JP2018198870A - Evaluation device, evaluation method, program, and information recording media - Google Patents

Abstract

A score probability is estimated from a position time series of players and objects belonging to two teams participating in a team sport who can obtain points by controlling an object until a score condition is satisfied. In an evaluation apparatus, a distance unit is configured to select an attacker belonging to a desired team from an attack segment consisting of a position time series while the desired team controls the object, and the other team. Calculate multiple types of distance time series with the defenders belonging to the team. The decomposition unit 103 decomposes the calculated plural types of distance time series into feature vectors. The learning unit 104 learns to estimate the score probability from the feature vector for the attack segment using whether or not the score is obtained in the attack segment as teacher data. [Selection] Figure 1

Description

  The present invention is suitable for evaluating a player's formation in team sports in which points are obtained by controlling an object until a scoring condition is satisfied, such as a goal-type collective ball game in which points are obtained by shooting a ball on a goal. The present invention relates to an evaluation apparatus, evaluation method, program, and information recording medium.

  Traditionally, in team sports where points are obtained by controlling objects until the scoring conditions are met, such as a ball-type collective ball game where points can be obtained by shooting the ball to the goal, player formation can be analyzed Widely done.

  Such team sports include basketball, soccer, rugby, American football, hockey, handball, etc., which employs a ball as an object, and ice hockey, which employs a pack as an object.

  Especially in the professional sports industry, it is systematized to analyze the movements of the own team and opponent teams, and evaluate and plan tactics.

  Due to the development of measurement technology, a series of systems has been developed from collecting the time series of the positions of each player and object from the progress of a game shot with a video camera to distributing various information. Coaches of each team can use this system to analyze formation and make plans based on their own experiences.

  Further, a technique for roughly analyzing the movement of the entire team by applying a machine learning technique used in natural language processing or the like has been proposed (Non-patent Documents 1 and 2).

  Non-Patent Document 1 discloses a technique for classifying attack formations by inputting optical flow images of the trajectories of all players into a recursive neural network (RNN).

  In Non-Patent Document 2, applying the latent Dirichlet distribution method (LDA), which is an unsupervised learning method, using a set of trajectory data of individual players of a certain length as a unit and not considering the arrangement, partially The attack formation was classified.

On the other hand, the inventor of the present application analyzed the positional relationship between players in evaluating whether or not the attacking team controlling the ball in the basketball can satisfy the scoring conditions and obtain points. As a result, at least
(1) The distance (ball-mark distance) between the main attacker closest to the ball among the players belonging to the attacking team and the defender closest to the main attacker among the players belonging to the defensive team,
(2) The distance (ball-help distance) between the main attacker and the defender closest to the main attacker among the players belonging to the defending team,
(3) The maximum distance between each of the sub-attacks other than the main attacker among the players belonging to the attacking team and the defender closest to each of the sub-attackers among the players belonging to the defending team ( (Path-mark distance)
The three distances have been found to play an important role in obtaining points. The accuracy rate when estimating the success or failure of whether or not points are obtained based on these parameters was about 54% (Non-patent Document 3).

  In addition, the inventor of the present application analyzes the global behavior mode of a nonlinear dynamic system by analyzing the spectrum of the Koopman operator, which is an infinite dimensional linear operator on an observable object. A dynamic mode decomposition algorithm was proposed (Non-patent Document 4). Based on this algorithm, if dynamic mode decomposition by kernel method is used, for a small number of time series obtained from a system with nonlinear latent dynamics, even if there is no knowledge of the explicit governing equation of the system, Modes with strictly dynamic meaning can be defined.

K. C. Wang, and R. Zemel.Classifying NBA offensive plays using neural networks.MIT Sloan Sports Analytics Conference, 2016 A. C. Miller, and L. Bornn.Possession Sketches: Mapping NBA Strategies.MIT Sloan Sports Analytics Conference, March 3, 2017 K. Fujii, K. Yokoyama, T. Koyama, A. Rikukawa, H. Yamada and Y. Yamamoto. Resilient help to switch and overlap hierarchical subsystems in a small human group.Scientific reports 6: 23911, April 5, 2016 Y. Kawahara.Dynamic Mode Decomposition with Reproducing Kernels for Koopman Spectral Analysis.Advances in neural information processing systems, pp. 911-919, 2016

  However, since the technique according to Non-Patent Document 1 processes the trajectories of all the players together, the interaction between the players cannot be expressed effectively. In sports, even if the player follows the same trajectory, it is important which moves first.

  On the other hand, since the technique according to Non-Patent Document 2 does not consider the time series between certain operation units, it cannot express the spatiotemporal interaction important for team play.

  Furthermore, with these methods, it has not been possible to quantitatively analyze how much the movement of the players in the team contributes to the success of the shot.

  Therefore, the player's interaction by estimating the score probability from the time series of the players and objects belonging to the two teams participating in the team sport who can get points by controlling the object until the scoring condition is satisfied There is a need for a technique for quantitatively analyzing the above.

  The present invention is to solve the above-mentioned problem, from the position time series of players and objects belonging to two teams participating in a team sport in which points are obtained by controlling the object until the scoring condition is satisfied, It is an object of the present invention to provide an evaluation device, an evaluation method, a program, and a non-transitory computer-readable information recording medium suitable for estimating a score probability.

The evaluation apparatus according to the present invention is
Of the two teams participating in team sports where points are obtained by controlling the object until the scoring condition is satisfied, the player and the object belonging to the two teams while the desired team is controlling the object A plurality of types of distance time series between the attacker belonging to the desired team and the defender belonging to the other team,
Decomposing the plurality of calculated distance time series into feature vectors,
Learning to estimate the score probability from the feature vector for the attack segment is performed using whether or not the score is obtained in the attack segment as teacher data.

