Keywords: F-transform •fuzzy feature extraction •trajectory representation •iterated correlation matrices •contraction dynamics •convergence prediction •soft computing

1 Introduction↩︎

Nonlinear iterative procedures often generate trajectories whose early behavior carries information about subsequent convergence. In many computational settings, however, such trajectories are inconvenient objects for analysis: they may have variable stopping times, their raw states may be high-dimensional, and their early observations may be short or incomplete. A useful representation should therefore convert a short trajectory segment into fixed-dimensional coordinates while retaining information relevant to convergence, comparison, visualization, and exploratory grouping.

This paper studies such a representation problem for iterated Pearson correlation matrices. Given an initial matrix \(P_0\in\mathbb{R}^{n\times n}\), the iteration generates a trajectory \[P_{k+1}=f(P_k),\qquad k\ge 0.\] Under the row-wise Pearson correlation update, each row is centered, normalized to unit Euclidean norm, and correlated with all other normalized rows. From the first iterate onward, the trajectory lies in the set of symmetric positive semidefinite matrices with unit diagonal. This update is elementary, but its global behavior remains mathematically subtle. Classical work on repeated correlations and CONCOR reported rapid numerical stabilization and convergence toward block-pattern structures, especially in psychometrics and social network blockmodeling [1]–[4].

Recent work has organized this iteration quantitatively. A large-scale empirical study reported stable laws across dimensions from 3 to 2000, including a distinguished first-step contraction, nearly monotone post-transient decay, a stable relationship between contraction ratio and step size, and empirically bounded convergence lengths under the tested initialization model [5]. A subsequent finite-step analysis converted part of this empirical structure into probabilistic bounds for contraction ratios conditioned on normalized step size [6]. These results provide global and stepwise information about the dynamics. They do not, however, provide a compact representation of early trajectories that is directly suitable for machine-learning comparison, geometric visualization, or exploratory regime analysis.

The present paper addresses this gap using the F-transform framework. The F-transform replaces a signal by local coefficients computed over a fuzzy partition [7], [8]. Zero-degree coefficients summarize local levels, while first-degree coefficients capture local linear trends [9]. This value–trend separation has been useful in fuzzy modeling, time-series processing, filtering, and similarity analysis [10]–[12]. Here, the F-transform is used not primarily as a reconstruction tool, but as a fuzzy feature-extraction mechanism for partial dynamical information.

The proposed representation uses two scalar observables derived from the matrix trajectory. The first is the Frobenius step size, which measures update magnitude. The second is the contraction ratio, which measures the local evolution of contraction. On a short post-transient window, the logarithms of these two signals are projected onto a three-node triangular fuzzy partition. Three zero-degree coefficients and one centered first-degree coefficient are retained for each signal, giving an eight-dimensional two-channel descriptor. The step-size channel describes where the trajectory is along its contraction path, while the ratio channel describes how contraction is locally changing.

The methodological question is not whether this descriptor gives the highest possible predictive accuracy. Rather, the question is whether a compact, interpretable, fuzzy-local representation preserves enough information to support useful downstream analysis. The answer is assessed through four complementary criteria: predictive content, robustness to random train–test splits, robustness to a shifted observation window, and geometric organization under PCA and clustering. The methodology is intended as a general fuzzy-local trajectory representation framework and is not restricted to iterated correlation dynamics.

The contributions of the paper are as follows.

1. A two-channel F-transform descriptor is defined for early contraction trajectories of iterated correlation matrices. The descriptor separates local level from local trend and contraction magnitude from contraction evolution.

2. The descriptor is evaluated against raw, statistical, and PCA-compressed baselines across 22 matrix dimensions and 1000 trajectories per dimension.

3. Repeated random-split and shifted-window experiments show that the descriptor retains a stable compactness–performance trade-off rather than depending on a single favorable data split or observation window.

4. PCA and \(k\)-means analyses show that the descriptor induces a reproducible low-dimensional geometry, with a three-cluster description preferred among the tested values under the mean silhouette criterion.

