# Dense Matchers for Dense Tracking

Tomáš Jelínek, Jonáš Šerých, Jiří Matas
CMP Visual Recognition Group, Faculty of Electrical Engineering,
Czech Technical University in Prague
<EMAIL_ADDRESS>
###### Abstract

Optical flow is a useful input for various applications, including 3D
reconstruction, pose estimation, tracking, and structure-from-motion. Despite
its utility, the problem of dense long-term tracking, especially over wide
baselines, has not been extensively explored. This paper extends the concept
of combining multiple optical flows over logarithmically spaced intervals as
proposed by MFT. We demonstrate the compatibility of MFT with two dense
matchers, DKM and RoMa. Their incorporation into the MFT framework optical
flow networks yields results that surpass their individual performance.
Moreover, we present simple yet effective ensembling strategies that prove to
be competitive with more sophisticated, non-causal methods in terms of
position prediction accuracy, highlighting the potential of MFT in long-term
tracking applications.

##  1 Introduction

Obtaining point-to-point correspondences is a classical task in computer
vision, useful for a wide range of applications including tracking, structure-
from-motion, and localization. Despite the extensive research in wide baseline
stereo methods, including those with a time baseline, the domain of dense
point correspondences in videos has not been explored until recently [38, 53].
The emergence of the TAP-Vid dataset [11] has further fueled interest in long-
term point-tracking methods.

Point-trackers usually [12, 27, 45, 11] track sparse sets of points. However,
dense correspondences are useful in various applications, such as video
editing, object tracking, and 3D reconstruction. While optical flow techniques
provide dense correspondences, they are typically limited to pairs of
consecutive frames.

Long-term dense tracking has been recently addressed by Neoral  _et al_. [38]
MFT tracker, which computes optical flow not only for consecutive frames but
also for pairs of more temporally distant frames, including flow computation
between the reference and every other frame of the video. At every frame,
optical flow is computed w.r.t. the previous, first, and a constant number of
logarithmically spaced frames. Such approach is linear in the number of frames
and thus not computationally prohibitive.

In the original MFT[38], all optic flow computations are based on RAFT [50],
which has performed well in both standard benchmarks [2, 36] and in
applications. However, the RAFT optical flow network was trained on pairs of
consecutive frames, which is likely sub-optimal for large baselines.

Recently, dense matchers such as DKM [14] and RoMa [15] have been published.
This development opens the possibility to apply the MFT framework with
different dense matchers, or to use RAFT for pairs of frames with short
temporal, and thus probably spatial, baseline. The only requirement of the MFT
“meta optic flow algorithm” is that the basis dense two view optic flow or
matcher provides confidence in its predictions.

In this paper, we evaluate the MFT approach with the DKM and RoMa matchers
instead of RAFT. We show that both of these matchers provide accurate matches,
but inaccurate occlusion predictions. Addressing the strengths and weaknesses
of optical-flow-based and dense-matching-based methods, we propose a combined
tracker, that outperforms the original MFT design.

In summary, our contributions are: (1) We show how to adapt dense matchers DKM
and RoMa for use in the MFT framework, and experimentally evaluate their
performance. (2) We show that the MFT algorithm outperforms both direct flow
between the first and the current frame, and the chaining of optical flows
computed on consecutive frames for RAFT, DKM, and RoMa. (3) Based on better
results of RoMa over DKM in our experiments, we propose a dense long-term
tracker that combines the strengths of RAFT-based MFT and RoMa-based MFT.

##  2 Related Work

#### Tracking, 3D Reconstruction, and SLAM

Object tracking algorithms [1, 26, 10] traditionally outputted the track of an
object specified in the first frame in the form of bounding boxes. Later, the
focus shifted towards segmentation-based tracking [29, 41, 34].

Modern model-free trackers based on differentiable rendering [54, 43], that
can simultaneously track and reconstruct any object specified in the first
frame are naturally able to provide point-to-point correspondences for the
tracked object; however, to the best of our knowledge, they can track a single
object only or require multi-camera input [33]. Additionally, recent methods
[56, 57, 55], involving differentiable rendering of neural radiance fields
(NeRFs) [37], show potential in creating deformable 3D models for point
tracking. Nonetheless, the extensive computational demands of these methods
limit their practical applicability in real-world scenarios.

The traditional SLAM methods [46] produced sparse point clouds. Later on,
semi-dense [16, 51] SLAM methods appeared. Some SLAM-based trackers, [17] can
densely estimate point positions in static scenes, and recent advances in
differentiable rendering opened the avenue for differentiable-rendering-based
monocular SLAMs [42] but their application remains constrained to static
scenes.

#### Optical Flow

estimation is a classical problem in computer vision, with the early works
[32, 20] relying on the brightness-constancy assumption. With the advent of
deep neural networks, the focus shifted towards learning-based approaches [13,
49, 23, 50, 21] trained on synthetic data.

Optical flow estimation in state-of-the-art methods, exemplified by RAFT [50]
and FlowFormer [21], is achieved through the analysis of a 4D correlation cost
volume, considering features of all pixel-pairs. These techniques excel in
densely estimating flow between consecutive frames, yet they encounter
challenges in accurately determining flow across distant frames, particularly
in scenarios with large displacements or significant object deformation.

Multi-step-flow algorithms [7, 6, 8] address the limitations of concatenation-
based approaches for long-term dense point tracking. These algorithms create
extended dense point tracks by merging optical flow estimates across variable
time steps, effectively managing temporarily occluded points by bypassing them
until their re-emergence. However, their dependence on the brightness
constancy assumption renders them less effective over distant frames.
Subsequent works in multi-step-flow, such as the multi-step integration and
statistical selection (MISS) approach by Conze et al. [4, 5], further refine
this process. This approach relies on generating a multitude of candidate
motion paths from random reference frames, with the best path selected through
a global spatial smoothness optimization process. However, this strategy makes
these methods computationally demanding. Although certain optical flow
techniques [24, 39, 22, 59, 31, 58] address occlusions and flow uncertainty,
most leading optical flow methods, influenced by standard benchmarks like
those in Butler  _et al_. [3] and Menze  _et al_. [35], do not detect
occlusions. Jiang  _et al_. [25], building on RAFT [50], has taken a different
approach in which they handle occlusion implicitly by computing hidden motions
of the occluded objects. However, the method still falls short in the context
of tracking dynamic, complex motions.

We now describe in greater depth three methods that are most relevant to our
paper: RAFT [38], DKM [14], and RoMa [15]. While the latter two are in fact
dense matchers, we will use the term interchangeably with long-ranged optical
flow estimation with occlusion prediction.

#### MFT

extends optical flow into dense long-term trajectories by constructing
multiple chains of optical flows and selecting the most reliable one [38]. The
flow chains consist of optical flow computed both between consecutive frames,
and between more distant frames, which allows for re-detecting points after
occlusions. The intervals between distant frames are chosen to be
logarithmically spaced.

MFT extends the RAFT optical flow method with two heads, estimating occlusion
and uncertainty for each flow vector. Like the optical flow, the uncertainty
and the occlusion are accumulated over each chain, and the non-occluded flow
chain with the least overall uncertainty is selected as the most reliable
candidate. The long-term tracks of different points thus chain possibly
different sequences of optical flows. This strategy on one hand takes into
account that changes in appearance and viewpoint gradually accumulate over
time, which makes it more reliable to chain flows on easier-to-match frames
rather than estimating matches directly between the template and the current
frame. On the other hand, short chains containing longer jumps with low
uncertainty result in less error accumulation.

#### DKM

proposed by Edstedt  _et al_. [14], a dense point-matching method, employing a
ResNet [19]-based encoder pre-trained on ImageNet-1K [44] for generating both
fine and coarse features. The coarse features undergo sparse global matching,
modeled as Gaussian process regression, to determine embedded target
coordinates and certainty estimates. Fine features are refined using CNN
refiners, following a methodology similar to Truong  _et al_. [52] and Shen
_et al_. [47]. DKM’s match certainty estimation relies on depth consistency,
necessitating 3D supervision. The process concludes by filtering matches below
a certainty threshold of $0.05$ weighted sampling for match selection. Edstedt
_et al_. [14] released outdoor and indoor models trained on MegaDepth [30])
and ScanNet [9] respectively.

#### RoMa

similarly to DKM, RoMa [15] is a dense matching method that provides pixel
displacement vectors along with their estimated certainty, building upon the
foundation set by DKM [14]. RoMa differentiates itself by employing a two-
pronged approach for feature extraction: using frozen DINOv2 [40] for sparse
features and a specialized ConvNet with a VGG19 backbone [48] for finer
details. Unique to RoMa is their transformer-based match decoder, which
matches features through a regression-by-classification approach, better
handling the multimodal nature of coarse feature matching. In contrast to DKM,
RoMa’s pipeline omits the use of dense depth maps for match certainty
supervision, relying instead on pixel displacements for match supervision.
Their model is trained on datasets like MegaDepth [30] and ScanNet [9],
similar to DKM.