  The program according to the present invention includes a calculation code, a decomposition code, and a learning code for causing the processor to execute the calculation, the decomposition, and the learning in a computer having a processor and a memory.

  The program can be distributed, sold, etc., recorded on a non-transitory computer-readable information recording medium.

  The program can be distributed, sold, etc. via a temporary transmission medium such as a computer communication network.

  According to the present invention, it is suitable for estimating the score probability from the position time series of players and objects belonging to two teams participating in team sports who can obtain points by controlling the object until the score condition is satisfied. An evaluation apparatus, an evaluation method, a program, and a non-transitory computer-readable information recording medium can be provided.

It is explanatory drawing which shows schematic structure of the evaluation apparatus which concerns on this embodiment. It is a flowchart which shows the flow of control of the evaluation process performed by the evaluation apparatus which concerns on this embodiment. It is explanatory drawing showing the error rate when a naive Bayes classifier is employ | adopted. It is explanatory drawing showing the error rate at the time of employ | adopting RVM. It is explanatory drawing which shows distribution of the feature vector at the time of employ | adopting various distances and positions.

  Hereinafter, embodiments of the present invention will be described. In addition, this embodiment is for description and does not limit the scope of the present invention. Accordingly, those skilled in the art can employ an embodiment in which each element or all elements of the present embodiment are replaced with equivalent ones. In addition, the elements described in each embodiment can be omitted as appropriate according to the application. As described above, any embodiment configured according to the principle of the present invention is included in the scope of the present invention.

  Further, in the following description, a ball-type collective ball game in which points are obtained by shooting a ball to a goal, particularly basketball, is taken as an example of analysis, but as described above, as an object of the present embodiment, scoring conditions Any team sport where points can be obtained by controlling the object until is satisfied can be employed.

  For example, in rugby and American football, a touchdown is achieved by bringing the ball into the enemy's end zone and points are earned. In ice hockey, points are scored by shooting a puck at a goal instead of a ball.

(Overview configuration)
FIG. 1 is an explanatory diagram showing a schematic configuration of the evaluation apparatus according to the present embodiment. Hereinafter, a description will be given with reference to FIG.

  The evaluation apparatus 101 according to the present embodiment includes a distance unit 102, a decomposition unit 103, and a learning unit 104. Further, it may be configured to further include a dividing unit 105 that executes data preprocessing.

  These units are realized by executing predetermined programs on a computer. The program includes a code corresponding to each unit for causing the processor of the computer to execute each function. The program is loaded from the information storage medium into a memory functioning as a temporary storage area in the computer, and the processor reads a code included in the loaded program and executes a process associated with the code.

  Note that a program such as an FPGA (Field Programmable Gate Array) or an ASIC (Application Specific Integrated Circuit) can be used as a design drawing of an electronic circuit. In this aspect, the evaluation apparatus 101 is realized by hardware called an electronic circuit configured based on the program.

  FIG. 2 is a flowchart showing the flow of control of the evaluation process executed by the evaluation apparatus according to the present embodiment. Hereinafter, a description will be given with reference to FIG. In the following, processing by the optional dividing unit 105 is also described. Furthermore, in order to facilitate understanding, examples of experiments in which the present invention is applied to basketball and results thereof will be referred to as appropriate.

  First, the evaluation apparatus 101 accepts as input a position time series representing how the positions of all players and objects belonging to the two teams in the match have changed during the match (step S201). In basketball, a game is played with five to five players, so that the ball position time series and the ten player position time series are accepted.

  In this experimental example, the camera system SportVU is used to photograph the actual international game of basketball, and the position time series obtained by measuring the position data of all players and the ball in units of 25 Hz is used.

  Next, in the evaluation apparatus 101, the dividing unit 105 divides the received position time series into an attack segment and a defensive segment depending on whether or not a desired team is controlling the object (step S202). . Typically, the elapsed time of the match is the time zone during which the desired team is attacking (attack time zone), the time zone during which the opponent team is attacking (defense time zone), and It is divided into other time zones (time zone for switching between defense and attack, etc.), and only the part belonging to the attack time zone is extracted from the position time series.

  The obtained attack segment can be appropriately selected according to the rules of team sports. In the case of basketball, while the team has a ball possession, it is considered to be an attack segment, but if the ball is stolen by the defending team without trying to shoot, or if the attack ends due to a foul, It may be excluded from the attack segment.

  In this experimental example, the length of time from when all the players of both teams stepped into the attacking half-court with ball possession until the attacking team hits the shot while maintaining the ball possession. Those with 5 seconds or more were designated as attack time zones.

  In addition, in the case of offense rebound in which the attacking player has shot the team but the shot has been missed and the attacking player has held the removed ball again, it is determined that the attack time period is continuing.

  This is because the offense rebound situation is likely to be created by the attacking team and the defensive team in many cases.

  As described above, when a ball is stolen by a defensive team or when an attack ends due to a foul, it is not handled as an attack time zone and is not subject to evaluation.

  In basketball, there is a rule called shot clock that you must shoot within 24 seconds after having a ball possession, but since the attack time zone was defined as above, the time length of the attack segment is May exceed 24 seconds. In this experiment example, the duration of the attack time zone was 5 seconds or more and 44.5 seconds or less.

  As for the method of dividing the accepted position time series into attack segments, various automatic divisions can be made by using the technology disclosed in Non-Patent Documents 1 and 2 and the modified technology according to the rules of team sports. Can be applied.

  In this way, an attack segment representing the position time series of each player and the object while the desired team controls the object and makes an attack is obtained.

  Next, in the evaluation apparatus 101, the distance unit 102 includes a plurality of types of distance time series between an attacker belonging to a desired team and a defender belonging to the other team from a plurality of position time series forming each attack segment. Is calculated (step S203).