The paper is organized as follows. Section 2 gives background on iterated correlation dynamics and F-transforms. Section 3 defines the two-channel representation. Section 4 describes the experimental pipeline and feature families. Section 5 presents the predictive, robustness, and geometric results. Section 6 discusses interpretation, limitations, and future work.

2 Background↩︎

2.1 Iterated Pearson correlation dynamics↩︎

Let \(P_k\in\mathbb{R}^{n\times n}\). The Pearson correlation update maps \(P_k\) to \(P_{k+1}\) by centering each row, normalizing each centered row, and forming the Gram matrix of the resulting row vectors. If \(Z_k\) denotes the row-centered and row-normalized version of \(P_k\), then \[P_{k+1}=Z_k Z_k^T.\] Thus, from the first update onward, \(P_k\) is symmetric positive semidefinite with diagonal entries equal to one. This places the trajectory inside the elliptope of correlation matrices.

Repeated correlation procedures have a long history in psychometrics and network analysis. McQuitty’s elementary linkage analysis [1], CONCOR-type blockmodeling [2], [3], and generalized association plots [4] all involve repeated or iterated correlation-like transformations. These procedures motivate the study of convergence behavior, but the present paper focuses on a more specific computational representation question: how to summarize the early contraction behavior of such iterations in a compact and interpretable way.

Beyond fuzzy methods, the problem of representing short dynamical trajectories connects to classical approaches in nonlinear dynamics. Takens’ embedding theorem [13] provides conditions for reconstructing attractor topology from delay coordinates, while manifold learning methods [14], [15] offer nonlinear dimensionality reduction. More recently, dynamical feature extraction has been explored through symbolic dynamics [16] and information-theoretic state partitioning [17]. The present work is complementary: rather than reconstructing the full attractor or learning a latent manifold, we construct a compact, interpretable fuzzy representation tailored to early contraction dynamics.

2.2 F-transform and fuzzy local summaries↩︎

Let \([a,b]\subset\mathbb{R}\). A family \(A=\{A_1,\dots,A_m\}\), with \(A_j:[a,b]\to[0,1]\), is a fuzzy partition with nodes \(c_j\) if each \(A_j\) is localized around \(c_j\), satisfies \(A_j(c_j)=1\), and fulfills the Ruspini condition \[\sum_{j=1}^{m}A_j(t)=1,\qquad t\in[a,b].\] For a bounded signal \(z:[a,b]\to\mathbb{R}\), the zero-degree F-transform coefficients are weighted local averages: \[\label{eq:f0-continuous} F_j^0[z]=\frac{\int_a^b z(t)A_j(t)\,dt}{\int_a^b A_j(t)\,dt},\qquad j=1,\dots,m.\tag{1}\] The first-degree F-transform gives a local linear approximation \[\label{eq:f1-local} F_j^1[z](t)=\beta_j^0+\beta_j^1(t-c_j),\tag{2}\] where \(\beta_j^0=F_j^0[z]\), and \[\label{eq:f1-continuous} \beta_j^1=\frac{\int_a^b z(t)(t-c_j)A_j(t)\,dt}{\int_a^b (t-c_j)^2A_j(t)\,dt}.\tag{3}\] For discrete signals, integrals are replaced by finite sums.

In this work, the F-transform is used as a feature extraction device. The zero-degree coefficients encode local levels, and the first-degree coefficient encodes local trend. The fuzzy supports make the descriptor less dependent on sharply localized time points and allow adjacent local summaries to share information.

3 Two-Channel Early Trajectory Representation↩︎

3.1 Contraction observables↩︎

For a trajectory \((P_k)\), define the Frobenius step size \[\label{eq:delta} \delta_k=\left\lVert P_{k+1}-P_k \right\rVert_F,\tag{4}\] where \[\left\lVert M \right\rVert_F=\left(\sum_{i=1}^n\sum_{j=1}^n M_{ij}^2\right)^{1/2}.\] Whenever \(\delta_k>0\), define the contraction ratio \[\label{eq:rho} \rho_k=\frac{\delta_{k+1}}{\delta_k}.\tag{5}\] The step-size sequence measures update magnitude, while the contraction-ratio sequence measures the local contraction law.

