1. Problem & Motivation: A Topology-First Guardrail
Many workflows jump straight into smoothing, imputation, or global trend estimation without checking whether the data behave as a single connected system. topologyR elevates local metric information (adjacent distances) into neighborhoods → base → full topology, then poses a decisive question: Is the induced topology connected? If yes, global continuity assumptions are supported; if no, you must segment and analyze components separately. This is the package’s central guardrail and value proposition.
Concretely, topologyR provides a pipeline to: (i) build subbases/bases from numeric data with tunable thresholds, (ii) derive the induced topology and quantify its complexity, (iii) run undirected/directed/coverage connectivity checks, and (iv) explore how threshold choices change structure.
2. Formal Definitions
Topological space. Given a set \(X\), a topology \(\tau\) on \(X\) is a family of subsets (the open sets) that contains \(\emptyset\) and \(X\) and is closed under arbitrary unions and finite intersections; \((X,\tau)\) is a topological space.
Base & subbase (intuitive). From data, we form neighborhoods under a threshold \(\tau\) (not the topology!), take finite intersections to form a base, and from unions of base elements obtain the full topology. In topologyR this is done explicitly from a numeric vector.
Connectivity (β₀). We study whether the induced topology connects all present indices under either undirected or directed reachability, or at least covers all indices (manual check).
3. topologyR Pipeline: What the package builds for you?
Neighborhoods under a tunable threshold (e.g., IQR-based)
Base via intersections (include empty/full sets for correctness)
Full topology from the base (closure under unions/finite intersections)
Connectivity checks: undirected DFS, directed DFS, manual coverage
Threshold exploration (mean/median diffs, SD, IQR/factor, DBSCAN-like) and factor sweep summaries/plots
A compact R-sketch of the core algorithm appears below (the package implements a robust version):
# Neighborhoods under τ
subbase <- lapply(seq_along(x), function(i) which(abs(x - x[i]) <= τ))
# Base: include ∅ and X
base <- list(integer(0), seq_along(x))
for (i in seq_along(x)) for (j in i:length(x)) {
S <- intersect(subbase[[i]], subbase[[j]])
if (length(S) > 0) base <- c(base, list(S))
}
base <- unique(base)
# From the base -> topology (unions/finite intersections)
# Connectivity checks:
# is_topology_connected(topology) # undirected DFS
# is_topology_connected2(topology) # directed DFS (sequential)
# is_topology_connected_manual(topology) # coverage 1..n4. Key Innovation
4.1. Revealing Topological Invariants to Validate Imputation Methods
4.1.1. Introduction: The Fundamental Problem
We often assume global continuity (smoothing, kriging, splines, etc.) before verifying whether continuity is even mathematically defensible. topologyR fixes this by elevating local distances into a topology and reading off invariants (notably connectedness) that decide method validity.
4.1.2. Conceptual Foundations
4.1.2.1. The Nature of The Problem. Sequential data hide global properties critical for method choice: system connectivity, structural complexity, and breakpoints that may invalidate continuity. These are not visible from pairwise distances alone.
4.1.2.2. The Topological Solution.
- Metric proximity → neighborhood relations
- Neighborhoods → subbase
- Intersections/unions → base and full topology
- Topology → invariants (connectivity) that govern method validity.
4.1.3. The Theoretical Framework: Connectivity ⇒ Method Validity
Case I: Connected Topology. Valid to use global continuous methods (polynomial/linear interpolation, cubic splines, kriging/geostatistics, moving averages, continuous kernels), assuming adequate sampling/noise.
Case II: Disconnected topology. Global continuity is invalid; you must switch to segment-wise or regime approaches (independent imputation per component, regime-switching, conditional-by-segment, hot-deck within component, finite mixtures).
4.1.4. Global vs Local Properties
Global properties (full-period means, secular trend, systemic volatility, cycle structure, cointegration, long memory, global neural synchrony, centennial climate trends, circulation patterns) require connectivity; a disconnected topology invalidates global continuous imputation across the break. Local properties (short-window volatility, local derivatives, local clustering, AR(1)/AR(2) at short lags, local inflection points, short-segment spectra) can be analyzed per component regardless of global connectedness.
4.1.5. Critical Applications by Domain
- Econometrics. Disconnections mark structural crises; avoid trends that cross discontinuities.
- Medicine/Neuroscience. Interventions can split regimes; test continuity before longitudinal pooling.
- Climatology. Extreme events can sever continuity; don’t extrapolate trends across regimes.