  The inventors have determined that the success or failure of the attack is determined by the distance between the player (attacker) belonging to the attacking team, the player (defender) belonging to the defending team, and the object in the positional relationship of each player and object in each attack segment I have found that it has contributed greatly. In this embodiment, the following four types are adopted as such an attacker-defender distance.

  (1) Ball-mark distance. This is the distance between the main attacker who is closest to the ball among the players belonging to the attacking team and the defender who is closest to the main attacker among the players belonging to the defensive team. , Considered the most important element.

  (2) Ball-help distance. This is the distance between the main attacker and the defender closest to the main attacker among the players belonging to the defensive team. The second closest defender to the main attacker is to help the defender who is closest to the main attacker not to be broken by the main attacker or to be prepared in case it is broken.

  (3) Pass-mark distance. This is the maximum distance between each of the sub attackers other than the main attacker among the players belonging to the attacking team and each of the defenders closest to each of the sub attackers among the players belonging to the defending team. This assumes a secondary attacker who has the highest degree of separating the defender, that is, a secondary attacker who is most likely to be a pass destination.

  (4) Pass-help distance. This is the maximum distance between each attacker and the second closest defender to each attacker. This is based on the assumption that the defender is the least capable of helping the defender closest to the secondary attacker who can be the pass destination.

  The inventors have found that these play a major role in the analysis of various ball games such as basketball. Instead of handling all position time series at once, rearranging the information into the limited types of information described above can speed up the calculation process by removing information that does not contribute much. , Can raise the accuracy of evaluation.

  By considering the time series of the above four types of distances, more accurately evaluate the probability of obtaining points based not only on the player who directly controls the object but also on the movement and interaction between other players. Will be able to.

  The distance unit 102 may obtain a distance time series other than the above four types according to the type of team sports, or may omit any of the above four types of distance time series.

  The simplest distance between the attacker and the defender is the Euclidean distance between the two, but in the case of basketball, etc., the distance between the attacker and the defender according to the type of team sport It can also be adopted.

  For example, as in Non-Patent Document 3, the following method can be used to calculate the distance between the attacker and the defender.

  First, the ideal defensive position for an attacker is defined as follows. That is, assuming a half-line that starts from a basketball ring (goal) and passes through the attacker, a position advanced 0.5 m beyond the attacker is defined as an ideal defense position.

  Then, the Euclidean distance between the ideal defensive position and the actual defensive position is defined as the attacker-defender distance. This can be further corrected as follows.

  For example, an attacker far from the ball is considered to be less likely to shoot.

Therefore, when another attacker Y passes a ball to an attacker X, the time until the attacker X catches the pass is:
(a) The attacker X continues to move at the current speed,
(b) The passed ball keeps moving at a constant speed (for example, 10m / s)
(c) Assuming that the defender P facing the attacker X continues to move toward the ideal defense position for the attacker X at a constant speed (for example, 3 m / s), the future position of the attacker and the defender The distance from the future position of the attacker is the distance between the attacker and the defender. This is called time correction.

Further, according to the distance R between the attacker X and the ring, the space correction of the distance L between the attacker X and the defender P can be performed. For example, the distance L
L '= L × 1, (R ≦ 1.0 [m])
= L × (12.0-R) / (12.0-1.0), (1.0 [m] <R <12.0 [m])
= L × 0, (R ≧ 12.0 [m])
In this way, a spatially corrected value L ′ is obtained. This multiplies the distance L by the success rate p of the virtual shoot according to the Euclidean distance R between the attacker and the ring.
p = 1, (R ≦ 1.0)
= (12.0-R) /11.0, (1.0 <R <12.0)
= 0, (R ≧ 12.0)

  The function for obtaining the virtual shot success rate p is not limited to the above. Any thing can be adopted as long as it gradually decreases according to the distance R.

  Now, if all the distances between 5 attackers and 5 defenders are handled as they are, 25 distance time series can be obtained. Further, if the two-dimensional position of each player is handled as it is, 20 position time series can be obtained. In this experimental example, the above four distance time series are targeted for the following processing by arranging the positional information in consideration of the positional relationship with the object (ball).

  Thus, when a plurality of distance time series are obtained for each attack segment, the decomposing unit 103 of the evaluation apparatus 101 decomposes the plurality of distance time series into feature vectors for each attack segment (step S204). ).

  As a technique for obtaining a feature vector, dynamic mode decomposition using a kernel method can be applied, but other techniques can also be adopted. Below, the process by the dynamic mode decomposition | disassembly using a kernel method is demonstrated.

  In general, in dynamic mode decomposition, an input vector is decomposed into modes in a feature space. Thereby, the feature vector showing the contribution of each mode is obtained for each input vector.

  In the present embodiment, a set of a plurality of distance time series in each attack segment becomes an input vector, and each set is projected onto a feature space to be decomposed into modes. The similarity between each attack segment set and each mode is the feature vector of each attack segment.

  For the dynamic mode decomposition, the technique disclosed in Non-Patent Document 4 can be applied, but in this embodiment, an improvement not disclosed in Non-Patent Document 4 is added. The improvements are also specified below.

In general, in dynamic mode decomposition, there is a discrete-time nonlinear dynamical system that exists potentially behind a state vector x ₁ , x ₂ , x ₃ ,…
x _{t + 1} = f (x _i )
think of. Here, x ₁ , x ₂ , x ₃ ,... Are state vectors in a state space M⊂R ^{d having} real number d dimensions, and f is a state transition function assuming a nonlinear dynamic system.

Here, consider the Koopman operator K acting on the scalar functions g ₁ , g ₂ , g ₃ ,... That project the state space M to the complex number C. The Koopman operator K is an infinite dimensional linear operator. With the Koopman operator K, the scalar function g _{i is}
(K g _i ) (x) = g _i ○ f (x)
To the new function K g _i . Here, ○ is an operator that composes two functions.

