PAX-TS: Multi-Domain High-Resolution Methods

• PAX-TS is a context-dependent label used in three distinct fields: time series forecasting explainability, TES-based PIXE instrumentation, and antiprotonic atom spectroscopy.
• In forecasting, it employs localized perturbations and aggregated change ratios to generate multi-granular, model-agnostic explanations of input-to-output relationships.
• In spectroscopy, PAX-TS implementations leverage TES microcalorimeter arrays to achieve high energy resolution, enabling precise X-ray measurements and tests of strong-field QED.

In the arXiv literature, PAX-TS appears in three distinct contexts: a model-agnostic post-hoc algorithm to explain time series forecasting models and their forecasts via localized perturbations (Kreuzer et al., 26 Aug 2025); a transition-edge sensor detector array used to detect X-rays in particle induced X-ray emission measurements with a 2 MeV proton beam (Palosaari et al., 2013); and the TES-Sensor implementation within PAX (antiProtonic Atom X-ray spectroscopy), a program aimed at high-precision x-ray spectroscopy of antiprotonic atoms for testing strong-field QED (Baptista et al., 15 Jan 2025). Context therefore determines the meaning of the term.

1. Terminological scope and disambiguation

The three attested uses of PAX-TS span machine learning explainability, PIXE instrumentation, and antiprotonic-atom spectroscopy. A compact disambiguation is therefore necessary.

Context	Domain	Description
(Kreuzer et al., 26 Aug 2025)	Time series forecasting	Model-agnostic post-hoc algorithm based on localized input perturbations
(Palosaari et al., 2013)	PIXE instrumentation	TES microcalorimeter setup for proton-induced X-ray emission measurements
(Baptista et al., 15 Jan 2025)	Fundamental spectroscopy	TES-Sensor implementation within antiProtonic Atom X-ray spectroscopy

A common misconception would be to treat PAX-TS as a single framework with a unified technical meaning. The literature provided here does not support that reading. Instead, the term is reused across substantially different experimental and algorithmic settings. This suggests that PAX-TS is a context-dependent label rather than a standardized cross-domain acronym.

2. PAX-TS as a forecasting explanation formalism

In time series forecasting, PAX-TS is defined for a pre-trained forecasting model

$$f \equiv M_\theta : \mathbb{R}^{d \times b} \to \mathbb{R}^{d \times h},$$

where $d \in \mathbb{N}$ is the number of channels, $b \in \mathbb{N}$ is the input window length, and $h \in \mathbb{N}$ is the forecast horizon. The observed past subsequence is

$x \in \mathbb{R}^{d \times b},$

the ground-truth future is

$y \in \mathbb{R}^{d \times h},$

and the forecast is

$\hat y = M_\theta(x).$

The stated objective is to explain how small, localized changes in $x$ affect some human-comprehensible property $\pi(\hat y)$ of the forecast, such as a particular time step, the maximum, the mean, or a trend (Kreuzer et al., 26 Aug 2025).

The central quantity is the change ratio

$$r_{\pi,\alpha} = \frac{\pi(\hat y') - \pi(\hat y)}{\Delta},$$

where $d \in \mathbb{N}$0, $d \in \mathbb{N}$1 is obtained from a perturbation operator $d \in \mathbb{N}$2, $d \in \mathbb{N}$3, and $d \in \mathbb{N}$4 is the perturbation scale. Averaging over a set of scales $d \in \mathbb{N}$5, while re-introducing the sign of $d \in \mathbb{N}$6, yields

$d \in \mathbb{N}$7

Higher-magnitude $d \in \mathbb{N}$8 indicates greater sensitivity of $d \in \mathbb{N}$9 to the perturbation location or type. Within this formalism, PAX-TS is entirely post-hoc and model-agnostic.