The response variable in the experiments is the convergence length \[\label{eq:nconv} N_{\mathrm{conv}}=\min\{k\ge 1:\delta_k<10^{-6}\}.\tag{6}\]

3.2 Early post-transient window↩︎

The main descriptor uses the fixed post-transient window \[\label{eq:window} T=\{2,3,4,5,6\}.\tag{7}\] This window starts after the distinguished first contraction event observed in earlier empirical studies [5]. For a trajectory with \(\delta_t>0\) for \(t=2,\dots,7\), define the logarithmic step-size signal \[\label{eq:x-signal} x(t)=\log\delta_t,\tag{8}\] and the logarithmic contraction-ratio signal \[\label{eq:y-signal} y(t)=\log\rho_t,\tag{9}\] for \(t\in T\). The ratio signal on \(T\) uses \(\delta_7\), since \(\rho_6=\delta_7/\delta_6\).

3.3 Fuzzy projection↩︎

Identify \(T\) with local coordinates \(\{0,1,2,3,4\}\). Use three triangular basic functions with centers \[c_1=0,\qquad c_2=2,\qquad c_3=4,\] and bandwidth \(h=2\): \[\label{eq:partition} A_j(t)=\max\left(0,1-\frac{|t-c_j|}{h}\right),\qquad j=1,2,3.\tag{10}\] These functions satisfy the Ruspini condition on the discrete window. The three-node design is the smallest symmetric fuzzy partition that gives left, central, and right local averages while retaining a centered first-order trend coefficient at the middle node.

For a discrete signal \(z:\{0,1,2,3,4\}\to\mathbb{R}\), define \[\label{eq:f0-discrete} F_j^0[z]=\frac{\sum_{t=0}^4 z(t)A_j(t)}{\sum_{t=0}^4 A_j(t)},\qquad j=1,2,3,\tag{11}\] and \[\label{eq:f1-discrete} \beta_2^1[z]=\frac{\sum_{t=0}^4 z(t)(t-c_2)A_2(t)}{\sum_{t=0}^4 (t-c_2)^2A_2(t)}.\tag{12}\] Applying these coefficients to \(x\) and \(y\) gives \[v_j=F_j^0[x],\quad u_j=F_j^0[y],\quad s_2=\beta_2^1[x],\quad r_2=\beta_2^1[y].\]

The descriptor therefore compresses two length-five logarithmic signals into eight interpretable coordinates. The zero-degree coefficients represent local levels, and the centered first-degree coefficients represent local trends. The first channel describes contraction magnitude, and the second channel describes contraction evolution.

3.4 Stability and trend interpretation↩︎

The descriptor has a simple stability property that is useful for interpreting it as a robust soft-computing representation.

In addition, the centered slope coefficient recovers affine trends exactly: if \(z(t)=a+bt\) on \(\{0,1,2,3,4\}\), then \(\beta_2^1[z]=b\). This follows by substitution into 12 ; the constant part cancels by symmetry around \(c_2=2\), while the linear part is normalized by the same quadratic weight in the denominator.

4 Experimental Framework↩︎

4.1 Dataset generation↩︎

The experiments were conducted for \[\begin{align} n\in\{&3,6,7,9,12,16,23,30,40,50,55,69,80,100,150,\\ &250,350,600,1054,1600,1700,2000\}. \end{align}\] For each dimension, 1000 independent initial matrices were sampled with entries uniformly distributed in \([-1,1]\). Each matrix was iterated under the Pearson correlation map until \(\delta_k<10^{-6}\), with a maximum iteration cap of 100. The first index satisfying the stopping condition was recorded as \(N_{\mathrm{conv}}\). For each trajectory, the early window \(T=\{2,3,4,5,6\}\) was extracted and all feature families described below were computed. Unless otherwise stated, all reported aggregate statistics were computed across all 22 tested matrix dimensions.

4.2 Compared feature families↩︎

The experiments compare the proposed descriptor with several alternative representations of the same early information.

1. Step-size F-transform descriptor: four coordinates \((v_1,v_2,v_3,s_2)\), using only the logarithmic step-size signal.