#### Long-Term Point Tracking

aiming to track a set of physical points in a video has emerged significantly
since the release of TAP-Vid [11]. The dataset’s baseline method TAP-Net [11]
computes a cost volume for each frame, employing a technique akin to RAFT’s
approach [50]. It focuses on tracking individual query points. PIPs [18] takes
this approach to an extreme by completely trading off spatial awareness about
other points for temporal awareness within fixed-sized temporal windows,
making it unable to re-detect the target after longer occlusions. TAPIR [12]
combines TAP-Net’s track initialization with PIPs’ refinement while removing
the PIPs’ temporal chunking, using a time-wise convolution instead. CoTracker
[27] models the temporal correlation of different points via a sliding-window
transformer, modeling multiple tracks’ interactions. While these methods are
designed for sparse tracking, they can provide dense tracks by querying all
points in the first frame.

Notably, differentiable rendering has been leveraged in recent approaches,
with OmniMotion representing 3D points’ motion implicitly using learned
bijections [53] enabling it to provide dense tracks. Alternative methods like
[33] which models the scene as temporally-parametrized Gaussians[28]. However,
these methods have their limitations, such as OmniMotion’s quadratic
complexity and the multi-camera requirement of [33].

##  3 Method

For a stream $\\{\mathcal{I}_{1},\,...,\,\mathcal{I}_{N}\\}$ of $N$ video
frames defined on a common image domain $\Omega$, we denote the optical flow
between frames $i$ and $j$ as $\mathcal{F}^{(i,\,j)}$. Moreover, we use
$\sigma^{(i,\,j)}\in\mathbb{R}_{+}^{\Omega}$ to denote the estimated flow
variance, and $\rho^{(i,\,j)}\in[0,\,1]^{\Omega}$ to represent the estimated
certainty of $\mathcal{F}^{(i,\,j)}$. Finally, occlusion score
$o_{\text{}}^{(i,\,j)}\in[0,\,1]^{\Omega}$ denotes the estimated probability
of pixels appearing in frame $i$ being occluded in frame $j$. To simplify
notation, although $\mathcal{F}^{(i,\,j)}$, $\rho^{(i,\,j)}$, and
$\sigma^{(i,\,j)}$ are 2D or 3D tensors, we will use these symbols to denote
their values at a specific point $\boldsymbol{p}=(x,\,y)$ in the image.
Moreover, for every point $\boldsymbol{p}_{i}$ in frame $i$, its predicted
position $\boldsymbol{p}_{j}$ in frame $j$ relates to the optical flow
$\mathcal{F}^{(i,\,j)}$ as follows:

$\boldsymbol{p}_{j}=\boldsymbol{p}_{i}+\mathcal{F}^{(i,\,j)}(\boldsymbol{p}_{i}).$
(1)

Let us denote by $\phi_{\text{RAFT}}$, $\phi_{\text{DKM}}$,
$\phi_{\text{RoMa}}$, $\phi_{\text{MFT}}$ the functions computed by RAFT, DKM,
RoMa, and MFT respectively. By RAFT we mean the MFT’s adaptation of RAFT with
additional uncertainty and occlusion heads [38]. The output vectors of these
methods are as follows:

$\displaystyle\phi_{\text{RAFT}}=(\mathcal{F}^{(i,\,j)},\,\sigma^{(i,\,j)},\,\rho^{(i,\,j)})$
(2) $\displaystyle\phi_{W}=(\mathcal{F}^{(i,\,j)},\,\rho^{(i,\,j)})$ (3)
$\displaystyle\phi_{\text{MFT}}=(\mathcal{F}^{(i,\,j)},\,\sigma^{(i,\,j)},\,o_{\text{}}^{(i,\,j)}),$
(4)

where $W$ is one of the wide-baseline methods, either DKM or RoMa.

###  3.1 MFT Flow Chaining

MFT [38] achieves long-term optical flow estimation by combining multiple
optical flows. These flows are obtained from $\phi_{\text{RAFT}}$ over
logarithmically spaced distances. When estimating the flow
$\mathcal{F}^{(1,\,j)}$, MFT utilizes a sequence of intermediate flows. This
sequence, denoted as $\mathcal{S}$, comprises flows
$\mathcal{F}^{(j-\Delta_{1},\,j)},\,...,\,\mathcal{F}^{(j-\Delta_{K},\,j)}$.
Here, $\Delta_{i}$ represents logarithmic spacing and is defined as $2^{i-1}$
for $i<K$, with $\Delta_{K}=j-1$. We limit the number of intermediate flows,
denoted by $K$, to a maximum of 5 and ensure that $j-\Delta_{K-1}>1$.

Additionally, MFT employs a scoring function for evaluating the quality of the
intermediate flows chaining for each image point $\boldsymbol{p}_{1}$ in the
reference frame $1$. The scoring function $s^{(j-\Delta_{k},\,j)}$ utilizes
chaining of estimated flow variances and occlusion scores over an intermediate
frame $i$:

$\displaystyle\sigma^{(1,\,i,\,j)}(\boldsymbol{p}_{1})$
$\displaystyle=\sigma_{MFT}^{(1,\,i)}(\boldsymbol{p}_{1})+\sigma^{(i,\,j)}(\boldsymbol{p}_{i}),$
(5) $\displaystyle o_{\text{}}^{(1,\,i,\,j)}(\boldsymbol{p}_{1})$
$\displaystyle=\max\\{o_{\text{MFT}}^{(1,\,i)}(\boldsymbol{p}_{1}),\,o_{\text{}}^{(i,\,j)}(\boldsymbol{p}_{i})\\},$
(6)

The point $\boldsymbol{p}_{i}$ is computed using $\mathcal{F}_{MFT}^{(1,\,i)}$
and the relation in Equation 1. The scoring function is then defined as
$s^{(j-\Delta_{k},\,j)}(\boldsymbol{p}_{1})=-\sigma^{(1,\,j-\Delta_{k},\,j)}(\boldsymbol{p}_{1})$.
If the chained occlusion score
$o_{\text{}}^{(1,\,j-\Delta_{k},\,j)}(\boldsymbol{p}_{1})$ exceeds an
occlusion threshold $\theta_{o}$, we set
$s^{(j-\Delta_{k},\,j)}(\boldsymbol{p}_{1})=-\infty$. This score is used to
select the best flow for every point $\boldsymbol{p}_{1}$, that is the flow
with the lowest estimated variance computed on chains that do not contain
occluded points.

MFT computes long-term flow for any point $\boldsymbol{p}_{1}$ in the
reference frame $1$ iteratively via chaining as

$\displaystyle\mathcal{F}_{MFT}^{(1,\,j)}(\boldsymbol{p}_{1})=\mathcal{F}_{MFT}^{(1,\,i_{M})}(\boldsymbol{p}_{1})+\mathcal{F}^{(i_{M},\,j)}(\boldsymbol{p}_{i_{M}}),$
(7)

where $i_{M}\in\\{j-\Delta_{k}\ |\ 1\leq k\leq K\\}$ such that the score
$s^{(i_{M},\,j)}(\boldsymbol{p}_{1})$ is maximal. Again, the point
$\boldsymbol{p}_{i_{M}}$ is obtained using
$\mathcal{F}_{MFT}^{(1,\,i_{M})}(\boldsymbol{p}_{1})$ and Equation 1.
$\mathcal{F}^{(i_{M},\,j)}$ is the flow obtained from an arbitrary method that
can also estimate its variance $\sigma^{(i_{M},\,j)}$ and occlusion score
$o_{\text{}}^{(i_{M},\,j)}$. The flow chaining is visualized in Figure 1.

The estimated variance and occlusion score for frame $j$ are then obtained
from the chain over frame $i_{M}$ as
$\sigma_{MFT}^{(1,\,j)}(\boldsymbol{p}_{1})=\sigma^{(1,\,i_{M},\,j)}(\boldsymbol{p}_{1})$,
respectively
$o_{\text{MFT}}^{(1,\,j)}(\boldsymbol{p}_{1})=o_{\text{}}^{(1,\,i_{M},\,j)}(\boldsymbol{p}_{1})$.
A pixel observed in frame $i$ is considered occluded in frame $j$ if its value
$o_{\text{MFT}}^{(i,\,j)}$ is above a threshold $\theta_{o_{\text{}}}$. In
practice, we set different thresholds for different backbone networks as we
discuss in Subsection 3.2.

⋮$\boldsymbol{p}_{1}$$\boldsymbol{p}_{j-\Delta_{K-1}}$⋮$\boldsymbol{p}_{j-\Delta_{1}}$⋮$\boldsymbol{p}_{j}$
$\mathcal{P}=\\{j-\Delta_{k}|\ 1\leq k\leq K\\}$
$i_{M}=\arg\max_{i\in\mathcal{P}}s^{(i,\,j)}(\boldsymbol{p}_{1})$
$\mathcal{F}_{MFT}^{(1,\,j-\Delta_{K-1})}$$\mathcal{F}_{MFT}^{(1,\,j-\Delta_{1})}$$\mathcal{F}^{(j-\Delta_{K},\,j)},$$s^{(j-\Delta_{K},\,j)}$$\mathcal{F}^{(j-\Delta_{K-1},\,j)},$$s^{(j-\Delta_{K-1},\,j)}$$\mathcal{F}^{(j-\Delta_{1},\,j)},$$s^{(j-\Delta_{1},\,j)}$$\mathcal{F}^{(i_{M},\,j)}$
Figure 1: Illustration of the MFT flow chaining as defined in Equation 7. The
optical flows are evaluated on the points in the outbound nodes of their
respective arcs.

