Espartaco

Marxist Philosophy of Science

“Is that to say we are against Free Trade? No, we are for Free Trade, because by Free Trade all economical laws, with their most astounding contradictions, will act upon a larger scale, upon the territory of the whole earth; and because from the uniting of all these contradictions in a single group, where they will stand face to face, will result the struggle which will itself eventuate in the emancipation of the proletariat.”

Karl Heinrich Marx · Marx-Engels Collected Works, Vol. VI, p. 290

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Tag: investigación reproducible

  • General Dynamic Parameter Models via Reference Anchoring

    General Dynamic Parameter Models via Reference Anchoring

    You can also find this library at CRAN and download it directly from R and RStudio.

    Also, we recommend viewing the mind map summary at the end of the article to better understand the relationship between the functions of the package.

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    R Library Review

    Meet gdpar

    General Dynamic Parameter Models via Reference Anchoring

    In the fleeting calculus of a two-second decision—overtaking a car on a narrow road—the human brain performs a remarkable statistical trick. It does not build a model of the approaching driver from scratch. Instead, it retrieves a baseline: the average driver, representing typical reaction times and modal aggression. In a split second, it reads the specific signals of the actual driver—relative speed, vehicle type, micro-movements—and estimates how this specific driver deviates from the baseline. The decision to overtake emerges from that synthesis.

    This cognitive recipe—population reference + individual deviation—is the philosophical bedrock of the R package gdpar (General Dynamic Parameter models via Reference Anchoring) by José Mauricio Gómez Julián. The package takes this intuition, formalizes it as a rigorous statistical decomposition, proves the conditions under which it is mathematically identifiable, ships a Stan-based Bayesian engine to estimate it, and layers on causal inference, geometry-adaptive sampling, and dependence-robust inference.

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    The Anatomy of Deviation

    Every layer of gdpar is an elaboration of a single, elegant equation. For each observation $i$ with covariates $x_i$:

    $$ \theta_i \;=\; \theta_{\text{ref}} \;+\; \Delta(x_i,\; \theta_{\text{ref}}) $$

    Read it as: the parameter of individual $i$ equals a population reference, plus a deviation that is itself a function of the individual’s covariates and of the reference itself.

    That final clause is where the architecture pivots from classical statistics. The deviation $\Delta$ does not merely depend on who you are (your covariates $x_i$); it depends on what the reference is. If you transplant the model to a new population, the deviation function behaves differently because $\theta_{\text{ref}}$ is one of its arguments. This structural dependence is the defining feature of “reference anchoring.” It distinguishes gdpar from random-effects or varying-coefficient models, where the deviation is structurally separate from the reference.

    So, what is the shape of $\Delta$? The package singles out a specific functional form called the Additive–Multiplicative–Modulated (AMM) decomposition:

    $$ \Delta(x,\theta_{\text{ref}}) \;=\; \underbrace{a(x)}_{\text{additive}} \;+\; \underbrace{b(x)\odot\theta_{\text{ref}}}_{\text{multiplicative}} \;+\; \underbrace{W(\theta_{\text{ref}})\,x}_{\text{modulated}} $$

    Three mechanisms, cleanly separated and independently interpretable:

    • $a(x)$ — A pure additive shift. Think of this as a traditional fixed-effect driven by covariates.
    • $b(x)\odot\theta_{\text{ref}}$ — A covariate-dependent scaling of the reference (using the Hadamard/elementwise product). This is where “the deviation depends on the reference” enters multiplicatively.
    • $W(\theta_{\text{ref}})\,x$ — Covariates are mixed through a matrix $W$ that is, itself, tuned by the reference. This is the explicit, structural reference-dependent channel.

    Standard models drop out as special cases. Set $\Delta \equiv 0$ and you have fixed-effects regression. Set $W \equiv 0$ and you have a hierarchical model with multiplicative interaction. Set $b \equiv 0$ and you have a varying-coefficient model. The AMM is the smallest natural family that contains all three and elevates the reference to an active argument of the deviation.

    ❦

    The Three Estimation Engines

    gdpar defines three complementary engines for estimating $\Delta$. Crucially, only one is executable in the current release—a deliberate choice to promise a mathematical scope that exceeds the executable surface, and to say so honestly.

    Path Engine Representation Status
    Path 1 Hierarchical Bayesian (Stan) Parametric AMM ✅ Operational
    Path 2 Varying-coefficient (splines) Smooth $\beta(z)$ 🚧 Conceptual
    Path 3 Hypernetwork / Neural Net Net generates $\theta_i$ 🚧 Conceptual

    Paths 2 and 3 are documented to “reference grade”—full asymptotic theory (contraction rates, Bernstein–von Mises) is developed in the Wiki—but they abort with gdpar_unsupported_feature_error if invoked. Path 1 places priors on every component ($\theta_{\text{ref}}, a, b, W$) and samples the joint posterior with HMC, yielding native, full-posterior uncertainty.

