Partial Trajectory Regression (PTR)

• Partial Trajectory Regression (PTR) is a trajectory-based model that analyzes variable-length process routes to predict wafer defect density in semiconductor manufacturing.
• It uses proc2vec and route2vec to embed heterogeneous process steps and applies counterfactual comparisons of partial trajectories to assign per-process attribution scores.
• PTR’s additive structure and empirical results, with improvements in correlation up to 0.87, provide actionable insights for diagnosing high-defect process segments.

Partial Trajectory Regression (PTR) is a trajectory-based regression and attribution framework for wafer defect root cause analysis in semiconductor manufacturing. It is designed for settings in which the input is not a fixed-length feature vector but a variable-length process route, or trajectory, composed of heterogeneous process steps and timestamps. PTR treats wafer histories as trajectories, learns process-level and route-level embeddings through proc2vec and route2vec, and defines per-process attribution scores by comparing counterfactual predictions on neighboring partial trajectories that differ by a single process step (Miyaguchi et al., 27 Jul 2025).

1. Definition, scope, and problem setting

PTR addresses wafer defect root cause analysis for a particular defect type: given wafer histories and defect density measurements at a process-limited yield evaluation point, the goal is to identify which upstream processes along each wafer’s route are most responsible for high defect density (Miyaguchi et al., 27 Jul 2025). In the formulation used for prediction, the training data are

$$\mathcal{D}_\text{train}\coloneq \{(y^{(n)},\xi^{(n)})\}_{n=1}^N,$$

where $y^{(n)}$ is the observed defect density for wafer $n$, and $\xi^{(n)}$ is its process trajectory,

$\xi = \left( (x_1,t_1), \ldots, (x_L,t_L) \right),$

with $x_k$ denoting the process token at step $k$ and $t_k$ the corresponding timestamp (Miyaguchi et al., 27 Jul 2025).

The framework is motivated by the mismatch between standard vector-based regression and fab process data. PTR is intended for routes that are variable-length, variable-content, sequential, temporal, and strongly heterogeneous across tools, recipes, and process families. The paper states four reasons why vector-based regression is inadequate: variable-length, variable-content routes; severe process heterogeneity; sequential and temporal structure; and a data regime with high nominal diversity but low effective sample size (Miyaguchi et al., 27 Jul 2025).

Within this formulation, a wafer’s trajectory is the chronological sequence of high-level process attributes and timestamps,

$\xi=\big((x_1,t_1), \ldots, (x_L,t_L)\big),$

where each $x_k$ is a process token representing tool, recipe, tool type, photo layer, route ID, and related attributes (Miyaguchi et al., 27 Jul 2025). PTR therefore reframes root cause analysis as trajectory regression rather than feature-vector regression.

2. Partial trajectories and regression formulation

PTR formalizes both prediction and attribution in trajectory terms. The forward problem is to learn a function

$y^{(n)}$0

mapping an arbitrary trajectory $y^{(n)}$1 to defect density $y^{(n)}$2 (Miyaguchi et al., 27 Jul 2025). To make this tractable, the method converts each process token $y^{(n)}$3 to an embedding, aggregates these embeddings along the route into a fixed-length representation, and predicts defect density from that representation.

A central concept is the partial trajectory. The partial trajectory up to step $y^{(n)}$4 is defined as

$y^{(n)}$5

and route2vec maps it to a representation $y^{(n)}$6 (Miyaguchi et al., 27 Jul 2025). The regression model is then defined on partial representations: $y^{(n)}$7 The same function $y^{(n)}$8 is used for both full and partial trajectories, so PTR can evaluate

$y^{(n)}$9

The paper states that, conceptually, $n$0 can be viewed as an estimate of the potential outcome, namely defect density, given a trajectory truncated at step $n$1 (Miyaguchi et al., 27 Jul 2025).

This definition gives the framework its name. PTR is not only a trajectory regression model for full routes; it is also a model that can evaluate neighboring prefixes of a route and compare their predicted outcomes. That comparison underlies the attribution mechanism.

3. Representation learning backbone: proc2vec and route2vec

PTR depends on two representation-learning components: proc2vec for individual process steps and route2vec for full or partial trajectories (Miyaguchi et al., 27 Jul 2025).