2. Full two-channel F-transform descriptor: eight coordinates \((v_1,v_2,v_3,s_2,u_1,u_2,u_3,r_2)\), using both step-size and ratio signals.

3. Raw early samples: the five values of \(x\), the five values of \(y\), or all ten values together.

4. Statistical summaries: mean, standard deviation, minimum, maximum, range, linear slope, and intercept for each channel.

5. PCA-compressed raw features: eight principal components extracted from the ten raw early samples, followed by the same regression model.

Table 1 summarizes the qualitative differences among the main representation types.

4.3 Prediction protocol↩︎

Random Forest regression was used to assess how much information each representation retains about \(N_{\mathrm{conv}}\). The regressor used 300 trees and otherwise fixed parameters across all compared feature families. The main comparison used an 80/20 train–test split with random state 42. The goal was representation comparison, not hyperparameter optimization.

To assess robustness to data partitioning and estimator initialization, the main predictive comparison was repeated over five random states \(0,1,2,3,4\). For each state, the train–test split and Random Forest initialization were changed consistently across feature families.

4.4 Shifted-window robustness↩︎

The main descriptor uses \(T=\{2,3,4,5,6\}\). To test sensitivity to this specific early window, the experiment was repeated with the shifted window \[T_{\mathrm{alt}}=\{3,4,5,6,7\}.\] The shifted-window experiment used the same F-transform construction and the same repeated-seed protocol, comparing the four main feature families: step-size F-transform, full F-transform, raw full trajectory, and statistical full summary.

4.5 Geometric organization↩︎

The geometric structure of the representation was analyzed using standardized eight-dimensional F-transform coordinates. Standardization was applied within each matrix dimension for PCA and clustering. PCA was used to quantify low-dimensional concentration and to generate two-dimensional visualizations. \(k\)-means clustering was applied in the standardized eight-dimensional representation space for \(k\in\{3,4,5\}\), using the silhouette score to compare cluster quality. Cluster separation was summarized by the range of cluster mean convergence lengths.

5 Results↩︎

5.1 Predictive information retained by the descriptor↩︎

Table 2 reports the aggregate predictive performance across the 22 tested dimensions. The strongest baselines are the fourteen-dimensional statistical full summary and the ten-dimensional raw full trajectory. The proposed eight-dimensional F-transform descriptor is slightly below these two baselines but remains very close, while substantially outperforming the four-dimensional step-size-only F-transform descriptor.

The principal interpretation is not that the F-transform descriptor dominates all alternatives. Rather, it preserves nearly the same predictive information as larger raw and statistical representations while providing explicitly interpretable fuzzy-local coordinates. Relative to the step-size-only descriptor, the mean \(R^2\) increases from 0.5001 to 0.6480. This shows that the contraction-ratio channel contributes nonredundant information about convergence behavior.

Although PCA-compressed raw features use the same dimensionality as the full F-transform descriptor, they do not explicitly preserve fuzzy-local level and trend structure. The stronger performance of the proposed descriptor therefore suggests that the improvement is not due only to dimensionality reduction, but also to the structured separation of contraction magnitude and contraction evolution.

Across dimensions, the strongest predictive model alternates primarily between the statistical-summary baseline and the raw full trajectory baseline, while the proposed F-transform descriptor remains consistently competitive. This reinforces the intended interpretation: the descriptor is a compact and interpretable representation, not an accuracy-optimized replacement for all larger baselines.

Figure 1 shows performance across matrix dimensions. The difficulty of the prediction task varies substantially with dimension for all feature families. The full F-transform descriptor remains close to the strongest baselines over the tested range, which supports its use as a compact representation rather than as an accuracy-optimized predictor.

5.2 Feature importance↩︎

Random Forest feature importances for the full F-transform descriptor show that the central step-size level \(v_2\) and the step-size slope \(s_2\) are the largest average contributors, and the ratio-channel coordinates have nonzero importance in the full descriptor.

Feature importance analysis further suggests that the central coordinates \((v_2,s_2,u_2,r_2)\) carry a substantial portion of the predictive signal, although the remaining local coefficients also contribute to the full descriptor’s overall performance.

