Change-points in cricket run-scoring with permuted trends: Applications through functional alignment and outlier detection in excitement quantification and bolstering advocacy

Abstract

Supervised statistical techniques - such as predicting runs or wickets or outcomes of specific deliveries, or optimal field placements, players’ worth and contribution - have been deployed to explore a cricket match with a regularity verging on obsession. While goals that fuel these interests are vital and worth pursuing, scant attention has been paid to test whether the (potentially) economic purpose that animates this line of work ignores how or why the sport enthralls, and, occasionally, overwhelms its audience. In this work, we erect a firm bridge that connects excitement patterns of different kinds of sports. The tool helps check whether a new sport could merit a permanent place in global platforms such as the Olympics. Fundamental to the construction are change-points which detect moments of deviation from an ongoing trend. Through novel statistics and permuted trends, we detect these points in utility rate differentials for cricket, and effective playing spaces for soccer. These points locate where interest gets generated. Through the proximities of those change-points, we construct excitement meters, which, applied on 124 T20 tournament-season combinations, unearth 14 functional outliers and two (phase and amplitude-based) alignment clusters, showing how T20 cricket matches attain an optimum balance between unpredictability and chaosless-ness. We, therefore, focus more on entire cricketing tournaments and on the overall health of the sport than on a specific match. This shift sheds vital light on how cricket fares similarly to other sports such as soccer that are already included in the Olympics. Using excitement measurement as an example, we point out other far-reaching potentials of this motivational broadening.

Research into numbers cricket matches generate have seldom been a chain of nuanced discoveries that generate subtle insights, but a slow accumulation of tweaks and refinements, fleshed out by the generational dictates of custom and anchored by a profit-bound sense of purpose. Much effort gets routinely expended on developing tools that predict scores or outcomes or those that quantify a player's contribution to a specific match. While the impulses to refine these techniques are understandable - marketing avenues, players’ pricing etc. are often at stake - the connection between these narrow, specific match-level markers and the overall palatability of the sport deserves a closer examination than it tends to get. The results of that neglect prove costly. The sport is kept from being appreciated by a larger and more diverse section of the population by, for instance, not finding a firm - that is, permanent - footing in global sporting events such as the Olympics. With the sport being provisionally accepted for inclusion in the LA 28 games, a key goal of this work is to test whether the sport, on average, generates games that are as exciting as those already included in the Olympics, such as soccer. An answer in the affirmative will supply the quantitative justification for its permanent inclusion and plug at once, both a conceptual and an empirical gap in the cricketing literature. In addition to the natural beneficiaries - the spectators - the insights we offer here will be of relevance to policymakers and experts outside the cricketing community who may want to test whether the sport is keeping up with other sports to query its permanent inclusion in large events such as the Olympics, experts inside the cricketing community, for instance, organizers of T20 tournaments who may be curious about whether and why some T20 tournaments are faring better than others, and cricket players and coaches who may want to decide which tournaments to participate in and which ones to sit out.

The article is organized as follows. In the opening section, we will look at the criteria behind including fresh spots within the Olympics and examine the potential benefits and challenges to cricket's inclusion. The next will scrutinize what scholars have understood excitement to mean in a cricketing context. In the following section, we offer an alternative and demonstrate how, through refinements of some of our previous published work in this area, a fresh formulation of interesting-ness opens up. One that takes into consideration not something eventual - such as the margin of victory - but the proximity of internal twists and turns. This becomes the vital bridge through which cricket - a potential candidate for the Olympic Games - may be compared, as we showcase, with soccer - already in the Games. The following two sections, through unearthing functional outliers and clusters - that is, through conducting unsupervised statistical tasks, demonstrate how T20 tournaments offer a nice balance of unpredictability and chaosless-ness. We close with some directions for the future.

Evaluating the criteria for integrating new sports into the Olympic framework: A focus on cricket and emerging disciplines

[A reader familiar with the game of cricket and the Olympics framework may skip this section and move on to “A comparison of different kinds of sports” without significant interruption to the flow.]

The International Olympic Committee (IOC) employs a multifaceted approach to evaluate and select new sports for inclusion in the Olympic Games. This process is guided by a set of criteria designed to ensure that the chosen sports align with the values and objectives of the Olympic Movement. One of the primary considerations is the universality of the sport, which refers to its global reach and popularity. The IOC aims to include sports that are practiced widely across different regions and cultures, thereby promoting inclusivity and diversity. Another critical factor is the sport's ability to uphold the principles of fair competition. The 2015 consensus emphasized the importance of ensuring that any restrictions on participation are necessary and proportionate to achieving fair competition. ¹ This principle is essential to maintain the integrity of the Olympic Games and to provide a level playing field for all athletes.

The debate over the inclusion of newer sports, such as cricket, highlights the complexities involved in this selection process. Cricket, despite its immense popularity in certain regions, faces challenges in meeting the universality criterion due to its limited presence in other parts of the world. Additionally, the format and duration of cricket matches pose logistical challenges for integration into the Olympic schedule. Measuring the appeal or interestingness of a cricket match is another aspect that the IOC must consider. This involves assessing factors such as spectator engagement, media coverage, and the sport's ability to attract a diverse audience. The integration of new sports requires a thorough ethical valuation of the sport itself, the behavior of its players, and its organizational structure. ² This holistic approach ensures that the selected sports not only meet the technical and logistical requirements but also align with the ethical standards of the Olympic Movement.

Furthermore, the multiculturalism of the Olympic Programme does not always represent the universalism proposed by the philosophy of Olympism. The challenge for leading institutions of the Olympic Movement is to maintain and expand this universalism while navigating the current situation of thick multiculturalism worldwide. ³ This underscores the need for a balanced approach that respects cultural diversity while promoting global inclusivity. In summary, the IOC's criteria for evaluating and selecting new sports for the Olympic Games are designed to ensure that the chosen sports align with the values of universality, fair competition, and ethical standards. The debate over the inclusion of newer sports like cricket highlights the complexities involved in this process, including the need to measure the sport's appeal and address logistical challenges. In this work, we deal primarily with the question of appeal. Dickson et al. (2016) ⁴ outline that major sport events must be inclusive and accessible, adding another layer of complexity to the evaluation process.

Benefits of cricket's inclusion

The inclusion of cricket in the Olympic Games would significantly enhance its visibility on a global stage. Cricket, traditionally popular in countries such as England, Australia, India, and South Africa, has a substantial following. However, its presence in the Olympics would introduce the sport to a broader audience, potentially increasing its popularity and participation worldwide. The International Olympic Committee (IOC) plays a crucial role in shaping the evolution of sports through its decisions, which can be seen as a paradigm of sport.^5–9 The IOC's inclusion of cricket would not only validate the sport's global appeal but also align with the committee's efforts to diversify the Olympic program. The inclusion would also address the issue of accessibility and fairness in sports. Postma et al. ² reflect on the challenges of cost, access, and unfair advantage in both traditional sports and esports. By including cricket, the IOC would be promoting a sport that has a well-established infrastructure and a significant number of players worldwide, thereby offering a level playing field to athletes from various backgrounds. This move would be in line with the IOC's commitment to safeguarding intrinsic sport values such as meritocracy and fair competition. Furthermore, the inclusion of cricket in the Olympics could help mitigate the historical exclusion and marginalization of certain groups within the sport. Malcolm ¹⁰ discusses the exclusion of black players from formal cricket structures and the status-emphasizing practices that maintained a degree of distance between black and white players. By bringing cricket into the Olympic fold, the IOC would be promoting inclusivity and provide a platform for players from diverse backgrounds to compete on equal terms. The appeal of cricket as an Olympic sport can also be measured by its potential to attract spectators, sponsors, and media attention. Batuev and Robinson ¹¹ highlight the importance of a clear competition structure in making a sport appealing to these stakeholders. Cricket, with its well-defined formats such as Test matches, One Day Internationals, and Twenty20, offers a structured and engaging competition that could captivate a global audience. The inclusion of cricket in the Olympics would not only enhance its visibility but also contribute to the sustainability of the sport in the global sports business. The performance of cricket players, particularly those who ascend to captaincy roles, demonstrates the sport's potential for high-level competition. Sadekar et al. ⁷ found that batsmen who become captains exhibit higher performance levels, both overall and during their tenure as captain. This pattern underscores the competitive nature of cricket and its suitability for the Olympic stage, where athletes are expected to perform at their best. The integration of cricket would also align with the philosophy of Olympism, which emphasizes universalism and the inclusion of sports that reflect diverse values and rules. ³ By selecting cricket, the IOC would be acknowledging the sport's historical significance and its role in various societies, thereby enriching the Olympic program with a sport that has a deep cultural and social impact.

