Normalized Temporal Profiles: Concepts and Applications

• Normalized Temporal Profiles (NTP) are techniques that re-scale temporal data to expose invariant patterns, applied in event cameras, network cascades, and multivariate time series.
• In event-camera applications, NTP (or TNT) converts optical-flow-induced shears into pure translations, allowing CNNs to exploit native translation equivariance.
• In avalanche dynamics and time series, NTP normalizes duration and base-level differences, enabling universal scaling, improved correlation, and enhanced pattern detection.

Searching arXiv for the cited papers on Normalized Temporal Profiles across the relevant domains. arxiv_search query: (Zhu et al., 2019) Motion Equivariant Networks for Event Cameras with the Temporal Normalization Transform “Normalized Temporal Profiles” (NTP) is a term used in technically distinct ways across several research literatures. In event-camera learning, it denotes the re-parameterization $(x,y,t)\mapsto (x/t,y/t,t)$, also called the Temporal Normalization Transform (TNT), which converts optical-flow-induced 3-D shears into translations under constant flow (Zhu et al., 2019). In cascade dynamics on networks, it refers to normalized average avalanche shapes, obtained by rescaling time by duration and amplitude by either the maximum or a duration-dependent factor $T^\alpha$, so that curves for different durations collapse onto a universal profile at criticality (Gleeson et al., 2016). In continuous multivariate time series, it denotes component-wise normalization by the initial value in a window, $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$, followed by similarity measurement through correlation of the normalized curves (Luo et al., 17 Sep 2025). Across these usages, the common theme is normalization of temporal structure to isolate invariances or universal behavior, although the underlying objects, assumptions, and analytical purposes differ substantially.

1. Terminological scope and core definitions

The term NTP does not denote a single standardized construct across all fields. In the cited literature, it names three different normalization procedures applied to time-indexed data.

Domain	Object being normalized	Definition
Event cameras	Event coordinates in a spatiotemporal volume	$\rho:(\mathbf x,t)\mapsto (\mathbf x/t,t)$
Avalanche dynamics on networks	Average activity of avalanches of duration $T$	Rescale by $u=t/T$ and by $\max \langle V(t\mid T)\rangle$ or $T^\alpha$
Continuous multivariate time series	Feature trajectories over a window	$$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$

In the event-camera setting, an event stream is written as $e_i=(x_i,y_i,t_i,p_i)$, where $T^\alpha$0 is the pixel location, $T^\alpha$1 the timestamp, and $T^\alpha$2 the polarity. In the avalanche setting, the primitive quantity is the instantaneous activity $T^\alpha$3 of an avalanche $T^\alpha$4, and the NTP is the duration-conditioned average shape after normalization. In the multivariate time-series setting, the primitive quantity is $T^\alpha$5, and the NTP is defined component-wise by division by the window-start value (Zhu et al., 2019, Gleeson et al., 2016, Luo et al., 17 Sep 2025).

A plausible implication is that NTP should be understood as a family resemblance term rather than a single formalism: each usage removes a different nuisance factor. In the three cases above, the nuisance factor is, respectively, constant optical flow, duration-dependent scale in critical cascades, and per-series level effects.

2. Event-camera NTP as the Temporal Normalization Transform

For event cameras, the NTP representation is introduced through a binary or trinary event function

$T^\alpha$6

Under a constant optical-flow field $T^\alpha$7, an object point at $T^\alpha$8 at time $T^\alpha$9 moves as

$$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$0

In the raw $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$1 volume this appears as a 3-D shear (Zhu et al., 2019).

The transform $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$2 rescales the spatial coordinates by the reciprocal of the timestamp: $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$3 equivalently $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$4 and $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$5. Intuitively, events that lie along a linear track in $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$6 due to constant flow become a vertical stack of points at constant $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$7 plus a uniform shift when the flow changes.

