# Descriptive Parameters and Its Hysteresis of the Group Separation and Recombination in Bicycle Points Races: Leader’s Velocity and Speed Difference between Leader and Main Group

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Participants

#### 2.2. Experimental Setup

#### 2.3. Group Definition

_{draft}, which indicates the ratio of air resistance when traveling alone and in drafting, is the function of distance (d

_{w}) between the rear end of the rear wheel of the front vehicle and the front end of the front wheel of the rear bicycle. Moreover, it is derived that the effect of reducing drag by drafting disappears when d

_{w}becomes larger than 3 m. This study defined that the cyclists belong to different groups when d

_{w}is larger than 3 m. d

_{w}was calculated by multiplying the time difference between the two cyclists, the average speed of the subsequent cyclist in a half lap, and a bike wheelbase of 1.6 m. The average speed of the half lap was calculated by the elapsed time of the half lap (125 m). In some cases, small groups occurred in addition to the leading group and the largest group, but in this study, only the leading group and the main group were accounted for.

#### 2.4. Analysis

#### 2.5. Candidates for Descriptive Parameter

_{leader}), the velocity of the cyclist in front of the main group (V

_{main}), the speed difference between the leader and front of the main group (ΔV = V

_{leader}− V

_{main}), and the time difference between the leader and front of the main group (ΔT) as the candidate descriptive parameters. Here, the main group is the group with the largest number of cyclists among the groups, defined by d

_{w}mentioned above. We analyzed these parameters in each state and the state transition from STR and DIV.

#### 2.6. Statistics

_{p}

^{2}) was calculated. Multivariate analysis of variance (MANOVA) was performed by Wilks’ lambda for V

_{leader}and ΔV comparing STR was changed to DIV, and STR remained, and DIV was changed to STR, and DIV remained. Software R (version 4. 2. 0, R development Core Team, Vienna, Austria) was used for statistical calculations [24].

## 3. Results

#### 3.1. Candidate Descriptive Variables in Each Global State

_{leader}, V

_{main}, ΔV, and ΔT in each state to compare each descriptive candidate variable in the four global states. The mean and standard deviation of V

_{leader}(m/s) in each state were DEN: 13.67 ± 0.73, STR: 14.51 ± 1.28, DIV: 14.43 ± 1.12, and ESC: 13.86 ± 0.71 (Figure 1a). The result of ANOVA show that the effect of the state was statistically significant (F(3, 236) = 9.239, p = 7.84 × 10

^{−6}, η

_{p}

^{2}= 0.106). Tukey’s multiple comparison test shows that V

_{leader}in STR and DIV was significantly higher than V

_{leader}in DEN and ESC (STR vs. DEN: t = 4.200, p = 2.199 × 10

^{−4}, r

^{2}= 0.137, STR vs. ESC: t = 3.567, p = 0.002, r

^{2}= 0.091, DIV vs. DEN: t = 3.861, p = 8.350 × 10

^{−4}, r

^{2}= 0.120, DIV vs. ESC: t = 3.196, p = 0.009, r

^{2}= 0.076).

_{main}, the mean and standard deviation in each state were DEN: 13.67 ± 0.73 (m/s), STR: 14.26 ± 1.39 (m/s), DIV: 14.42 ± 1.11 (m/s), and ESC: 14.18 ± 0.78 (m/s) (Figure 1b). The result of ANOVA shows that the effect of the state was significant (F(3, 236) = 4.478, p = 0.004, η

_{p}

^{2}= 0.054). The multiple comparisons showed a significant difference between the DEN and STR or DIV, and no other significant differences were observed (DEN vs. STR: t = 2.828, p = 0.025, r

^{2}= 0.067, DEN vs. DIV: t = 3.588, p = 0.002, r

^{2}= 0.106).

_{p}

^{2}= 0.045); however, no significant differences were observed in the multiple comparison results.

^{−16}, η

_{p}

^{2}= 0.809). Multiple comparison results showed a significant difference in all combinations (DEN vs. STR: p = 0.004, others: p < 0.001).

#### 3.2. Group’s Separation and Recombination and the Candidate for Descriptive Parameters

_{leader}, V

_{main}, ΔV, and ΔT when transitioning to each of the defined four states from STR or DIV. Table 1 shows the results of ANOVA using Welch’s method and multiple comparisons using the Steel–Dwass method. In this test, the change to DEN from DIV is excluded because the number of samples of the change was small (N = 2). There was no change in ESC for STR. Therefore, when staying in each state and transitioning to the adjacent state, we analyzed the changes in each candidate’s parameters.

#### 3.3. Separation from the Stretch State

_{leader}had a significant difference in ANOVA by Welch’s method depending on which state transitioned (F(2, 25.1) = 3.644, p = 0.041, η

^{2}= 0.049) (Table 1). In multiple comparisons, there was a significant difference between the change of STR to DEN or DIV (t = 2.426, p = 0.040, r

^{2}= 0.197); however, there was no significant difference in V

_{leader}between staying in STR and when transitioning to different states, that is, STR to STR and STR to DEN or DIV (Figure 2a).