###  3.2 Integration of DKM and RoMa

As we mentioned in the Introduction, we make the conjecture that training RAFT
for optical flow prediction on consecutive video frames is suboptimal for wide
baselines. We therefore integrate DKM and RoMa, capable of handling wider
baselines. However, integrating these methods with MFT poses certain
challenges due to their incompatible outputs.

In the first place, neither RoMa nor DKM provides an occlusion score $o$, but
only an estimate of the flow prediction certainty $\rho$. We therefore
artificially set their occlusion scores as $o=1-\rho$. Furthermore, although
$\sigma$ and $\rho$ both represent the quality of estimated optical flow, they
are not directly comparable. But in order to integrate them into the MFT
framework, we need to converse between them.

Through empirical analysis, we established a flow certainty threshold
$\theta_{\rho}$. When $\rho$ exceeds this threshold, we deem the optical flow
reliable, assigning $\sigma=0$. Conversely, when $\rho$ is below this
threshold, $\sigma$ is set to $1000$, correlating higher uncertainties with
increased variances in predicted flow. Additionally, we observed that while
$o_{\text{MFT}}$, $o_{\text{DKM}}$, and $o_{\text{RoMa}}$ fundamentally
represent the same concept, their respective occlusion thresholds
$\theta_{o_{\text{RAFT}}}$ and $\theta_{o_{\text{RoMa}}}$ vary. In our
experiments in Section 4, we use

$\displaystyle\theta_{o_{\text{RAFT}}}=0.02,\
\theta_{o_{\text{DKM}}}=\theta_{o_{\text{RoMa}}}=0.95.$ (8)

For a visual comparison between the original MFT and the integration of RoMa
into MFT, see Figure 2.

###  3.3 Ensembling

We observed that, in terms of occlusion prediction, MFT’s modification of RAFT
achieves higher accuracy compared to RoMa. Conversely, RoMa exhibits better
performance in optical flow prediction relative to RAFT. Based on these
findings, we developed an integrated approach that combines the strengths of
both methods. Specifically, our method utilizes occlusion data from RAFT,
while RoMa is employed for position prediction, with both processes executed
in parallel within the MFT framework. As detailed in Section 4, our most
effective strategy involves employing RAFT for occlusion score prediction and
RoMa for position prediction, provided the point is not predicted as occluded;
in cases of occlusion, RAFT’s predictions are preferred.

(a) Reference frame

(b) RAFT-based MFT Strategy.

(c) RoMa-based MFT Strategy.

(d) DKM-based MFT Strategy.

(e) Direct matching between frames #0 and #140 using RAFT.

(f) Direct matching between frames #0 and #140 using RoMa.

(g) Combined RAFT and RoMa strategy.

(h) Selective RoMa position prediction.

Figure 2: Visual comparison of selected dense tracking methods: (a) reference
frame #0; (b)-(h) predicted positions of points in frame #140. All blue points
are invisible in frame #140; blue points in (b)-(h) thus indicate false
matches. Green points are visible both in frame #0 and frame #140. Red points
highlight the points on the body of the lioness. Different shades are used to
identify different points. The sequence is available at
https://cmp.felk.cvut.cz/~serycjon/MFT/visuals/ugsJtsO9w1A-00.00.24.457-00.00.29.462_HD.mp4.

##  4 Experiments

|  | main metrics |  |
---|---|---|---|---
base | strategy |  | AJ |  $<$$\delta^{x}_{avg}$ | OA |  | $<\\!1$ | $<\\!2$ | $<\\!4$ | $<\\!8$ | $<\\!16$
| direct |  | 38.4 | 50.8 | 65.6 |  | 29.0 | 44.1 | 54.6 | 60.4 | 65.7
RAFT | chain |  | 38.7 | 55.0 | 69.5 |  | 25.2 | 43.8 | 59.4 | 70.4 | 76.3
| MFT |  | 47.4 | 67.1 | 77.7 |  | 34.0 | 57.3 | 74.3 | 82.8 | 86.9
| chain |  | 27.3 | 63.5 | 48.2 |  | 36.4 | 56.2 | 69.4 | 76.0 | 79.6
DKM | direct |  | 34.0 | 60.7 | 52.8 |  | 37.0 | 54.5 | 65.3 | 70.9 | 76.0
| MFT |  | 47.8 | 72.0 | 70.2 |  | 43.0 | 65.8 | 79.0 | 84.5 | 87.8
| direct |  | 37.7 | 63.7 | 57.6 |  | 37.5 | 55.9 | 67.8 | 75.5 | 81.5
RoMa | chain |  | 40.3 | 63.1 | 60.7 |  | 36.8 | 55.3 | 68.1 | 75.5 | 79.8
| MFT |  | 48.8 | 72.7 | 71.7 |  | 43.0 | 65.5 | 79.2 | 85.5 | 90.1
Table 1:  TAP-Vid DAVIS evaluation of different optical flow combination
strategies. The MFT strategy outperforms both simple chaining and direct
matching for all base optical flow methods on all the metrics.

In this section, we evaluate our proposed method. Initially, we compare the
MFT framework with direct optical flow prediction and simple optical flow
chaining. Subsequently, we explore RoMa’s optical flow prediction performance
within the MFT framework depending on whether it predicts the point as
occluded or non-occluded, which serves as a foundational finding for our most
effective ensembling strategy. The final part of our experimentation serves as
a comparison of different ensembling strategies, justifying the design of our
most effective architecture, and comparing it to other tracking methods.

#### Evaluation setup

Our experiments were conducted on all 30 tracks of the TAP-Vid-DAVIS dataset
[11] with a resolution of 512×512 using the first evaluation mode. This
approach aligns with the methodology described in MFT [38]. It is important to
stress that in the dataset, the tracks are annotated only sparsely with more
focus on the foreground objects rather than the static background.

#### Evaluation metrics

In assessing the performance of our approach, we employ three key metrics as
defined by the TAP-Vid benchmark. The Occlusion Accuracy (OA) evaluates the
accuracy of classifying the points as occluded. We measure the quality of the
predicted positions, using average displacement error, denoted as
$<$$\delta^{x}_{avg}$. This metric calculates the fraction of visible points
with a positional error below specific thresholds, averaged over thresholds of
1, 2, 4, 8, and 16 pixels. These accuracies for individual thresholds are
denoted as $<i$ with $i$ representing the threshold. Additionally, the Average
Jaccard (AJ) as defined in [11] index is used to collectively assess both
occlusion and position accuracy.

###  4.1 MFT Chaining

A key aspect of our analysis involves contrasting the performance of RAFT,
DKM, and RoMa within the MFT framework against direct optical flow prediction
with the first frame serving as a reference, and chaining of the optical flows
computed on consecutive video frames. The results presented in Table 1 clearly
show that for each base method (RAFT, DKM, RoMa), the MFT strategy
consistently outperforms the other strategies in all metrics by a large
margin. These results underscore the effectiveness of MFT in handling complex
motion trajectories over extended periods, surpassing the limitations of
direct prediction and simple chaining methods. A key observation exemplified
in Figure 2 is that RoMa is substantially less prone to predict mismatches in
the background than RAFT.

The results in Table 1 also show that RoMa within the MFT paradigm achieves
arguably the best results in position prediction, while RAFT outperforms all
other methods in the occlusion classification accuracy. This finding serves as
a foundation for our ensemble strategies in Subsec. 3.3. Due to the
consistently better performance of RoMa over DKM in the evaluation benchmark
in all, average Jaccard, average displacement error, and occlusion accuracy we
from now on focus our experiments on RoMa even if DKM runs slightly faster.

