---
title: 'PAX-TS: Multi-Domain High-Resolution Methods'
url: https://www.emergentmind.com/topics/pax-ts
type: topic
---

# PAX-TS: Multi-Domain High-Resolution Methods

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 [2508.18982]; a **transition-edge sensor detector array** used to detect X-rays in **particle induced X-ray emission** measurements with a **2 MeV proton beam** [1310.1733]; 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 [2501.08893]. 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 |
|---|---|---|
| [2508.18982] | Time series forecasting | Model-agnostic post-hoc algorithm based on localized input perturbations |
| [1310.1733] | PIXE instrumentation | TES microcalorimeter setup for proton-induced X-ray emission measurements |
| [2501.08893] | 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 [2508.18982].

The central quantity is the **change ratio**
$$
r_{\pi,\alpha} = \frac{\pi(\hat y') - \pi(\hat y)}{\Delta},
$$
where $\hat y' = M_\theta(x')$, $x'$ is obtained from a perturbation operator $p(\cdot)$, $\Delta = \mathrm{dist}(x,x')$, and $\alpha$ is the perturbation scale. Averaging over a set of scales $\mathcal{A}$, while re-introducing the sign of $\alpha$, yields
$$
\bar r_\pi = \frac{1}{|\mathcal{A}|} \sum_{\alpha \in \mathcal{A}} \mathrm{sign}(\alpha)\, r_{\pi,\alpha}.
$$
Higher-magnitude $\bar r_\pi$ indicates greater sensitivity of $\pi(\hat y)$ 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** [2508.18982]. The first is a **localized Gaussian-smoothed index perturbation**. For any channel $c$ and input index $t \in \{1,\dots,b\}$,
$$
w_p(i; t,w,s) =
\begin{cases}
\exp\!\left[-s\left(\frac{i-t}{w}\right)^2\right], & |i-t| \le w, \\
0, & \text{otherwise,}
\end{cases}
$$
and
$$
w_a(i; t) = \frac{\min(x_{c,i},x_{c,t})}{\max(x_{c,i},x_{c,t})+\epsilon}.
$$
The perturbed input is then
$$
x'_{c,i} = x_{c,i} + \alpha \cdot w_p(i; t,w,s)\cdot w_a(i;t)\cdot x_{c,i}.
$$
Here $w$ is a window width, $s$ controls Gaussian softness, and $\alpha \in \mathbb{R}$ is the upward or downward scale.

The second perturbation family is **summary-statistic scaling**. Denoting
$$
\mu_c = \frac{1}{b}\sum_{i=1}^{b} x_{c,i},
$$
the first-moment scaling is
$$
x' = x + \alpha \cdot \mu,
$$
and the second-moment scaling is
$$
x' = (x-\mu)\cdot \sqrt{1+\alpha} + \mu.
$$
The third perturbation family is **trend-drift adjustment**. With seasonality length $k$, one first deseasonalizes
$$
x_d = \frac{1}{k}\cdot \mathbf{1}_k * x.
$$
After fitting a line $x_{d,i} \approx a + m\cdot(i-1)$, the slope is adjusted by
$$
m'=
\begin{cases}
m+\alpha m, & m\ge 0,\\
m-\alpha m, & m<0.
\end{cases}
$$
Three intercept options—**left-fixed, symmetric, right-fixed**—define $a'$, and the adjusted series is
$$
x'_i = x_i - a + a' - (m-m')\cdot(i-1).
$$

The structured analysis loop is given as: compute $\hat y \leftarrow M_\theta(x)$; for each scale $\alpha \in \mathcal{A}$ form $x' = p(x;\alpha,\phi)$, compute $\Delta = \mathrm{dist}(x,x')$, evaluate $\hat y' = M_\theta(x')$, and for each property $\pi \in \Pi$ compute
$$
r_{\pi,\alpha} = \frac{\pi(\hat y')-\pi(\hat y)}{\Delta};
$$
then aggregate to $\bar r_\pi$ [2508.18982].