Since the Koopman operator K is a linear operator, eigenvalue decomposition is possible.
(K φ _j ) (x) = λ _j φ _j (x), (j = 1, 2, 3,…)

Here, λ _j ∈C is a j-th Koopman eigenvalue, and φ _j is a Koopman eigenfunction corresponding to the eigenvalue λ _j .

A function vector that concatenates functions g ₁ , g ₂ , g ₃ ,…, g _p vertically
g = [g ₁ , g ₂ , g ₃ ,…, g _p ] ^T
When g ₁ , g ₂ , g ₃ ,…, g _p are in the space spanned by the eigenfunction φ _j , the vector value g (x) can be expanded as follows: it can.
g (x) = Σ _{j = 1} ^∞ φ _j (x) ψ _j

Here, ψ ₁ , ψ ₂ , ψ ₃ ,... Are vector coefficients called Coupman modes. From the above results, the following holds.
g ○ f (x) = Σ _{j = 1} ^∞ λ _j φ _j (x) ψ _j

That is, the lambda _j, a parameter characterizing the time evolution of the corresponding Koop engineer mode [psi _j, lambda _j of the phase determines the frequency, the amplitude of the lambda _j determines the growth rate dynamics.

Dynamic mode decomposition is a finite-length observation data string
y ₀ , y ₁ , y ₂ ,…, y _τ ∈ R ^P ;
And a data matrix with this
Y = [y ₀ , y ₁ , y ₂ ,…, y _τ ]
When,
y _t = g (x _t ), (t = 0, 1, 2,…, τ)
From this, it is a general approach for estimating λ _j and ψ _j .

Here, a matrix obtained by arranging the observation data strings shifted by one
A = [y ₀ , y ₁ , y ₂ ,…, y _τ-1 ],
B = [y ₁ , y ₂ , y ₃ ,…, y _τ ],
And a matrix defined by B and a pseudo inverse matrix A ^† of A
P = BA ^†
think of.

In basic dynamic mode decomposition,
(1 / τ) Σ _{t = 0} ^τ-1 | y _{t + 1} -P y _t | ²
Is obtained by eigenvalue decomposition to estimate λ _j and ψ _j .

In order to calculate this, a gram matrix is created in this embodiment. First, the gram matrix G _xx of the Gaussian kernel k (y _i , y _k ) at y _i and y _j of the matrix A is calculated. As the variance parameter of the Gaussian distribution, the median value of the distance between samples of the data matrix Y is adopted.

Similarly, a gram matrix G _xy of a Gaussian kernel in each time vector of the matrices A and B is calculated.

At this time, the feature map from the state space M to the regenerated kernel Hilbert space H is φ,
M _τ = [φ (x ₀ ), φ (x ₁ ), φ (x ₂ ),…, φ (x _τ-1 )],
M ₊ = [φ (x ₁ ), φ (x ₂ ), φ (x ₃ ),…, φ (x _τ )]
If you write Hermitian transpose of matrix X as X ^* ,
G _xx = M _τ ^* M _τ ,
G _xy = M _τ ^* M ₊
It becomes.

The Koopman operator K _H to the feature map φ is
(K _H φ) (x) = φ ○ f (x)
It is defined as Practically, φ need not exist in the reproduction kernel Hilbert space, but it is sufficient to simply assume that φ exists in the Hilbert space.

  Based on the above assumption, dynamic mode decomposition is made robust by projecting data in the proper orthogonal decomposition (POD) direction.

  First, using the matrix H obtained by subtracting 1 / τ from each element of the unit matrix I, the Gram matrix G is converted into a centralized Gram matrix.

The centering of a matrix may be described by drawing a line on the matrix name as described above. However, in this application, the character “ ^c ” is added to the right side of the matrix name as follows as appropriate. write. That is,
G ^c = HGH

Assuming that the eigenvalues and eigenvectors can be truncated based on the magnitude of the eigenvalues, adopt p (p ≦ τ) eigenvalues,
G ^c ≒ V ^c S ^c V ^{c *}
And consider the matrices V ^c , S ^c .

Then, the main orthogonal direction in the feature space is β _{j in the} matrix V ^c .
ν _j = M _τ H (S ^c ) ^-1/2 _jj β _j
Can be expressed as

further,
U = [ν ₁ , ν ₂ ,…, ν _p ] = M _τ HV ^c (S ^c ) ^-1/2
After all,
M ₊ = K _H M _τ
The projection onto the space spanned by ν ₁ , ν ₂ ,…, ν _p is
F = U ^* K _H U
= (S ^c ) ^-1/2 V ^{c *} HM _τ M ₊ HV ^c (S ^c ) ^-1/2
Is obtained as follows.

here,
G _xy = M _τ ^* M ₊
So F above
F = T ^-1 Λ T
The eigenvalue decomposition is performed as follows. The matrix Λ is a diagonal matrix composed of Koopman eigenvalues and represents time evolution.

Then the centralized Koopman mode writes the _jth row of the matrix T ^-1 as b _j ,
ψ _j = U b _j = M _τ HV ^c (S ^c ) ^-1/2 b _j
Is obtained as follows.

In the present embodiment, since it is possible to obtain a feature vector is an object, the matrix U representing the principal component direction of the kernel, its components ν _1, ν _2, ..., to the space spanned in [nu _p the projection F of K _H, utilized in the creation of the kernel.

  Hereinafter, a method for obtaining the similarity in the feature space between attack segments will be described. The kernel obtained here reflects the Koopman eigenvalue, the Koopman eigenvector, and the Koopman mode obtained by spectrally decomposing the Koopman operator, and can be called a Koopman spectrum kernel.