###  4.2 RoMa Visibility

While RoMa demonstrates high accuracy in position prediction, its capability
in occlusion detection is relatively limited in comparison to RAFT. However,
the quality of occlusion prediction is vital for scoring the optical flows as
described in Subsec. 3.1, and thus for computing new flows. We hence
conjecture that if we only use the RoMa’s optical flow predictions that are
predicted as not occluded, we can achieve even better tracking results. The
results, as shown in Tab. 2, indicate a marked improvement in tracking
accuracy when measured only on points predicted as non-occluded.

predicted |  $<$$\delta^{x}_{avg}$ | $<\\!1$ | $<\\!2$ | $<\\!4$ | $<\\!8$ | $<\\!16$
---|---|---|---|---|---|---
occluded | 47.4 | 18.7 | 32.7 | 52.0 | 62.6 | 71.1
visible | 77.2 | 46.9 | 70.9 | 84.5 | 89.8 | 93.7
any | 72.7 | 43.0 | 65.5 | 79.2 | 85.5 | 90.1

Table 2:  TAP-Vid DAVIS evaluation of MFT-RoMa separated by the occlusion prediction. Using only the points predicted as not occluded leads to improved position accuracy on all error thresholds.  | MFT base |  | main metrics |  | visibility
---|---|---|---|---|---
| position | occlusion |  | AJ |  $<$$\delta^{x}_{avg}$ | OA |  | precision | recall
(1) | RAFT | RAFT |  | 47.4 | 67.1 | 77.7 |  | 78.0 | 91.5
(2) | RoMa | RoMa |  | 48.8 | 72.7 | 71.7 |  | 74.5 | 85.3
(3) | RoMa | RAFT |  | 50.2 | 72.7 | 77.7 |  | 78.0 | 91.5
(4) | RAFT/RoMa | RAFT |  | 51.6 | 73.4 | 77.7 |  | 78.0 | 91.5
| TAPIR |  |  | 56.2 | 70.0 | 86.5 |  |  |
| CoTracker |  |  | 61.0 | 75.9 | 89.4 |  |  |
Table 3:  TAP-Vid DAVIS evaluation of combinations of two trackers. We run
MFT-RAFT and MFT-RoMa independently in parallel, using the two outputs for the
final position and occlusion prediction. RAFT-based MFT (1) has good occlusion
accuracy (OA), RoMa-based MFT (2) has good position accuracy
$<$$\delta^{x}_{avg}$. Using MFT-RAFT to predict occlusion and MFT-RoMa to
predict position (3) achieves better AJ. The best results (4) are achieved
when the position is predicted by MFT-RoMa, but only when it predicts visible
(see Tab. 2).

###  4.3 Ensembling Strategies

In the concluding part of our experimental analysis, we compare various
ensembling strategies within the MFT framework, building on the insights from
the previous sections. The results, detailed in Table 3, demonstrate the
effectiveness of the ensemble strategy.

#### RAFT-based MFT Strategy

For comparison we show the original MFT strategy, utilizing RAFT for both
position and occlusion predictions. This approach, while achieving the highest
occlusion accuracy among all ensembling strategies tested, exhibits suboptimal
performance in position precision.

#### RoMa-based MFT Strategy

Substituting RAFT entirely with RoMa, we observed an improvement in position
prediction accuracy. However, this modification led to a significant decrease
in occlusion prediction accuracy, highlighting the trade-offs between these
two aspects.

#### Combined RAFT and RoMa Strategy

Our next strategy involved a simple combination of RAFT and RoMa: RAFT for
occlusion prediction and RoMa for position prediction. This hybrid approach
resulted in enhanced performance across all metrics, outperforming the
aforementioned individual strategies.

#### Selective RoMa Position Prediction

However, further refinement was achieved by integrating findings from
Subsection 4.2. We found that RoMa’s position predictions are more accurate
for points it identifies as visible. Therefore, we devised a strategy where
MFT-RoMa’s position predictions are used only if the points are marked as
visible; otherwise, RAFT’s predictions are utilized. This selective strategy
led to improvements in both position prediction accuracy and occlusion
accuracy. We visually compare this strategy with other two best-performing
strategies and MFT with RAFT in Figure 3.

#### Comparison with Point Trackers

We observe that our approach closely rivals or exceeds the performance of
established sparse point tracking methods like CoTracker and TAPIR in the
average position accuracy while achieving worse performance in the occlusion
prediction accuracy. It is noteworthy that our method attains these results
within a strictly causal framework, contrasting with CoTracker and TAPIR,
which utilize attention-based temporal refinement strategies. Moreover, it is
important to highlight that, unlike our approach, CoTracker and TAPIR are
designed as sparse trackers.

##  5 Conclusion

We have showcased the benefits of employing the MFT framework over direct
optical flow computation and optical flow chaining. We have also demonstrated
the flexibility of the MFT paradigm which can be readily used together with
different optical flow computation methods. Without complex architectural
modifications and using simple ensemble strategies, we were able to
demonstrate position prediction accuracy on the Tap-Vid dataset competing with
that of state-of-the-art sparse trackers that utilize non-causal tracking
refinement.

#### Limitations and Future Work

Our current approach does not take into account the speed of the baseline
optical flow networks. The main limitation is the need for two optical flow
networks to operate concurrently within the ensemble strategy. Exploring co-
training strategies that enable a single network to deliver similar
performance could be a viable solution. A key task is to bridge the existing
gap in occlusion prediction accuracy between our method and the state-of-the-
art. We also put forward the need for new datasets featuring dense annotations
of point tracks in both the foreground and background.

#### Acknowledgments

This work was supported by Toyota Motor Europe and by the Grant Agency of the
Czech Technical University in Prague, grant No.SGS23/173/OHK3/3T/13.

(a) RAFT-based MFT Strategy

(b) RoMa-based MFT Strategy

(c) Combined RAFT and RoMa Strategy

Figure 3: Images show the first frames of two selected TAP-Vid DAVIS
sequences. Dots represent ground-truth tracking points, with shades of green
showing the improvement in $<$$\delta^{x}_{avg}$ achieved by the Selective
RoMa Position Prediction ensemble over methods (a)-(c), shades of red show the
converse.

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# Quantifying individual and team performance in cricket

Onkar Sadekar  Department of Network and Data Science, Central European
University, Vienna, Austria.  Sandeep Chowdhary  Department of Network and
Data Science, Central European University, Vienna, Austria.  M. S. Santhanam
Department of Physics, Indian Institute of Science Education and Research, Dr.
Homi Bhabha Road, Pune 411008, India  Federico Battiston  Department of
Network and Data Science, Central European University, Vienna, Austria.

###### Abstract

Advancements in technology have recently allowed to collect and analyze large-
scale fine-grained data about human performance, drastically changing the way
we approach sports. Here we provide the first comprehensive analysis of
individual and team performance in cricket, one of the most popular sports in
the world. We investigate temporal patterns of individual success, finding
that the best performance of a player can happen at any time in their career,
surrounded by a burst of comparable top performances. We show that long-term
performance can be predicted from early observations and that temporary
exclusions of players from teams are often due to declining performances but
are also associated with strong comebacks. We demonstrate that teams are more
likely to endure hot streaks of winning performances compared to random
expectations, but also sequences of consecutive losses. We investigate the
crucial contributions of specialists such as openers, all-rounders and wicket-
keepers and show that a balanced performance from multiple individuals is
required to ensure team success. Finally, we quantify how transitioning to
captaincy in the team improves the performance of batsmen, but not that of
bowlers. Our work emphasizes how individual endeavours and team dynamics
interconnect and influence collective outcomes in sports.

##  I Introduction

The origin of sports can be traced back to early civilizations, notably with
the inception of the Olympic Games in ancient Greece Roche (2000); Guttmann
(2002); Bryson (2008); Horne and Whannel (2020). These early sporting events
were not just about physical achievements but also played a role in cultural
and societal bonding. As societies progressed, sports evolved, reflecting
changes in social structures and becoming more organized and diverse. The
recent advancements in digital technology have significantly altered the
landscape of sports, introducing a new era of sports analytics Nevill  _et
al._ (2008). Enhanced data acquisition capabilities have resulted in large-
scale data sets which lend themselves to quantitative analysis of athletes and
teams. This evolution has not only heightened the excitement for fans but also
provided researchers and sports strategists with valuable insights into the
physical and cognitive skills essential in competitive sports Radicchi (2011);
Pappalardo  _et al._ (2019); Chowdhary  _et al._ (2023); Zappalà  _et al._
(2023).

In the realm of baseball, the influence of sports analytics has been
particularly noteworthy, a phenomenon popularized by the ‘Moneyball’
revolution Lewis (2004). The strategic use of data analytics in baseball has
led to a profound shift in team management and player evaluation, where data-
driven decisions have often trumped traditional scouting methods Brown  _et
al._ (2017); Pappalardo and Cintia (2018). NBA and NFL have also embraced this
analytical wave and now leverage data to optimize player performance, team
strategies, and even in-game decisions Mason and Foster (2007). In the NBA,
for instance, analytics have led to an increased emphasis on three-point
shooting and defensive metrics, reshaping the style of play Dawson (2023);
Neuhaus and Thomas (2024). On the other hand, more intellectual games like
chess are also seeing an elevated level of sophistication and achievements
owing to the introduction of advanced chess engines Roring and Charness
(2007); Vaci and Bilalić (2017).

Cricket, the world’s second most-followed sport after soccer according to
multiple sources noa (2023, 2024), enjoys widespread popularity primarily in
Commonwealth countries, including India, Australia, New Zealand, the United
Kingdom, South Africa, the West Indies, Sri Lanka, and Pakistan. The
availability of match data, driven by amateur and professional enthusiasts,
has fostered various analyses, ranging from quantifying the hot-hand effect
Ram  _et al._ (2022) to predicting match outcomes using machine learning
Subasingha  _et al._ (2019), assessing cricket-related injuries J Orchard  _et
al._ (2002); Finch  _et al._ (1999); Orchard  _et al._ (2005), and modelling
the game with state-of-the-art techniques Swartz  _et al._ (2009); Kimber and
Hansford (1993); Arora  _et al._ (2014); K. (1905); Mehta  _et al._ (1983).
Despite this, there remains a substantial gap in the understanding of
individual performances and their contributions to team success in cricket. In
this work, we track players’ careers, unveiling universal patterns of
performance and identifying correlates of team-level success.