3. Perturbation operators, analysis loop, and multi-granular explanations

PAX-TS in forecasting consists of three building-block perturbations plus a structured analysis loop (Kreuzer et al., 26 Aug 2025). The first is a localized Gaussian-smoothed index perturbation. For any channel $b \in \mathbb{N}$0 and input index $b \in \mathbb{N}$1,

$b \in \mathbb{N}$2

and

$b \in \mathbb{N}$3

The perturbed input is then

$b \in \mathbb{N}$4

Here $b \in \mathbb{N}$5 is a window width, $b \in \mathbb{N}$6 controls Gaussian softness, and $b \in \mathbb{N}$7 is the upward or downward scale.

The second perturbation family is summary-statistic scaling. Denoting

$b \in \mathbb{N}$8

the first-moment scaling is

$b \in \mathbb{N}$9

and the second-moment scaling is

$h \in \mathbb{N}$0

The third perturbation family is trend-drift adjustment. With seasonality length $h \in \mathbb{N}$1, one first deseasonalizes

$h \in \mathbb{N}$2

After fitting a line $h \in \mathbb{N}$3, the slope is adjusted by

$h \in \mathbb{N}$4

Three intercept options—left-fixed, symmetric, right-fixed—define $h \in \mathbb{N}$5, and the adjusted series is

$h \in \mathbb{N}$6

The structured analysis loop is given as: compute $h \in \mathbb{N}$7; for each scale $h \in \mathbb{N}$8 form $h \in \mathbb{N}$9, compute $x \in \mathbb{R}^{d \times b},$0, evaluate $x \in \mathbb{R}^{d \times b},$1, and for each property $x \in \mathbb{R}^{d \times b},$2 compute

$x \in \mathbb{R}^{d \times b},$3

then aggregate to $x \in \mathbb{R}^{d \times b},$4 (Kreuzer et al., 26 Aug 2025).

This procedure yields multi-granular explanations. At high granularity, $x \in \mathbb{R}^{d \times b},$5 selects a single $x \in \mathbb{R}^{d \times b},$6, producing time-step importance. At medium granularity, $x \in \mathbb{R}^{d \times b},$7 returns $x \in \mathbb{R}^{d \times b},$8, producing a $x \in \mathbb{R}^{d \times b},$9 heatmap of input-to-output correlations. At low granularity in the multivariate case, perturbing one channel at a time and returning each output channel’s full-window mean or individual steps yields a $y \in \mathbb{R}^{d \times h},$0 cross-channel correlation matrix.

For multivariate forecasts, the channel-wise response is

$y \in \mathbb{R}^{d \times h},$1

and the aggregate matrix is

$y \in \mathbb{R}^{d \times h},$2

Thus $y \in \mathbb{R}^{d \times h},$3 summarizes how perturbing input channel $y \in \mathbb{R}^{d \times h},$4 impacts output channel $y \in \mathbb{R}^{d \times h},$5 on average.

4. Pattern classes, benchmark evidence, and multivariate interpretation

The forecasting study reports that, when visualizing the $y \in \mathbb{R}^{d \times h},$6 temporal-dependency heatmaps, six recurring patterns emerge (Kreuzer et al., 26 Aug 2025). These are Diagonals, Diagonals (End), Last-Timestep, Bipolar Regions, Fully Correlated, and Other. The criteria are expressed directly on the normalized matrix $y \in \mathbb{R}^{d \times h},$7 of $y \in \mathbb{R}^{d \times h},$8 values. For example, Diagonals requires

$y \in \mathbb{R}^{d \times h},$9

with small off-diagonals, while Last-Timestep requires

$\hat y = M_\theta(x).$0

and all earlier rows to remain below that threshold on average. Bipolar Regions requires two contiguous submatrices with opposite signs, each of area at least $\hat y = M_\theta(x).$1.

Empirically, Diagonals and Bipolar Regions indicate the best forecast errors, with normalized error $\hat y = M_\theta(x).$2 and $\hat y = M_\theta(x).$3 respectively, whereas Last-Timestep and Fully Correlated correspond to poor performance with $\hat y = M_\theta(x).$4–$\hat y = M_\theta(x).$5. Other lies in between. The paper also states that explanations of high-performing and low-performing algorithms differ on the same datasets, and that these differences highlight that PAX-TS explanations effectively capture a model’s behavior.

