neighborhood structure

Usage

complete_topology(datos)

Arguments

Value

A list with four components:

R

List. The equivalence relation defined on the data, where each element represents a relationship between vertices based on their degrees and values.

subbase

List. The neighborhoods of each vertex, where each element contains the indices of neighboring vertices in the graph.

base

List. The base generated from intersections of neighborhoods, including the empty set, full set, and all non-empty intersections.

topology

List. The complete topology generated by taking unions of base elements, satisfying the axioms of topological spaces.

Examples

data <- c(1, 2, 3, 4, 5)
result <- complete_topology(data)

Topology Analysis Functions for Time Series Data

Description

Functions for analyzing topological properties of time series data

Usage

is_topology_connected(topology)

Arguments

Details

This module provides three main functions for analyzing the connectivity of topological structures, particularly focused on economic time series:

• is_topology_connected: Uses an undirected graph approach

• is_topology_connected2: Uses a directed graph approach

• is_topology_connected_manual: Uses a manual checking approach

Created as part of research on economic cycle analysis. Check if a topology is connected using undirected graph approach

Value

logical scalar. Returns TRUE if the topology is connected (all elements can be reached from any starting point through the undirected graph representation), FALSE otherwise. Returns FALSE if the topology is empty.

Author(s)

José Mauricio Gómez Julián

Examples

topology <- list(c(1,2,3), c(3,4,5))
is_topology_connected(topology)

Check if a topology is connected using directed graph approach

Description

Check if a topology is connected using directed graph approach

Usage

is_topology_connected2(topology)

Arguments

Value

logical scalar. Returns TRUE if the topology is connected through directed graph traversal (considering sequential connections between elements), FALSE otherwise. Returns FALSE if the topology is empty.

Examples

topology <- list(c(1,2,3), c(3,4,5))
is_topology_connected2(topology)

Check if a topology is connected using manual checking approach

Description

Check if a topology is connected using manual checking approach

Usage

is_topology_connected_manual(topology)

Arguments

Value

logical scalar. Returns TRUE if all elements from 1 to the maximum value in the topology are present in at least one set, FALSE otherwise.

Examples

topology <- list(c(1,2,3), c(3,4,5))
is_topology_connected_manual(topology)

Create a topology with completely disconnected sets

Description

This function generates a topology where each set in the topology is a singleton (contains only one element), resulting in a completely disconnected topological structure. Each vertex exists in isolation, with no meaningful connections between sets.

Usage

simplest_topology(datos)

Arguments

Details

The subbase contains individual elements. The base consists of singleton sets. The topology is formed by these singleton sets. No meaningful topological relationships are established between elements.

Value

A list with four components:

R

List. The equivalence relation defined on the data vertices.

subbase

List. Sets of individual vertices forming the subbase.

base

List. Singleton sets forming the base of the topology.

topology

List. The complete topology consisting of disconnected singleton sets.

Examples

data <- c(1, 2, 3, 4, 5)
result <- simplest_topology(data)

Visualize and Compare Different Threshold Methods

Description

This function creates a comprehensive visualization of different threshold methods used in topology analysis. It generates three distinct plots to help understand the relationships between different threshold methods and their effects on topological structure.

Usage

visualize_topology_thresholds(data, plot = TRUE)

Arguments

Details

Visualize Topology Thresholds

Value

A data.frame with one row per threshold method containing:

method

Character. Name of the threshold calculation method.

threshold

Numeric. The calculated threshold value.

base_size

Integer. Number of sets in the resulting topological base.

If plot = TRUE, generates three plots as side effects: (1) Bar chart comparing threshold values by method, (2) Bar chart comparing base sizes by method, (3) Scatter plot showing threshold vs base size relationship.

Examples

# Generate sample data
data <- rnorm(100)

# Visualize threshold comparisons
results <- visualize_topology_thresholds(data)