    A Tale of Two Posteriors: EB vs. FB

    Within Path 1, gdpar offers two inferential regimes. Full Bayes (FB) via gdpar() samples the joint posterior, remaining most faithful to the cognitive analogy. Empirical Bayes (EB) via gdpar_eb() estimates the hyperparameters by maximizing a marginal likelihood via a Laplace approximation, then samples the remaining parameters conditionally.

    The EB vs FB Comparator

    Rather than forcing a choice, gdpar treats them as parallel routes. It ships a dedicated comparator, gdpar_compare_eb_fb(), which quantifies agreement on $\theta_{\text{ref}}$ and the reduced parameter vector $\xi$. The Wiki develops the theory to first-class depth: EB and FB lower-level posteriors agree asymptotically (Theorem 7A), while EB intervals under-cover by $O(n^{-1})$ (Proposition 7B). If you have ever wondered if EB is “good enough” for your data, gdpar lets you answer that empirically.

    Distributional Regression: Every Parameter is a Slot

    gdpar is not constrained to modeling the mean. A probability distribution has multiple parameters—location, scale, shape, tail index, zero-inflation probability—and each one can carry its own AMM decomposition. The package indexes these by $k = 1, \dots, K$:

    $$ \theta_i^{(k)} = \theta_{\text{ref}}^{(k)} + \Delta^{(k)}(x_i, \theta_{\text{ref}}^{(k)}), \qquad k = 1, \dots, K $$

    The built-in roster covers Gaussian, Poisson, negative binomial, Bernoulli, Beta, Gamma, Student-$t$, Tweedie, ZIP, ZINB, and hurdle families. Zero-inflated and hurdle models receive an especially elegant treatment: both the zero-inflation probability $\pi_i$ and the count parameter $\theta_i$ are anchored to their respective references—a dual deviation design.

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    The Causal Bridge

    Because the AMM form produces individual parameters, individual treatment effects emerge naturally. gdpar_causal_bridge() implements a T-learner: fit the anchored model separately under treatment and control, then read the conditional average treatment effect (CATE) at $x_i$ as the difference of the anchored individual predictions:

    $$ \widehat{\tau}(x_i) = \widehat{\mu}_1(x_i) – \widehat{\mu}_0(x_i) $$

    A second layer, gdpar_compare_meta_learners(), benchmarks the AMM-based learner against external meta-learners via pluggable adapters: grf::causal_forest on the R side and EconML’s CausalForestDML on the Python side (via reticulate). The framework’s causal claims are benchmarked, not asserted.

    Mechanics & Clockwork

    Several engineering decisions elevate gdpar from a theoretical exercise to a serious computational environment:

    • Stan Code Generator: Composes programs from canonical pieces—AMM blocks for $p=1$ and $p \geq 1$, EB marginal/conditional blocks, distributional-$K$ blocks—selected by the resolved $(K, p, \text{family}, W, \text{parametrization}, \text{group})$. The $W$ basis supports B-splines with Stan-side Cox–de Boor evaluation, ensuring differentiability inside HMC.
    • Identifiability Pre-flight: Before any sampling, gdpar_check_identifiability() runs a Gram-matrix check (Proposition 1C), a per-coordinate cross-component check (C4-bis) for $p > 1$, and a per-group anti-aliasing check (C7). If your design is non-identifiable, you find out before the sampler burns your CPU, accompanied by a structured gdpar_identifiability_error naming the dependent directions.
    • Data-Driven Reparametrization: Treats the parametrization of $b(x) \odot \theta_{\text{ref}}$ as a pre-fit decision. A short pilot computes an information ratio, dispatching to CP, NCP, or—gdpar‘s root-cause resolution—a linear reparametrization that samples the product $\theta_{\text{ref}} \cdot b$ directly, sidestepping bilinear funnels altogether.

    Opt-in Power Tools

    Two advanced capabilities are switched off by default, documented as thoroughly as the core path.

    1. Geometry-Adaptive Sampling

    Hierarchical AMM posteriors can be geometrically hostile—funnels, near-determinism, heavy tails. The opt-in geometry engine climbs a ladder of Riemannian metrics: Euclidean → Fisher/SoftAbs → sub-Riemannian → relativistic/Finsler. A certifying orchestrator diagnoses the pathology, selects a metric, tunes the integrator, and emits a certificate. If full sampling is certified infeasible, a Laplace fallback provides a plug-in posterior with ELPD on par with mgcv-REML or INLA-Laplace.

    2. Dependence-Robust Inference

    gdpar does not model temporal or spatial dependence in its point structure; instead, it makes the inference robust to dependence (a working-independence + sandwich-variance stance in the spirit of Liang & Zeger, 1986). You receive diagnostics (Durbin–Watson, Ljung–Box, Moran’s $I$) and robust SEs via block bootstrap—moving or circular blocks in time (with the Politis–White flat-top automatic block length), tiled randomized-origin blocks in space. Point estimates remain pristine; only the uncertainty is made honest.