The potential to expand cricket's reach through its inclusion in the Olympic Games is significant. Cricket, a sport with deep historical roots and widespread popularity, particularly in countries like India, Australia, and England, has the capacity to attract a global audience and enhance the diversity of the Olympic program. The inclusion of cricket could serve as a catalyst for increasing its visibility and popularity in regions where it is less prominent, thereby fostering a more inclusive and varied sporting environment. Cricket's unique appeal lies in its strategic depth and the variety of roles within the game, such as opening batsmen, all-rounders, and wicket-keepers, each contributing distinct skills and dynamics to the sport. ⁷ This diversity within the game itself can attract a broad spectrum of participants and spectators, enhancing the sport's reach. Furthermore, the fast bowling of black players in the West Indies has historically been a central aspect of cricket, highlighting the sport's role in broader social and cultural negotiations. ¹⁰ This aspect underscores cricket's potential to engage diverse communities and promote cultural exchange through the Olympic platform. This also aligns with the broader goals of inclusivity and accessibility in sports. The Convention on the Rights of Persons with Disabilities (CRPD) provides a framework for the inclusion and integration of disabled people in sport, emphasizing the importance of inclusive approaches in heterogeneous groups. ¹² By incorporating cricket, the Olympics can further these goals, offering new opportunities for participation and representation. Moreover, the legal and organizational structures within international sport, such as the Court of Arbitration for Sport (CAS), play a crucial role in ensuring accountability and compliance with fundamental principles. ¹³ The inclusion of cricket would necessitate adherence to these structures, thereby promoting fair play and equitable treatment of athletes. This legal framework can help address potential challenges and ensure that cricket's integration into the Olympics is conducted in a manner that upholds the integrity of the sport and the values of the Olympic movement. The historical success of the United States in leveraging the Olympic Games for national promotion and cultural exchange demonstrates the potential benefits of including cricket. ¹⁴ The Olympics provide a unique platform for showcasing sports to a global audience, and cricket's inclusion could similarly enhance its international profile and foster greater engagement across different regions.

Additionally, the legacy of the Olympic Games extends beyond the duration of the event itself, with long-term impacts on host cities and participating sports. ¹⁵ Cricket's inclusion could contribute to this legacy, promoting sustained interest and investment in the sport, and encouraging the development of infrastructure and programs that support its growth.

Challenges to cricket's inclusion

Logistical issues related to the inclusion of cricket in the Olympic Games encompass several critical aspects, including venue requirements and scheduling complexities. Cricket, traditionally played over extended periods, poses unique challenges for Olympic integration. The standard formats of cricket, such as Test matches, One Day Internationals (ODIs), and Twenty20 (T20), vary significantly in duration, with Test matches lasting up to five days, ODIs taking approximately eight hours, and T20 matches lasting around three hours. This variability necessitates careful consideration of scheduling to ensure compatibility with the Olympic timetable, which typically spans two weeks. The venue requirements for cricket are also distinct. Cricket grounds are larger than most other sports venues, requiring substantial space for the pitch, outfield, and spectator areas. The infrastructure needed to support cricket matches includes specialized equipment such as sight screens, pitch covers, and boundary ropes, which may not be readily available at existing Olympic venues. Additionally, the maintenance of cricket pitches demands expertise in turf management, which could complicate venue preparation and upkeep during the Games. Scheduling cricket matches within the Olympic framework presents further logistical challenges. The extended duration of cricket matches, particularly Test matches, conflicts with the condensed schedule of the Olympics. To address this, shorter formats like T20 could be considered, as they align more closely with the time constraints of the Olympic schedule. However, even T20 matches require careful planning to avoid overlap with other events and ensure adequate rest periods for athletes. ¹⁰ The logistical issues extend beyond the physical and temporal aspects to encompass legal and administrative considerations. The Olympic Charter outlines specific obligations for athletes and sports organizations, which must be adhered to for cricket's inclusion. These include compliance with anti-doping regulations, competition manipulation prevention, and adherence to the rules of the International Federation (IF) governing cricket. Ensuring that cricket meets these criteria requires coordination between the International Olympic Committee (IOC) and cricket's governing bodies such as the International Cricket Council (ICC).^13,15

Adherence to universal Olympic rules is a fundamental criterion for the inclusion of any sport in the Olympic Games. The International Olympic Committee (IOC) mandates that the statutes of a National Olympic Committee (NOC) must comply with the Olympic Charter at all times, and this compliance is essential for the recognition of any national federation by the NOC. This requirement ensures that all sports included in the Olympic program adhere to a standardized set of rules and regulations, promoting fairness and consistency across all competitions. The Olympic Charter also includes provisions for arbitration in case of disputes arising in connection with the Olympic Games. Rule 61.2 of the Olympic Charter specifies that disputes related to the Games must be arbitrated, and this rule applies to any party with a valid legal claim connected to the Games. ¹³ This arbitration clause is crucial for maintaining the integrity of the Olympic Games, as it provides a mechanism for resolving conflicts in a structured and legally binding manner. The inclusion of new sports, such as cricket, in the Olympic program must also consider the potential bureaucratization and cultural separation that could arise from such inclusion. The experience of snowboarding, which faced bureaucratization and cultural separation following its inclusion in the Olympics, serves as a cautionary example. ¹¹ The skateboarding community has expressed concerns that similar consequences could occur if skateboarding were included in the Olympic program, highlighting the need for careful consideration of the impact of Olympic inclusion on the sport's culture and community.

Interestingness of cricket matches: Suspicions and previous work

External engagement and expectations

Spectator engagement metrics, such as attendance and viewership, are crucial in evaluating the appeal and interestingness of cricket matches, especially in the context of their potential inclusion in the Olympic Games. These metrics provide quantifiable data that can be used to assess the popularity and marketability of the sport, which are essential factors for the International Olympic Committee (IOC) when considering new sports for the Olympic program. Attendance at cricket matches is a direct indicator of the sport's ability to draw crowds to the stadium. High attendance figures suggest a strong local and regional interest in the sport, which can be a compelling argument for its inclusion in the Olympics. Sadekar et al. ⁷ highlight the importance of understanding the dynamics of sports attendance, noting that factors such as the quality of the teams, the significance of the match, and the overall fan experience play significant roles in attracting spectators. Additionally, the clustering of teams based on their performance metrics, such as average runs scored and wickets taken, can provide insights into which matches are likely to draw larger crowds. ⁶ Viewership, on the other hand, extends the reach of cricket beyond the stadium, capturing the interest of a global audience through television and online streaming platforms. The trend in viewership numbers can reflect the sport's international appeal and its potential to engage a diverse audience. According to Ekstrøm & Jensen, ⁸ the granularity of the running score difference throughout a match can influence viewership, as closely contested games tend to maintain higher viewer interest. This is supported by the findings in Vecer et al. ⁵ which indicate that the excitement of a match increases as the teams’ strengths become more evenly matched, thereby enhancing the viewing experience.

Moreover, the integration of new sports into the Olympic framework requires a consideration of the intrinsic values and meanings that these activities hold for both players and spectators. Postma et al. ² discuss the intrinsic values of esports, which can be paralleled to traditional sports like cricket. The differentiation based on these values can help in understanding the broader impact of the sport on its audience, beyond mere attendance and viewership numbers. The moral and virtuous aspects of sports also play a role in spectator engagement. Cricket, like other sports, promotes virtues such as temperance, justice, fidelity, and courage, which resonate with audiences and contribute to the sport's overall appeal. These virtues can enhance the spectators’ connection to the sport, fostering a deeper engagement that goes beyond the physical presence at matches or viewership statistics.

Internal game dynamics

Equally, the internal game dynamics, combining elements such of pace and unpredictability, play a crucial role in defining the interestingness of cricket matches. The pace of a game refers to the speed at which events unfold, including the frequency of scoring, the rate of wickets falling, and the overall tempo of play. Unpredictability, on the other hand, involves the degree to which outcomes are uncertain and can change rapidly, adding to the excitement and engagement of spectators. Cricket, traditionally known for its slower pace compared to other sports, has evolved with formats like Twenty20 (T20) that significantly increase the game's tempo. T20 cricket, characterized by its fast-paced nature, has gained immense popularity due to its ability to deliver quick and thrilling matches. The rapid scoring and frequent changes in momentum make T20 matches highly unpredictable, keeping audiences on the edge of their seats. ¹⁴ The unpredictability in cricket is further enhanced by the diverse conditions under which the game is played. Factors such as pitch conditions, weather, and player form contribute to the variability in match outcomes. For instance, a pitch that favors bowlers can lead to unexpected collapses of batting line-ups, while favorable weather conditions can assist batsmen in scoring runs more freely. This inherent unpredictability is a key aspect that adds to the interestingness of cricket matches. The strategic depth of cricket, involving decisions on field placements, bowling changes, and batting orders, introduces a layer of complexity that can influence the pace and unpredictability of the game. Captains and coaches constantly adapt their strategies based on the evolving match situation, which can lead to sudden shifts in momentum and unexpected results. This strategic element is particularly pronounced in longer formats like Test cricket, where the game can change dramatically over the course of several days. ⁹ The integration of technology in cricket, such as the Decision Review System (DRS), has also impacted game dynamics. DRS allows teams to challenge umpire decisions, adding an element of suspense and uncertainty to the game. The use of technology to review close calls can lead to dramatic reversals in match situations, further enhancing the unpredictability and interestingness of cricket matches.^11,13 The individual brilliance of players can significantly influence the pace and unpredictability of a cricket match. Exceptional performances, such as a rapid century or a hat-trick of wickets, can alter the course of a game in a matter of minutes. These moments of individual excellence contribute to the overall excitement and appeal of cricket, making it a sport that captivates audiences worldwide. ² The tools we propose later in this work can be updated – and we show how -to incorporate excitements generated from nun-numerical sources, such as, from a DRS decision. Excitements are also shaped by the cultural and historical context of the sport. Cricket's rich tradition and the passionate following it commands in countries like India, England, and Australia add to the game's allure. The historical rivalries and the significance of certain matches, such as The Ashes series, bring an additional layer of excitement and unpredictability to the sport. These cultural elements enhance the interestingness of cricket matches, making them more than just a contest of skill but also a celebration of heritage and national pride.^4,15 While Test matches offer a deep and traditional cricket experience, their lengthy duration may limit their suitability for the Olympics. ODIs provide a balanced and engaging format that fits well within the Olympic schedule, while T20 matches offer high-intensity action that can captivate a diverse audience. The selection of cricket formats for the Olympics should prioritize those that enhance spectator engagement and align with the IOC's goals of promoting sports participation and Olympic values. ²

