Robust baseball pitch reconstruction using artificial neural networks with noisy data
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Ryan D. DeBoskey
,
Veeraraghava Raju Hasti
and
Venkateswaran Narayanaswamy
Abstract
The use of high-fidelity camera-vision and Doppler tracking systems in Major League Baseball (MLB) has created an influx of advanced analytics that have transformed game and personnel strategy. However, due to the cost and complexity of these systems, advanced statistics are severely limited in most amateur games. In this work, we develop a robust pitch reconstruction methodology using artificial neural networks (ANN) and a three-point reconstruction technique, requiring the knowledge of only three baseball spatiotemporal locations. The ANN models were trained to predict the initial baseball speed and spin rate, which are then used as initial conditions to integrate the full pitch trajectory. We use numerically simulated baseball pitches to train two ANN models, each with clean and noisy training inputs, respectively. We then performed a robustness analysis to test ANN performance with increasingly noisy data to simulate low-fidelity camera tracking systems. Probability distributions of predicted model output are calculated using the Monte Carlo method to quantify model uncertainty. We show that the ANN models accurately predict the true baseball trajectory from both high-fidelity and noise-injected testing data. The results demonstrate the effectiveness of ANN models for quick and robust pitch trajectory reconstruction using minimal input data, even in the presence of noise. The present methodology provides a first step towards enabling pitch tracking and advanced analytics in a wider variety of baseball games.
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Research ethics: Not applicable.
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Informed consent: Not applicable as there were no human subjects and IRB approval is not applicable.
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Author contributions: R. DeBoskey – Conceptualization, Methodology, Software, Visualization, Writing – original draft, Writing – reviewing and editing. V. Hasti – Conceptualization, Methodology, Software, Resources, Supervision, Writing – reviewing and editing. V. Narayanaswamy – Supervision, Resources, Writing – reviewing and editing.
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Use of Large Language Models, AI and Machine Learning Tools: None declared.
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Conflict of interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Research funding: This research did not receive any specific grants from funding agencies in the public, commercial, or not-for-profit sectors. Ryan DeBoskey was supported by the National Defense Science and Engineering Graduate Fellowship.
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Data availability: Data will be made available upon reasonable request.
Appendix A: Baseball trajectory model
Equation (1) is the governing equation for baseball trajectory, derived from Newton’s second law of motion. The equation is a simplified version derived from Aguirre-López et al. (2018), which assumed the centrifugal and Coriolis body forces are both negligible. The centrifugal and Coriolis forces are approximately two orders of magnitude smaller than the magnus force and are generally negligible to trajectory calculation (Robinson and Robinson 2017). The drag force, f D , was given by Giordano (1996):
where v is the velocity vector, v is the velocity magnitude, and v d = 35 m/s and Δ = 5 m/s are calibrated model constants. The magnus force, f M , is present and plays a prominent effect in the physics of ball trajectory in many sports (Kray et al. 2014; Sayers and Hill 1999). The force is oriented orthogonal to the spin axis and velocity, resulting in the well-known curving of baseball pitches. By Bernoulli’s principle, the net force was given by the cross product (×) of velocity about the spin axis:
where B = 0.00041 is a non-dimensional scaling factor given by Giordano (1996) for baseballs. We assumed a constant acceleration due to gravity, g = −9.81 m/s2, in the z-direction. We solved Equation (1) using 4th-order Runge–Kutta time integration (Lomax et al. 2001) with a fixed time step, Δt = 2.083 ms.
Several simplifying assumptions were made in the current work. First, we neglected the orientation of the baseball seams. Extremely complicated physics, including the seam-shifted wake phenomenon (Smith and Smith 2021), arise from the transition and break-up of the boundary that develops over the ball due to the presence and orientation of the seams on an otherwise smooth sphere (Higuchi and Kiura 2012). This results in small fluctuations to the baseball trajectory, which may not be negligible compared to magnus forces (Borg and Morrissey 2014; Higuchi and Kiura 2012; Smith and Smith 2021).