    ⚠️ Honest Limitations

    The Wiki is admirably forthright about scope. Only Path 1 is executable in 0.1.0. Dependence is not modelled—only the inference is made robust. The package’s mathematical scope exceeds its executable surface by design. Read the “Implementation status” notes carefully before relying on a feature.

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    TL;DR

    gdpar takes one of the most natural ideas in human prediction—predict an individual as a deviation from a population reference, where the deviation itself depends on the reference—and transforms it into a fully specified, identifiability-checked, Stan-powered Bayesian regression framework. It is theoretically rigorous, computationally serious, and unusually honest about what it does and does not yet do. If your work involves individual heterogeneity, distributional regression, or causal effect estimation with principled uncertainty, gdpar demands a careful look.

    🔗 Repository: github.com/IsadoreNabi/gdpar
    📚 License: GPL (≥ 3) · Version reviewed: 0.1.0
    ✍️ Author: José Mauricio Gómez Julián

    MIND MAP SUMMARY

  • Marx, Adam Smith, and the Law of Large Numbers

    Marx, Adam Smith, and the Law of Large Numbers

    A new research uses probability theory — and sixty years of U.S. economic data — to test one of the most consequential (and most overlooked) assumptions in political economy.

    The Assumption Hiding in Plain Sight

    If you’ve ever read a Marxist analysis of how profits equalize across industries, you’ve probably encountered something called the average rate of profit. The idea is straightforward: competition between capitalists drives different rates of profit in different sectors toward a common, system-wide average. This is one of the pillars of Marx’s theory of value in Capital, Volume III.

    But there’s a quieter assumption underneath this one — so quiet that most discussions never mention it explicitly. To arrive at a uniform profit rate, Marx first assumes a uniform rate of surplus value across all productive sectors. In plainer terms: he assumes that the degree to which workers are exploited — the ratio of unpaid labor to paid labor — is roughly the same everywhere, whether you work in steel manufacturing, food processing, or textiles.

    Adam Smith proposed this idea before Marx. Smith argued that if one job were obviously more exploitative (in the sense of yielding far more unpaid surplus per dollar of wages paid), workers and capital would flow toward or away from it until the differences vanished. Marx adopted this observation and, as scholar Jonathan Cogliano notes, elevated it to “the status of a central economic law” within his framework.

    Yet the assumption has been challenged from multiple directions — Marxist and non-Marxist alike. Is it actually justified? Or is it a convenient simplification that distorts our understanding of how capitalism works?

    José Mauricio Gómez Julián, of the Universidad Latina de Costa Rica, decided to approach the question from an unexpected angle: probability theory. His paper, published in Ciencia Económica (2022), asks whether the mathematical law that would need to hold for this assumption to be valid actually does hold — and then checks the answer against six decades of real-world data from the United States.

    The Mathematical Backbone: The Law of Large Numbers

    If you’ve taken any statistics course, you’ve likely met the Law of Large Numbers (LLN). It tells us that as you observe more and more instances of something random — coin flips, dice rolls, stock returns — the average of those observations settles down toward the true expected value.

    There are two versions:

    • The Weak Law (WLLN): With enough observations, the sample average is probably close to the expected value.
    • The Strong Law (SLLN): With enough observations, the sample average is almost certainly equal to the expected value — a much stronger guarantee.

    Gómez Julián’s insight is this: if you think of each productive sector of the economy as a random variable representing that sector’s rate of surplus value, then the LLN tells you what happens to the average across sectors as the number of sectors grows large. In mathematical language:

    • Strong Law: The probability that the average surplus-value rate across sectors equals the global expected value, in the limit, is exactly 1.
    • Weak Law: The probability that the average deviates from the global expected value by more than any tiny amount shrinks to zero.

    If either version holds, you get the result Marx needs: across a sufficiently large number of sectors, the rates of surplus value converge to a common value — uniformity, or at least a powerful tendency toward it.

    The Catch: Independence and Identical Distributions

    Here’s where things get interesting — and where the classical LLN hits a wall.

    The textbook version of the LLN requires two conditions:

    1. Independence: The random variables (sectoral surplus-value rates) must be statistically independent of each other.
    2. Identical distribution: Each variable must follow the same probability distribution.

    Neither condition holds for the real economy. And Gómez Julián is admirably upfront about this. Sectors are deeply intertwined — the steel industry depends on mining, manufacturing depends on steel, services depend on consumer spending powered by manufacturing wages. The idea that one sector’s surplus-value rate has no relationship to another’s is economically unrealistic. Furthermore, different industries have different cost structures, different labor intensities, and different technologies. There is no reason their surplus-value rates should follow the same statistical distribution.