One approach to quantitative analysis in cricket is the use of Markov models to predict win expectancy and scoring intensity. These models can simulate the progression of a cricket match by considering the probabilities of different outcomes based on current game states. For instance, Vecer et al. ⁵ outline the use of a Poisson model to estimate the probabilities of a draw, win, or loss during a match, which can provide insights into the excitement levels of the game. Another involves the evaluation of scoring patterns at different venues. Ayub et al. ⁶ demonstrate the significance of venue-specific scoring patterns by analyzing matches at various levels of granularity, including team clusters and innings stages. This analysis helps in understanding how different conditions affect the performance and appeal of cricket matches. The concept of hot and cold streaks in team performance is also relevant. Sadekar et al. ⁷ discuss the fluctuations in team performance, indicating that winning teams tend to have a larger effective team size, meaning more players contribute significantly to the team's success. This metric can be used to assess the consistency and overall appeal of cricket matches. The accuracy of projected scores is another important aspect of quantitative analysis. Ayub et al. ⁶ describe the computation of projected scores using machine learning techniques such as k-nearest neighbors (kNN), Random Forest, and Ridge Regression. The mean absolute error (MAE) in these projections can be used to evaluate the reliability of scoring predictions, which is essential for assessing the potential excitement of cricket matches. Dynamic programming models have also been employed to analyze cricket game progression. Clarke's model, for example, proposes an optimal scoring rate for the first innings and estimates the total number of runs that would be scored. For the second innings, the model considers the probability of winning based on wickets in hand, overs remaining, and runs yet to be scored. This approach provides a comprehensive framework for understanding the dynamics of cricket matches and their potential appeal. ⁶ The evaluation of the excitement index (ETI) for cricket matches can provide valuable insights. Ekstrøm and Jensen ⁸ summarize ETI at the team level to identify groups of teams more or less likely to give an exciting game. Our change point-based proposals, described in the following sections, use a different construction and is not tied to specific teams’ profiles, that is we allow for even low-ranked teams to generate interesting matches. These metrics can be used to quantify the interestingness of cricket matches and support the case for their inclusion in the Olympics. The inflation of performance across decades, with a 40% increase in the total team score from the 1980s to the 2020s, indicates a significant evolution in the sport. ⁷ This trend reflects changes in playing styles, strategies, and overall competitiveness, which are essential factors in assessing the appeal of cricket. By analyzing these historical trends, researchers can gain insights into how the sport has evolved and how it continues to captivate audiences.

A comparison of different kinds of sports through established change-proximities: The guiding motivation

Cricket and football are extremely different. While a football game unfolds over ninety minutes (with the potential addition of extra-time and penalties), a 50-over cricket contest stretches, in theory, much longer: nearly an entire day. In the former, the action occurs simultaneously - both teams poised to seize the ball at once, whereas in the latter, events unfold sequentially - one team takes to batting while the other bowls. Their styles of play, therefore, differ significantly. So too do the terms we employ to articulate them. In cricket, we measure runs and wickets, while in football, it is all about goals and fouls. The distinction could not be more pronounced. Yet, there lies a possibility of crafting a universal excitement meter - one that allows us to juxtapose a cricket match alongside a football match and observe them through a unified perspective, facilitating comparisons, if desired.

We collect some initial ideas from Bhaduri ¹⁶ in this section and use the next sections to extend them. Recent non-linear modifications, for instance those outlined in Bhaduri and Predescu ¹⁷ or Smith and Bhaduri, ¹⁸ may be used to understand interactions. First, it is vital to grasp that excitement thrives as a dynamic notion - something that morphs as the match unfolds, and any measure that is stagnant or final - like the margin of victory (whether through runs, wickets, or goal differences) or the disparity in ball possession, for example - may not capture its essence fully. It is crucial to recognize that predictability quashes excitement; unpredictability ignites it. A shift from the mundane is essential to spark excitement. The more often these shifts occur and the greater the intensity of these deviations, the higher the overall thrill. It is with pointing these moments of transformation that our innovations begin.

Utility rate differentials in cricket

The ultimate showdown of a tournament that graces the stage once every four years holds immense significance, doubtless. Still, the 14th of July, 2019, etched itself into the annals of cricketing legend far beyond its anticipated fate. England and New Zealand, both longing for the elusive Cricket World Cup, clashed in a stunning duel, and bustling Lords Cricket Stadium stood witness to a peculiar finish. New Zealand, batting first, amassed a humble total of 241, which England matched – exactly - in their pursuit. Curiously, the super-over designed to differentiate teams in such scenarios, also ended up tied. The saga ended with England claiming victory through a seldom-discussed rule concerning boundary counts. For detailed ball-by-ball statistics, we visited www.cricsheet.org. ¹⁹

The escalating intricacies of current regulations and fresh strategies make the task of detecting changes in cricket especially challenging. A case in point is the advent of “powerplays.” These are segments of play - some predetermined at the outset of an innings, some subject to the whims of the batting team - that limit the number of fielders stationed near the boundary. This typically entices the batting side to ramp up their scoring tempo. Such powerplays are modern marvels, unknown to cricket enthusiasts a mere fifteen years ago. Consequently, for modern spectators, fluctuations in scoring during these intervals are anticipated and should not escalate the thrill. Also, we note that the striking of boundaries or the loss of wickets is recognized to have varying effects at different phases of an innings. The notion of resource allocation was first introduced by Duckworth and Lewis ²⁰ and refined later by Stern, ²¹ to tackle this dilemma. Structurally akin to conventional run-rates, it is measured through S(u, t)/R(u, t) where S(u, t) represents the total runs amassed by the end of t overs and u wickets, while R(u, t) denotes the resources expended to reach that juncture. These resources, are “assets”, in a way, remaining to the batting side at any point during their innings. Stern (2016) details a 51 × 11 table: 51 rows corresponding to the 50 possible overs and 11 columns representing the 10 potential wickets. The first cell, number (1,1), stands at 100. This is logical since, at the beginning of the innings, each batting side possesses their complete “wealth”: their full allotment of 50 overs and 10 wickets. Cells (2,1) and (2,2), however, contain values 99.1 and 92.6, respectively. This indicates that if a team completes the first over without losing any wickets, their remaining resources will be 99.1, while if they lose one, it will drop to 92.6. Resources consumed at any stage can be calculated as {100-resources remaining}. For example, New Zealand, at the conclusion of their first over, notched up five runs without losing any wickets. Their resource utilization rate at this point is thus 5/(100–99.1) = 5.556. Conversely, had they lost one wicket, the rate would have been 5/(100–92.6) = 0.676, highlighting their diminished efficiency in resource consumption. Observe how a traditional metric like the standard run-rate would fail to capture this distinction: it would yield a flat 5 in both scenarios. Figure 1(c) illustrates the difference in these rates (New Zealand - England) as a function of the number of balls bowled (both teams playing 50 overs, totaling 300 balls, plus a handful of extras such as no-balls and wides).

Figure 1.

The pertinent “elements” (the utility disparity in cricket, the effective spatial variance in football) being monitored, in panels (c) and (a), throughout “duration” (measured by balls delivered and frame count). The interestingness-es may parallel with the peaks and troughs on the former, while extremes (illustrated through the horizontal limits) on the latter. Scaled intervals of these “jolts” are cataloged in panels (d) and (b), correspondingly, in a cumulative manner, with the curve ascending by one unit each time a jolt occurs. Consequently, the ultimate height on the vertical axis illustrates the total number of jolts experienced. To facilitate a comparison, we adjust the bands in (a) so that these peaks on (d) and (d) become aligned for comparison. Variations detected through our analytical tools – represented as red dividers on (b) and (d) – disrupt the flow substantially, refreshing the pace of these jolt-generations.