We categorize players into batsmen and bowlers, adjusting their performance
metrics for inflation, and examine the emergence of phenomena such as the
random impact rule and hot streaks. These phenomena, extensively studied in
diverse fields like science Sinatra  _et al._ (2016); Liu  _et al._ (2018,
2022), arts Fraiberger  _et al._ (2018); Janosov  _et al._ (2020); Williams
_et al._ (2019), and sports Bar-Eli  _et al._ (2006); Gilovich  _et al._
(1985); Chowdhary  _et al._ (2023), provide a framework for understanding
patterns in cricket. Additionally, we investigate performance predictors,
finding that early career achievements are indicative of overall career
trajectories. We also explore factors contributing to player exclusion from
teams. Building on these insights we assess the impact of leadership roles and
pathways to captaincy and analyze the roles and contributions of specialists,
such as openers, all-rounders, and wicket-keepers. Lastly, we apply
established metrics from the literature to quantify patterns in team success.
Collectively, our work presents the first comprehensive analysis of individual
and team performance in cricket.

Figure 1: Time of top performance in a player’s career. (a) Distribution of
the match number ($N^{*}$) of the top performance for a player’s career. The
inset shows the distribution of the time of the top performance in a career
$N^{*}$ normalized by the career length $N$. Real data (pink circles) is
consistent with a randomized null model which removes temporal correlations
(black dotted line) suggesting that the best performance can occur at any time
within the career. (b) Distribution of differences between the locations of
the best ($N^{*}$), second best ($N^{**}$), and third best ($N^{***}$)
performances of a player normalized by their career lengths. Higher values
near zero compared to the null-model indicate that the best performances are
clustered, indicating the presence of hot streaks in cricket careers.

##  II Results

### Data

In this work, we conduct a comprehensive analysis of a dataset comprising
approximately 4,500 One-Day International (ODI) cricket matches, extracted
from howstat.com, an open-access repository for cricket statistics. This
dataset encompasses records of all ODI matches played from their inception in
1971 until October 2023 (see Methods). Our data extraction focused on key
metrics, including individual player performance (quantified in terms of runs
scored and wickets taken), team scores, match winners, venues, and match
dates. Although the primary aggregation of data is at the team level for each
match, the dataset is also suitable for an in-depth exploration of individual
player trajectories and careers. We have 607 batsmen and 594 bowlers in our
dataset (see Methods). This allows for a multifaceted analysis of performance
trends, the impact of various factors on player and team success, and the
evolution of the sport over more than five decades.

Team scores per match have generally increased over the decades (see SM1).
This observed variation is potentially due to a confluence of factors,
including modifications in game regulations (fielding rules for example),
advancements or changes in the equipment used (notably the cricket bat), and
the reduction in the dimensions of the playing field 65-75 metres compared to
80-85 metres in stadiums earlier as spectators are interested in high scoring
matches. To facilitate a robust and equitable comparison of player and team
performances across distinct time periods, we account for the inflation in
run-scoring by implementing a normalization procedure on all batting
statistics. Specifically, we adjusted these statistics such that the average
team score is constant over the examined time period, following a well-known
procedure originally introduced to correctly assess the impact of scientific
papers published in different years Radicchi  _et al._ (2008) (see SM1). This
approach ensures that the comparative analysis of players’ performance from
different eras is conducted in an unbiased manner which controls for
background temporal trends.

###  II.1 Temporal patterns of top performances

Figure 2: Early career performance and effect of drop and re-entry (a), (b)
Scatter plots showing the correlation between early career and full career
performance for batsmen and bowlers respectively. The $R^{2}$ values indicate
that performances in different stages of the career are correlated. (c), (d)
Average runs scored and average number of wickets taken by individual players
(who were dropped for at least 3 matches) before they were dropped and after
they were reinstated in the team. We observe a significant dip in the
performance prior to the drop and a strong comeback in performance after re-
entering the team.

The progression of a player’s skill level enhances their likelihood of
surpassing previous performance peaks. Conversely, as players age, their
athletic prowess may diminish, potentially impeding their ability to exceed
past achievements. This contrast of skill development against physical decline
poses a critical question: at which point in their careers do players deliver
their best performance? We address this question by making use of tools and
methodologies from data science and extreme value theory already deployed in
the realms of arts, culture, and more recently in sports Coles (2001);
Beirlant  _et al._ (2004); Sinatra  _et al._ (2016); Janosov  _et al._ (2020);
Chowdhary  _et al._ (2023). We define $N^{*}$ as the match number where a
player achieves their career-best performance. For batsmen, this is marked as
their highest run score. For bowlers, it corresponds to the highest number of
wickets taken. In instances of identical runs or wickets, the performance
involving respectively fewer balls played or fewer runs conceded is
considered. The probability distribution function $P(N^{*})$ for all players
is shown in fig. (1)(a). We observe that $P(N^{*})$ is exponentially
distributed, suggesting a much higher likelihood of peak performance occurring
earlier rather than later in a career. However, this analysis does not account
for variations in career lengths. When normalizing the timing of peak
performance relative to overall career length, we observe a uniform
distribution, as shown in fig. (1)(a) (inset). This pattern, previously dubbed
as the ‘random impact rule’ for scientific careers Sinatra  _et al._ (2016),
suggests that the timing of peak performance is generally unpredictable and
can occur at any point in a career. Indeed, Sachin Tendulkar, one of the most
prolific batsmen across generations has $N^{*}/N=0.95$, but for Virat Kohli
$N^{*}/N=0.3$. Similarly among the bowlers, Muttiah Muralitharan, the highest
wicket-taker in ODI has $N^{*}/N=0.43$, but Waqar Younis has $N^{*}/N=0.81$.
To assess the robustness of our results, we compared them against a randomized
null model which shuffles the order of performances for each player (see
Methods), denoted by a dotted line in fig. (1). We find no qualitative
difference between the empirical and randomized data distributions.

Across a range of domains such as science Liu  _et al._ (2018, 2022), arts
Williams  _et al._ (2019); Janosov  _et al._ (2020), and sports Ram  _et al._
(2022); Chowdhary  _et al._ (2023), individual careers display hot streaks,
where exceptional performances tend to occur in bursts which are clustered in
time. To extend this inquiry to the realm of cricket, we designate $N^{*}$,
$N^{**}$, and $N^{***}$ as the match numbers corresponding to a player’s best,
second-best, and third-best performances, respectively. A small temporal gap
between $N^{*}$, $N^{**}$, and $N^{***}$ would indicate a concentration of top
performances within a specific timeframe, indicating the existence of a hot
streak. Conversely, a larger difference among these time stamps would suggest
a lack of correlation in the timing of peak performances. For e.g. Rohit
Sharma, who holds the record for the highest individual score in a match
achieved this feat in his $121^{st}$ match ($N^{*}$=121). However, his second-
highest score came in his $105^{th}$ match ($N^{**}=105$,
$|N^{*}-N^{**}|/N=0.06$. In fig. (1)(b) we plot the distribution of the
normalized number of matches between the best and second-best performances, as
well as between the best and third-best performances. Additionally, we employ
a null model (same as before) where we reshuffle the timestamps of
performances and analyze the corresponding differences to benchmark our
findings. Our analysis reveals that the probability distributions of
$|N^{*}-N^{**}|$ and $|N^{*}-N^{***}|$ are consistently higher around smaller
values ($\approx 0$) than those derived from the randomized null model.
Moreover, the probability of performing well in consecutive matches grows
exponentially as evidenced by the recurrence time distribution (see SM2).

In conclusion, while the best performance of a cricket player’s career may
manifest at any point within their career, these further observations indicate
that peak performances in cricket tend to occur in bursts over short periods
of time.

###  II.2 Long-term performance from early career observations

Early identification of talent and skills can provide key advantages in many
competitive settings, from firm growths Golder and Tellis (1997) to
information spreading Bak-Coleman  _et al._ (2022). Also in sports, nurturing
talent in young players can lead to higher returns Baker  _et al._ (2018);
Till and Baker (2020); Zappalà  _et al._ (2024). Here we investigate the
relationship between a player’s initial performance and their overall career
trajectory to capture whether it is possible to see hints of future
performance based on early-career display.