    So does this kill the argument? Not at all. In fact, it’s the most intellectually interesting part of the paper.

    Non-Classical Varieties: When the Rules Relax

    Over the past several decades, mathematicians and econometricians have developed non-classical versions of the LLN that weaken or entirely drop the independence and identical-distribution requirements. Gómez Julián surveys several of these:

    • Li, Rao, and Wang (1995) showed the LLN holds for random variables arranged on a lattice structure under certain conditions — a structure that, as it happens, economic data naturally exhibits.
    • Adler and Rosalsky (1987) proved the law for weighted sums of independent, identically distributed random variables belonging to a normalized sum, generalizing the classical case.
    • Chen and Sung (2016) extended those results further: the variables no longer need to be identically distributed. They only need to be “stochastically dominated” by a single random variable, with certain weighting conditions.
    • Sung (2011) showed that the strong law can hold even when variables are dependent on each other, provided their probability moments (roughly, their averages and variability) satisfy certain finiteness conditions.

    The crucial point: these results collectively tell us that the LLN’s convergence conclusion can survive even when the classical assumptions are substantially violated — which is exactly the situation with sectoral surplus-value rates.

    Gómez Julián argues that the economic dynamics described by Smith — workers and capital moving between sectors in response to unequal advantages — are precisely the kind of compensatory dependence mechanism that these non-classical versions accommodate. The variables aren’t independent, but their dependence is structured in a way that still drives convergence.

    What the Data Actually Shows

    The theoretical argument is compelling, but Gómez Julián doesn’t stop there. He turns to sixty years of U.S. data (1960–2020), sourced from the Bureau of Economic Analysis (BEA), to see what the empirical evidence says.

    He calculates sectoral surplus-value rates using macroeconomic data on gross operating surplus (representing surplus labor time) and employee compensation (representing necessary labor time), following a standard operationalization of Marx’s categories. After carefully determining which sectors qualify as “productive” in the Marxist sense — a nontrivial task, since the service sector includes activities with very different relationships to surplus-value production — he arrives at 36 productive sectors.

    Here’s what the statistical analysis found:

    Finding 1: No Identical Distributions

    A probability distribution fitting exercise (using the Bayesian Information Criterion) revealed that the 36 sectors’ surplus-value rates follow a patchwork of different distributions — Log-Normal, Cauchy, Uniform, Weibull, and Logistic — with none following a normal distribution. The identical-distribution requirement of the classical LLN is not met.

    Finding 2: No Statistical Independence

    A Pearson correlation analysis across all 630 possible sector pairs yielded a mean correlation of about 0.08 and a median of about 0.14. While these may look small, a deeper cut reveals that roughly 40% of sector pairs have correlations of 0.3 or above — a level that’s practically meaningful. The sectors are not independent. This makes intuitive sense: industries are connected through supply chains, labor markets, and shared macroeconomic conditions.

    Finding 3: Differences Tend Toward Zero

    This is the key finding. When Gómez Julián computed the differences between each sector’s surplus-value rate and the global average (both the mean and the median), he found that these differences exhibit a strong tendency toward reciprocal nullification — positive differences roughly cancel out negative ones. The sum of all differences relative to the global mean was essentially zero (on the order of 10⁻¹⁴). The mean of differences relative to the global median was 0.0012 — vanishingly small.

    Distributional fitting on these differences revealed they follow a Cauchy distribution (when measured against the global mean) or a uniform distribution (against the global median), with the medians of these distributions sitting very close to zero.

    In plain language: sectors deviate from the average in different directions, and those deviations largely cancel each other out.

    Why This Matters

    Gómez Julián’s paper makes two types of contributions that are worth distinguishing:

    For Marxist political economy: If the uniformity assumption holds — even approximately, even as a tendency rather than an iron law — then a large body of research on the long-run behavior of the average rate of profit, both within countries and across the global economy, is on sounder footing than critics have suggested. Researchers studying the tendency of the rate of profit to fall (or not) can continue to work without needing to explicitly model sector-by-sector differences in exploitation rates, at least for aggregate, long-run analyses.

    For probability theory and economics: The paper demonstrates a productive intersection between a specific question in political economy and the deep mathematics of convergence theorems. It shows that the non-classical LLN theorems aren’t just abstract curiosities — they have direct relevance to understanding real economic phenomena. The structured dependence between economic sectors isn’t a bug that invalidates the mathematical framework; it’s a feature that the right version of the framework already accounts for.

    A Few Honest Caveats — And Why They No Longer Apply

    The original 2022 paper was unusually transparent about its limitations, and that transparency is one of its strengths. Rather than forcing the data into inappropriate statistical procedures, it openly acknowledged where the available inferential tools broke down.

    At the time, three important caveats remained.

    First, formal hypothesis testing had to be abandoned.