  Such a kernel can be calculated from the trace of the synthesis matrix of order q consisting of the system trajectory. In this experimental example, three types of kernels are used: the Koopman trace kernel, the Koopman determinant kernel, and the Koopman principal component angle kernel. doing.

  Here, the Koopman trace kernel is a kernel when the order q = 1, and directly reflects the property of time evolution of the dynamical system.

  On the other hand, the Koopman determinant kernel is a kernel when the order q is equal to the dimensionality of the dynamic system, and the characteristics of the dynamic system are more reflected than the Koopman trace kernel.

  The Koopman principal component angle kernel is a special case of the Koopman trace kernel, and can be easily calculated without reflecting the Koopman eigenvalue.

  In any case, it is a feature of the present invention to compare arbitrary nonlinear dynamic systems without assuming a special model in the background. Hereinafter, each kernel will be described in detail.

Now, after the dynamic mode decomposition, the trace kernels of the dynamic systems DS _i and DS _j can be defined as follows.
k (DS _i , DS _j ) = Σ _{t = 0} ^∞ e ^-κt g _i (x _{i, t} ) ^T W g _j (x _{j, t} )
= φ _i (x _{i, 0} ) ^T Σ _{t = 0} ^∞ e ^-κt Λ _i ^t Φ _i ^T W Φ _j Λ _j ^t (x _{j, 0} )

here,
g _i is an observation function, and Λ _i is a diagonal matrix in which Koopman eigenvalues λ ₀ , λ ₁ , λ ₂ ,…, λ _p for the i-th attack segment are arranged in diagonal elements,
Φ _i is a matrix [φ ₀ , φ ₁ , φ ₂ ,…, φ _p ] in which Coupman modes φ ₀ , φ ₁ , φ ₂ ,…, φ _p for the i-th attack segment are arranged,
W is an arbitrary semi-definite matrix including a unit matrix and the like.

Also, in order to converge the above equation, the exponent discount μ (t) = e ^-κt , (k> 0)
Is multiplied.

Since the above equation is an infinite sum, direct calculation is difficult, but as shown in Non-Patent Document 4, the matrix
M = Σ _{t = 0} ^∞ e ^-κt Λ _i ^t Φ _i ^T W Φ _j Λ _j ^t
For the Sylvester equation
M = e ^-κ Λ _i ^T M Λ _j + Φ _i ^T W Φ _j
Against
Φ = U b _j = M _τ HV ^c (S ^c ) ^-1/2 T ^-1
By substituting and solving, the solution M is obtained in closed form.

Then, the trace kernel
k (DS _i , DS _j ) = φ _i (x _{i, 0} ) ^T M φ _j (x _{j, 0} )
It is determined as follows.

If we denote the left eigenvector of F as a and the column vector consisting of the first column of the matrix M _t as M _{τ, 0} ,
φ (x ₀ ) = a ^* (M _τ HV ^c (S ^c ) ^-1/2 ) ^* M _{τ, 0}
Can be used to calculate the trace kernel for each i and j.

In order to create an independent trace kernel from the initial values φ _i (x _{i, 0} ) and φ _j (x _{j, 0} ) of the latent dynamics, the expected values of both may be used. That is,
k (DS ₁ , DS ₂ ) = E {x _{i, 0} , x _{j, 0} } φ _i (x _{i, 0} ) ^T M φ _j (x _{j, 0} )
= tr (σ {φ _i (x _{i, 0} ), φ _j (x _{j, 0} )} M)
Here, σ {φ _i (x _{i, 0} ), φ _j (x _{j, 0} )} is the eigenvalue of all initial values φ (x ₀ ) of i, j = 1, 2,. This is a matrix with variance calculated for each index 1, 2,…, p.

Next, the Koopman determinant kernel will be described. As with the Koopman determinant kernel,
k (DS _i , DS _j ) = det (Σ _{t = 0} ^∞ g _i (x _{i, t} ) g _j (x _{j, t} ) ^T )
= det (Σ _{t = 0} ^∞ e ^-κt Φ _i Λ _i ^t φ _i (x _{i, 0} ) φ _j (x _{j, 0} ) ^T Λ _j ^t Φ _j ^T )
= det [Φ _i M Φ _j ],
M = e ^-κ Λ _i ^T M Λ _j + φ _i (x _{i, 0} ) φ _j (x _{j, 0} )
It can be asked. Here, in consideration of the generality of the discussion, a unit matrix is adopted as the matrix W.

Note that the determinant kernel independent of the initial value is multilinear and depends on the covariance matrix σ {φ _i (x _{i, 0} ), φ _j (x _{j, 0} )} as well as higher-order statistics Therefore, it cannot be calculated except in special cases such as a single output system.

Finally, the Koopman principal component angle kernel will be described. First, for DS _i , the inner product of the linear subspace in the feature space
A ^* A = T _i ^-1 U _i ^* G _xxi U _i T _i
think of. When the rank of the matrix F a (rank) and r _i, A ^* A is the r _i order square matrix.

A similar matrix B ^* B is _created for DS _j .

Furthermore, as the inner product of the linear subspace of DS _i and DS _j ,
A ^* B = T _i ^-1 U _i ^* G _xxij U _j T _j
think of. Here, G _xxij is _centered on the data _obtained by connecting observation data Y _i (distance time series of i-th attack segment) and Y _j (distance time series of j-th attack segment) in series in time series. This is a matrix obtained by cutting the upper right part of the centralized gram matrix so as to be n _i rows and n _j columns after creating the grammar matrix. Here, n _i is the time series length of Y _i , and n _j is the time series length of Y _j .

  Then, using these matrices, the following generalized eigenvalue problem is solved.