In figures (2) (a) and (b) we conduct an analysis correlating early and
overall career statistics. Our analysis considers players who have
participated in a minimum of 50 matches. We focus on their mean performance
during the initial 25 matches. For batsmen, we look at the average runs scored
per match in this early phase, whereas for bowlers, we look at the average
number of wickets taken per match within the same period. Concurrently, we
examine the full-career performance of these players. The scatter plots in
figure (2) reveal a correlation between early and full career average
performance metrics. We calculate the linear regression coefficients
$R^{2}=0.49$ for (a) batsmen and $R^{2}=0.66$ for (b) bowlers. This finding
suggests that a player’s initial performance may serve as an indicator of
their subsequent career performance. However, notable differences emerge when
comparing batsmen and bowlers. For illustration, consider Ricky Ponting
(batsman) and Shane Warne (bowler) from the Australian team. Ricky Ponting
averaged 32.45 runs per match in his early career and later improved his
career average to 37.5 runs per match. In contrast, Shane Warne picked up 2.12
wickets per match in his early career, but ultimately his performance dropped
to 1.53 wickets per match over his full career. In general, approximately 55%
of batsmen are observed to improve upon their early career averages, whereas
this figure drops to about 45% for bowlers. Such disparities may be attributed
to the distinct elements inherent in the roles of batsmen and bowlers.
Batsmen, for instance, often leverage accumulated experience and refined
skills to augment run-scoring, while bowlers may depend more on innovation and
strategic creativity to increase their wicket tally. This distinction
potentially accounts for the observed trend of a higher proportion of batsmen
surpassing their initial career performance compared to bowlers.
Alternatively, our observations can also be explained if some players are
simply better than the rest. Thus, their performance in various stages of
their careers will correlate with their full career performance. This effect
is also known as the Q-model in the context of scientific careers Sinatra  _et
al._ (2016). Further analyses supporting this hypothesis are presented in SM3.

Figure 3: Leadership affects player performance. (a) Boxplot showing the
average runs scored and average wickets taken by captain and non-captain
players. The median trend for batsmen and bowlers is opposite with captain-
batsmen performing better than their non-captain counterparts. PDFs of
performance for captains who are (b) batsman and (c) bowlers during various
phases of their career. The dotted black lines denote the median of the
distribution. The performance of captain-batsmen improves during captaincy,
while it declines for captain-bowlers.

###  II.3 Effect of drop and re-entry

In the team-oriented sport of cricket, fluctuations in team composition
frequently arise due to injuries and variations in player performance. While
injuries often occur unpredictably, a decline in performance typically
manifests more gradually and may not be immediately apparent. A fitting
inquiry in this context is whether it is possible to discern patterns in a
player’s performance preceding their expulsion from the team. Additionally, it
is important to determine whether a player’s performance improves post-
reinstatement to the team.

Figures (2) (c) and (d) show the average performance trajectories of batsmen
and bowlers, respectively, both prior to their removal from the team and
subsequent to their rejoining. We consider all players who experienced a
temporary exclusion from the team for a minimum of three matches before making
a return. We observe a consistent downtrend in performance during the pre-
removal phase, with average runs and wickets demonstrating a monotonic
decline, amounting to an approximate 19% reduction for both batsmen and
bowlers. Notably, the lowest performance levels are recorded in the match just
before the player’s exclusion. By contrast, players tend to exhibit a strong
comeback performance after reinstatement to the team. In particular, the
initial performance post-return exhibits a substantial elevation, with batsmen
registering an approximate 36% improvement and bowlers showing a 30%
enhancement compared to their final pre-removal performance. This enhanced
performance appears to be stable in subsequent matches, consistently
displaying higher performance compared to the pre-removal, suggesting that
short temporal exclusion may improve performance in the longer run.

###  II.4 Role of leadership

In team sports, the composition of an effective team encompasses skilled
players under proficient leadership. Case studies across various domains,
including sports, science, and business, have illustrated scenarios, where
moderately weak teams achieve success under adverse conditions, guided by
strategic leadership Guimerà  _et al._ (2005); Zhang  _et al._ (2022); Hancock
_et al._ (2023). Here we explore the influence of captaincy on a player’s
performance. Specifically, we investigate whether the added responsibility of
leadership alters player performance and if such an impact persists throughout
their careers.

We consider captains who have led their team in a minimum of 50 matches (see
Methods). Within our dataset, we identify 71 captains, comprising 69% batsmen
and 31% bowlers, indicating a higher propensity for batsmen to be chosen as
captains – nearly twice as often as bowlers. In fact, many of the most
successful captains including Allan Border, Graeme Smith, M S Dhoni, Ricky
Ponting, and Sourav Ganguly have been batsmen. However, only a handful of
captains such as Imran Khan, Shaun Pollock, and Wasim Akram have been bowlers.
Despite these differences, captain-bowlers remain in this position for longer
than the batsmen on average (see SM4).

Figure (3) (a) presents a comparison of the career average performances
between captains and non-captains (referred to as players). A detailed
analysis reveals a notable variation in performance levels between captain-
batsmen and captain-bowlers, compared to their non-captain counterparts.
Quantitatively, the average runs scored by captain-batsmen ($32$) exceed those
of players ($25$) by approximately 20%. Conversely, captain-bowlers secure, on
average, 21% fewer wickets than non-captain-bowlers ($0.9$ versus $1.1$). We
have considered only the matches where players get a chance to perform. This
disparity suggests differing pathways to captaincy, where batsmen may need to
consistently outperform peers, while bowlers’ captaincy seems less dependent
on individual performance.

To further elucidate this phenomenon, we divide the careers of captains into
three phases: pre-captaincy, during captaincy, and post-captaincy, for both
batsmen (fig (3) (b)) and bowlers (fig (3) (c)). We observe that batsmen
typically enhance their performance upon assuming captaincy, whereas bowlers
exhibit a decline. Closer examination of fig (3) (c) reveals a bimodal
distribution during the captaincy phase for bowlers, indicating a dichotomy in
skill sets among captain-bowlers. Post-captaincy, the trajectories diverge
significantly for batsmen and bowlers. For batsmen, average performance
markedly decreases post-captaincy, even falling below their pre-captaincy
levels. In contrast, bowlers tend to recover and often improve upon their
captaincy phase performance. These observations suggest a close correlation
between individual performance and captaincy tenure for batsmen, where they
often lead through exemplary performance. Conversely, bowlers may experience
captaincy as a burden, reflected in their individual statistics. This could
account for the observed differences in performance between captains and non-
captains among batsmen and bowlers in fig. (3) (a).

###  II.5 Contribution of specialists

Figure 4: Contribution of specialists. (a) Runs and strike rates of opener and
non-opener batsmen. The solid black lines denote the mean of the
distributions. Openers score more runs at a slower pace, while non-openers
play aggressively at the cost of lower contribution to team scores. (b)
Fractional contribution $f_{c}$ of players to the team scores for batsman,
bowler, and all-rounder. All-rounders contribute significantly higher to the
team. (c) Fractional dismissals $d_{i}$ per match by wicketkeeper and other
fielders over the entire career. Wicketkeepers exhibit a very high tendency to
contribute to team dismissals as compared to the fielders.

In team sports, the strategic composition of a team is crucial for its
effective functioning. The concept of utilizing specialists – individuals who
perform highly specific roles – extends beyond sports into various domains of
human activity, including business organizations, scientific research,
software development and even hunter-gatherer societies. In this study, we
examine three distinct specialist roles within cricket, encompassing different
facets of the game: namely, opening batsmen, all-rounders (contributing with
both bat and ball), and wicket-keepers.

Figure (4) (a) elucidates the variations in batting patterns between opening
batsmen and their non-opening counterparts. Batsmen are categorized as openers
if they occupy the first or second position in the batting lineup, while
positions 3 to 6 are considered non-openers (see Methods). We focus on the
pace at which they accumulate these runs. The run-scoring rate is quantified
as the number of runs per 100 balls faced, commonly referred to as the strike-
rate in cricket terminology. This metric is frequently used to assess a
player’s defensive or aggressive playing style. Virender Sehwag (average
strike-rate=94) and Brendon McCullum (average strike-rate=97) are prime
examples of explosive openers who scored runs at a brisk pace in their career.
However, our analysis reveals that openers average a strike rate of around 63
runs per 100 balls, whereas non-openers exhibit a higher average of 68 runs
per 100 balls. This suggests that non-openers score approximately 9% more
rapidly than their opening counterparts. However, the average score for
openers stands at approximately 31 runs, in contrast to 26 runs for non-
openers, indicating a reduction of about 17% for batsmen coming later in the
batting order. Collectively, these findings illustrate a strategic balance
between defensive and attacking approaches contingent on a player’s batting
position. Openers tend to adopt a more conservative approach, possibly due to
the initial uncertainty of the pitch conditions, while subsequent batsmen
often adopt a more aggressive strategy, building upon the foundational efforts
of the openers. Further analysis reveals that this disparity may be partially
attributed to the differing number of balls faced by players based on their
batting order (see SM5).