The paper motivates embeddings by pointing to heterogeneous processes, low effective sample size, and the need for similarity-aware representations. Process steps are encoded as strings built from high-level attributes such as equipment, recipe, tool type, photo layer, and route: $n$2 (Miyaguchi et al., 27 Jul 2025).

For proc2vec, a kernel matrix $n$3 is constructed over the $n$4 unique process tokens, where $n$5 is the similarity between tokens $n$6 and $n$7, computed using a custom variant of a substring kernel (Miyaguchi et al., 27 Jul 2025). Kernel-based multidimensional scaling, or equivalently kernel PCA, then yields the embedding of token $n$8: $n$9 These vectors are the proc2vec embeddings (Miyaguchi et al., 27 Jul 2025).

Given the proc2vec embeddings, route2vec encodes a trajectory by the recurrence

$\xi^{(n)}$0

with a simple linear Cell proposed for the limited-data regime: $\xi^{(n)}$1 where

$\xi^{(n)}$2

This makes $\xi^{(n)}$3 the embedding of the partial trajectory up to step $\xi^{(n)}$4, and $\xi^{(n)}$5 the embedding of the full route (Miyaguchi et al., 27 Jul 2025).

The design is intentionally simple. The paper states that the recurrence is additive, has no gates, and avoids deep RNN parameters, which helps numerical stability and avoids overfitting given small sample sizes (Miyaguchi et al., 27 Jul 2025). A plausible implication is that the representation design is optimized less for expressive sequence modeling than for stable attribution in a sparse industrial dataset.

4. Counterfactual attribution and additive structure

PTR’s key contribution is its attribution mechanism based on two neighboring counterfactual partial trajectories (Miyaguchi et al., 27 Jul 2025). For a trajectory $\xi^{(n)}$6 and step $\xi^{(n)}$7, the model compares the prefix just before the step and the prefix that includes the step:

• before step $\xi^{(n)}$8: representation $\xi^{(n)}$9
• after step $\xi = \left( (x_1,t_1), \ldots, (x_L,t_L) \right),$0: representation $\xi = \left( (x_1,t_1), \ldots, (x_L,t_L) \right),$1

The PTR attribution score for process $\xi = \left( (x_1,t_1), \ldots, (x_L,t_L) \right),$2 is defined as

$\xi = \left( (x_1,t_1), \ldots, (x_L,t_L) \right),$3

The interpretation given in the paper is explicit: $\xi = \left( (x_1,t_1), \ldots, (x_L,t_L) \right),$4 is the predicted defect density if the route stops just before process $\xi = \left( (x_1,t_1), \ldots, (x_L,t_L) \right),$5, and $\xi = \left( (x_1,t_1), \ldots, (x_L,t_L) \right),$6 is the predicted defect density if process $\xi = \left( (x_1,t_1), \ldots, (x_L,t_L) \right),$7 is included with its actual timestamp and identity (Miyaguchi et al., 27 Jul 2025). The difference quantifies the causal intervention of process $\xi = \left( (x_1,t_1), \ldots, (x_L,t_L) \right),$8 in the sense of Rubin’s potential outcome framework.

These attributions are local to a specific wafer. Every wafer has its own trajectory, its own sequence of embeddings $\xi = \left( (x_1,t_1), \ldots, (x_L,t_L) \right),$9, and its own process-wise scores $x_k$0 (Miyaguchi et al., 27 Jul 2025). Aggregation across wafers can be used for global process analysis, but the definition is fundamentally per-wafer.

The paper proves an additive property: $x_k$1 For the full trajectory,

$x_k$2

This telescoping structure enables cumulative attribution plots in which

$x_k$3

traces the predicted defect density as processes are added one by one (Miyaguchi et al., 27 Jul 2025).

The additive structure is a defining feature of PTR. Unlike feature-attribution methods that require a synthetic or averaged baseline, PTR uses the empty route $x_k$4 as the baseline and assigns each attribution directly to a time-aligned process step (Miyaguchi et al., 27 Jul 2025). This suggests a form of interpretability grounded in the actual execution order of the wafer rather than in a permutation-based or coordinate-wise decomposition.

5. Architecture, training procedure, and variable-length handling

The PTR architecture is divided into three modules: an embedding module, a regression module, and an attribution module (Miyaguchi et al., 27 Jul 2025).