The optical-flow action on the event function is written as

$$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$8

so that a motion by $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$9 translates points by $\rho:(\mathbf x,t)\mapsto (\mathbf x/t,t)$0 in raw space. Ordinary 3-D convolutions are not equivariant to $\rho:(\mathbf x,t)\mapsto (\mathbf x/t,t)$1. After applying $\rho:(\mathbf x,t)\mapsto (\mathbf x/t,t)$2, the same flow becomes a pure translation in the $\rho:(\mathbf x,t)\mapsto (\mathbf x/t,t)$3 domain: $\rho:(\mathbf x,t)\mapsto (\mathbf x/t,t)$4 Hence

$\rho:(\mathbf x,t)\mapsto (\mathbf x/t,t)$5

where $\rho:(\mathbf x,t)\mapsto (\mathbf x/t,t)$6 is a translation by $\rho:(\mathbf x,t)\mapsto (\mathbf x/t,t)$7 in the spatial coordinates of the $\rho:(\mathbf x,t)\mapsto (\mathbf x/t,t)$8-space. Because ordinary convolution $\rho:(\mathbf x,t)\mapsto (\mathbf x/t,t)$9 is equivariant to translation,

$T$0

so convolution after TNT commutes with a change in flow. The central claim is therefore not invariance but equivariance: constant optical-flow transformations become translations, and standard 3-D convolutions can exploit their native translation-equivariance without having to learn all possible motions.

3. Discretization, network integration, and empirical behavior in event-based classification

The event-camera implementation begins with the “discretized event volume,” attributed to Zhu et al. ’18. A set of $T$1 events $T$2 is linearly interpolated into a 3-D tensor $T$3 via

$T$4

where $T$5. Before TNT, a single 2-D landmark $T$6 is predicted via a small CNN+heatmap; $T$7 is then subtracted from all $T$8 to center the object and restore translation invariance. Temporal normalization is applied to the centered events by replacing each $T$9 by $u=t/T$0, followed by re-discretization into a second volume $u=t/T$1. The classification CNN receives $u=t/T$2 and consists of two 3-D convolution layers, both stride $u=t/T$3 in space and followed by average-pooling, then two fully-connected layers, $u=t/T$4-hidden $u=t/T$5 classes; no special modifications to convolution beyond standard 3-D conv are required (Zhu et al., 2019).

Training uses cross-entropy loss on the final 10-way softmax, $u=t/T$6 k iterations, and batch size $u=t/T$7. When training the landmark regressor, random 2-D translations are added as data augmentation. Timestamps are rescaled to $u=t/T$8 with $u=t/T$9 so that TNT does not blow up near $\max \langle V(t\mid T)\rangle$0. The reported datasets are N-MNIST, described as real DAVIS recordings of MNIST under 3 fixed motions, and N-MOVING-MNIST, described as synthetic via ESIM with 30 motion directions at $\max \langle V(t\mid T)\rangle$1 increments. Four named training regimes are specified: “all”/“all”, “1”/“all”, “all”/“sim”, and “1”/“sim”; the results table also includes a regime labeled “1/train”.

Regime	Baseline	TNT	TNT + regress
all/all	0.991	0.981	0.981
1/all	0.437	0.468	0.485
1/train	0.442	0.464	0.481
all/sim	0.396	0.592	0.566
1/sim	0.207	0.318	0.324

The pattern reported is that when training and testing on the same small set of motions, all methods do well, whereas with limited training motions but many test motions, TNT yields large gains of approximately $\max \langle V(t\mid T)\rangle$2–$\max \langle V(t\mid T)\rangle$3 points over baseline. TNT + regress further improves slightly when only a single training motion is available. The stated advantages are that TNT converts shear-deformations into translations, makes CNNs equivariant to constant optical flow, yields strong generalization to unseen motion directions, especially in data-scarce regimes, and is simple to integrate because it only requires resampling into a second event-volume. The stated limitations are equally specific: it assumes global constant optical flow over the window, the $\max \langle V(t\mid T)\rangle$4 factor can blow up as $\max \langle V(t\mid T)\rangle$5, translation invariance still has to be restored via a landmark or heuristic centering, and polarity reversal and missing edges parallel to motion remain unresolved and must be learned by the network.

4. Average avalanche shapes as NTPs at criticality

In cascade dynamics on networks, NTP refers to the normalized temporal profile of avalanches of fixed duration. Let $\max \langle V(t\mid T)\rangle$6 be the instantaneous activity of a single avalanche $\max \langle V(t\mid T)\rangle$7, for example the number of newly activated nodes or spikes at time $\max \langle V(t\mid T)\rangle$8 after initiation. If $\max \langle V(t\mid T)\rangle$9 is the subset of avalanches whose lifetimes are exactly $T^\alpha$0, or lie in a narrow bin around $T^\alpha$1, then the average avalanche shape of duration $T^\alpha$2 is

$T^\alpha$3

At the critical point of the dynamics, the average avalanche shapes for different durations can be rescaled so that they collapse onto a single universal curve. The paper states this both as normalization by the maximum,

$T^\alpha$4

plotted against $T^\alpha$5, and as the scaling ansatz

$T^\alpha$6

for large $T^\alpha$7, $T^\alpha$8 (Gleeson et al., 2016).