5.3 Robustness under repeated random splits↩︎

Table 3 reports the repeated-seed comparison over five train–test splits and estimator initializations. The full F-transform descriptor remains close to the strongest raw and statistical baselines, with mean \(R^2=0.645\). Its average within-dimension variability is also comparable to those baselines.

The repeated-seed experiment supports two conclusions. First, the full descriptor’s performance is not an artifact of a single favorable train–test split. Second, adding the contraction-ratio channel improves not only mean performance but also stability relative to the step-size-only descriptor.

5.4 Robustness to shifted observation window↩︎

Table 4 reports the shifted-window experiment using \(T_{\mathrm{alt}}=\{3,4,5,6,7\}\). Predictive performance increases for all feature families, as expected because the shifted window observes a slightly later part of the contraction trajectory. The full F-transform descriptor again remains close to the raw and statistical baselines and continues to outperform the step-size-only descriptor.

The shifted-window result is important because it shows that the descriptor is not tuned narrowly to the specific window \(\{2,3,4,5,6\}\). The same construction remains effective when the early observation window is moved forward by one step. The full descriptor also has the lowest average within-dimension variability in this shifted-window experiment, which further supports the stability of the two-channel construction.

5.5 Geometric organization of the descriptor↩︎

Prediction measures only one aspect of representation quality. A useful trajectory descriptor should also organize trajectories geometrically. To test this, \(k\)-means clustering was applied to standardized eight-dimensional F-transform descriptors for \(k=3,4,5\). Table 5 reports mean silhouette scores and the mean range of cluster mean convergence lengths across dimensions.

The highest mean silhouette occurs for \(k=3\). The silhouette score is moderate rather than strong, so the result should not be interpreted as proof of sharply separated dynamical classes. Nevertheless, in high-dimensional stochastic trajectory data, silhouette values near 0.25–0.30 can still indicate reproducible coarse geometric organization rather than random structure. The consistent ranking \(k=3>k=4>k=5\) indicates that the representation induces a reproducible coarse geometry. Larger \(k\) increases the separation between extreme cluster mean convergence lengths, but at the cost of weaker cohesion and separation in the descriptor space.

5.6 Low-dimensional concentration↩︎

PCA applied to standardized eight-dimensional F-transform descriptors shows substantial low-dimensional concentration. Across dimensions, the first principal component explains on average \(55.94\%\) of variance, and the second explains \(28.31\%\). Together they explain \(84.26\%\) of total variance, with values ranging from \(78.4\%\) to \(89.2\%\).

This concentration supports the interpretation that the descriptor is not merely a collection of unrelated features. Most variation in early contraction behavior is organized along a small number of dominant directions, which explains why two-dimensional projections remain informative for visualization and exploratory clustering.

6 Discussion↩︎

6.1 What the descriptor contributes↩︎

The proposed descriptor is best understood as a compact fuzzy coordinate system for early nonlinear trajectory information. Its value is not that it universally outperforms raw or statistical baselines in predictive accuracy. Instead, it preserves nearly the same predictive information as larger feature families while making the structure of the representation explicit: local level and local trend are separated, and contraction magnitude and contraction evolution are encoded in distinct channels.

This distinction directly addresses the limitations of simpler summaries. Statistical summaries reduce a trajectory to global quantities and may suppress local organization. Raw samples preserve time-point values but do not explicitly encode fuzzy locality or local trend. The two-channel F-transform descriptor provides a middle ground: it is low-dimensional, interpretable, robust to small signal perturbations, and useful for prediction and geometry-based analysis.

6.2 Role of the contraction-ratio channel↩︎

The comparison between the step-size-only and full descriptors shows that the contraction-ratio channel is essential. The mean \(R^2\) gain from 0.5001 to 0.6480 indicates that local contraction evolution contains information not captured by contraction magnitude alone. This agrees with earlier empirical and finite-step analyses in which the ratio sequence \(\rho_k\) played a central role in describing the dynamics [5], [6].

6.3 Why compact fuzzy trajectory coordinates matter↩︎

Early-stage trajectory representations are useful when full trajectories are expensive to store, when convergence is not yet complete, or when many runs must be compared across sizes or initializations. The proposed descriptor converts a short segment into fixed-dimensional coordinates that can be used for convergence-length approximation, indexing of dynamical behavior, visualization, and exploratory grouping.

