{"id":"55c4a246-b50f-4dca-8011-c063fbdd62f7","arxiv_id":"2501.04712","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":6,"one_line_summary":"A soccer pressing intensity metric is defined as the probability that any defender reaches an attacker or the ball, using a time-to-intercept formula with a logistic transform.","lead":"This paper defines a frame-level pressing intensity metric for soccer, turning time-to-intercept values from an extended pitch-control model into probabilities. The metric is intuitive, but the paper includes no validation and the central formula mixes position and velocity units.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central intercept-time equations are dimensionally inconsistent as written, so the pressure probabilities in Eqs. 4-5 are not computable from tracking data.","rationale":"The reader's weakest-assumption identification is exactly the load-bearing issue: Eqs. (2)-(3) are dimensionally inconsistent. My independent read of the formulas confirms that d_j and tau_beta are not physically meaningful as written, and since Eqs. (4)-(5) are monotone functions of T_i,j, every reported pressure value is uninterpretable. I did not find a separate, more central flaw: the independence assumption in Eq. (5) is stated explicitly and can be accepted as a modelling choice for a first-order pitch-pressure metric, and the lack of validation is secondary to the formula defect. The open-source code could potentially rescue the idea, because the paper's stated implementation may use a corrected form; but the scientific record under review, i.e., the equations in the manuscript, does not specify a computable metric. Therefore the REJECT verdict stands. If a code inspection shows a missing-horizon typo, a revised manuscript that fixes the units and adds any validation against pressing events or existing pressing metrics could be considered; this does not change the current verdict.","tokens_in":4149,"tokens_out":5461,"duration_ms":54641,"concrete_test":"Run the released pressing-intensity function from the unravelsports package ([14]) on a synthetic tracking frame with known SI units (r_i=(0,0), v_i=(0,0), r_j=(10,0), v_j=(2,0), vmax=5, tau_r=0.5) and compare the returned intercept time with the value obtained from the literal Eqs. (2)-(3). If the code returns a physically plausible time in seconds because it uses an explicit horizon such as r_j + v_j * dt or a corrected tau_beta, then the paper misstates its own algorithm and the equations need correction; if the code reproduces the paper's expressions literally, the defect is present in both the paper and the implementation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Section 2.1. Equation (2) defines d_j = r_j(t) + v_j(t): a position plus a velocity. For a moving target, the future intercept point should be r_j(t) + v_j(t) * Delta_t for some explicit time horizon, but no Delta_t appears; the expression is not a distance. Equation (3) then sets tau_beta(t) = ||u|| * beta / pi with u = (r_i(t) + v_i(t)) - r_i(t), which algebraically collapses to v_i(t); hence tau_beta has units of speed (m/s), not time (s). Both terms feed directly into T_i,j = tau_r + tau_i,j + tau_beta in Eq. (1), which is then used as a time argument in the logistic function (4) and combined in Eq. (5). Every pressure value therefore inherits the unit error. The paper claims a full open-source implementation ([3], [14]), so the formulas may be an uncareful transcription of working code; but as published, the central metric is not executable or interpretable. No amount of validation can fix a formula that never evaluates to a well-defined probability with physical units.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Pressing Intensity, a frame-level metric that quantifies defensive pressure on each attacking player by converting an estimated defender-to-attacker interception time into a probability via a logistic function and combining probabilities over defenders with a product rule. The metric is implemented in an open-source Python package and illustrated with tracking-data snapshots, and extensions to pass-lane pressure and smart pressing are sketched. The central mathematical object, however, is not well defined as written because the defining equations contain unit inconsistencies.","tokens_in":4389,"tokens_out":7109,"duration_ms":67771,"significance":"If the dimensional issues were corrected, the metric would be a plausible and interpretable addition to pressing measurement in sports analytics, and the open-source implementation plus the conceptual link to Spearman's pitch control are strengths. The paper is not circular: the metric is defined directly from the equations and no parameters are fitted to pressing outcomes. Nevertheless, the current formulation does not evaluate