The derivation proceeds by mapping, under unidirectional and locally tree-like assumptions, a cascade on a network to a continuous-time Markov branching process. Each particle represents an exposed-vulnerable node waiting to activate; a particle dies after an exponential(1) lifetime and is replaced by $T^\alpha$9 children with probability $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$0. The generating function is

$$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$1

If $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$2 is the extinction probability by time $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$3 of a process started from one particle, then the backward Kolmogorov equation is

$$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$4

Branching-process theory yields the closed-form expression

$$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$5

The same account also rewrites the evolution of the unconditioned average number of particles $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$6 as

$$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$7

where $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$8 and $$\mathrm{NTP}_i^{(d)}(t)=X_i^{(d)}(t)/X_i^{(d)}(t_0)$$9 are determined by $e_i=(x_i,y_i,t_i,p_i)$0. At criticality, $e_i=(x_i,y_i,t_i,p_i)$1 and $e_i=(x_i,y_i,t_i,p_i)$2, so that the process neither grows nor decays; conditioning on extinction at time $e_i=(x_i,y_i,t_i,p_i)$3 then produces the universal shape $e_i=(x_i,y_i,t_i,p_i)$4.

5. Heavy-tailed asymmetry, universality classes, and tests for criticality

At the critical point, $e_i=(x_i,y_i,t_i,p_i)$5. If the offspring distribution has a finite second moment, $e_i=(x_i,y_i,t_i,p_i)$6, then

$e_i=(x_i,y_i,t_i,p_i)$7

and one finds $e_i=(x_i,y_i,t_i,p_i)$8 with $e_i=(x_i,y_i,t_i,p_i)$9, a symmetric parabola. If instead $T^\alpha$00 with $T^\alpha$01, so that $T^\alpha$02, then

$T^\alpha$03

and

$T^\alpha$04

The scaling exponent is therefore

$T^\alpha$05

and the universal shape function, up to an overall constant, is

$T^\alpha$06

Its maximum lies at

$T^\alpha$07

so the profile is left-skewed (Gleeson et al., 2016).

The paper further shows that on a random configuration-model network the branching-process tail exponent depends on both topology and dynamics. In undirected networks,

$T^\alpha$08

where $T^\alpha$09 is the degree-distribution exponent and $T^\alpha$10 characterizes how the single-seed-activation probability $T^\alpha$11 decays with degree $T^\alpha$12, $T^\alpha$13. In directed networks with in/out-degrees independent,

$T^\alpha$14

For threshold models, $T^\alpha$15 for Centola-Macy and $T^\alpha$16 for Watts. The stated condition for asymmetric, left-skewed NTPs is $T^\alpha$17, equivalently $T^\alpha$18, where $T^\alpha$19 in the undirected case.

Three numerical examples are reported. In an information-spreading model on directed networks, scale-free out-degree with exponent $T^\alpha$20 yields a left-skewed collapse, whereas $T^\alpha$21-regular out-degree yields the symmetric parabola $T^\alpha$22. In a neuronal-avalanche binary-firing model, scale-free out-degree with $T^\alpha$23 collapses onto $T^\alpha$24, while regular networks again give a parabolic collapse. In a behavior-adoption Centola-Macy threshold model on undirected networks, $T^\alpha$25 with $T^\alpha$26 produces left-skewed collapse, whereas $T^\alpha$27-regular networks produce symmetric collapse. The proposed empirical protocol is to collect avalanche time series, bin by duration $T^\alpha$28, compute $T^\alpha$29, rescale time by $T^\alpha$30, rescale amplitude by $T^\alpha$31, adjust $T^\alpha$32 until the curves collapse best, and confirm criticality by plotting the unconditioned average activity $T^\alpha$33, which is flat at criticality and shows exponential decay or growth away from criticality. The paper presents this as a more sensitive diagnostic than relying solely on power-law size distributions.