    The reason was purely statistical rather than economic. Classical inferential procedures—Student’s t tests, Wilcoxon tests, and most conventional non-parametric alternatives—are built on assumptions that the data simply did not satisfy. Sectoral surplus-value rates are neither independent nor identically distributed. They are linked through supply chains, technological change, capital mobility, and macroeconomic shocks. Even bootstrap procedures could not fully solve the problem because ordinary resampling may weaken dependence between resamples while leaving the internal dependence structure fundamentally unchanged. Consequently, the 2022 paper relied primarily on descriptive statistics together with probability-theoretic arguments instead of formal significance testing.

    Second, the classification of productive sectors inevitably involved theoretical judgment.

    Although the paper carefully justified the inclusion and exclusion of economic activities using Marxian categories and modern national accounting, reasonable scholars could still debate where certain services belong within the circuit of capital.

    Third, the empirical evidence came exclusively from the United States.

    The descriptive regularities were remarkably strong, but demonstrating that the same convergence mechanism operates under different institutional settings naturally remained an empirical question.

    Those were genuine limitations in 2022.

    Today, however, the first—and arguably the most important—of them has largely been overcome.

    A much more comprehensive methodological paper (Gómez Julián, 2026; SSRN 5172185) develops an entirely new inferential framework specifically designed for exactly the type of data that made the original analysis difficult: dependent, heterogeneous, and unbalanced observations. Instead of trying to force classical statistical tests to work outside the assumptions under which they were derived, the newer paper constructs hypothesis testing from the ground up for this class of problems.

    The key innovation is recognizing that the convergence of sectoral surplus-value rates is fundamentally a law-of-large-numbers problem under dependence, not an independent-samples problem. The framework therefore combines three complementary asymptotic structures—triangular arrays (TAC), correlation-weighted sums (WSC), and mixingale processes (MPC)—which respectively model hierarchical dependence, contemporaneous intersectoral dependence, and temporal dependence. Rather than treating these as competing approaches, the paper proves conditions under which they become metrically equivalent and therefore support the same inferential conclusions.

    The inferential consequences are substantial.

    Instead of abandoning significance testing because dependence invalidates classical procedures, the new framework explicitly extends the Neyman-Pearson paradigm to dependent observations, derives dependence-aware confidence regions, establishes rigorous Type I error control under strong-mixing assumptions, and integrates Bayesian and frequentist inference into a single coherent architecture. Robust procedures—including fixed-b heteroskedasticity-and-autocorrelation-robust inference, block bootstrap techniques that preserve dependence, adaptive conformal inference, composite and Whittle likelihoods, and hierarchical Bayesian estimation—serve as mutually reinforcing validation mechanisms rather than isolated alternatives.

    In other words, what had been acknowledged as a methodological limitation in the 2022 paper became the central research question of the later work.

    Rather than concluding that inference was impossible under dependence, the subsequent research asks a more fundamental question: what should hypothesis testing look like when dependence is the normal state of the data rather than an exception? The result is a unified inferential framework specifically intended for datasets that violate the assumptions of classical statistics—precisely the situation encountered with sectoral surplus-value rates.

    The other caveats remain, although they are considerably less problematic than before. The classification of productive sectors continues to depend on theoretical interpretation, because that issue belongs to political economy rather than statistics. Likewise, expanding the empirical analysis to additional countries remains a desirable avenue for future research. Yet the principal statistical objection—that no valid inferential procedure existed for dependent sectoral data—has now been directly addressed through a purpose-built mathematical framework.

    Looking back, the 2022 paper can therefore be read as identifying an important statistical obstacle, while the later work attempts to remove it. Together, the two papers form a coherent research program: first demonstrating that the convergence hypothesis is theoretically plausible and descriptively supported, and then developing the inferential machinery required to test that hypothesis rigorously without relying on unrealistic assumptions of independence or identical distributions.

    Gómez Julián, J.M. (2022). Sobre la validez del supuesto de uniformidad en las tasas de plusvalía sectorial desde la teoría de las probabilidades. Ciencia Económica, 11(17). DOI: 10.22201/fe.24484962e.2022.11.17.2

    Gómez Julián, J.M. (2026). Hypothesis Testing for Dependent Variables with Unbalanced Data: A Unified Framework: Theory, Robustness, and Software. SSRN Electronic Journal. DOI: 10.2139/ssrn.5172185.

  • topologyR: Turning Time Series into Shapes to Test What Your Models Quietly Assume

    topologyR: Turning Time Series into Shapes to Test What Your Models Quietly Assume

    You can also find this library at CRAN and download it directly from R and RStudio.

    LISTEN TO THIS POST AS A PODCAST:

    There is a habit so embedded in quantitative work that most practitioners never think to question it. You have a time series — quarterly GDP, an EEG channel, a temperature record — and at some point you fit a smooth curve through it, interpolate a missing value, or estimate a “long-run trend.” All of these moves rest on a single, seldom-checked assumption: that the data form one continuous whole, that a single smooth function can legitimately pass through every point.