The benchmark uses 7 algorithms—Naïve, DLinear (MLP), MultiPatchFormer (Transformer), SegRNN (RNN), TimeMixer (MLP), iTransformer (Transformer), and TimesFM (LLM decoder)—across 10 datasets: M4-Hourly, Weather, Transactions, CIF, Rain, M4-Yearly, COVID (6 variates), ETTh1 (7 variates), Exchange, and Illness (7 variates). The setup fixes forecast length $\hat y = M_\theta(x).$6, input length $\hat y = M_\theta(x).$7, scales $\hat y = M_\theta(x).$8, window $\hat y = M_\theta(x).$9, softness $x$0, and season length $x$1 by dataset. The metrics are MAE, MSE, OWA. Averaged over 3 seeds, TimeMixer, MultiPatchFormer, iTransformer consistently outperform Naïve (Bonferroni-Dunn $x$2), while multivariate datasets remain challenging: none beats Naïve by a large margin. Reported inference times per dataset are iTransformer $x$3, MultiPatchFormer $x$4, TimeMixer $x$5, and TimesFM(LLM) $x$6.

A multivariate ETTh1 example further illustrates the framework. On the 7-channel ETTh1 dataset, the cross-channel algorithm produces a $x$7 matrix $x$8. Visualized as a directed graph, iTransformer shows two strong effects: input channel 5 $x$9 output channel 0 and input channel 5 $\pi(\hat y)$0 output channel 2. A finer heatmap $\pi(\hat y)$1 of size $\pi(\hat y)$2 shows that perturbing channel 5 at its last few steps produces strong positive and negative shifts in each $\pi(\hat y)$3, especially near the end of the horizon, whereas channel $\pi(\hat y)$4 shows only weak, near-zero effects. The paper interprets this as evidence that the model exploits cross-channel information rather than only autoregressive patterns.

5. PAX-TS as TES-based PIXE instrumentation

In the spectroscopy literature, PAX-TS denotes a transition-edge sensor microcalorimeter setup for particle induced X-ray emission measurements using a 2.015 MeV $\pi(\hat y)$5 beam from a Pelletron 1.7 MV accelerator (Palosaari et al., 2013). The detector system uses a 160-pixel transition-edge sensor microcalorimeter array fabricated at NIST—Boulder. In the first proof-of-principle measurement, 12 pixels were connected and 9 yielded good data. The superconducting bilayer is a Mo/Cu proximity-effect film with $\pi(\hat y)$6, the absorber material is Bi, the absorber geometry is $\pi(\hat y)$7, and the absorber thickness is $\pi(\hat y)$8. The electrical readout is TDM-SQUID multiplexing, specifically 1-row time-division–multiplexing (NIST electronics). The measured intrinsic noise performance is $\pi(\hat y)$9 FWHM at $$r_{\pi,\alpha} = \frac{\pi(\hat y') - \pi(\hat y)}{\Delta},$$0 for the best pixel.

The cryogenic platform is a cryogen-free adiabatic demagnetization refrigerator (ADR, Denali Model 102, High Precision Devices, Inc.) with base temperature $$r_{\pi,\alpha} = \frac{\pi(\hat y') - \pi(\hat y)}{\Delta},$$1 and a regulated operating point of $$r_{\pi,\alpha} = \frac{\pi(\hat y') - \pi(\hat y)}{\Delta},$$2 with $$r_{\pi,\alpha} = \frac{\pi(\hat y') - \pi(\hat y)}{\Delta},$$3 (rms) stability over 12 h (with readout on). Pre-cooling stages at 60 K and 3 K are provided by a Cryomech PT407 RM pulse-tube cooler, with vibrations decoupled by a remote valve and flexible bellows. Magnetic shielding consists of two nested A4K cylinders at 60 K and 1 K, plus a superconducting Al can at 50 mK around the detector snout. A custom-built snout terminates in the sample chamber and couples proton-induced X-rays through an AP3.3 ultra-thin polymer/Al grid vacuum window (Moxtek Inc.). Inside the snout, the X-rays pass through three infrared filters (1 $$r_{\pi,\alpha} = \frac{\pi(\hat y') - \pi(\hat y)}{\Delta},$$4m PET + 1 $$r_{\pi,\alpha} = \frac{\pi(\hat y') - \pi(\hat y)}{\Delta},$$5m Al each) at 60 K, 1 K and 50 mK, blocking room-temperature black-body radiation while transmitting X-rays down to $$r_{\pi,\alpha} = \frac{\pi(\hat y') - \pi(\hat y)}{\Delta},$$6.