For CONCOR-type iterations and related repeated-correlation procedures, such a descriptor can support early monitoring of convergence behavior without requiring the full trajectory to be retained. The present experiments are performed on controlled random matrix ensembles rather than real network case studies, so this operational interpretation should be understood as a methodological motivation rather than as a fully developed application workflow.

The representation may also be interpreted through the lens of fuzzy information granulation. Early trajectory segments are replaced by coarse fuzzy-local summaries that preserve relevant dynamical structure while reducing detail. In this sense, the descriptor forms a low-order information granule of trajectory behavior, connecting the construction to broader granular computing principles [18], [19].

6.4 Geometric interpretation↩︎

The PCA and clustering results suggest that early contraction trajectories have reproducible geometric organization in the descriptor space. The preference for \(k=3\) under the mean silhouette criterion should not be overinterpreted as a definitive taxonomy of dynamical regimes. It does, however, show that the descriptor creates neighborhoods associated with measurable differences in convergence length. Together with the PCA concentration result, this suggests that the descriptor captures a low-dimensional organization of early contraction behavior rather than only providing a regression feature vector.

6.5 Limitations↩︎

The experiments use one initialization law, with entries independently sampled from the uniform distribution on \([-1,1]\), and one Pearson correlation update rule. The conclusions therefore should not be read as universal for all nonlinear normalization dynamics. The descriptor is deliberately local and low-dimensional, so it may omit finer-scale information present in longer trajectories. The paper also does not claim a new F-transform theorem; the contribution lies in the construction and validation of the two-channel descriptor for early contraction dynamics.

The present study is not intended as a final applied CONCOR case study. Rather, it provides a controlled representation benchmark for early contraction dynamics, establishing whether fuzzy-local trajectory coordinates preserve predictive and geometric information before extending the framework to empirical network blockmodeling data.

6.6 Future work↩︎

Future work may develop analytic relationships between descriptor coordinates and contraction laws, test the descriptor on empirical CONCOR and blockmodeling datasets, compare alternative fuzzy partitions and higher-degree F-transforms, and study whether descriptor clusters correlate with fixed-point types once the fixed-point classification is complete.

7 Reproducibility↩︎

All experiments are reproducible from the stated algorithmic choices. For each matrix dimension, 1000 independent matrices were generated with entries uniformly distributed in \([-1,1]\). The same Pearson correlation update, tolerance \(10^{-6}\), maximum iteration cap, early observation window, and feature-construction procedures were applied across dimensions. The main prediction benchmark used an 80/20 train–test split and Random Forest regression with 300 trees. Robustness experiments repeated the benchmark over five random states. PCA and \(k\)-means analyses used standardized eight-dimensional F-transform descriptors within each dimension. The complete implementation is archived at Zenodo [20].

8 Conclusion↩︎

This paper introduced a two-channel F-transform descriptor for early contraction trajectories in iterated correlation matrices. The descriptor maps logarithmic step-size and contraction-ratio signals on a short post-transient window to eight interpretable fuzzy-local coordinates. These coordinates separate local level from local trend and contraction magnitude from contraction evolution.

Across 22 matrix dimensions, the descriptor preserves predictive information about convergence length close to larger raw and statistical summaries while substantially improving over a step-size-only F-transform descriptor. Repeated random-split and shifted-window experiments show that the result is stable. PCA and clustering further show that the descriptor induces a reproducible low-dimensional geometry, with the first two principal components explaining about \(84\%\) of total variance and a three-cluster description favored among the tested values by the mean silhouette criterion.

The results support the use of low-order F-transform coefficients as compact soft-computing coordinates for partial nonlinear trajectory information. More broadly, they show that short early observations of an iterative matrix system can be transformed into interpretable fuzzy descriptors useful for convergence characterization, trajectory comparison, and exploratory geometric analysis.

Data availability↩︎

The code, computational pipeline, and archived research materials supporting this study are deposited in Zenodo [20].

References↩︎