    But what if they don’t? What if your series is, structurally, two or three disjoint pieces glued together by the calendar — pieces between which no continuous function can travel? In that case, the spline you just fitted is not an approximation of reality; it is a mathematical fiction painted over a fracture.

    topologyR is an R package that lets you check this before you model. It takes a numeric time series, converts it into a graph, converts that graph into a topological space, and then asks the one question that determines whether global continuous methods are even valid: is this space one connected piece, or several?

    It sounds abstract. It is abstract — but the consequence is concrete. The package is the work of José Mauricio Gómez Julián, and it is open-source, with a GitHub repository, a detailed Wiki, and a companion research paper archived on Zenodo. What follows is a tour of what the package does, why it matters, and where it fits in the broader landscape of topological data analysis.

    The Hidden Assumption

    Think about what happens when you impute a missing value in a time series using a cubic spline. The spline assumes that the points on either side of the gap belong to the same continuous process — that the missing value lies somewhere along a smooth bridge between them. If the series has actually undergone a structural break, a regime change, or a discontinuity between those points, the spline will happily produce a number, and that number will be wrong in a way no confidence interval can capture.

    This is not a niche problem. It appears in econometrics (trend estimation across business cycles), in neuroscience (coherence across brain-state transitions), in climatology (warming trends across regime shifts). The methodological error is always the same: assuming continuity without first verifying that continuity is mathematically possible.

    topologyR’s contribution is to make that verification explicit, parameter-free, and exact.

    From Numbers to Shapes: The Pipeline in Three Steps

    The package’s workflow has an elegant, almost architectural logic. You feed it a series of numbers; it returns a topological verdict. Between input and output, three transformations occur.

    Step 1: The Series Becomes a Graph

    The first move is borrowed from network science: the visibility graph. Imagine your time series plotted as a mountain range — each observation is a peak or a valley at a given time. Two points are connected by an edge if you could stand on one and see the other, with no taller peak blocking the line of sight.

    topologyR implements two flavours. The Horizontal Visibility Graph (HVG) connects two points if every point between them is strictly lower than the shorter of the two — a horizontal line of sight. It runs in linear time and captures the skeleton of the series’ ups and downs. The Natural Visibility Graph (NVG) is more generous: it connects two points if every intermediate point lies below the straight line joining them, regardless of the heights of the endpoints. It is denser, richer, and runs in O(n log n) expected time. The NVG always contains the HVG as a subgraph.

    Both are parameter-free. There is no threshold to tune, no bandwidth to select, no ε to agonise over. The graph is determined entirely by the data’s own geometry. This matters enormously: it eliminates the single largest source of arbitrariness in the entire pipeline.

    Step 2: The Graph Becomes a Topology

    Here is where topologyR departs from ordinary network analysis. A graph tells you who is adjacent to whom. A topology tells you something deeper: what the neighbourhood structure of the entire space looks like — which collections of points form coherent open regions, and how those regions combine.

    The construction follows a method introduced by Nada, El Atik, and Atef in 2018. For each vertex v in the graph, you form its closed neighbourhood — the vertex itself plus all its direct neighbours. This family of closed neighbourhoods serves as a subbase. You then close it under finite intersections to obtain a base, and close the base under arbitrary unions to obtain the full topology.

    If those words feel heavy, think of it this way: the subbase is a rough draft of “who belongs with whom.” Intersecting neighbourhoods refines the draft — “the points that both neighbourhoods agree on.” Taking unions completes the picture — “every region that can be assembled from these building blocks.” The result is a genuine topological space, complete with open sets satisfying the standard axioms, sitting on top of your time series like a scaffolding you didn’t know was there.

    Step 3: The Topology Reveals Its Connectivity

    Now comes the decisive question. A topological space is connected if it cannot be split into two non-empty open pieces — if there is no clean fracture running through it. For finite spaces, there is a beautiful theorem, due to McCord (1966) and Stong (1966), that makes this check exact and tractable. The specialization preorder orders the points by how their neighbourhoods nest inside one another, and the connected components of the resulting structure are precisely the topological connected components.

    The crucial practical point: this computation works directly on the base — the refined building blocks — without ever needing to enumerate the full topology (which can be exponentially large). It runs in polynomial time, and the components it returns are exact, not approximate.

    The Decision Rule

    Everything so far converges on a single, actionable verdict. topologyR hands you a connectedness decision, and that decision has a direct methodological consequence:

    • If the induced topology is connected, then your data are consistent with a single continuous process. Global continuous methods — splines, kriging, polynomial interpolation, moving-average imputation, kernel methods — are mathematically supported. You may proceed.
    • If the induced topology is disconnected, then no single continuous function can cover the entire series. Global continuous methods are invalid by construction. You must segment the series along the connected components the package identifies, and model each piece independently — with regime-switching models, component-wise imputation, or finite mixtures.