to a well-defined probability, and the manuscript offers no quantitative validation or comparison with existing metrics, so the practical claims are unsupported.","major_comments":[{"comment":"The vector d_j is defined as r_j(t) + v_j(t), adding a position (meters) to a velocity (meters/second). Because there is no time horizon multiplying v_j, d_j is not a position and d = d_j - d_i is not a length. Consequently tau_i,j = ||d||/v_max is not a time, and the intercept-time interpretation fails at the first step.","section":"Section 2.1, Eq. (2)"},{"comment":"In the definition of u, (r_i(t) + v_i(t)) - r_i(t) simplifies algebraically to v_i(t), so ||u|| has units of speed. Since beta is an angle, tau_beta(t) = ||u|| beta/pi has units of m/s rather than seconds. Equation (1) therefore adds a reaction time, a term with units of speed, and tau_i,j whose units are already broken; T_i,j is not a valid duration.","section":"Section 2.1, Eq. (3)"},{"comment":"Because T_i,j is not a well-defined time, p_i,j is not a well-defined probability and P_j inherits this problem. The paper's central claim that Eqs. (1)-(5) produce a frame-level pressure field is therefore not supported by the manuscript as written; please provide corrected equations, for example with an explicit prediction horizon for the attacker's position, and verify the resulting units.","section":"Section 2.2-2.3, Eqs. (4)-(5)"},{"comment":"The treatment of the ball carrier switches from the product in Eq. (5) to a maximum over defenders, described only as an improvement. This changes the metric without a formal definition or rationale, and makes the reported pressure on the ball carrier inconsistent with the general formula.","section":"Section 2.4"},{"comment":"The manuscript contains no quantitative validation, error analysis, sensitivity analysis, or comparison with existing pressing metrics such as PPDA [13], StatsBomb pressure events [6], or the models of [5] and [10]. The included figures are illustrative snapshots; as a statistical application paper this leaves the practical usefulness unestablished.","section":"Sections 2-3"}],"minor_comments":[{"comment":"There are typos: 'intepretable', 'inherit lack', and 'the the Pressing Intensity' in the implementation note; please proofread the manuscript.","section":"Abstract/Introduction"},{"comment":"The text says Figure 2 shows unfiltered pressure and Figure 4 shows filtered pressure, but Figure 2's caption reads 'Filtered Pressing Intensity' and no Figure 4 for filtering exists; Section 4.1 uses Figure 4 for pass lanes. The figure numbering should be corrected.","section":"Section 3.1 and Figures"},{"comment":"The notation switches between 'Formula' and 'Equation', and T is used both as the time horizon in Eq. (4) and inside T_i,j; please use distinct symbols such as T_horizon to avoid ambiguity.","section":"Section 2.1-2.2"},{"comment":"The active-pressing threshold of 2 m/s is introduced without justification or sensitivity analysis, and it conflicts with the abstract claim that every movement influences pressure; please discuss its effect and how it was chosen.","section":"Section 3.1"},{"comment":"No information is given about the tracking data source, sampling frequency, coordinate system, or velocity estimation method; these details are needed for reproducibility beyond the open-source code reference.","section":"General reproducibility"}],"recommendation":"reject","confidential_remarks":"The stress-test concern is on target: the dimensional inconsistency in Eqs. (2)-(3) is the central defect. I also note that the paper's only implementation references are the author's own package [14] and related paper [3], which do not provide independent support for the pressure formula. If the authors correct the equations and add a validation exercise with real tracking data, the manuscript might merit another review."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, the short version: this paper has a decent idea buried under a load-bearing unit error. The author adapts Spearman's time-to-intercept model to moving targets by replacing the target position with r_j+v_j, adds a speed gate, and wraps the result in a logistic function. That is a sensible way to get a frame-level pressing intensity. I'd believe a working implementation exists; the formulas look like an uncareful transcription.\n\nThe problem is that Equation 2 adds a position vector to a velocity vector. d_j = r_j(t)+v_j(t) has units of meters plus meters per second. Equation 3 then defines u = (r_i(t)+v_i(t))-r_i(t), which is just v_i(t), so tau_beta has units of m/s. Both feed into Equation 1, so T_i,j is not a time. Every probability in Equations 4-5 inherits the inconsistency. Unless there's a hidden normalization or a typo (say, v_j times a time horizon), the metric as defined is not executable and the paper's central output is uninterpretable.