Effective playing spaces in football (soccer)

Goal-scoring in football is not as common as run-scoring in cricket. In addition, balls – that may be taken to record time – do not get bowled. Hence, we require a different tool to track the flow of a match seamlessly. Effective Playing Spaces (EPS) offer a metric. At any moment, a convex hull crafted by all players except the goalkeeper for one of the two teams, can form a ten-sided geometric decagon, where each player stands as a vertex, a point of connection. The area contained within this imaginary closed shape represents that team's EPS at that moment.^22,23 As the game unfolds and formations shift, this area transforms. When players cluster closely, the area contracts (i.e., the shape diminishes), when they spread out, it expands (i.e., the shape grows). To calculate this area, the (X,Y) coordinates of each player must be monitored. Such tracking data, gathered at high frequency – one frame every 0.04 s – is accessible online for select matches (courtesy of Metrica Sports). Figure 1(a) illustrates the difference between these areas (home team - away team) as a function of these captured moments.

Detecting interesting changes

These fresh tools - the URDs and EPSs - will be vital in crystallizing our instincts regarding what defines something as captivating. In cricket, it could be the dramatic highs and lows of the URDs - the peaks and troughs (regardless of their magnitude) shown in Figure 1(c). A high peak, signaling an ensuing downturn, indicates a phase advantageous to England, while a deep trough -foreshadowing an upswing - signifies a phase favorable to New Zealand. We extract these extremes, these crucial junctures to craft our shock sequence - the description that summarizes the moments when captivating events occurred. They predominantly gather towards the end of the fifty-over match, confirming the battle remained fierce until the final moments. Some are anticipated. The peak, for instance, around ball 260 (illustrated as point B), marks the moment when Neesham was dismissed. Others, however, are nuanced. The instance around ball 30, for example, aligns with the phase when Archer and Woakes bettered their bowling accuracy.

In football, however, the drama unfolds especially when one team teeters on the brink of scoring, even if the ball ultimately doesn’t grace the net. Such instances will show up as significant EPS differentials. The attacking team, to broaden their tactical choices, often disperses across the field (boosting their EPS), while the defending team generally congregates near their goalkeeper (diminishing their EPS), striving to thwart the attempt. The greater the frequency of these thrilling moments - characterized by pronounced EPS differentials - the more our excitement escalates. The red lines in Figure 1(a) mark the upper 10% of these extremes. Point C, for instance, marks a moment when the “away” team found the net despite the tight formation by the “home” team to repel the assault.

Exciting moments characterized in this manner - through the twists of the URD or the extremes of the EPS differentials - appear as tiny peaks of height one on panels d and b, respectively, in Figure 1. The “times” have been compressed onto the unit interval - a common maneuver in point process asymptotics. ²⁴ It serves to recalibrate the analysis to reflect our default anticipation: a cricket match is meant to unfold over a longer duration, thus ensuring a fair comparison. With $t_i$ as the (scaled) time of the $i^{th}$ exciting moment and n denoting the grand total,

$$Z=-2\sum _{i=1}^n\log \left({\displaystyle \frac{t_i}{t_n}}\right)$$

provides a statistic to check a possibly consistent rhythm of event generation. If they are sprayed uniformly (technically, homogeneously), with 126 shocks as in out cricketing example, 95% of the time the Z values will fall between 214.89 and 287.88 (Bhaduri ²⁵ ). However, our observed Z was 197.04, implying a significantly non-steady rate of shock generation over the full history. The precise moments when the rates shift - when shocks begin to surge or diminish substantially - can be uncovered by calculating Z not once, but in succession, each time incorporating the subsequent shock from, and reassessing the stability. We control the false positives following Benjanini and Hochberger ²⁶ 's recommendation for multiple testing. A change-point marks the inception of the first sign of instability. Once one is identified, we restart the process to pick the next. These change-points are shown as the red vertical dividers on Figure 1. Z is sub-optimal in general. Its optimal conditions arise when the magnitude of change is substantial and occurs early in the process. If the change is subtle or delayed, Z lacks a sufficient volume of post-change data to recognize that a shift has occurred. $Z_B$ mirrors, therefore, the fractions around 1 to give

$$Z_B=-2\sum _{i=1}^{n-1}\log \left(1-{\displaystyle \frac{t_i}{t_n}}\right)$$

Shocks that appeared to occur with decreasing frequency (observed through Z) will seem to happen with increasing frequency (observed through $Z_B$ ). One type of rotation may not always be the most effective; $Z_B$ has its own challenges. ²⁵ Through our previous works (Bhaduri, ²⁵ Ho et al. ²⁷ ), we offered combinations of these two versions that demonstrate better change-detection power nearly always. Two of them are

$$R=max(Z,Z_B)$$

and

$$L=min(Z,Z_B)$$

Change-points identified by at least one of them are illustrated as vertical strips on Figure 1. This way of detecting substantial changes have found applications in a variety of settings: legal battles, ²⁸ social surveys,^29,30 Bayesian profiling, ³¹ network science applications in finance, ³² public health, ³³ and other kinds of solo or group sports such as mountaineering. ³⁴

Change proximities and magnitudes

With $\hat{\tau }=\{{\hat{t}}_1,{\hat{t}}_2,\cdots ,{\hat{t}}_m\}\mathrm{and}\tilde{\tau }=\{{\tilde{t}}_1,{\tilde{t}}_2,\cdots ,{\tilde{t}}_n\}$ representing the set of change times from the cricketing and the football scenario, the Hausdorff distance defined through

$$H(\hat{\tau },\tilde{\tau })={\displaystyle \frac{1}{T}\Vert \hat{\tau }-\tilde{\tau }{\Vert }_{\mathrm{\infty }}}$$

where

$$\Vert \hat{\tau }-\tilde{\tau }{\Vert }_{\mathrm{\infty }}:=max\left\{\underset{\hat{t}\in \hat{\tau }}{max}\underset{\tilde{t}\in \tilde{\tau }}{min}|\hat{t}-\tilde{t}|,\underset{\tilde{t}\in \tilde{\tau }}{max}\underset{\hat{t}\in \hat{\tau }}{min}|\hat{t}-\tilde{t}|\right\}$$

measures the extent of overall proximity between the two. The smaller the value, the greater the resemblance of times when shock-generation rates altered. In our cricket-football scenario, it stands at just 0.04, a small number, implying the timings of these changes were largely indistinguishable. Context plays a crucial role. The distance of this football match is next evaluated against all 44 other matches within the cricket World Cup, extending beyond just the final. The left panel of Figure 2(b) shows the proximity had never been quite so striking. A parallel distance assessment can also be conducted between the two innings of a single cricket match, prompting us to explore other analytical tools beyond just $Z,Z_B,R,L,$ (details elaborated in Bhaduri ²⁵ ). Figure 2(a) showcases these distances. Regardless of the analytical tool selected for any given innings, with predominantly cooler shades throughout the heat-map, we confirm our intuition that the final remained tightly contested almost to the end.

Figure 2.

(a) heatmap illustrating the Hausdorff distance between the sets of change points derived from the English and Kiwi innings. We explore fifteen detection tools, spotlighting the top four that are examined in this study. (b) How closely does our football match resemble each of the 45 cricket matches from the 2019 World Cup concerning the locations of changes (depicted in the first violin plot on the Hausdorff metric) and the magnitudes of changes (shown in the second, on the MRC difference)? Values nearing 0 signify a higher level of similarity.

Two processes may exhibit similar change positions, but the magnitude of changes could vary. To gauge this divergence, there exist several avenues. Bhaduri^28,33 proposes one approach using point process bootstrapping. Another method could involve recording the rate of event generation across the pre- and post-change segments, measuring the differences, and averaging them across the number of phases, culminating in the mean rate of change (MRC). The right panel of Figure 2(b) illustrates the MRC discrepancies between the football match and each of the 45 cricket matches throughout the World Cup.

The tree shown in Figure 3 summarizes the entire tournament through a similar process of Hausdorff distance computation: each innings is represented as a leaf. If one's transformation patterns resonate closely with another's, that is, if both innings (not necessarily from the same match) undergo twists and turns around similar times, they will be situated closely on this tree. We discover four primary clusters (represented by the four distinct colors). Observe that some of these findings are intuitive. The blue branch indicates that New Zealand's innings, batting second in match 3, mirrored that of West Indies’, who batted second in match 2. They both achieved similar scores, and the matches unfolded over a comparable number of overs. However, this is not true for the England-Afghanistan branch. Despite the stark differences in the runs made, had these innings been part of the same match, it would have generated quite an intriguing spectacle.

Figure 3.