This procedure yields **multi-granular explanations**. At **high** granularity, $\pi$ selects a single $\hat y_{c,t^*}$, producing time-step importance. At **medium** granularity, $\Pi=\{\pi_1,\dots,\pi_h\}$ returns $\hat y_{c,1},\dots,\hat y_{c,h}$, producing a $b \times h$ 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 $d \times d$ **cross-channel correlation matrix**.

For multivariate forecasts, the channel-wise response is
$$
r_{c,t,c',j,\alpha} = \frac{\hat y'_{c',j} - \hat y_{c',j}}{\Delta},
$$
and the aggregate matrix is
$$
C_{c,c'} = \frac{1}{|\mathcal{A}|\, b\, h}
\sum_{\alpha \in \mathcal{A}} \sum_{t=1}^{b}\sum_{j=1}^{h}
\left|r_{c,t,c',j,\alpha}\right|.
$$
Thus $C \in \mathbb{R}^{d \times d}$ summarizes how perturbing input channel $c$ impacts output channel $c'$ on average.

## 4. Pattern classes, benchmark evidence, and multivariate interpretation

The forecasting study reports that, when visualizing the $b \times h$ temporal-dependency heatmaps, **six recurring patterns emerge** [2508.18982]. These are **Diagonals**, **Diagonals (End)**, **Last-Timestep**, **Bipolar Regions**, **Fully Correlated**, and **Other**. The criteria are expressed directly on the normalized matrix $R \in \mathbb{R}^{b \times h}$ of $\bar r$ values. For example, **Diagonals** requires
$$
\frac{\sum_{i=1}^{\min(b,h)} |R_{i,i}|}{\min(b,h)} > \theta_d
$$
with small off-diagonals, while **Last-Timestep** requires
$$
\frac{1}{h}\sum_{j=1}^{h}|R_{b,j}| > \theta_l
$$
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 $A_{\min}$.

Empirically, **Diagonals** and **Bipolar Regions** indicate the best forecast errors, with normalized error $e_{\mathrm{norm}} \approx 0.42$ and $0.29$ respectively, whereas **Last-Timestep** and **Fully Correlated** correspond to poor performance with $e_{\mathrm{norm}} \approx 0.96$–$0.97$. **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 $h=20$**, **input length $b=20$**, **scales $\mathcal{A}=\{\pm 0.01,\pm 0.05,\pm 0.1\}$**, **window $w=2$**, **softness $s=1$**, and season length $k$ by dataset. The metrics are **MAE, MSE, OWA**. Averaged over 3 seeds, **TimeMixer, MultiPatchFormer, iTransformer consistently outperform Naïve (Bonferroni-Dunn $p<0.05$)**, while **multivariate datasets remain challenging: none beats Naïve by a large margin**. Reported inference times per dataset are **iTransformer $\approx 1\,\mathrm{s}$**, **MultiPatchFormer $\approx 9\,\mathrm{s}$**, **TimeMixer $\approx 21\,\mathrm{s}$**, and **TimesFM(LLM) $\approx 1\,400\,\mathrm{s}$**.