4.1.6. Fundamental Decision Rule
- Global analyses: connectivity is necessary for continuous methods.
- Local analyses: use continuous methods within each component.
- Mixed: segment globally; apply local continuous tools inside each segment.
4.1.7. Conclusions On the Fundamental Contribution
topologyR transforms imputation from heuristic to mathematically grounded procedure by revealing the underlying topological structure that justifies (or forbids) continuity assumptions before application — crucial in policy, biomedical, and climate contexts.
4.1.8. Positioning vs General-Purpose TDA
Focus: 1D series, β₀ (connectedness), explicit decision rule. Complement PH stacks (Ripser/GUDHI/TDAstats/scikit-TDA) for multi-scale/higher-β structure (no explicit imputation rule).
4.1.9. Limitations and Risks
Parameter Sensitivity (threshold choice) → mitigate via factor sweeps; still empirical.
Sampling/noise can mimic (dis)connection → treat “connected ⇒ continuous” as prima facie.
Expressivity limited to β₀.
Scalability: exhaustive builds can be \(O(n^2)\).
4.1.10. Global conclusions & practical guidance
Use topologyR as a pre-model governance tool for binary validity decisions; complement with PH for subtle multi-scale features that don’t break global connectedness.
4.1.11. References
Alvarado, E., Beckelhymer, D., Dorrington, J., Lam, T., Majhi, S., Noory, J., Sánchez Muniz, M., & Strømmen, K. (2025). Detecting the Indian Monsoon using Topological Data Analysis (arXiv:2504.01022). https://arxiv.org/abs/2504.01022
Bauer, U. (2021). Ripser: Efficient computation of Vietoris–Rips persistence barcodes. Journal of Applied and Computational Topology, 5(3), 391–423. https://doi.org/10.1007/s41468-021-00071-5
Chung, M. K., et al. (2023). Unified topological inference for brain networks in temporal dynamics. Frontiers in Neuroscience, 17, 1140289. https://doi.org/10.3389/fnins.2023.1140289
Flammer, M., et al. (2023). Persistent homology-based classification of chaotic multivariate time series: Application to electroencephalograms. SN Computer Science, 4, 396. https://doi.org/10.1007/s42979-023-02396-7
Gidea, M. (2017). Topological data analysis of financial time series (arXiv:1703.04385). https://arxiv.org/abs/1703.04385
Guo, H., et al. (2020). Empirical study of financial crises based on topological data analysis. Physica A: Statistical Mechanics and its Applications, 551, 124198. https://doi.org/10.1016/j.physa.2019.124198
Kang, Y., et al. (2024). High-order brain network feature extraction and characterization via persistent homology. Frontiers in Neuroscience, 18, 1378837. https://doi.org/10.3389/fnins.2024.1378837
Kelley, J. L. (2017). General topology (Dover ed.; original work published 1955). Dover Publications.
Maria, C., Boissonnat, J.-D., Glisse, M., & Yvinec, M. (2014). The GUDHI library: Simplicial complexes and persistent homology. In H. Hong & C. Yap (Eds.), Mathematical Software – ICMS 2014 (pp. 167–174). Springer. https://doi.org/10.1007/978-3-662-44199-2_28
Otter, N., Porter, M. A., Tillmann, U., Grindrod, P., & Harrington, H. A. (2017). A roadmap for the computation of persistent homology. EPJ Data Science, 6, 17. https://doi.org/10.1140/epjds/s13688-017-0109-5
scikit-TDA. (n.d.). scikit-TDA documentation. https://docs.scikit-tda.org/
Tralie, C., Saul, N., & Bar-On, R. (2018). ripser.py: A lean persistent homology library for Python. Journal of Open Source Software, 3(29), 925. https://doi.org/10.21105/joss.00925
Tymochko, S., et al. (2020). Using persistent homology to quantify a diurnal cycle in tropical cyclone convection. Pattern Recognition Letters, 133, 137–143. https://doi.org/10.1016/j.patrec.2020.02.003
Ver Hoef, L., et al. (2023). A primer on topological data analysis to support image segmentation and feature extraction for environmental science. AI for the Earth Systems, 2(1), e220039. https://doi.org/10.1175/AIES-D-22-0039.1
Wadhwa, R. R., Williamson, D. F. K., Dhawan, A., & Scott, J. G. (2018). TDAstats: R pipeline for computing persistent homology in topological data analysis. Journal of Open Source Software, 3(28), 860. https://doi.org/10.21105/joss.00860
Wang, Z., et al. (2023). Automatic epileptic seizure detection based on persistent homology. Computational and Mathematical Methods in Medicine, 2023, 9165842. https://doi.org/10.1155/2023/9165842
Xu, X., Gao, Y., Zhong, S., Li, P., & Wang, Y. (2021). Topological data analysis as a new tool for EEG processing. Frontiers in Neuroscience, 15, 761703. https://doi.org/10.3389/fnins.2021.761703
5. Scope & Typical Use Cases
- Economic & other time series: connectivity regimes, structural breaks.