It should be noted that the size of the generalized eigenvalue λ _ij is finally adjusted so that r _ij = min (r _i , r _j ) from the largest.

Then, the principal component angle kernel is obtained by the obtained generalized eigenvalue λ _ij .
k (DS _i , DS _j ) = λ _ij
It can be expressed as

Now, when various kernels are obtained in this way, the matrix
X _{i, j} = k (Y _i , Y _j )
Can be determined. Of this matrix,
X _i = [x _{i, 1} , x _{i, 2} ,…, x _{i, n} ]
Is used as the feature vector in the i-th attack segment.

  When the feature vectors in each attack segment are obtained in this way, the learning unit 104 of the evaluation apparatus 101 scores from the feature vectors for each attack segment using teacher data as to whether or not points have been obtained in each attack segment. Learning for estimating the probability is performed (step S205).

  Various supervised machine learning techniques can be employed for learning. Note that, when predicting the success or failure of a score based on the movement of the player, it is desirable to calculate the posterior probability for the prediction rather than the identification result because the ability of each player is also related, for example, the accuracy of the shot. Therefore, in this embodiment, two types of naive Bayes classifiers and related vector machines are employed.

First, the naive Bayes classifier simply assumes that the feature vectors are independent of each other and uses the Bayes theorem.
P (Y | X) = P (Y) P (X | Y) / P (X)
Is a classifier that applies

  In this embodiment, Y corresponds to a predictor (score success / failure), and X corresponds to a feature vector. In the learning by the naive Bayes classifier, learning for predicting the probability of Y when a new (unknown) feature vector X is given is performed.

  In the calculation of the right-hand side, it is sufficient to obtain Y having the highest likelihood for a certain X. At this time, since X is a constant, P (Y) P (X | Y) may be obtained.

  P (Y) is a prior distribution, and P (X | Y) is a conditional probability (likelihood).

In the naive Bayes classifier, with respect to the conditional probability P (X | Y),
P (X | Y) = Π _{i = 1} ^N P (x _i | Y)
Assume a simple conditional independence. That is, it is assumed that each feature variable x _i is conditional and independent of other feature variables x _j (where i ≠ j).

In this example, it is considered that P (x _i | Y) follows the following multivariate Gaussian distribution.

However, μ _Y and σ ² _Y are the mean value and variance, which are parameters of the Gaussian distribution, and it is necessary to estimate them.

In this embodiment, since the prior distribution P (Y) is assumed to be a uniform distribution (constant),
Π P (x _i Y)
It is necessary to calculate μ _Y and σ ² _Y when is the maximum.

  The objective function M to be maximized can be expressed as follows.

Here, s _kij is the j-th element of the i-th input data of the correct data where Y = k.

  A necessary condition for the objective function M to be maximized is that the partial derivative of M with respect to μ and σ is zero. If it calculates based on this, a solution will be obtained as follows.

This makes it possible to predict the most probable Y value for the new x _i .

  The simple Bayes classification constructed in this way is very simple in design and assumption, but often works with sufficient accuracy in complex real-world situations. Furthermore, there is an advantage that less training example data is required to estimate parameters (means and variances of variable groups) indispensable for classification.

  Next, a relation vector machine (RVM) will be described. RVM has the advantage of being able to calculate posterior probabilities for prediction, which is a requirement in this analysis, while taking over many of SVM (Support Vector Machine) widely used for identification problems. In addition, cross-validation is required in SVM, but RVM is not required, so RVM can perform faster prediction.

In this example,
y (x, w) = σ (w ^T φ (x))
As described above, in order to use a linear combination of basis functions converted by a logistic sigmoid function σ, it is necessary to determine a weight parameter w and a superparameter α related to the accuracy. Therefore, first, α is initialized.

  Then, the posterior distribution is approximated by a Gaussian distribution, and the marginalized likelihood is calculated using this. Then, α is updated so as to maximize the obtained marginalization likelihood.

  These processes are repeated until an appropriate convergence condition is satisfied.

According to Laplace approximation, the mean and covariance matrix of the approximate posterior distribution of α is
w ^* = A ^-1 Φ ^T (ty),
Σ = (Φ ^T B Φ + A) ^-1
Can be obtained as follows.

here,
A is a diagonal matrix in which the elements α ₁ , α ₂ ,.
t is a target variable that takes a binary value of 0 or 1.
B is an N × N diagonal matrix whose n-th diagonal component is given by y _n (1 − y _n ),
Φ is a design matrix in which each element is φ (x _i ).

The marginal likelihood is calculated using the obtained Laplace approximation, and the derivative of the log marginalized likelihood with respect to α _i is 0.
α _i ^new = γ _i / (ω _i ^* ) ²
Is obtained. However,
γ _i = 1-α _i Σ _ii
It is.

The approximate expression of the log marginalized likelihood that is the objective function is
ln (p | α, β) =-(1/2) [N ln (2π) + ln | C | + h (t) ^ (T) C ^-1 h (t)],
h (t) = Φ w ^* + B ^-1 (t-y),
C = B + Φ A Φ ^T
It can be expressed.

  If the super parameter α is updated so as to maximize this objective function, the corresponding weight parameter w is determined.

  In this way, when the learning is finished, this processing is finished. After learning is completed, the distance time series of the unknown attack segment (attack segment used for learning) or unknown (attack segment used for learning) as an input vector to the classifier or vector machine obtained from the learning result If the feature vector is given, the probability that a point is obtained in the attack segment can be output.

(Experimental result)
If the estimated score probability exceeds 50%, the score is obtained if the estimated score probability exceeds 50%, and if it is less than 50%, a score is obtained. The correct answer rate was obtained by experiments, assuming that no answer was correct. FIG. 3 is an explanatory diagram showing the error rate when the naive Bayes classifier is employed. FIG. 4 is an explanatory diagram showing the error rate when RVM is employed. Hereinafter, description will be given with reference to these drawings.