The embedding module consists of proc2vec and route2vec. The regression module applies

$x_k$5

but the paper states that in experiments this was implemented as a linear model, described as an MLP without hidden layers, for numerical stability (Miyaguchi et al., 27 Jul 2025). The attribution module then computes the scores $x_k$6.

Variable-length trajectories are handled directly by the recurrence: $x_k$7 No padding or masking is required, because each wafer has its own sequence length $x_k$8, and the model maintains $x_k$9 only up to that wafer’s terminal step (Miyaguchi et al., 27 Jul 2025).

A notable feature of the training procedure is the use of all partial prefixes as supervised inputs. The loss is

$k$0

with the paper noting a minor typographical issue in the typeset equation and clarifying the intended form (Miyaguchi et al., 27 Jul 2025). For each wafer $k$1, all partial prefixes $k$2 are used during training, all paired with the same outcome $k$3.

This training design is essential to PTR’s attribution semantics. Because the model is trained to predict the final defect density from any prefix, the differences $k$4 can be interpreted as the marginal effect of adding one more process step under the learned route summary (Miyaguchi et al., 27 Jul 2025). The paper also states that the embedding and recurrence layers are frozen in the current study and that only the projection or regression layer is learned, again to avoid overfitting (Miyaguchi et al., 27 Jul 2025).

6. Empirical results, interpretation, and disambiguation

The experiments use data from a state-of-the-art FEOL process at the NY CREATES Albany NanoTech fab. The dataset contains $k$5 wafers, each with a process history spanning hundreds of processes, and defect density measured at a PLY evaluation point (Miyaguchi et al., 27 Jul 2025).

The paper compares three process encodings inside the route2vec-plus-regression framework: a constant scalar, one-hot encoding, and kernel embedding via proc2vec. The reported correlation between predictions and true defect density is 0.27 for the constant encoding, 0.52 for one-hot, and 0.61 for kernel embedding (Miyaguchi et al., 27 Jul 2025). It further states that t-SNE visualization of proc2vec embeddings shows clear clustering by equipment type, including Wet Process, RTP, Inspection, Lithography, RIE, Implant, Furnace, and CMP (Miyaguchi et al., 27 Jul 2025). With additional process features and a more optimized prediction model, correlation improves to 0.87, though exact architecture details are omitted due to space (Miyaguchi et al., 27 Jul 2025).

A principal qualitative result is the cumulative attribution plot for a particular high-defect wafer. The curve

$k$6

shows two major jumps, labeled A and B, corresponding to process segments with unusually long waiting times (Miyaguchi et al., 27 Jul 2025). The paper states that investigating those processes suggested they were likely root causes, which the authors present as evidence that PTR can produce actionable root cause analysis.

The limitations are stated explicitly. The causal interpretation of $k$7 depends on assumptions: $k$8 must approximate the conditional expectation of defect density given trajectory history; $k$9 must be a sufficient summary of the past; and unobserved confounders must not systematically bias the model (Miyaguchi et al., 27 Jul 2025). Additional limitations include the simple linear route2vec cell, the use of only high-level MES attributes, and the fact that multiple rework and branching routes are represented through a single linear accumulation path (Miyaguchi et al., 27 Jul 2025).

The acronym “PTR” is not unique in the broader literature. In offline reinforcement learning, PTR refers to “Prioritized Trajectory Replay,” a trajectory-level replay and prioritization mechanism (Liu et al., 2023). In autonomous driving, PTR refers to “PlanTRansformer,” a unified Gaussian Mixture Transformer for prediction and planning (Selzer et al., 3 Feb 2026). In offline robot policy learning, PTR refers to “Posterior-Transition Reweighting,” a reward-free sample reweighting method (Zhang et al., 17 Mar 2026). The semiconductor-manufacturing usage considered here is therefore distinct: PTR in (Miyaguchi et al., 27 Jul 2025) denotes Partial Trajectory Regression, a framework for trajectory-based defect prediction and per-process attribution rather than replay, planning, or reweighting.

A plausible implication is that this disambiguation matters operationally. In manufacturing and root cause analysis contexts, “PTR” denotes a counterfactual trajectory-regression framework built around partial prefixes and additive per-step attribution; in other research areas the same acronym names methods with unrelated objectives and mathematical structure (Miyaguchi et al., 27 Jul 2025, Liu et al., 2023, Selzer et al., 3 Feb 2026, Zhang et al., 17 Mar 2026).