- Signal/biological data: neighborhood graphs, coverage checks.
- Didactic: bases/subbases, topology construction, graph↔︎topology bridges.
6. Design Principles
Correctness (include ∅ and \(X\); track coverage), transparency (step-wise functions), explorability (fast heuristics & visuals for threshold selection).
7. Technical Approach: Algorithms & Connectivity Engines
Core Algorithm and Methods (undirected DFS; directed sequential DFS; manual coverage) as sketched in §3 above.
Connectivity Functions (Package Internals).
is_topology_connected()builds a symmetric adjacency from set co-membership and runs DFS over present elements.is_topology_connected2()directs edges along sorted, consecutive elements within each set and DFSs from the minimum.is_topology_connected_manual()checks whether each original index 1..n appears in at least one set.
8. API at A Glance (Quick Start)
library(topologyR)
# Small example vector
x <- c(1, 2, 3, 4, 5)
# Build a complete topology and check connectivity
topo <- complete_topology(x)
undirected <- is_topology_connected(topo$topology)
directed <- is_topology_connected2(topo$topology)
manual <- is_topology_connected_manual(topo$topology)
# Threshold exploration and factor sweep
ths <- calculate_thresholds(x)
res <- analyze_topology_factors(x, factors = c(1, 2, 4, 8, 16))
viz <- visualize_topology_thresholds(x) # optional visuals9. Threshold Heuristics & Factor Sweep
You can derive τ from mean/median adjacent differences, SD, IQR / factor, or a DBSCAN-like heuristic; then run an IQR-factor sweep (e.g., 1, 2, 4, 8, 16) and track base size / set sizes to pick a stable regime.
Internally, the factor sweep computes \(\tau=\mathrm{IQR}(x)/f\), builds subbases, forms the base (with ∅ and \(X\)), and summarizes base size and min/max set sizes per \(f\).
set.seed(1)
x <- cumsum(rnorm(200))
ths <- calculate_thresholds(x)
resf <- analyze_topology_factors(x, factors = c(2, 4, 8, 16), plot = FALSE)
print(ths)
print(resf)10. Performance & Scalability
Exact neighborhood/base construction can grow as \(O(n^2)\); prefer robust thresholds + factor sweeps, and consider down-sampling beyond moderate \(n\). For first-pass diagnostics on very large data, use the manual coverage check and then refine with directed DFS as needed.
11. Limitations & Validity Notes
Threshold Sensitivity: false (dis)connections if τ is mis-set → mitigate with grids, but keep domain judgment.
Sampling/noise: sparse sampling may mimic disconnections; light overlaps may mimic connection → treat “connected ⇒ continuous” as prima facie, not absolute.
Expressivity: emphasis on β₀; higher-order features require PH stacks.
Assumptions: sequential indices 1..n; very large topologies can be memory-intensive.
12. End-To-End Example: Imputation Decision
library(topologyR)
x <- as.numeric(AirPassengers)
# Transparent, robust default for τ and illustrative factor sweep
ths <- calculate_thresholds(x)
res <- analyze_topology_factors(x, factors = c(2,4,8,16), plot = FALSE)
tau <- IQR(x)/4
topo <- complete_topology(x) # for small n; otherwise use thresholded subbases
is_conn <- is_topology_connected(topo$topology)
if (is_conn) {
message("Connected: using global continuous imputation.")
# e.g., stats::spline(...), global kriging, smoothers
} else {
message("Disconnected: segment-wise imputation & regime models.")
# split by connected components; impute/model per segment
}13. Positioning vs. General-Purpose TDA
topologyR is a focused, decision-oriented pre-model tool for 1D series (β₀/connectedness). For multi-scale/higher-β structure or early-warning signals, complement with Ripser/GUDHI/TDAstats/scikit-TDA.
(Key external sources listed in the Wiki’s references subsection 4.1.11. of the library)
14. Installation
remotes::install_github("IsadoreNabi/topologyR")
# or local build
# setwd("/path/to/topologyR"); library(devtools); library(roxygen2); document(); install()