\n\nWhat's good: the motivation is clear, the writing is concise, and using pitch control as a basis for pressing is a legitimate direction. The paper also openly acknowledges limitations like the infinite-field assumption and the noise in raw pressure. That honesty is worth something.\n\nWhat's missing: no validation against labeled pressing events or existing proxies like PPDA or StatsBomb's radius-based pressure. No comparison to other pressing metrics. No error analysis. The speed threshold is set arbitrarily. So even if the units were fixed, the paper would need a serious empirical section to support its claims.\n\nWho should read it? Sports analytics people interested in pressing metrics, but the current draft would mislead rather than inform. I'd tell the author: fix the units, add a real validation, and resubmit. As is, I would not cite it, and I wouldn't bring it to a reading group as a model of good practice. But the topic is legitimate and the flaw is fixable, so I'd send it to referees rather than desk-reject.","headline":"The central intercept-time formula has mismatched units, so the paper's main metric is not computable as written—but the underlying idea is a reasonable extension of pitch-control work.","tokens_in":4929,"tokens_out":3630,"would_cite":false,"duration_ms":31891,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that pressing intensity in soccer can be measured frame by frame as the probability that any defender intercepts an attacker or the ball, based only on positions, velocities, reaction time, and a logistic transform.","keywords":["pressing intensity","soccer analytics","time to intercept","pitch control","positional tracking data","defensive pressure","logistic function","frame-level metric"],"falsifier":"Take a single tracking frame and recompute the pressure values after converting all velocities from meters per second to kilometers per hour; a literal implementation of the direction-penalty term changes the intercept time by the conversion factor, so the pressure values change for an identical physical scene. A valid metric must give the same answer under both unit systems.","tokens_in":3881,"feed_emoji":"⚽","tokens_out":6656,"duration_ms":65032,"temperature":0.7,"pith_summary":"This paper tries to establish that pressing in soccer can be quantified at the level of a single tracking frame: the pressure on any attacker is the probability that at least one defender can intercept that attacker or the ball before the play moves on. The authors build a time-to-intercept for every defender-attacker pair from positions, velocities, a reaction-time delay, a maximum running speed, and a direction penalty, then convert those times into probabilities with a logistic curve. Combining the probabilities across defenders gives one readable number, the pressing intensity, which they argue captures how every player's movement creates pressure somewhere on the field and is more informative than event-based proxies such as passes-against-defensive-actions.","feed_headline":"Tracking data now yields a per-frame pressing intensity number","feed_subtitle":"Player speeds, directions, and reaction times become a per-frame probability of defensive pressure.","key_machinery":"The carrying object is the modified time-to-intercept $T_{i,j}(t) = \\tau_r + \\tau_{i,j}(t) + \\tau_\\beta(t)$, where $\\tau_{i,j}(t) = \\|\\vec{d}\\| / v_{\\max}$ is the time to cover the distance between the defender's post-reaction position and the attacker's extrapolated position, and $\\tau_\\beta(t)$ penalizes defenders running away from the target. Each intercept time is passed through a logistic function with width $\\sigma = 0.45$ and horizon $T = 1.5$ seconds to give a probability, and the probabilities are combined as $P_j = 1 - \\prod_i (1 - p_{i,j})$, a product rule that the paper notes assumes independence across defenders. This mechanism converts raw tracking coordinates into a continuous pressure field whose values are directly interpretable as the chance that pressure arrives in time.","core_discovery":"The central claim is that Equations 1 to 5 define a valid, interpretable measure of pressing intensity. For each attacking target $j$, the pressure is $P_j = 1 - \\prod_i (1 - p_{i,j})$, where $p_{i,j}$ is the logistic function of the time $T_{i,j}$ estimated for defender $i$ to intercept target $j$. The intercept time is built from a reaction-time delay, the distance between extrapolated positions divided by a maximum speed, and a penalty for defenders whose velocity points away from the target; the logistic scale is set with $\\sigma = 0.45$ and a horizon of $T = 1.5$ seconds. The authors state that this captures how every player's movement influences pressure