88 innings (=44 × 2, with one match succumbing to rain) from the 2019 Cricket World Cup united through the closeness of their change points. Innings that are nearby underwent changes at identical times and of identical magnitudes. “EnglandB2,M43”, for example, should be interpreted as the English innings, when they took to the crease second, during encounter 43.

A football clash with a 6-2 scoreline or a cricket showdown where one team triumphs by 250 runs might appear lopsided at first glance. Yet, there may be phases within the games where excitements were similarly dispersed. Regardless of the sport, we often overlook that the metrics deemed conclusive can only be finalized at the match's end. They fail, frequently, to express the thrill in its purest essence online, i.e., as the game goes on. Our sequential change-detection procedure corrects this oversight. By monitoring shifts in real time and assessing the magnitude of these variations, we present a novel lexicon - one that bridges games that might seem worlds apart. A way that may be generalized to other sports as soon as the appropriate tracking systems - such as the URDs or the EPSs - are identified.

The methods recalled in this section (Bhaduri ¹⁶ ) will be used to elevate match-level analysis to tournament-level ones in the two sections to follow. In summary, therefore, in this section we:

recalled how continuous monitoring of games such as cricket and soccer could generate more realistic indications of excitement than a glance at the final score line.

saw how it is worthwhile to locate crucial change-points that serve as apt proxies of turning points in online tracks of certain metrics of interest: run-scoring utilities in a cricketing context, and effective playing fields in a soccer context.

checked out the statistics offered in Bhaduri ¹⁶ : Z, Z_b, R, L, etc. estimate these change points accurately. Power analysis behind these proposals may be collected from Bhaduri (2018). ²⁵

recalled how the Hausdorff distance may be used to come up with one number that reveals how exciting a specific cricket match was. This is the distance found between the change point sets of the two innings. How this value may be compared to a similar one from a soccer match to contrast interestingness profiles.

saw how these change point-induced distances (one distance found for one match) helps paint the excitement profile of an entire tournament like the 2019 cricket World Cup (Figure 3) through clustered trees where a leaf represents an innings.

In the following two sections, we exploit the motivation from this earlier match-level analysis to construct tournament-level functional objects like excitement histograms (for instance, the cluster leaves may be interpreted to be tournaments instead of specific matches or innings). Excitement score differences (i.e., change point-induced Hausdorff distances) from all the matches in a given tournament will combine to generate the excitement histogram for that tournament. If the distribution assigns higher relative frequencies to lower values of the distance, the matches in the tournament may be taken to have been exciting, in general. Such functional structures will be our objects of analysis. Specifically, in the next section on outliers, we demonstrate how among several popular T20 tournaments, the number of outlying tournaments is not large. Technical ideas such as functional depth will be used for this purpose. This suggests a reasonably predictable potential Olympic experience. In the following section, through phase-amplitude decompositions of the histograms using warping functions, we show this predictability is not fatal enough to disengage viewers. The presence of substantially different groups still provides necessary vitality.

Outlying-ness of cricketing tournaments: Applications of functional depth on change proximities to characterize excitement predictability of tournaments

We extend the work shown in the previous section to several tournaments, not just one specific World Cup. We have argued above how, in keeping with the format of the Olympics, the more recent twenty over (that is, T20) versions could lend more appeal. We have, therefore, sampled eight popular T20 tournaments that have been running fairly irregularly: The ICC T20 league (T20), the Indian Premier League (IPL), The Big Bash (BBL), the Caribbean Premier League (CPL), Cricket South Africa T20 Challenge (CTC), T20 Blast (NTB), the Bangladesh Premier League (BPL), and the Pakistan Premier League (PSL), each across several seasons, highlighted in the following tables, with the goal of checking whether the interestingness profile for any one, during any specific season, stands substantially out from the rest. Outliers, in any context, by definition, are fewer than the more mainstream observations but the presence of several of them may raise suspicions around the presence of a solid central theme. In our context, the presence of many outlying tournaments (as opposed to individual games) may elevate worries about potential gaps between viewers’ expected excitement and the actual, and, as a corollary, about the viability of the possible inclusion of the sport. A small number of outlying observations, is, therefore, desirable.

A note about “observations” is in order. In this section, and the next, by an “observation”, we indicate a fresh data structure: functional data (Tucker et al. ³⁵ ) which are different from standard random vectors on which we routinely impose standard probability distributions. Variation - almost everyone would concur - is the lifeblood upon which the discipline of statistics thrives. As one begins a formal study, one stumbles upon straightforward data constructs where variation is both easily articulated and computed. These constructs could just be numbers. As scenarios grow intricate or as prerequisites become subtle, these data constructs evolve, and the very essence of variation - let alone its computation – becomes a slippery concept to grasp. Relative frequency distributions - graphed through histograms similar to those in Figure 4(a) - serve as prime examples. These are functional data. Each tournament produces not one solitary number, but an entire histogram - a full representation of how the excitement scores (collected from the component matches) are spread across a spectrum. We observe most of the tournaments generate pronounced right tails which point to an encouraging pattern concerning the interestingness health of these tournaments: the Hausdorff distance between change points across innings and matches tend to be smaller, rather than larger which suggests how a random tournament is, on an average (across the component matches), more prone to being exciting rather than boring. Formal tests may be constructed to check power law assumptions:

$$P(X=k)\propto {\displaystyle \frac{1}{k^{\alpha }}}$$

Figure 4.

a (left panel): functional outliers picked with bootstrapped depth metrics. Fourteen tournaments – shown in red – exhibit substantially different natures of variation in their excitement profiles. b (right) Function clustering (post alignment) exhibit two major types of T20 tournaments, shown in green and red.

For some $\alpha$ , where X represents the (average Hausdorff distance-based) excitement for a randomly chosen tournament. High p-values over large ranges for nearly all the tournaments validate this skew more objectively. The data sources for this section are similar to the earlier ones.^26,36

To test potential outlying-ness of some of these tournaments, we use the concept of statistical depth.^37,38 Given simpler constructs such as random vectors $\{X_1,X_2,\dots ,X_N\},$ the depth of an example $X_i$ is defined as

$$D_F(X_i)=min\{F_N(X_i),1-F_N(X_i)\}$$

where F is the empirical cumulative distribution function

$$F_N(x)={\displaystyle \frac{1}{N}\sum _{n=1}^{N}I(X_n\le x)}$$

with a shallow depth, closer to 0, suggesting a potential outlier, and a larger depth, closer to 0.5, suggesting an observation like the median. The limits are altered at times to define the depth as

$$D_N(X_i)=1-|0.5-F_N(X_i)|$$

when the largest depth is 1, and the smallest, 0.5. The definitions and intuitions generalize for more complex constructs such as histograms in Figure 4. Given a sample of functions $\{X_n(t),t\in [a,b],n=1,2,\dots ,N\},$ we define the empirical cumulative distribution function at point t by

$$F_{N,t}(x)={\displaystyle \frac{1}{N}\sum _{n=1}^{N}I(X_n(t)\le x)}$$

and the functional depth, by integrating the univariate depths:

$$F{D}_N(X_i)=\underset{a}{\overset{b}{\int }}\{1-|0.5-F_{N,t}(X_i(t))|\}dt$$

Analogous to random variables, smaller functional depths are suggestive of functional outliers. To check for significance, that is, to establish critical cut-offs from the null distributions of these depth statistics, on whose basis outlying-ness can be judged, we have applied Febrero et al. ³⁸ 's strategy of bootstrapping the observed functional curves in the following way:

we let $\{X_n(t),t\in [a,b],n=1,2,\dots ,N\},$ be the original data family and $D=D(\{X_n(t),t\in [a,b],$ $n=1,2,\dots ,N\})$ to be the observed sample depth statistic.

we calculate 1000 bootstrapped resamples $\{X_n*(t),$ $t\in [a,b],n=1,2,\dots ,N\},$ using normal perturbations to the original tournaments: that is $X_i*(t)=X_i(t)+Z(t)$ where Z(t) is normally distributed with 0 mean and covariance matrix $\gamma {\mathrm{\Sigma }}_x$ where ${\mathrm{\Sigma }}_x$ is the covariance matrix of the original data and $\gamma$ is a smoothing parameter.

we take $D*=D(\{X_n*(t),t\in [a,b],n=1,2,\dots ,N\})$ to be the depth estimate from the j-th bootstrap resample.

We compute $d(D,D*),$ the sequence of (Euclidean) distances between the observed and bootstrapped depths and define the bootstrap confidence ball of level $1-\alpha$ as $CB(\alpha )=X:d(D,X)\le d_{\alpha }$ , with $d_{\alpha }$ being the $(1-\alpha )$ -th quantile of the distances between the bootstrapped and the observed depths.