A multivariate ETTh1 example further illustrates the framework. On the **7-channel ETTh1 dataset**, the cross-channel algorithm produces a **$7 \times 7$ matrix $C$**. Visualized as a directed graph, **iTransformer shows two strong effects: input channel 5 $\to$ output channel 0 and input channel 5 $\to$ output channel 2**. A finer heatmap $R_{5\to 0}$ of size $20 \times 20$ shows that perturbing channel 5 at its last few steps produces **strong positive and negative shifts** in each $\hat y_{0,j}$, especially near the end of the horizon, whereas channel $1 \to 0$ 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 $H^+$** beam from a **Pelletron 1.7 MV accelerator** [1310.1733]. 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 **$T_c \approx 100\,\mathrm{mK}$**, the absorber material is **Bi**, the absorber geometry is **$350\,\mu\mathrm{m} \times 350\,\mu\mathrm{m}$**, and the absorber thickness is **$2.5\,\mu\mathrm{m}$**. The electrical readout is **TDM-SQUID multiplexing**, specifically **1-row time-division–multiplexing (NIST electronics)**. The measured intrinsic noise performance is **$\sim 3\,\mathrm{eV}$ FWHM at $5.9\,\mathrm{keV}$ 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 $\sim 30\,\mathrm{mK}$** and a **regulated operating point of $65\,\mathrm{mK}$ with $<15\,\mu\mathrm{K}$ (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 $\mu$m PET + 1 $\mu$m Al each) at 60 K, 1 K and 50 mK**, blocking room-temperature black-body radiation while transmitting X-rays down to **$\sim 200\,\mathrm{eV}$**.

The measurement geometry places the **sample tilted $45^\circ$ toward the detector**, with **beam incidence at $0^\circ$** and the **TES at $90^\circ$ relative to beam**. Backscattered ions are stopped by an **$80\,\mu\mathrm{m}$ 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
$$
\Omega = 1.3 \times 10^{-5} \times 4\pi \,\mathrm{sr}.
$$
Planned upgrades move the array to **5 cm** for a **$\times 36$ solid-angle gain** and activate **all 160 pixels** for a **$\times 13$ gain**, giving a total **$\times 468$ increase in collecting power**.

The instrumental energy resolution is expressed as
$$
\mathrm{FWHM}(E)=2.355\sqrt{S_n^2 + E \cdot e_n^2},
$$
where $S_n$ is the energy-independent noise term and $e_n$ describes the energy-dependent term from thermodynamic fluctuation noise. For the best pixel at **$E=5.9\,\mathrm{keV}$**, the observed resolution is **$3.06\,\mathrm{eV}$ FWHM**. Under the assumption that the dominant noise is the amplifier-and-Johnson term,
$$
S_n \approx \left(\frac{3.06}{2.355}\right)^2 \approx (1.30\,\mathrm{eV})^2 \approx 1.69\,\mathrm{eV}^2,
$$
with the second term contributing negligibly at $5.9\,\mathrm{keV}$.

The spectral results emphasize what this resolution enables. For a **$^{55}$Fe Mn $K_\alpha$ line**, the best-pixel resolution is **$3.06\,\mathrm{eV}$ FWHM**; for a **2 MeV proton-induced Mn spectrum** combined over **9 pixels**, the resolution is **$4.45\,\mathrm{eV}$ FWHM**, with **$\approx 0.7\,\mathrm{eV}$ broadening**, likely from environmental interference. For a **bulk Cu sample**, the TES resolves **Cu $K\alpha_{1/2}$, Cu $K\beta$, and small Ni and Co impurity lines at $\sim 7.47\,\mathrm{keV}$ and $6.93\,\mathrm{keV}$** 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 $K\alpha_1$** 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 **$\approx 3\,\mathrm{eV}$ 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** [2501.08893]. 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 **$\ge 100 \times$** to avoid pile-up in the TES array, with **spill duration $\sim 10\,\mathrm{s}$ with $\sim 100$ microbunches**.

The detection system is a **large-area Transition-Edge-Sensor (TES) microcalorimeter**. The prototype is a **$\sim 9\,\mathrm{cm}^2$ Sn-absorber on Si** with **$\sim 50\,\mathrm{eV}$ FWHM resolution at $50$–$250\,\mathrm{keV}$**. The final detector is specified as **four “microsnouts,” 240 pixels total**, each pixel **$1.3\,\mathrm{mm} \times 1.3\,\mathrm{mm}$**, with **optimized absorber thickness $\sim 0.4$–$0.5\,\mathrm{mm}$** and **quantum efficiency $\approx 40\%$**. The readout is **microwave SQUID-multiplexing**, cooled by a **$^3$He-backed adiabatic demagnetization refrigerator**. Calibration uses a permanently mounted **$^{57}$Co** source with **122 keV** and **136 keV** $\gamma$-lines of **sub-eV uncertainties**, intended to counteract TES nonlinearity.