on the field, works at the individual frame level, and offers an improvement over rule-based and event-based pressing measures because it uses full positional tracking data.","pith_inferences":["One testable extension is to compare the per-frame pressure values with expert labels of pressured touches; a stable optimal threshold would validate the logistic calibration.","The unit inconsistency in the intercept-time formula implies the metric should be repaired and re-validated before use; if fixed, the product structure could transfer to any sport with similar intercept geometry.","The sideline example suggests a natural modification: adding a repulsive 'touchline potential' that shortens the effective intercept time when the attacker is near the boundary; this is an editorial extension, not yet in the paper."],"forward_implications":["Coaches and analysts get a continuous pressure value for every attacker at every tracking frame, instead of only for players within a fixed radius of the ball.","The speed-threshold filter turns the raw values into a measure of active pressing, suppressing pressure from defenders who are nearby but not moving toward the ball.","Because the pressure is defined for every defender-attacker pair, a team can sum or average it to describe whole-team pressing behavior and compare it across matches.","The same intercept-time machinery can be pointed at pass lanes, giving a pressing intensity for space rather than only for players.","Boundary behavior remains incomplete: near the sideline the model can report zero pressure in situations where defenders are herding an attacker out of play."],"supporting_citations":[{"why":"Supplies the pitch-control time-to-intercept logic and the logistic transform that the new metric adapts.","marker":"[12]"},{"why":"Provides the simplified reaction-time-then-full-speed intercept formula that is extended here to moving targets.","marker":"[11]"},{"why":"Contributes the velocity-direction penalty parameter that becomes the $\\tau_\\beta$ term in the intercept time.","marker":"[8]"},{"why":"Serves as the open-source implementation of the algorithm, making the proposed metric reproducible and testable.","marker":"[14]"}],"fun_headline_variants":["Per-frame pressing intensity from player tracking data","Tracking data now yields per-frame defensive pressure values","New metric turns player velocities into per-frame pressing scores","Pressing intensity quantified intuitively for each frame of play","Time-to-intercept plus logistic transform gives per-frame pressing"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire metric rests on the intercept-time formula producing a true time in seconds; if the direction penalty and the position-plus-velocity extrapolation are not unit-corrected, every pressure probability inherits an arbitrary scale.","fun_headline_variants_meta":{"raw":{"variants":["Per-frame pressing intensity from player tracking data","Tracking data now yields per-frame defensive pressure values","New metric turns player velocities into per-frame pressing scores","Pressing intensity quantified intuitively for each frame of play","Time-to-intercept plus logistic transform gives per-frame pressing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000661,"raw_usage":{"total_tokens":2994,"prompt_tokens":890,"completion_tokens":2104,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":2031}},"tokens_in":506,"tokens_out":2104,"duration_ms":13301,"temperature":1.0,"reasoning_tokens":2031,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:05:07.840273+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a single tracking frame and recompute the pressure values after converting all velocities from meters per second to kilometers per hour; a literal implementation of the direction-penalty term changes the intercept time by the conversion factor, so the pressure values change for an identical physical scene. A valid metric must give the same answer under both unit systems.","supporting_citations":[{"cited_title":"Physics-based modeling of pass probabilities in soccer","cited_arxiv_id":null,"evidence_quote":"Supplies the pitch-control time-to-intercept logic and the logistic transform that the new metric adapts."},{"cited_title":"Laurieontracking, 2020","cited_arxiv_id":null,"evidence_quote":"Provides the simplified reaction-time-then-full-speed intercept formula that is extended here to moving targets."},{"cited_title":"Analytics handbook, 2020","cited_arxiv_id":null,"evidence_quote":"Contributes the velocity-direction penalty parameter that becomes the $\\tau_\\beta$ term in the intercept time."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Serves as the open-source implementation of the algorithm, making the proposed metric reproducible and testable."}],"review_version":1}