The above strategy applied on functional histograms generated with the Hausdorff distances between games on specific tournaments detect fourteen tournaments as outliers (Figure 4(a)). We note the shift in our interpretations which follow from a shift in the nature of our data. If one imagines erecting a vertical strip at a specific point on the excitement distance axis, say at 30, and record all the colors that fall on that slice (i.e., examine non-functional, i.e., numerical data), a red color need not look extreme. But as this slice varies, i.e., as we move the 30 strip along the horizontal axis to other locations, collectively, in sum, a red function would look extreme in terms of its pattern, relative to the rest. Interesting questions may be raised at this point around potential reasons why these specific tournaments are flagged as substantial outliers. We note there need not necessarily be any shared commonality: neither in terms of organizers nor the times (whether in recent years, outliers are becoming more common); those are detected through functional clusters, outlined in the next section. For our purpose, so we do not deviate from the chief goal of this work, we notice this phenomenon and take it to be a crucial ingredient behind reporting how these tournaments are, in terms of excitement, predictable, but not predictable enough. Details are offered in Table 1.

Table 1.

Summary of excitement histograms from 124 tournament-season combinations. Functionally outlying tournaments are shown in bold.

Season	Min	Q1	Median	Mean	Q3	Max
Ipl 2007/08	3	17	23	22.15517	26	73
Ipl 2009	2	16	23	21.54386	26	67
Ipl 2009/10	2	17	21	20.2	25.25	53
Ipl 2011	0	14	20	20.89041	25	65
Ipl 2012	0	7.25	18	16.51351	23	44
Ipl 2013	0	11	19	17.73684	24	44
Ipl 2014	2	10	20.5	18.96667	24	91
Ipl 2015	1	11	19	22.59322	25	128
Ipl 2016	3	11.75	18	19.31667	23.25	74
Ipl 2017	2	15.5	24	22.62712	26	87
Ipl 2018	2	10.5	20.5	19.25	24	72
Ipl 2019	2	9.75	19	18.01667	26	39
Ipl 2020/21	3	9.5	22	19.13333	25.25	51
Ipl 2021	2	16	22	21.71667	27	70
Ipl 2022	2	10	21	19.31081	26	56
Ipl 2023	2	12.25	20	20	24.75	118
Ipl 2024	2	9.5	20	18.97183	25	63

Season	Min	Q1	Median	Mean	Q3	Max
Bbl 2011/12	4	12.5	19	20.14815	24.5	65
Bbl 2012/13	3	10.75	23	22.75	26.25	87
Bbl 2013/14	3	16	24	25.20588	27	95
Bbl 2014/15	3	17	22	23.5	27.75	56
Bbl 2015/16	2	11.5	20	18.59375	24	35
Bbl 2016/17	1	13.5	21	19.51429	25	62
Bbl 2017/18	4	12.5	21	19.25581	24.5	41
Bbl 2018/19	2	17.5	23	25.83051	26.5	106
Bbl 2019/20	2	10	21	21.13115	26	95
Bbl 2020/21	1	11	19	20.5082	26	100
Bbl 2021/22	3	16	20	19.81667	24.25	55
Bbl 2022/23	1	12	21	21.67213	24	87
Bbl 2023/24	2	14	21	23	27	98

Season	Min	Q1	Median	Mean	Q3	Max
T20 2005/06	8	8.75	12	12 . 25	15.5	17
T20 2006	8	12.25	16.5	16.5	20.75	25
T20 2006/07	6	16.5	24	35	42.5	86
T20 2007	15	16.5	18	18	19.5	21
T20 2007/08	5	10.25	19	21.13158	27	56
T20 2008	16	19	23	22	25	27
T20 2008/09	3	6.5	19	15.78571	24.25	27
T20 2009	2	17	22	24.63636	26	116
T20 2009/10	1	12	22	21.89474	26	54
T20 2010	3	17	21	26.08571	26	101
T20 2010/11	16	21.5	22	22 . 36364	23	27
T20 2011	2	21	22.5	20 . 4	26.75	30
T20 2011/12	2	14.5	23	24.26667	26	75
T20 2012	12	17	21.5	21 . 25	26	30
T20 2012/13	0	11.25	22.5	25.17391	28.75	117
T20 2013	2	8	20	17.73333	24	37
T20 2013/14	0	14	22	22.63934	27	94
T20 2014	7	20	22	20 . 6	25	29
T20 2014/15	8	21	24.5	25	29	41
T20 2015	3	17	24	23.11111	27	54
T20 2015/16	1	8	17.5	17.85714	23	74
T20 2016	2	8.5	22.5	28.5	31.25	111
T20 2016/17	4	15	21.5	21.30769	27.75	47
T20 2017	7	20	25	22 . 30769	26	32
T20 2017/18	1	11	21	21.85366	27	81
T20 2018/19	0	10	22	24.21154	26.5	122
T20 2018	1	13	20.5	25.5	28.25	102
T20 2019	2	13	22	24.90411	28	103
T20 2019/20	2	12	21	23.0069	26	102
T20 2020	2	7	14	24 . 5	23	99
T20 2020/21	4	11	18	18.2973	26	39
T20 2021	2	12	21	22.76423	26.5	108
T20 2021/22	1	13	21	24.43655	28	93
T20 2022	1	13	21	22.16942	26	95
T20 2022/23	1	16	23	28.43939	31	127
T20 2023	2	16.25	23.5	27.37356	31.75	118
T20 2023/24	1	16	23	25.87946	31	97
T20 2024	1	15	22	27.74699	32.25	125
T20 2024/25	1	15.5	22	25.98551	29	113

Season	Min	Q1	Median	Mean	Q3	Max
Cpl 2013	4	16	19	21.43478	26.5	59
Cpl 2014	3	15.5	23	20.81481	27	36
Cpl 2015	1	13.5	21	23.09375	25.25	117
Cpl 2016	3	11	21	21.51724	25	64
Cpl 2017	3	20.25	23.5	27.52941	28.75	75
Cpl 2018	2	12	19	19.48485	24	61
Cpl 2019	2	19	22	23 . 41176	25.75	79
Cpl 2020	4	17	21	24.54545	29	86
Cpl 2021	1	15	19	18.66667	23	34
Cpl 2022	4	12	21	21.06452	25	72
Cpl 2023	1	16.75	21	21.78125	26.25	53
Cpl 2024	6	18.5	22	20	23.5	29

Season	Min	Q1	Median	Mean	Q3	Max
Ctc 2011/12	3	8.5	25	22.8	29	67
Ctc 2012/13	7	15	22	20.48	26	37
Ctc 2013/14	4	10.5	20	20.14815	23.5	71
Ctc 2014/15	3	11	22	25.27586	26	118
Ctc 2015/16	3	9.25	22	19.625	28.25	46
Ctc 2016/17	0	12	19	21.59259	24.5	85
Ctc 2017/18	3	7	20	25.28	35	111
Ctc 2018/19	3	13.25	20.5	26.68182	29	115
Ctc 2020/21	2	10	18.5	16.1875	21	28
Ctc 2021/22	6	13.5	22	20.67742	25.5	60
Ctc 2022/23	3	14.25	21	30.57143	31.25	130
Ctc 2023/24	6	15	20	17.75	22.75	25

Season	Min	Q1	Median	Mean	Q3	Max
Ntb 2014	1	12	20	21.72131	25	130
Ntb 2015	1	11	21	21.54839	27	116
Ntb 2016	1	10	21	22.20661	26	124
Ntb 2017	1	11	21	25.65289	27	126
Ntb 2018	1	15	20.5	23.56349	26	123
Ntb 2019	2	12.75	20	21.73214	25.25	104
Ntb 2020	3	15	23	24.07865	26	123
Ntb 2021	2	13.75	21	21.66667	26	95
Ntb 2022	0	14	21	22.08462	26	112
Ntb 2023	3	17	22.5	22.86364	26	72
Ntb 2024	1	13	21	22.77236	26	108

Season	Min	Q1	Median	Mean	Q3	Max
Bpl 2011/12	1	11	19	18.76471	24	47
Bpl 2012/13	2	9.25	17	18.23684	27	42
Bpl 2015/16	1	14	22	21.29032	26	49
Bpl 2016/17	3	10	19	17.15556	23	36
Bpl 2017/18	4	12.75	22	20.11364	26	71
Bpl 2018/19	1	14	20	18.88889	24	44
Bpl 2019/20	3	13	20	19.32609	24	43
Bpl 2021/22	2	13.5	21	19.25	24.25	36
Bpl 2022/23	3	12.25	21.5	19.1087	26	41
Bpl 2023/24	7	16	21	20.44444	25	36

Season	Min	Q1	Median	Mean	Q3	Max
Psl 2015/16	7	17	20	20.3913	24.5	34
Psl 2016/17	4	10.5	21.5	22.33333	24.25	98
Psl 2017/18	1	9	17	16.9697	23	44
Psl 2018/19	3	16	21	19.41176	24.75	43
Psl 2019/20	4	14	20.5	24.10714	24.25	102
Psl 2020/21	2	17.5	21.5	20 . 2777	24	31
Psl 2021	2	7.5	14	19 . 15	23.25	62
Psl 2021/22	3	5.25	17.5	16.23529	23.75	37
Psl 2022/23	4	10.25	20	19.14706	25	38
Psl 2023/24	3	14.25	21	19.25	26	29