All-rounders are players who make significant contributions in both aspects of
the game – batting and bowling (see Methods). In contrast, specialized batsmen
and bowlers are characterized by their significant contribution in only one of
these areas. To quantitatively assess the impact of each player type, we
introduce the fractional contribution $f_{c}$ as,

$f_{c}=\frac{1}{2}\times\left(\frac{\text{runs scored}}{\text{total team
runs}}+\frac{\text{wickets taken}}{\text{total team wickets}}\right)$ (1)

, so that a player’s maximum possible contribution ($f_{c}$) is normalized
within the range of 0 to 1. It is important to note that, given this
framework, specialized batsmen and bowlers have an upper limit of $f_{c}=0.5$
in best-case scenario. Figure (4) (b) presents the distribution of the
contributions of all players across all matches. Among these, batsmen
demonstrate the lowest mean contribution ($f_{c}\approx 0.06$), with bowlers
exhibiting a slightly higher average contribution of $f_{c}\approx 0.1$.
Notably, all-rounders surpass both groups with an average contribution of
$f_{c}\approx 0.11$. Furthermore, there are instances where all-rounders
singularly account for more than 50% of the team’s performance, evidenced by
instances of $f_{c}>0.5$. We also note that non-specialists contribute
significantly less than the specialists in both departments (see SM6).

In summary, while bowlers generally contribute more towards the team’s
collective efforts than batsmen, all-rounders emerge as pivotal players, often
driving the team’s success with contributions that exceed 50% of the total
team scores. This finding underscores the vital role of all-rounders in
cricket. Future research could delve into a more detailed examination of the
all-rounder’s role and their strategic impact on the dynamics of the game.

We now concentrate on the impact of fielding (see Methods) in cricket.
Although most players are predominantly recognized for their batting or
bowling abilities, fielding is an integral aspect of a team’s overall success.
Indeed, a commonly reiterated phrase in cricket states, ‘catches win matches’,
emphasizing the significance of player dismissals through effective fielding.
Among various fielding roles, the wicket-keeper is considered to hold a
pivotal position (see Methods). Typically, a batsman can be dismissed in three
primary ways: (a) if the bowler successfully hits the stumps or the batsman
obstructs the stump with their leg, (b) if a fielder catches the ball hit by
the batsman before it touches the ground, or (c) if the stump is dislodged by
the fielders before the completion of a run between the wickets 111There are
other forms of dismissals such as ‘retired hurt’, ‘obstructing fielding’, etc.
but we do not consider them since the frequency of such dismissals is low..
The first method relies exclusively on the bowler’s skill, whereas the latter
two require strategic collaboration between the bowler and fielders. Our
interest lies in the last two dismissal types. Specifically, we define $d_{i}$
as the number of dismissals executed by a fielder of type (b) and (c) per
match. Figure (4) (c) presents a boxplot depicting the distribution of
dismissal fractions attributed to wicket-keepers and fielders. Our
observations reveal that wicket-keepers play a vital role in dismissals,
accounting for approximately 0.7 dismissals per match. In contrast, fielders
contribute to a relatively minor number of dismissals, averaging around 0.3.
The strategic positioning of wicket-keepers behind the wickets significantly
facilitates these outcomes with well-known captains such as M.S. Dhoni (441
dismissals in 340 matches) winning multiple accolades for India and Kumar
Sangakkara (477 dismissals in 391 matches) guiding Sri-Lanka to the finals of
the 2011 World Cup.

Figure 5: Patterns in team success (a) Occurrences of hot streaks (wins
denoted by red) or cold streaks (losses denoted by blue) of a given length
$L_{streak}$ compared to a null model. $N(l\geq L_{streak}$ denotes the number
of times when the team had the same result for $l$ or more matches, averaged
over teams. We observe that teams demonstrate both hot and cold streaks. The
fluctuations in the numbers calculated as the standard error are smaller than
the symbols. (b) Differences between effective team size $S_{eff}$ for winning
and losing teams. Winning teams have a larger $S_{eff}$ on average indicating
that a larger number of players contribute significantly in winning teams than
in losing teams.

In summary, team success depends on the successful coordination of different
types of – often specialized – contributions. Openers, by adopting a cautious
approach assessing pitch conditions and scoring at a lower rate, lay a strong
foundation for the team. All-rounders enhance the team’s score by applying
their skills across both facets of the game. Lastly, wicket-keepers, by
contributing to a substantial number of total dismissals, enhance the bowlers’
efforts and, consequently, the overall team performance.

###  II.6 Patterns of team success

So far we have predominantly focused on individual contributions to team
success in cricket. We now broaden our perspective to analyze team success,
setting aside specific individualistic factors that contribute to victory.
Given that each cricket match culminates in a definitive outcome (see Methods
for more information on what it means to win), our interest lies in discerning
patterns of wins and losses for teams.

Figure (5) (a) outlines the occurrence of hot streaks within teams. Similar to
the bursts of exceptional performance observed in individual players (as
discussed in sec.(II.1)), we assess the propensity of teams to consecutively
win (or lose) a series of matches. A familiar example is Australia’s 21 match
winning streak in 2003 which culminated in them winning the World Cup. We
quantify this tendency by defining the ratio $\frac{N_{data}}{N_{rand}}$,
where $N_{data}$ represents the frequency of a team winning (or losing) $l$ or
more consecutive matches. In contrast, $N_{rand}$ is the analogous frequency
within a randomized null model (see Methods). Thus, the ratio
$\frac{N_{data}}{N_{rand}}$ indicates the likelihood of a team experiencing a
streak of at least $l$ consecutive wins or losses compared to a random
outcome. Figure (5) (a) reveals that teams are more prone to winning or losing
matches in sequence than would be expected by chance. Notably, the probability
of winning 7 or more consecutive matches is nine times higher than chance,
while the likelihood of losing 7 or more consecutive matches is about three
times the random expectation. This observation suggests that teams exhibit hot
streaks in their winning performances. Intriguingly, we also observe cold
streaks, where teams undergo multiple successive losses.

We then characterise the relation between collective contributions and match
outcomes. Specifically, we employ the concept of effective team size, a well-
established metric in the team analysis literature Klug and Bagrow (2016);
Delice  _et al._ (2019). Effective team size quantifies the essential number
of players in a team, effectively measuring the redundancy in team
composition. Mathematically, we denote the contribution of the player to the
team score as $f_{c}$, such that $\sum_{c}f_{c}=1$, as per Eq. (1). The
effective team size $S_{eff}$ is defined in such a way that it equals the
actual team size if contributions are evenly distributed among all players and
reduces to 1 if a single player is solely responsible for the entire
performance. A common approach to calculate $S_{eff}$ is to use the formula
$S_{eff}=2^{H}$, where $H=-\sum_{c}f_{c}\log_{2}f_{c}$. Statistically,
$S_{eff}$ represents the entropy of the distribution of contributions $f_{i}$
from team players.

As illustrated in Figure (5) (b) via a boxplot, the distributions of $S_{eff}$
for winning and losing teams differ notably. The median $S_{eff}$ for winning
teams ($\approx 7.3$) is approximately 5% greater than that for losing teams
($\approx 6.9$). This indicates that successful teams require a broader array
of individual contributions. Indeed in the 2015 World Cup, Australia, the
winner, had $S_{eff}=6.77$ in the finals, as compared to New-Zealand who had
$S_{eff}=5.6$. Surprisingly, in the controversial final of the 2019 World Cup,
despite being the losing side, New-Zealand had $S_{eff}=8.1$, while England
had $S_{eff}=7.7$. Moreover the data also reveal a redundancy of 3 to 4
players in most teams. However, our supplementary analysis also reveals that
the top performers from the winning team contribute more to their team than
their top performing counterparts from the losing team (see SM7).

In conclusion, our analysis reveals that teams tend to win or lose matches in
clusters. However, for a team to secure victory, it is imperative that
multiple individuals make substantial contributions. A lack of such collective
effort often results in teams losing matches.

##  III Discussion

In the last few years, the advent of detailed sports analytics has
revolutionised our understanding of human behaviour in sports. It offers
insights into physical performance, cognitive processes, and team dynamics,
moving beyond traditional training methods. This shift towards a data-driven
approach in sports reflects a broader societal trend in optimizing human
potential, combining historical practices with modern scientific
methodologies.

In this study, we quantified the individual and team performance in cricket,
examining various aspects of the game. Due to the absence of quantifiable
performance metrics such as the Elo rating system in chess, we validated the
use of runs scored and wickets taken as indicators of batting and bowling
performance, respectively. Our analysis revealed a significant increase in
average runs scored over time. To negate this inflation in run-scoring and
facilitate fair comparisons, we implemented a normalization of all
performances involving runs. This rescaled data corroborated the “random
impact rule” observed in other domains, indicating that a player’s peak
performance can occur at any point during their career. However, this best
performance often co-occurs with other comparable exemplary performances,
pointing to the existence of hot streaks in individual cricket players. Our
exploration into the predictors of individual performance unveiled a strong
correlation between early career achievements and overall career trajectory.
Additionally, we confirmed that players are often excluded from teams due to
declining performances, yet they typically exhibit a sustained enhancement in
performance following their comeback.