    This is the package’s value proposition: a reproducible, topology-first workflow that decides, before you touch a model, whether global continuity is a justified assumption or a silent error.

    Global versus Local

    A subtlety worth flagging: the rule depends on what you are trying to learn. Global properties — a secular trend, a Hurst exponent, total neural synchronisation, a centennial warming signal — depend on relationships among all points and require topological connectivity to be valid. Local properties — instantaneous volatility in a small window, point-to-point rates of change, low-order autocorrelation — are defined on restricted neighbourhoods and remain valid within each connected component, regardless of whether the whole series is one piece or several. The package gives you the component structure to make that distinction operational.

    Time Has an Arrow: Directed Topologies and Irreversibility

    So far, the construction has treated the visibility graph as undirected — time flows, but the edges don’t care which way. That discards information. Time series are inherently directional: time runs from past to future, and many real systems are irreversible — they behave differently forwards and backwards. Economic expansions creep upward over years; recessions collapse in quarters. Neurons fire and recover on different timescales. The undirected graph cannot see this asymmetry.

    topologyR’s directed mode fixes this. With directed = TRUE, each visibility edge is oriented from the earlier time point to the later one, producing a directed acyclic graph (a DAG) in which the time index is a natural topological order. From this directed graph, the package extracts two neighbourhood structures: the forward neighbourhood (who can I see ahead of me?) and the backward neighbourhood (who behind me can see me?).

    Applying the Nada construction to each yields two topologies: a forward topology τ⁺ and a backward topology τ⁻. The pair (X, τ⁺, τ⁻) forms what Kelly (1963) called a bitopological space — a set equipped with two topologies rather than one. The divergence between them is a direct, topological measurement of temporal irreversibility.

    Irreversibility Indices

    In a perfectly reversible process — symmetric dynamics, no privileged direction — the two topologies coincide: τ⁺ ≅ τ⁻. They have the same number of connected components, the same base size, the same connectivity. In an irreversible process, they pull apart.

    topologyR quantifies this with several indices. The component irreversibility measures the normalised difference in the number of connected components between the forward and backward topologies: zero means symmetric, one means maximally asymmetric. The base irreversibility does the same for the sizes of the topological bases. The asymmetry direction — the signed difference in component counts — tells you which way the arrow points: a positive value means the forward topology is more connected (fewer components) than the backward one.

    That last point has a concrete physical interpretation. Consider a time series with gradual expansions and abrupt contractions — the classic shape of a business cycle, where GDP creeps up over years and drops in a quarter. During a gradual rise, forward visibility is relatively unobstructed: looking ahead from a point on the upslope, you can see far. After an abrupt drop, backward visibility is blocked: looking back from the trough, the cliff face hides earlier points. This asymmetry means the forward topology should be more connected than the backward topology — fewer forward components, more backward fragmentation. The package predicts, and the data confirm, a positive asymmetry direction for such series.

    The Alexandrov Layer and the Resolution Hierarchy

    There is a third topology lurking in the directed graph, and it is older than the Nada construction by several decades. The Alexandrov topology τ_A, introduced by Alexandrov in 1937, is the topology whose open sets are the upsets of the reachability relation — the sets that, once you enter them, contain everything reachable downstream. For each vertex, its minimal open set is the collection of all vertices reachable from it via directed paths.

    topologyR computes this efficiently: a reverse-order bitset propagation that processes vertices from last to first, OR-ing reachability sets together in O(nm/64) time, reusing the same high-performance bitset infrastructure as the Nada engine.

    The relationship between the Alexandrov and Nada topologies is precise and informative: τ_A is always a subset of the forward Nada topology. The Alexandrov base captures pure order structure — “who can reach whom” — while the Nada intersection closure generates additional sets that are not upsets, catching finer-grained structure. The difference in base sizes, |B_Nada| − |B_A|, tells you exactly how much extra topological information the Nada pipeline extracts beyond the raw ordering. A large gap means the closure operations are doing real work; a small gap means the order structure already tells most of the story.

    Under the Hood: Performance Without Compromise

    Topological enumeration is, in the worst case, exponential — the number of open sets can in principle double with every additional element. This is an inherent mathematical fact, not a software limitation. But topologyR is engineered so that the decision you actually need — connectedness — never requires that enumeration.

    The connectivity computation works on the base alone, via the specialization preorder, in O(n² · ⌈B/64⌉) time. The C++ backend (via Rcpp) represents every subset as a packed array of 64-bit words, so set operations reduce to machine-level bitwise instructions. A compile-time template dispatch selects single-word operations for series up to 64 points, two-word for up to 128, three-word for up to 192 — zero loop overhead, branch-free. Beyond that, a runtime fallback handles arbitrary sample sizes, and OpenMP parallelisation is available where the build supports it.