The measurement geometry places the sample tilted $$r_{\pi,\alpha} = \frac{\pi(\hat y') - \pi(\hat y)}{\Delta},$$7 toward the detector, with beam incidence at $$r_{\pi,\alpha} = \frac{\pi(\hat y') - \pi(\hat y)}{\Delta},$$8 and the TES at $$r_{\pi,\alpha} = \frac{\pi(\hat y') - \pi(\hat y)}{\Delta},$$9 relative to beam. Backscattered ions are stopped by an $d \in \mathbb{N}$00 PET filter inside the chamber, which also filters low-energy X-rays. The sample–TES absorber distance is 30 cm, and the stated solid angle for 12 pixels is

$d \in \mathbb{N}$01

Planned upgrades move the array to 5 cm for a $d \in \mathbb{N}$02 solid-angle gain and activate all 160 pixels for a $d \in \mathbb{N}$03 gain, giving a total $d \in \mathbb{N}$04 increase in collecting power.

The instrumental energy resolution is expressed as

$d \in \mathbb{N}$05

where $d \in \mathbb{N}$06 is the energy-independent noise term and $d \in \mathbb{N}$07 describes the energy-dependent term from thermodynamic fluctuation noise. For the best pixel at $d \in \mathbb{N}$08, the observed resolution is $d \in \mathbb{N}$09 FWHM. Under the assumption that the dominant noise is the amplifier-and-Johnson term,

$d \in \mathbb{N}$10

with the second term contributing negligibly at $d \in \mathbb{N}$11.

The spectral results emphasize what this resolution enables. For a $d \in \mathbb{N}$12Fe Mn $d \in \mathbb{N}$13 line, the best-pixel resolution is $d \in \mathbb{N}$14 FWHM; for a 2 MeV proton-induced Mn spectrum combined over 9 pixels, the resolution is $d \in \mathbb{N}$15 FWHM, with $d \in \mathbb{N}$16 broadening, likely from environmental interference. For a bulk Cu sample, the TES resolves Cu $d \in \mathbb{N}$17, Cu $d \in \mathbb{N}$18, and small Ni and Co impurity lines at $d \in \mathbb{N}$19 and $d \in \mathbb{N}$20 that are not separable in an Amptek SDD (130 eV resolution at 5.9 keV). A satellite peak on the high-energy flank of Cu $d \in \mathbb{N}$21 is also observed, consistent with multivacancy satellite-line theory. The setup is summarized as demonstrating for the first time multiplexed TES microcalorimeters in proton-induced X-ray emission spectroscopy, with $d \in \mathbb{N}$22 resolution at 6 keV and the capacity to resolve impurity and satellite lines unobservable with standard detectors.

6. PAX-TS within antiprotonic-atom X-ray spectroscopy

A third use of PAX-TS appears in PAX (antiProtonic Atom X-ray spectroscopy), where it denotes the TES-Sensor implementation of a program designed to test strong-field quantum electrodynamics effects by performing high-precision x-ray spectroscopy of antiprotonic atoms (Baptista et al., 15 Jan 2025). The experiment uses a 100 keV antiproton beam extracted from the ELENA ring at CERN. A special “waterfall” slow-extraction (microbunching) reduces instantaneous rates by $d \in \mathbb{N}$23 to avoid pile-up in the TES array, with spill duration $d \in \mathbb{N}$24 with $d \in \mathbb{N}$25 microbunches.