6. Scale-invariant NTPs in continuous multivariate time series and community detection

In “FTSCommDetector,” NTP is defined for an entity $T^\alpha$34 observed over a window $T^\alpha$35, with raw multivariate observation $T^\alpha$36, by

$T^\alpha$37

For the special case $T^\alpha$38 and $T^\alpha$39 equal to price, this reduces to the NAV profile,

$T^\alpha$40

Pairwise behavioral similarity is then measured by the Pearson correlation of the normalized curves: $T^\alpha$41 If a raw trajectory is rescaled by a positive constant $T^\alpha$42, $T^\alpha$43, then $T^\alpha$44. The paper states that this removes level effects, such as low-priced versus high-priced stocks, and isolates pure behavioral co-movement. It further states that because correlation is homogeneous of degree zero, dividing by the window-start value guarantees that differences in volatility scale or price level do not bias the similarity score (Luo et al., 17 Sep 2025).

The practical computation is given in pseudocode. For each entity and feature, the base value is the value at $T^\alpha$45, and the normalized series is formed as $T^\alpha$46, with $T^\alpha$47 added for numeric stability. A NAV correlation matrix is then computed, followed by a modularity null model: $T^\alpha$48 The NTP array is fed into the evaluation metrics IntraCorr and InterDissim, and the matrix $T^\alpha$49 is used as edge features in the dynamic-connectivity module of the Temporal Coherence Architecture. The static graph $T^\alpha$50 is built once per window via thresholded correlation, then each edge $T^\alpha$51 is augmented with $T^\alpha$52. In each TransformerConv layer, attention incorporates the edge term through

$T^\alpha$53

During training, an NAV-based composite score

$T^\alpha$54

is used for early stopping, and final quality is reported in terms of IntraCorr and InterDissim of NTP profiles.

The theoretical properties stated for this NTP are Theorem 2, “Behavioral Coherence Optimality,” and an information-theoretic justification. The theorem states scale-invariance for any positive scalars $T^\alpha$55 and a portfolio-risk interpretation: grouping entities to maximize

$T^\alpha$56

is equivalent to minimizing the variance of an equal-weight portfolio over each cluster, since cluster variance is proportional to $T^\alpha$57. The information-theoretic justification states that NTP correlation extracts pure co-movement information unpolluted by marginal variance differences and, when combined with dual-scale encoding, helps the model focus on complementary short- versus long-term patterns.

The comparison given in the paper places NTP-correlation against raw Euclidean distance, Dynamic Time Warping, and feature-wise Pearson correlation on $T^\alpha$58. Its listed strengths are exact invariance to scaling of any individual series, robustness to cross-sectional volatility differences, cheap computation linear in $T^\alpha$59, and alignment with financial intuition through normalized returns. Its listed limitations are that ratios can be unstable if the base value $T^\alpha$60 is extremely small or noisy, mitigated by adding $T^\alpha$61, and that absolute magnitude signals are discarded because only relative shape matters. Empirically, FTSCommDetector is reported to achieve IntraCorr up to $T^\alpha$62 and InterDissim up to $T^\alpha$63 on SP100, with gains ranging from $T^\alpha$64 to $T^\alpha$65 over the strongest baselines across SP100, SP500, SP1000, and Nikkei 225. IntraCorr varies by only approximately $T^\alpha$66 when the sliding-window length $T^\alpha$67 ranges from $T^\alpha$68 to $T^\alpha$69 days. Over $T^\alpha$70 windows, the number of clusters discovered by maximizing NTP-correlation remains stable, with mean approximately $T^\alpha$71 for SP100, except during major market shocks. Case studies reported include the GameStop event from January to June 2021, where NTP clustering reveals 6 behavioral groups cutting across GICS sectors, and an AI valuation reset in January 2025, where NTP clusters separate AAPL from MSFT despite identical sector labels. Ablations indicate that removing NAV/NTP from dynamic edge features or from evaluation costs approximately $T^\alpha$72–$T^\alpha$73 in IntraCorr.

A plausible implication of these three usages is that NTP functions as a normalization primitive whose scientific role depends on the target symmetry. In event cameras, it converts motion-induced shear into translation-equivariant structure; in avalanche analysis, it exposes universality classes through collapse of duration-conditioned shapes; in multivariate time series, it factors out per-series scale so that correlations emphasize relative temporal behavior.