Secondly, the batch simulations assume a fixed spin axis, ϕ = 45°, fixed launch angle α = 1°, and constant release point, x o = (0, 0, 0). In addition to the speed and spin rate, the spin axis is very important for pitch classification (the constant angles of ϕ = 45° and α = 1° used in the present study correspond roughly with a curveball from a right-hand pitcher). The magnus force is directed orthogonal to the spin axis and velocity (via Eq. (10)) which causes the breaking direction of the pitch. With differing amounts of break and velocity, the pitcher must adjust the launch angle accordingly between pitches. The fixed angles and constant release point were used in the present work to reduce dimensionality to demonstrate the use of an ANN as a proof-of-concept for pitch reconstruction. Higher dimensional prediction output (necessary to predict spin axis, launch angle, and initial condition) requires a much larger ANN model and substantially more training data (curse of dimensionality), which is outside the scope of the present work. The spatiotemporal release point of the baseball must be known to determine the relative deflection of the pitch. We assumed a known release point in this paper, which in practice must be determined by the system to fully reconstruct trajectory. This introduces an additional data point (and associated uncertainty) to collect and process within a real-world pitch tracking system.
Lastly, we sampled the training data for the ANN at a fixed constant interval (t1 = 0.467 s, t2 = 0.483, and t3 = 0.5s). The selection of time interval in the present work was arbitrary, where t3 is the (approximate) time required for an 80 mph pitch to reach home plate without consideration of the pitcher’s extension from the rubber. Sampling at the same three time points (relative to release) cannot be realized in practice, particularly for a low-cost vision tracking system. The relative time from release would need to be varied for a real-world system, providing another layer of complexity.
Appendix B: ANN sensitivity study
We performed a sensitivity study to determine the training batch size, activation function for the hidden layers, and number of hidden layers to be used by the final model. Table 8 summarizes the range of hyperparameters tested in this sensitivity study. We used minimization of the mean-squared training and testing loss as the parameter to evaluate the ANN performance. The sensitivity study was performed only on the Quiet ANN model, as a larger number of training epochs was required to train this model. The decrease in time required to train the Noisy ANN model was indicative of a less complex mapping due to lower-fidelity input data.
Summary of ANN hyperparameter sensitivity study.
| Hyperparameter | A | B | C | D |
|---|---|---|---|---|
| Batch size | 10 | 20 | 30 | 40 |
| Activation function | sigmoid | tanh | relu | leakyrelu |
| Hidden layers | 1 | 2 | 3 | 4 |
Figures 13–15 show the results of the sensitivity analysis for batch size, activation function, and number of hidden layers, respectively. For all plots, the logarithm of the testing and training loss was taken to better visualize the relative differences between model parameters. The ANN training time was largely insensitive to batch size. Although the loss decreases significantly with fewer epochs, a smaller batch size required more computation as the network parameters were updated more frequently. A batch size of 20 is taken in the final model architecture, as a compromise between number of epochs and speed of training. The ANN was insensitive to activation function and we used the ReLu activation function, R(ζ), in the hidden layers, represented mathematically as:
Sensitivity study on batch size showing its effect on the training and testing MSE loss (Eq. (3)) versus epoch for the Quiet ANN model.
Sensitivity study on activation function showing its effect on the training and testing MSE loss (Eq. (3)) versus epoch for the Quiet ANN model.
Sensitivity study on number of hidden layers showing its effect on the training and testing MSE loss (Eq. (3)) versus epoch for the Quiet ANN model.
The ANN requires 2 hidden layers or greater to accurately model the nonlinear relationship of velocity and spin rate. Once the model is larger than 2 hidden layers it was relatively insensitive to an increasing number of layers. To ensure that the model was able to capture the nonlinear trends, the final model architecture contains 3 hidden layers.
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