PAX focuses on circular Rydberg transitions
$$
(n,\ell=n-1)\to(n-1,\ell=n-2)
$$
in antiprotonic atoms, including **p-Ne: $6h_{11/2}\to 5g_{9/2}$ at $29\,173.9\,\mathrm{eV}$**, **p-Ar: $6h_{11/2}\to 5g_{9/2}$ at $97\,002.1\,\mathrm{eV}$**, **p-Xe: $10m_{19/2}\to 9l_{17/2}$ at $170\,495.2\,\mathrm{eV}$**, and **p-W: $12o_{23/2}\to 11n_{21/2}$ at $180\,553.1\,\mathrm{eV}$**. The gross Dirac binding energy for a hydrogen-like nucleus of charge $Z$ is given as
$$
E^{\rm D}_{n\kappa}
= m_{p}c^2
\Biggl[1 + \Bigl(\tfrac{Z\alpha}{n - |\kappa| + \sqrt{\kappa^2-(Z\alpha)^2}}\Bigr)^2\Biggr]^{-1/2},
$$
with corrections
$$
E_{n\kappa}
= E_{n\kappa}^{\rm D}
+ \Delta E_{\rm VP}^{(1)} + \Delta E_{\rm SE}^{(1)}
+ \Delta E_{\rm VP}^{(2)} + \Delta E_{\rm SE-VP}
+ \Delta E_{\rm FNS} + \cdots .
$$
The one-loop vacuum-polarization potential is
$$
V_U(r)
= -\frac{2Z\alpha^2}{3\pi r}
\int_1^\infty dt\,
\sqrt{t^2-1}\Bigl(\tfrac{1}{t^2}+\tfrac{1}{2t^4}\Bigr)e^{-2m_e r t},
$$
with corresponding shift
$$
\Delta E_{\rm VP}^{(1)} = \langle \psi_{n\kappa}|V_U|\psi_{n\kappa}\rangle.
$$

The stated methodological advantage is the **elimination of nuclear uncertainties**. Circular Rydberg antiprotonic wavefunctions have nodes **$\gg 100\,\mathrm{fm}$** 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 $\sim 0.8\,\mathrm{eV}$** while **second-order QED is $3.9\,\mathrm{eV}$**, whereas in **H-like U** the **finite-nuclear-size shift is $\sim 198\,\mathrm{eV}$** and is comparable to **first-order QED at $\sim 257\,\mathrm{eV}$**. It also states that **strong-interaction level shifts and broadening are negligible for $n \gtrsim 6$–10**.

The field strengths and precision goals are correspondingly extreme. For **p-Ne at $n=6$**, the peak Coulomb field is **$\approx 10^{21}\,\mathrm{V/m}$**, or **$\sim 10^3 E_S$**; for **p-Pb at $n=11$**, it is **$\sim 10^{20}\,\mathrm{V/m}$**, or **$\sim 10^2 E_S$**, with **$E_S=1.3\times 10^{18}\,\mathrm{V/m}$**. Earlier germanium-detector measurements at LEAR achieved **$\Delta E/E \sim 4\times 10^{-2}$**, whereas the PAX-TES goal is **intrinsic $\Delta E \sim 50\,\mathrm{eV}$ at $100\,\mathrm{keV}$**, corresponding to **$\Delta E/E \sim 5\times 10^{-4}$**; with careful calibration, the overall target accuracy is **$\sim 10^{-5}$–$10^{-6}$**. The program is therefore positioned as a route to testing **vacuum polarization**, **second-order QED corrections**, and, if deviations at the level **$\delta E/E \gtrsim 10^{-5}$** were found, possible contributions from **new interactions** such as **a light $U(1)_B$ 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.

Source: https://www.emergentmind.com/topics/pax-ts