  The vertical axis of the graphs in both figures indicates the incorrect answer rate, and the value obtained by subtracting this from 1 is the correct answer rate.

The horizontal axis represents the type of input vector.
Use only (1) of the attacker-defense distance, use (1)-(2), use (1)-(3), use (1)-(4), etc. The ones increased in order correspond to the horizontal axis "1""2""3""4"
The total distance between players (25 distance time series) corresponds to the horizontal axis “25”
The attacker-defender distance (1)-(4) is used, but the Euclidean distance is used as it is without time correction or spatial correction, which corresponds to the horizontal axis `` Euclid ''
The one using all the positions of the player as they are (20 position time series) corresponds to the horizontal axis “xy”.

The □ mark (Ktr) in each graph represents the Koopman trace kernel.
The triangle (Kdet) is the Koopman determinant kernel,
○ mark (Kpa) is each Koopman principal component kernel,
▽ mark (w / o K) is a simple feature vector for comparison.
The case where each is employed and learning is shown is shown.

  According to this figure, when the distance time series of (1)-(4) was adopted by the naive Bayes method, the accuracy rate was 64.1% (error rate 35.9%), and the performance was the best.

  That is, the accuracy rate 53.8% (error rate 46.2) of the technique disclosed in Non-Patent Document 3 (the three types of distance time series (1)-(3) are directly subject to learning and dynamic mode decomposition is not performed). %) Is greatly improved.

  That is, the feature vector obtained by the dynamic mode decomposition considering the latent dynamics shows generally higher performance (including the highest performance) than the simple feature vector not considering the time series.

  From this, it can be seen that the characteristics of the data structure of time series are important when predicting the ease of scoring from the movements of all the players.

  Also, with the input vector given to the dynamic mode decomposition, if only the single attack-defender distance (1) related to the shot or the two-dimensional position of all the players is used, Performance is worse than the case. From the former, it can be seen that the distance between attackers and defenders who are not directly related to the shot is also important. From the latter, it can be seen that the attacker-defender distance contributes more to the score than the player's two-dimensional position itself.

  In the above experimental example, the naive Bayes classifier based on the feature vector using the four types of attacker-defender distance shows the highest performance.

The feature vector obtained here is the main attacker controlling the ball
(1) Shoot,
(2) Dribbling,
(3) Reflects the ease of scoring for yourself and allies when you pass,
(4) It is thought that it reflects the ease of scoring after the pass ally dribbles and removes the defender after passing.

  In this experiment, even if the distances (1)-(3) were adopted, a good percentage of correct answers was obtained, but what kind of distances should be adopted depends on the mode of team sports. Considered different.

  However, in team sports that use objects, attack methods similar to shooting, dribbling, and passing are generally adopted, and the distances (1)-(4) above are useful in many team sports. Attacker-defender distance.

Furthermore, in order to visualize the similarity of feature vectors, a distance matrix is used using a kernel.
dist (DS _i , DS _j ) = k (Y _i , Y _i ) + k (Y _j , Y _j )-2 k (Y _i , Y _j )
Was created, and its distribution was made two-dimensional by the multidimensional scaling method. FIG. 5 is an explanatory diagram showing the distribution of feature vectors when various distances and positions are employed. Hereinafter, a description will be given with reference to FIG.

  Figure a shows the case where the four types of attacker-defender distances (1)-(4) are adopted.Figure b shows the case where the attacker-defender distance of (1) only is adopted. FIG. C shows a two-dimensional distribution of features when two-dimensional position information is used instead of distance.

  As can be seen from these figures, the distribution of the four types of attacker-defender distance is the most widespread and expressive.

(Summary)
As described above, the evaluation apparatus according to the present embodiment is
Of the two teams participating in team sports where points are obtained by controlling the object until the scoring condition is satisfied, the player and the object belonging to the two teams while the desired team is controlling the object A distance unit for calculating a plurality of types of distance time series between an attacker belonging to the desired team and a defender belonging to the other team
A decomposition unit for decomposing the plurality of calculated distance time series into feature vectors;
A learning unit for learning to estimate a score probability from a feature vector for the attack segment is configured by using whether or not a score is obtained in the attack segment as teacher data.

In the evaluation apparatus according to the present embodiment,
The decomposition unit can be configured to obtain the feature vector by performing dynamic mode decomposition of the plurality of types of distance time series by a kernel method.

In the evaluation apparatus according to the present embodiment,
The decomposition unit calculates a Koopman trace kernel or a Koopman determinant kernel by solving a Sylvester equation in which an exponential discount is applied to a Koopman eigenvalue, a Koopman eigenfunction, and a Koopman mode, and the calculated Koopman trace kernel or The Koopman determinant kernel can be configured to be the similarity of the attack segments.

In the evaluation apparatus according to the present embodiment,
The decomposition unit creates a Koopman mode based on a matrix obtained by cutting out the upper right part of the centralized Gram matrix of observation data in which time series are connected in series, and uses a matrix obtained by synthesizing the inner product of the created Koopman mode By solving the generalized eigenvalue problem, the Koopman principal component angle kernel can be calculated, and the calculated Koopman principal component angle kernel can be configured as the similarity of the attack segment.

In the evaluation apparatus according to the present embodiment,
As the plurality of types of distance time series,
(1) A time series of distances between the main attacker closest to the object and the defender closest to the main attacker,
(2) a time series of distances between the main attacker and the defender closest to the main attacker among the defenders;
(3) a time series of maximum values of the distances between each of the sub-attackers other than the main attacker and each of the defensive players closest to the defensive player, and
(4) The time series of the maximum distance between each of the attackers and each of the attackers and the second closest defender among the defenders can be adopted.