We investigated the relationship between player performance and captaincy. Our
data suggested that batsmen who ascend to captaincy roles demonstrate higher
performance levels, both overall and during their tenure as captain, in
contrast to non-captain-batsmen. This pattern was not mirrored among bowlers,
highlighting the differential impact of captaincy on distinct player types.
Focusing on the role of specialists, our findings indicated that opening
batsmen typically score more runs but at a slower rate, whereas subsequent
batsmen tend to adopt more aggressive strategies. All-rounders were observed
to contribute significantly more to team performance compared to batsmen. In
the realm of fielding, wicketkeepers emerged as pivotal in effecting
dismissals, thereby bolstering the impact of bowlers on team success.
Additionally, our analysis identified both hot and cold streaks within teams,
underscoring that victories and defeats tend to occur in clusters. Winning
teams were characterized by more evenly distributed player performances, as
indicated by higher effective team sizes.

Our results might be subject to certain limitations. Firstly, the volume of
One Day International (ODI) matches and the resulting dataset size are not
fully extensive. Future research could benefit from incorporating data from
diverse cricket formats such as Test matches, T20 games, and franchise leagues
like the IPL, which might yield deeper insights into varied player performance
patterns. Supplementing our analysis with data from various professional
levels may reveal further patterns in players’ mobility and skill levels.
Additionally, our dataset focuses on end-match statistics, omitting the
nuanced temporal dynamics within individual games. While the final scorecard
offers substantial information, some facets of performance may only become
apparent with a more granular data analysis.

All in all, our work reveals intriguing patterns of individual and team
performance in cricket. We hope that our analysis motivates further research
into synergistic individual efforts and team dynamics.

##  IV Methods

###  IV.1 Data

In our study, we extracted data on 4566 One Day International (ODI) matches in
men’s cricket, involving 2882 unique players, sourced from howstat.com. This
dataset spans from 1971 to 2023. For each match, we extracted information
including match-date, teams, runs scored, wickets taken, and overs played by
each team, along with player names and their contributions in terms of batting
and bowling. This included details like batting position, number of balls
faced, number of fours and sixes hit, number of overs bowled, number of runs
conceded, number of maiden overs, and number of wickets taken. Furthermore,
data on the captain and wicketkeeper for each match, as well as the winner of
the match, were also collected.

###  IV.2 Cricket Terms

####  IV.2.1 Gameplay

A One Day International cricket game consists of each team playing 50 overs,
with 11 players per side and one over comprising 6 balls 222The number of
overs per side in a match was originally 60 before all teams adopted the
50-over format. However, the number of such matches is low, so we do not
account for this factor.. The playing field is circular, featuring a
20.11-meter pitch at its centre, with stumps or wickets at both ends. The
gameplay is similar to baseball with few key differences and some terminology
changes. The bowler or the thrower projects the ball towards the batsman or
hitter, aiming for it to bounce once before reaching them. The batsman’s goal
is to hit the ball hard so that it crosses the boundary for 4 or 6 runs. If
the bowl doesn’t cross the boundary, the batsmen run between the wickets for
additional runs until the players from the other team collect the ball and
throw it back towards the wickets. The bowler’s objective is to hit the stumps
or induce the batsman to hit the ball such that a fielder can catch it before
it touches the ground, resulting in the batsman being declared out. The
innings of a team batting first concludes when either 50 overs are bowled or
10 batsmen are declared out. The aim for the second team is to score at least
1 run more than the first team. The match ends when the second team scores
more runs than the first team in 50 overs or the second team fails to reach
the target either due to all players declared out or 50 overs being done. The
team with a higher score is declared as the winner.

####  IV.2.2 Type of Players

Batsman: We consider players as batsmen who have played at least 25 matches
and batted at position 7 or above in at least 50% of their matches. We have
607 batsmen in our dataset after applying this criteria. Like in baseball, the
job of the batsman is to score as many runs as possible by hitting the ball
out of bounds or by running between the wickets if the ball remains within
bounds.

Bowlers: We classify players as bowlers who have played at least 25 matches
and bowled in at least 50% of the matches they played. We have 594 bowlers in
our dataset after applying this criterion. The main task of the bowlers is to
dislodge the stumps or force the batsman to hit the ball towards the fielder.

All-rounders: We consider all-rounders as players who have played at least 25
matches and are classified as both batsmen and bowlers using the above
definitions.

Captains: A captain for a match is the leader of the team who directs the team
and makes strategic in-game decisions. In our analysis, we consider players as
captains if they have captained the team in at least 50 matches.

Openers and Non-openers: We characterize openers as players who have batted at
first or second position at least 10 times in their career, while non-openers
as players who have batted at positions three to six at least 10 times in
their career.

Wicket-keepers: The dataset contains an attribute for each player specifying
if they were the wicket-keepers in the match. The wicket-keepers ensure that
the bowl doesn’t go beyond them in case the batsman fails to hit the ball.
They also try to dislodge the stumps if the batsman fails to make it to the
wickets while scoring runs.

Fielders: All players on the bowling side who are not bowling or wicket-
keeping are classified as fielder in our analysis. There are two main
responsibilities of the fielder. First, they have to catch a ball hit by the
batsman to dismiss them before the ball touches the ground. Second, in case
the first scenario is not possible, the fielder has to collect the ball and
throw it at either wickets to catch the batsman short of completing the run
(thus, dismissing them) or stopping further runs.

###  IV.3 Null Models

####  IV.3.1 Best individual performance

In fig (1), we characterize the existence of random impact rule and hot
streaks. In order to establish a baseline for these measures, we shuffle the
timestamps of individual performances for each player 5000 times keeping the
actual values of performance the same. In this way, we are breaking the
temporal correlation between different performances. For each shuffling, we
find the time stamp for the best ($N^{*}$), second best ($N^{**}$), and third
best ($N^{***}$) performances of a player. We plot the PDF of this randomly
shuffled data to compare the results from our dataset.

####  IV.3.2 Hot and cold streaks for teams

Figure (5) (a) showcases the existence of hot and cold streaks for teams. We
consider all the teams that have played at least 200 matches with a definite
result. By keeping the number of wins constant, we shuffle the timestamps of
the match results $10^{4}$ times for each team. For each reshuffling, we
compute the number of consecutive wins (hot) and losses (cold) denoted by
$L_{streak}$. By taking the average across $10^{4}$ reshufflings, we compute
$N_{rand}$ to highlight the significance of $N_{data}$ – the number of streaks
of length $l$ or higher in the actual dataset.

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## Acknowledgements

The authors warmly thank Aanjaneya Kumar for his initial involvement in the
project and discussions of preliminary results. F.B. acknowledges support from
the Air Force Office of Scientific Research under award number
FA8655-22-1-7025.

## Competing interests

The authors declare that they have no competing interests.

## Additional information

Supplementary information is provided as a separate file.

## Materials and correspondence

Correspondence should be addressed to<EMAIL_ADDRESS>
## Supplemental material for “Quantifying individual and team performance in
cricket”

##  SM1 Inflation of runs over time

Figure S1: Inflation of performance across decades. We observe a 40% increase
in the total team score from the 1980s to the 2020s. To ensure a fair
comparison for players from different eras we normalize all the scores such
that the average team score over the years is constant.

##  SM2 Distribution of inter-event match difference

Figure S2: Distribution of recurrence times for (a) batsmen and (b) bowlers.
$\Delta N_{x}$ denotes the distribution of the number of matches between
consecutive instances of a player showcasing a performance of $x$ or higher
for various values of $x$. For batsmen, $x$ denotes the runs scored, while for
bowlers, $x$ indicates the number of wickets taken. An exponential fit
reproduces this behaviour accurately, implying that these extreme events are
rare (e.g. a batsman takes on average 15 matches to score 100 or more runs
after scoring 100 runs).

##  SM3 Correlators of career performance

Figure S3: Correlation of mean performance over different time periods to the
full career performance for (top row) batsmen and (bottom row) bowlers. We
notice that (a),(d) random, (b), (e) mid, and (c),(f) late career performances
are correlated ($R^{2}>0.6$) to the full career performances giving evidence
that intrinsic talent influences player performance.

##  SM4 Length of leadership

Figure S4: Distribution of leadership duration for captains who have led their
team for at least 50 matches for batsmen and bowler captains. On average the
batsmen lead their team in 215 matches, while the bowlers lead their team in
approximately 240 matches before stepping down.

##  SM5 Effect of batting position on runs scored and balls faced.

Figure S5: Number of balls faced and number of runs scored for each batting
position. We notice that batsmen at positions 1,2, and 3 face more balls and
score more runs than players coming in to bat at latter positions.

##  SM6 Contribution of non-specialists to team

Figure S6: Cross-category contribution of various types of players in batting
and bowling. (a) A bowler contributes 1% of team runs, while an all-rounder
contributes 5% of team runs on average. A batsman contributes 7% of team runs
as seen in the main text. (b) A batsman contributes to 0.3% of team wickets,
while an all-rounder contributes to 5% of team wickets. A designated bowler
contributes 10% of team wickets as seen in the main text.

##  SM7 Individual contributions to team success

Figure S7: Individual contributions to the team in absolute (top) and relative
scales (bottom) for winning and losing teams. We notice that the top
performers (in absolute scale) for the winning team always outperform the top
performers of the losing team. On the other hand, when we look at the relative
contributions, top batsmen from the winning team contribute higher than their
losing counterparts. However, the top bowler from the losing team contributes
more to his team than that of the top bowler from the winning team.