The detection system is a large-area Transition-Edge-Sensor (TES) microcalorimeter. The prototype is a $d \in \mathbb{N}$26 Sn-absorber on Si with $d \in \mathbb{N}$27 FWHM resolution at $d \in \mathbb{N}$28–$d \in \mathbb{N}$29. The final detector is specified as four “microsnouts,” 240 pixels total, each pixel $d \in \mathbb{N}$30, with optimized absorber thickness $d \in \mathbb{N}$31–$d \in \mathbb{N}$32 and quantum efficiency $d \in \mathbb{N}$33. The readout is microwave SQUID-multiplexing, cooled by a $d \in \mathbb{N}$34He-backed adiabatic demagnetization refrigerator. Calibration uses a permanently mounted $d \in \mathbb{N}$35Co source with 122 keV and 136 keV $d \in \mathbb{N}$36-lines of sub-eV uncertainties, intended to counteract TES nonlinearity.

PAX focuses on circular Rydberg transitions

$d \in \mathbb{N}$37

in antiprotonic atoms, including p-Ne: $d \in \mathbb{N}$38 at $d \in \mathbb{N}$39, p-Ar: $d \in \mathbb{N}$40 at $d \in \mathbb{N}$41, p-Xe: $d \in \mathbb{N}$42 at $d \in \mathbb{N}$43, and p-W: $d \in \mathbb{N}$44 at $d \in \mathbb{N}$45. The gross Dirac binding energy for a hydrogen-like nucleus of charge $d \in \mathbb{N}$46 is given as

$d \in \mathbb{N}$47

with corrections

$d \in \mathbb{N}$48

The one-loop vacuum-polarization potential is

$d \in \mathbb{N}$49

with corresponding shift

$d \in \mathbb{N}$50

The stated methodological advantage is the elimination of nuclear uncertainties. Circular Rydberg antiprotonic wavefunctions have nodes $d \in \mathbb{N}$51 from the nucleus, so overlap with the finite-size core is strongly suppressed. The paper gives p-Xe as an example where the finite-nuclear-size shift is $d \in \mathbb{N}$52 while second-order QED is $d \in \mathbb{N}$53, whereas in H-like U the finite-nuclear-size shift is $d \in \mathbb{N}$54 and is comparable to first-order QED at $d \in \mathbb{N}$55. It also states that strong-interaction level shifts and broadening are negligible for $d \in \mathbb{N}$56–10.

The field strengths and precision goals are correspondingly extreme. For p-Ne at $d \in \mathbb{N}$57, the peak Coulomb field is $d \in \mathbb{N}$58, or $d \in \mathbb{N}$59; for p-Pb at $d \in \mathbb{N}$60, it is $d \in \mathbb{N}$61, or $d \in \mathbb{N}$62, with $d \in \mathbb{N}$63. Earlier germanium-detector measurements at LEAR achieved $d \in \mathbb{N}$64, whereas the PAX-TES goal is intrinsic $d \in \mathbb{N}$65 at $d \in \mathbb{N}$66, corresponding to $d \in \mathbb{N}$67; with careful calibration, the overall target accuracy is $d \in \mathbb{N}$68–$d \in \mathbb{N}$69. The program is therefore positioned as a route to testing vacuum polarization, second-order QED corrections, and, if deviations at the level $d \in \mathbb{N}$70 were found, possible contributions from new interactions such as a light $d \in \mathbb{N}$71 gauge boson or dark-sector induced vacuum refractive index changes.

Taken together, these three usages show that PAX-TS denotes either a localized-perturbation explanation framework for forecasting, a TES-based PIXE setup, or a TES implementation for antiprotonic-atom spectroscopy. The shared label does not imply shared method or apparatus; the commonality lies instead in the deployment of technically specific, high-resolution analysis procedures within their respective domains.