Grouping of cricketing tournaments: Amplitude-phase warpers to cluster change-induced histograms

The previous section established there are about fourteen tournaments that are significant outliers, in a functional sense. The presence of a few outlying tournaments, but, simultaneously, more than one type of tournament (in terms showed their excitement profile) would help allay concerns around the sport's inclusion in large events such as the Olympics. This is because outliers and clusters help strike a neat balance between unpredictability and predictability. While a few substantial outliers could suggest predictability, that is, the laying out of a reasonably solid expectation baseline of how a tournament is likely to unfold, the presence of major groups (of tournaments) would point to the fact that this predictability has not become extreme to the point where viewers find it useless to participate. The goal of this section is to test whether more than one type of groups exist. We stress - since this may be open to misinterpretation - the aim here is not to establish some tournaments are doing “well” in the sense of being more interesting (i.e., containing more interesting matches) and some, not that well. It is to check whether there is substantial variation in the ways the excitement values are sprayed out over the component matches. If there is a significant variation - and there are two groups - it may not necessarily imply that within one group, the excitement scores are high (i.e., the Hausdorff distances are small) while for the other, these are low. All it points to is - depending on the type of hypothesis test we are conducting (below) - there is non-ignorable differences in certain aspects of the functional nature of their histograms, such as the amplitude or the phase, described below.

Analyzing variations among histograms poses a challenge because one may be uncertain about where to focus (on the minimal scores to the left of the graph or the substantial scores to the right, and so forth). Those acquainted with functional data analysis will remember that condensing to a single figure via some norm-based distance typically proves unhelpful, particularly when clustering these curves. In response, statisticians have recently introduced a robust methodology: the technique of curve alignment or curve registration. We first ease the visual burden: we align the curves with one another, that is, adjust them so their peaks coincide, and then observe the fluctuations or variations along one axis (for example, the vertical axis) - a task we recognize from elementary statistics.

The panels in Figure 5, for instance, depict the histograms before and after alignment. Variation in the heights of the aligned or registered histograms, that is, the discrepancies in the y-axis values on the “after” alignment panel 5b, is referred to as amplitude variation and encapsulates all pertinent “information” (in a manner akin to how sufficient statistics or increasing filtrations of sigma algebras embody information) regarding the vertical dimensions of variation. Such as the heights of the peaks, the depths of the troughs, and so on. There are nuances involved. This process of “pushing” curves to make them conform, in essence, is termed warping and must be executed with precision since the total area beneath them must always sum to one.

Figure 5.

(a) left: original histograms of Hausdorff distances: the“before” alignment state. (b) right: the “after” peak alignment state, post-warping and registration.

Formally, a warp function $\gamma ():[0,1]\to [0,1]$ transforms its input based on (often accepted) convex or concave (or a blend of both) alterations. For example, the identity warp $\gamma (x)=x$ keeps its input completely intact. A key element we require to gauge the amplitude-based “distance” or “height”-based gap between two functions is the concept of a square-root velocity function (SRVF). The elegance this provides ensures we create a legitimate metric within the realm of amplitudes. Given a differentiable function $f()$ , its SRVF $q_f()$ is given by

$$q_f(t)=sign(f^{\mathrm{\prime }}(t))\sqrt{|f^{\mathrm{\prime }}(t)|}$$

(1)

and gauges, in a sense, how swiftly it changes. In our scenario, each tournament produces an $f()$ , a histogram whose functional essence we unearth using kernel density estimates (the curves in Figure 5(a)). The amplitude distance (that is, the variation in the y-axis on the aligned chart) between histogram one, that is $f_1$ , which comes from the first tournament, and histogram two, that is $f_2$ , which comes from the second, is expressed by

$${\delta }_A(f_1,f_2)=\underset{\gamma \in \mathrm{\Gamma }}{min}||q_{f_{1}o\gamma }-q_{f_2}|{|}_2,$$

(2)

where $\mathrm{\Gamma }$ represents the full space of all conceivable warp functions, and $\Vert {\Vert }_2$ is the usual L-2 norm. In simpler terms, to discuss the disparity between two histograms with regards to their peaks and valleys, we maintain one unchanged (technically, the SRVF of one remains intact; this is function $f_2$ in (2)) and strive to mold the other (that is, $f_1$ ), so it resembles this “base” one. Some warp function $\gamma ()$ should neatly reshape this adaptable one to the fixed one, with the goal of minimizing the peak-based discrepancies, and this minimum value is taken as the amplitude gap or the peak-based separation between the two histograms.

This distance function (2) provides other advantages, in addition to offering a legitimate metric. It aids the formulation of a solid definition of a functional center: the “average” functional entity (remember, the data structures we are engaging with are not numbers) - these are referred to as Karcher means (Tucker and Yarger ³⁵ ):

$$\overline{[f]}=argmi{n}_{[f]}\sum _{i=1}^n{\delta }_A([f],[f_i]{)}^2$$

(3)

where is $[f]$ the equivalence class of f, i.e., the collection of all other functions that can be derived from a seed f through some form of warping. Observe how this aligns with the slightly less conventional definition for means of numbers:

$$\overline{X}=argmi{n}_{\theta }\sum _{i=1}^n(X_i-\theta {)}^2$$

the value that minimizes the (squared loss) variance. In our example with the T20 tournaments, this “central” histogram, computed via (3), is shown as the black central curve in Figure 6(b) (note the distinction from the original data mean – i.e., the unaligned pointwise averages shown on the left, in 6a) and illustrates how an average tournament's excitement scores are distributed across a range.

Figure 6.

Summaries like mean and standard deviations. (a) left: Observe how pointwise averages out of unaligned histograms create strange results such as frequencies potentially becoming negative. (b) right: the “average” histogram after alignment which show how a typical tournament's interestingness score is likely to hover around 25.

Next, to detect possible grouping among tournaments, we deploy a sophisticated form of inference: identifying discontinuities or clusters among these functional objects, while still employing the Karcher mean, which serves as a functional average. Given any well-defined “average,” it is inherent, definitionally, that certain “objects” will occur “below” it, with others “above” it. It may therefore, be natural to claim the tournaments whose post-alignment histograms were “lower” than the average differ from those whose histograms were “above.” This dichotomy – a result of the very definition of an average – need, however, to be strengthened in terms of statistical significance. What renders this split-checking an inferential investigation is whether any observed differences are statistically substantial, i.e., whether the variation might have resulted from random chance. We frame the problem as follows (Tucker and Yarger ³⁵ ):

If we assume functions $f_1,f_2,\dots ,f_n$ are absolutely continuous and that they are generated through

$$f_i=\mu +\delta I(i>k*)+\epsilon$$

(4)

the hypothesis test that detects a structural shift concerns choosing between

$$H_0:\delta =0vs{H}_a:\delta \ne 0$$

(5)

We analyze a sequential series of standardized scores:

$$S_{n,k}={\displaystyle \frac{1}{\sqrt{n}}(k{\mu }_k-k{\mu }_n)}$$

where ${\mu }_n$ is the overall Karcher mean and ${\mu }_k$ , the Karcher mean with the current k-many curves. Again, we note the similarity of form with our familiar z-score. The test statistic

$$T_n={max}_{1\le k\le n}||S_{n,k}|{|}^2$$

by documenting the maximum of these scores, assesses whether, at any given moment, the partial mean significantly deviated from the overall mean. Such a deviation would indicate the formation of a cluster that is substantially distinct from the group formed so far.

The cluster detection criteria is, therefore, a change-point one. The null distribution of this maximum statistic ensures the quality of the cluster detection exercise, i.e., insignificance of subsequent binary-segmented maximums, point to the unnecessity of further subdivisions. As different tools for clustering functional data become available, one can implement RAND- or Jaccard-type similarity fractions to compare the agreement between two clustering methods.

Inference may also be effectively performed on phase transitions (indeed, it has been recently revealed ³⁵ that the variability among functional entities can be elegantly partitioned into two fundamental components: amplitude and phase fluctuations, analogous to the analysis of variance, wherein total variation decomposes into between-group and within-group sums of squares) to investigate whether a considerable amount of effort is necessitated to align the peaks.

While querying this congruence of the shape of the functions, especially these warps, the partial mean sequence $S_{n.k}$ maximized at k = 29, i.e., $S_{124,29}=0.1070,$ a snapshot of the full vector being {0. 0.0039, 0.0073, 0.01189,…., 0.1045, 0.1070, 0.1000,…., 0.0056, 0.0021, 0}, the corresponding p-value being 0.0757, confirming significance at 10% level of significance. These 29 tournaments correspond to the IPLs and all except the last (i.e., the 2023/24 season) BBL seasons. Excitement scores from these tournaments are, therefore spread similarly, while the other tournaments form another group with similar varying scores. Figure 7 offers a graphical report. The proximity of these warping functions to the diagonal (for instance, the solid red line, Figure 8(a)) signifies the lesser effort required to transform the corresponding curve into the (Karcher) mean, i.e., how that specific tournament was close to achieving the average excitement profile one can expect to witness out of these T20 tournaments.