In the evaluation apparatus according to the present embodiment,
The learning unit can be configured to learn using a naive Bayes classifier.

In the evaluation apparatus according to the present embodiment,
The learning unit can be configured to learn by a related vector machine.

Moreover, the evaluation apparatus according to the present embodiment is
A division unit that divides a time series of positions of the player and the object belonging to the two teams in the team sport into the attack segment and the defensive segment depending on whether or not the desired team controls the object. Further comprising
The plurality of types of distance time series can be calculated from the divided attack segments.

Further, in the evaluation method according to the present embodiment, the evaluation device is
Of the two teams participating in team sports where points are obtained by controlling the object until the scoring condition is satisfied, the player and the object belonging to the two teams while the desired team is controlling the object A plurality of types of distance time series between the attacker belonging to the desired team and the defender belonging to the other team,
Decomposing the plurality of calculated distance time series into feature vectors,
Learning to estimate the score probability from the feature vector for the attack segment is configured using teacher data as to whether or not points are obtained in the attack segment.

The program according to the present embodiment is stored in a computer.
Of the two teams participating in team sports where points are obtained by controlling the object until the scoring condition is satisfied, the player and the object belonging to the two teams while the desired team is controlling the object A plurality of types of distance time series between the attacker belonging to the desired team and the defender belonging to the other team,
Decomposing the plurality of calculated distance time series into feature vectors,
A process for learning to estimate a score probability from a feature vector for the attack segment is executed using teacher data as to whether or not points have been obtained in the attack segment.

  Further, the program of the invention of the present embodiment can be recorded on a non-transitory computer-readable information recording medium.

  Various embodiments and modifications can be made to the present invention without departing from the broad spirit and scope of the present invention. The above-described embodiments are for explaining the present invention and do not limit the scope of the present invention. In other words, the scope of the present invention is shown not by the embodiments but by the claims. Various modifications within the scope of the claims and within the scope of the equivalent invention are considered to be within the scope of the present invention.

  According to the present invention, it is suitable for estimating the score probability from the position time series of players and objects belonging to two teams participating in team sports who can obtain points by controlling the object until the score condition is satisfied. An evaluation apparatus, an evaluation method, a program, and a non-transitory computer-readable information recording medium can be provided.

101 Evaluation equipment
102 Distance section
103 Disassembly part
104 Learning Department
105 Division

Claims (11)

Of the two teams participating in team sports where points are obtained by controlling the object until the scoring condition is satisfied, the player and the object belonging to the two teams while the desired team is controlling the object A distance unit for calculating a plurality of types of distance time series between an attacker belonging to the desired team and a defender belonging to the other team
A decomposition unit for decomposing the plurality of calculated distance time series into feature vectors;
An evaluation apparatus comprising: a learning unit that learns to estimate a score probability from a feature vector for the attack segment using teacher data as to whether or not a score is obtained in the attack segment. 2. The evaluation device according to claim 1, wherein the decomposition unit obtains the feature vector by performing dynamic mode decomposition on the plurality of types of distance time series by a kernel method. The decomposition unit calculates a Koopman trace kernel or a Koopman determinant kernel by solving a Sylvester equation in which an exponential discount is applied to a Koopman eigenvalue, a Koopman eigenfunction, and a Koopman mode, and the calculated Koopman trace kernel or 3. The evaluation device according to claim 2, wherein the Koopman determinant kernel is set as the similarity of the attack segment. The decomposition unit creates a Koopman mode based on a matrix obtained by cutting out the upper right part of the centralized Gram matrix of observation data in which time series are connected in series, and uses a matrix obtained by synthesizing the inner product of the created Koopman mode 3. The evaluation device according to claim 2, wherein a Koopman principal component angle kernel is calculated by solving a generalized eigenvalue problem, and the calculated Koopman principal component angle kernel is used as the similarity of the attack segment. . As the plurality of types of distance time series,
(1) A time series of distances between the main attacker closest to the object and the defender closest to the main attacker,
(2) a time series of distances between the main attacker and the defender closest to the main attacker among the defenders;
(3) a time series of maximum values of the distances between each of the sub-attackers other than the main attacker and each of the defensive players closest to the defensive player, and
(4) The time series of the maximum values of the distances between each of the attackers and each of the attackers and the second closest defender among the defenders is adopted. Evaluation device. 2. The evaluation device according to claim 1, wherein the learning unit learns using a naive Bayes classifier. 2. The evaluation device according to claim 1, wherein the learning unit learns using a related vector machine. The position time series of the player and the object belonging to the two teams in the team sport is divided into either the attack segment or the defensive segment depending on whether or not the desired team controls the object. Further comprising a dividing section
2. The evaluation apparatus according to claim 1, wherein the plurality of types of distance time series are calculated from the divided attack segments. Evaluation device
Of the two teams participating in team sports where points are obtained by controlling the object until the scoring condition is satisfied, the player and the object belonging to the two teams while the desired team is controlling the object A plurality of types of distance time series between the attacker belonging to the desired team and the defender belonging to the other team,
Decomposing the plurality of calculated distance time series into feature vectors,
A learning method for estimating a score probability from a feature vector for the attack segment using teacher data as to whether or not a score is obtained in the attack segment. On the computer,
Of the two teams participating in team sports where points are obtained by controlling the object until the scoring condition is satisfied, the player and the object belonging to the two teams while the desired team is controlling the object A plurality of types of distance time series between the attacker belonging to the desired team and the defender belonging to the other team,
Decomposing the plurality of calculated distance time series into feature vectors,
A program for executing a process of learning to estimate a score probability from a feature vector for the attack segment using teacher data as to whether or not a score is obtained in the attack segment.   11. A non-transitory computer-readable information recording medium on which the program according to claim 10 is recorded.

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