    The practical upshot: you can run the connectivity decision on series with thousands of points without ever touching the exponential regime. Safety limits (max_base_sets, max_open_sets) cap the intersection and union closures with informative termination flags, so if a computation does hit resource limits, you know exactly where and why — and the connectivity result remains valid as long as the base closure completes.

    A Real-World Test: Reading the Business Cycle

    The paper accompanying the package applies the framework to quarterly U.S. real GDP growth from 1992 to 2024 — 129 observations spanning over three decades. The bitopological analysis recovers a positive asymmetry direction: the forward topology is more connected than the backward one, exactly as predicted for a series with gradual expansions and abrupt contractions.

    The undirected topology partitions the series into six connected components, each corresponding to a distinct macroeconomic regime. Strikingly, the COVID-19 contraction and its rebound — the deepest and fastest swing in the sample — are classified as a single topological episode: one connected component spanning the collapse and recovery, reflecting the fact that the visibility structure treats the V-shaped episode as one structural unit rather than two separate events.

    This is the kind of insight the package is designed to produce: not a forecast, not a parameter estimate, but a structural classification that tells you where the legitimate boundaries in your data lie — and, critically, whether a global model is appropriate at all.

    Where It Sits: Complementary, Not Competing

    It is important to be clear about what topologyR is not. It is not a general-purpose topological data analysis (TDA) engine. Packages like GUDHI, Ripser, TDAstats, and scikit-TDA compute persistent homology — multi-scale features across all dimensions, capturing higher-order structures (loops, voids) via Betti numbers β₁, β₂ and their persistence across scales. That is a richer and harder enterprise.

    topologyR has a narrower and more focused aim: it zeroes in on β₀ — connectedness — for one-dimensional series, using graph-induced topologies, and it turns that single invariant into an actionable decision rule for method selection. Think of it as a pre-model governance tool: a rigorous gatekeeper that runs before you choose your modelling strategy, telling you whether the continuity assumptions your favourite methods require are actually justified by the data’s structure.

    The two approaches are complementary. For early-warning detection, precursor signals, or multi-channel structure, persistent homology is the right tool. For the binary question “can I legitimately fit a global continuous model to this series?”, topologyR gives a direct, interpretable, and mathematically exact answer. A natural hybrid workflow uses topologyR as a pre-test and persistent homology for deeper multi-scale analysis.

    Honest Limitations

    No tool is universal, and topologyR is transparent about its boundaries:

    1. Graph choice matters. HVG and NVG produce different graphs, and therefore potentially different topologies. The NVG, being denser, tends to produce fewer connected components. The package encourages comparing both and interpreting the difference — the gap itself is diagnostic.
    2. Sampling and noise. Sparse sampling can mimic disconnection; minor overlaps can mimic connection. The connectedness verdict should be treated as prima facie evidence, not absolute truth — especially near the boundary.
    3. β₀ only. The approach focuses on connectedness. It will not capture loops, voids, or higher-order patterns that persistent homology can detect. If your question is about cycles or multi-scale structure rather than fragmentation, you need the heavier machinery.
    4. Enumeration is exponential; connectivity is not. This is handled honestly: the connectivity decision is polynomial and scalable; full topology enumeration (needed for pairwise connectedness in the bitopological sense) is capped by safety limits with transparent reporting.

    The Bigger Picture

    What makes topologyR more than a clever technical exercise is its epistemological stance. It transforms a step that is normally a tacit habit — assuming continuity — into an explicit, testable, mathematical procedure. In doing so, it removes arbitrariness from one of the most consequential decisions in applied quantitative work: the choice between global and segmented methods.

    The package’s central theorem — that the Nada construction extends to directed graphs and yields a bitopological space whose asymmetry quantifies irreversibility — is formalised in Lean 4 against Mathlib, so the mathematical foundation is not merely asserted but machine-checked. The implementation is CRAN-compliant, passes R CMD check --as-cran cleanly, and ships with 68 unit tests covering visibility graphs, topology generation, connectivity, directed topology, Alexandrov topology, and bitopological analysis.

    For anyone who works with time series and has ever fitted a spline, run a kriging model, or estimated a trend — which is to say, for most of applied quantitative science — topologyR offers something rare: a way to check, before you model, whether the smoothness you are about to assume is a property of your data or a story you are telling yourself.

    Links and Credits

    The package is authored by José Mauricio Gómez Julián and released under the MIT licence. It requires R ≥ 4.0.0 with Rcpp and ggplot2. The companion paper, “Bitopological Spaces from Directed Graphs: Extending the Nada Construction to Capture Temporal Irreversibility,” develops the full mathematical theory, including the central theorem, the Alexandrov sublayer, specialization preorder, pairwise connectedness, polynomial-time algorithms, and the Lean 4 formalisation.

    If you use topologyR in your research, please cite the repository release.