Figure 7.

Gaps, i.e., distances, in excitement amplitude and phases (i.e., variations according to the corresponding metrics) among the 124 tournaments, shown in panels (a, left) and (b, right), respectively. Substantial separations are more vivid in the phase (b) case. A smaller p-value, 0.0757, confirms this.

Figure 8.

(a) left: warping functions for the 124 tournaments. Closeness to the diagonal suggests less stressful warping. These warpers arrange themselves above and below the diagonal according to which curves need to be concavely or convexly transformed to the mean. (b) right: Principal component-type analyses can be conducted both on the phase (top) and amplitude (bottom) domain. We find above three domains to be sufficient.

The previous section on functional outliers and the present one on the cluster-finding jointly point to the palatability of shorter versions of cricket, such as T20 series for the Olympics: the previous section, by establishing there are not too many modern tournaments whose excitement profiles are hugely different, the current one by showing how, despite that prevailing order, there is still sufficient variation among the tournaments to retain viewers’ interest. In these sections, we have compared several tournaments of the same sport: cricket, whose inclusion in the Olympics is being advocated for. The neat tradeoff highlighted above (in addition to the qualitative persuasion, logistical easiness outlined in the opening sections) inspires confidence in the sport's success in the Games. The opening motivation, however, compared two different kinds of sports: cricket and soccer, with one match of each type sampled at comparable levels - both finals, that is, near the end of the tournament. Those with access to soccer (or other sports that proceeded in similar fashion - like hockey - already in the games) players’ tracking data over several matches may extend that motivating match-level cricket-soccer comparison to tournament-level ones. This may be an interesting agenda in its own right, and open up between-game equivalence metrics, to establish whose justifiability, many games - that is, tournaments - may be called for.

Reflections and directions for the future

At the close of a cricket match - and this happens with other sports as well - spectators return with feelings like “that match was quite exciting” or “that was boring and one-sided” without being able to pinpoint precisely what guided their decision-making. These impressions are nonetheless vital to work out sponsorship deals for large tournaments and navigate issues around advocacy: whether it is viable long-run to include a particular sport in the Olympics, for instance. In this work, we have introduced - and documented the applicability of - a way to quantify the interestingness of a cricketing tussle, that does not depend on some eventual margin of victory - either the runs or the wickets or the overs left - but instead, interrogates and uses the evolving nature of twisting fortunes thereby mirroring the excitement a viewer feels as the game goes on. The foundations were laid solidly in an earlier work (Bhaduri ¹⁶ ) as a way to query the potential equivalence - in terms of excitement - of sports that, on paper, look hugely different: cricket and soccer, for example (one proceeds in a sequence, the other does not, while one is a contact sport, the other is not, etc.). In this work, we refine the internal workings and examine how the strategy “scales up”, so to speak, as we look at a much larger set of matches, over several tournaments, over several seasons. In addition to this generalization, what is especially novel in the current work is the way the strategy has been put to use. We find T20 cricket tournaments - which, among all cricketing formats, may align most closely with the Olympics’ logistical constraints – are at par, in terms of excitement with sports such as soccer that are already part of the Games. Furthermore, we bring out how these cricket tournaments exhibit sufficiently non-ignorable, but simultaneously, a bounded amount of excitement fluctuations which should offer a strategic guarantee that organizers of large events may be looking for. Similar such work conducted in other domains such as weather science, finance, computer algorithms, etc. should inspire confidence: Bhaduri, ³⁹ Zhan et al., ⁴⁰ Bhaduri and Zhan, ⁴¹ Bhaduri and Ho, ⁴² Bhaduri et al., ⁴³ Bhaduri et al., ⁴⁴ Ho and Bhaduri, ⁴⁵ Ho et al., ⁴⁶ Ho and Bhaduri, ⁴⁷ Tan et al., ⁴⁸ Bhaduri.^49–51

In this work, therefore, we have accomplished the following:

Without relying on final fixed scorelines, we established how ideas from change-point detection may be used to describe the interestingness of a specific instance of a given sport (for instance, one cricket match) by tracking a metric of interest (such as utility transitions in cricket, or effective playing spaces in soccer). This is achieved by deploying novel statistics that enjoy high change-detection power as ingredients of a sequential testing algorithm.

We developed a bridge - the interestingness meter - through which games of different kinds may be compared in terms of their excitement generating capabilities. We saw how a cricket and a soccer match, both sampled at comparable stages within their respective tournaments, become comparable in terms of the excitement they generate.

We formalized a way, mirroring and generalizing the match-level analysis in (ii) above, in which several tournaments (each consisting of several matches) may be described or compared according to how interesting they proved to be. This description is unpacked in two specific ways: through identifying potentially outlying tournaments of a specific sport such as cricket, and by checking for groups of tournaments that generate similar excitement profiles. Each tournament's excitement histogram is treated as a functional object, and recent statistical techniques such as functional depths and curve registration have been used to accomplish the first and second task, respectively. These provide perfect entry points for applied researchers who may be curious of carrying out standard statistical tasks (change-point estimation, clustering, outlier detection) in non-standard (i.e., functional) settings which may not be connected to sports.

The findings from (iii) above are used to emit firm advocacy: at a time when cricket's permanent inclusion in the large sporting events such as the Olympics is being debated. We show how the T20 format - a specific version of the sport - is worth including in the Olympics both because of these tournaments achieving a neat balance of predictability (not too many outlying tournaments) and chaos (the existence of more than one cluster) that should prove appealing to both organizers and viewers and because of its excitement similarity to other sports such as soccer that are already included in the Games.

We note with our formulation of interesting-ness, the phenomenon need not correlate with specific game-related particulars such as whether it was a high-scoring tussle. With the way of measuring interestingness we have outlined, the possibility of ordering tournaments opens up. In addition, a case may be made for including factors that are non-internal (like the runs or the utilities) to a specific match's progression but could still sway emotions: how many high-profile players are taking part, the location where the game is being played, the difference in ICC rankings of the teams involved. We are studying ways to incorporate these various data formats - some continuous, some ordinal, some modal, some distributional - especially through the symbolic framework introduced by Beranger at al. ⁵² to offer a more nuanced format for excitement discussions, with flexibilities to control the degree of nuance.

While the techniques shown in this work serve cricket - through measuring its interestingness, through using that interestingness to order tournaments and installing firmer advocacy on platforms such as the Olympics - they generalize to other spots as soon as a proper tool (such as runs or utilities in cricket) for them may be found, on which to study the twists and turns. Bhaduri ¹⁶ has begun this work with the convex hull generated by ten players in soccer.

Some observations are in order. What makes measuring excitement complex is the unlabeled – technically, “unsupervised” nature of the problem. Unlike, say, score, wicket or outcome prediction, where the actual score or winner becomes known eventually - which makes possible the calculation of an “error” or “residual” that one can minimize to suggest good prediction systems - in our case, there is no final true “answer” (that is, no way to know whether a match was truly, i.e., objectively more exciting than another) against which to check our recommendation. While opinions from experts may help erect a “ground-truth” (much like how handwriting experts are consulted to check the accuracy of automatic text recognition systems), those may still be subjective and open to interpretations and biases. This is precisely where opportunities lie for the future. One can query several parallel methods for interesting-ness comparisons, and average out the excitement scores - in a spirit similar to bagging: bootstrapping followed by aggregating. While the approach is common in supervised settings, Agarwal ⁵³ show how it may be updated to unsupervised contexts such as ours, and how it could promise similar benefits, such as variance reduction. In this work, we have looked at substantial shifts in run-scoring and utilities since, eventually, games are decided through runs, instead of, say, wickets taken. But there are no conceptual roadblocks that would prevent one from constructing similar change-point systems based on the evolving nature of some other criteria. This approach, along with other tools showcased here to detect structural breaks (Z, Zb, etc.) will trigger these other excitement scores which, subsequently, one can average, that is “bag”, along with ours, to reduce uncertainties in excitement quantification. The assumption behind these strategies is that temporal similarities of structural breaks from the two innings as they evolve will generate excitement. While this is reasonable and valid if one wants to examine continued or sustained excitement, another type of excitement may result eventually: when two teams begin very differently in terms of their breakpoint locations - potentially giving off the impression of being a one-sided affair, but towards the end, go neck-and-neck and make for a thrilling finish. We have not pursued this latter type of interesting-ness since the initial one-sided-ness may trigger viewers to disengage (turn off the television sets, leave the stadium, etc.). Interesting research may, however, be conducted in this vein with the techniques supplying another class of excitement scores with which to bag the scores from our strategies.