_{main}did not have a significant difference (F(2, 22.1) = 1.351, p = 0.279, η

^{2}= 0.055) (Figure 2b), and ΔV had a significant difference in ANOVA (F(2, 21.7) = 8.143, p = 0.002, η

^{2}= 0.298). In multiple comparisons of ΔV, STR to DIV or DEN and STR to STR or DIV had a significant difference ((STR to DEN vs. STR to DIV) t = 4.337, p = 4.292 × 10

^{−5}, r

^{2}= 0.439; (STR to STR vs. STR to DIV) t = 4.019, p = 1.724 × 10

^{−4}, r

^{2}= 0.439) (Figure 2c).

^{−5}, η

^{2}= 0.0492). Multiple comparisons: ((STR to DEN vs. STR to STR) t = 2.443, p = 0.039, r

^{2}= 0.919; (STR to DEN vs. STR to DIV) t = 3.859, p = 3.350 × 10

^{−4}, r

^{2}= 0.383) (Figure 2d).

#### 3.4. Recombination of the Group from a Divided State

_{leader}(F(2, 13.7) = 0.079, p = 0.924, η

^{2}= 0.003) (Figure 2e) and V

_{main}(F(2, 13.8) = 3.011, p = 0.082, η

^{2}= 0.110) have no significant differences (Figure 2f), and ΔV (F(2, 14.6) = 5.045, p = 0.022, η

^{2}= 0.133) and ΔT (F(2, 14.0) = 4.829, p = 0.025, η

^{2}= 0.147) have significant differences (Table 1). The multiple comparisons show a significant difference between DIV to STR or DIV for ΔV (t = 2.560, p = 0.028, r

^{2}= 0.107) (Figure 2g).

^{2}= 0.256; (DIV to DIV vs. DIV to ESC) t = 2.758, p = 0.016, r

^{2}= 0.122) (Figure 2h).

#### 3.5. Effect of Velocity Difference between a Leader and Main Group for Group Separation or Recombination

_{leader}, and the vertical axis is ΔV, and they are plotted separately as the cases of transition from STR to STR and DIV (Figure 3a) and from DIV to STR and DIV (Figure 3b).

_{leader}= 14 m/s, ΔV = 0 (Figure 3a). In contrast, when the group recombines from DIV to STR, it is distributed below the area partitioned by V

_{leader}= 14 m/s, ΔV = 0 compared to when the group stayed in DIV (Figure 3b). The MANOVA results for V

_{leader}and ΔV showed significant differences for STR to STR or DIV (Wilks’ Λ = 0.718 (F(2, 104) = 9.361, p = 1.828 × 10

^{−4}, η

^{2}= 0.282)) and for DIV to DIV or STR (Wilks’ Λ = 0.848 (F(2, 108) = 4.632, p = 0.012, η

^{2}= 0.152)), respectively. In some cases, it remained stretched with a relatively high ΔV of 2.0 m/s (Figure 3a). In this situation, the group was on the verge of separating and might transition to a divided state at the next measurement point. Even in the stretched peloton configuration state, the group defined as “group definition” was formed. Therefore, even if the distance between the leading group and the main group widened due to the large ΔV, it might remain in a stretched state with the presence of a small chasing group in front of the main group.

_{leader}and ΔV was regarded as the descriptive parameter for state transition between STR to DIV and DIV to STR; however, each V

_{leader}and ΔV was not considered as the descriptive parameter independently.

#### 3.6. Hysteresis in Separation and Recombination in the Points Race

_{leader}and ΔV, respectively. Figure 3c shows that even if the same V

_{leader}is 14 m/s, the probability of “to DIV” is different between the separation and recombination, that is, from STR to DIV and from DIV to STR, respectively. Moreover, Figure 3d shows a similar hysteresis; for example, the probability of “to DIV” is different between STR to DIV and DIV to STR at ΔV = 1.75 m/s.

_{leader}and ΔV but depends on the current state of the group.

## 4. Discussion

_{leader}) was higher than in other states (Figure 1a). The speed of a leading cyclist may be too high for the follower to sustain when approaching maximal sustainable output when the group stretches or separates. According to Trenchard et al. [11], group separation in a cycling race occurs when a drafting or following cyclist cannot sustain the pace of the leading rider even by drafting. Therefore, group stretch and separation are a consequence of increases in competition speeds that approach cyclists’ maximal sustainable outputs. The time differences between the leader and the front of the main group (ΔT) became larger following the state transition from dense to stretched, divided, and escape and dense state. This suggests that time difference could be regarded as an order parameter corresponding to the global state transition defined by Okumura, Yokoyama, and Yamamoto [2].

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Candidate descriptive parameters for four global states: DEN, STR, DIV, and ESC. The box plots show the comparison of distributions in each parameter: V

_{leader}(

**a**), V

_{main}(

**b**), ΔV (

**c**), and ΔT (

**d**) in each state, respectively.

**Figure 2.**Separation and recombination. The box plots show candidate descriptive parameters V

_{leader}(

**a**,

**e**), V

_{main}(

**b**,

**f**), ΔV (

**c**,

**g**), and ΔT (

**d**,

**h**), respectively, when transitioning to each of the defined four states from STR or DIV. The distribution of the variables to each state from STR (

**a**–

**d**). The distribution of the variables to each state from DIV (

**e**–

**h**).

**Figure 3.**The relationship between V

_{leader}and ΔV in the transition from STR and DIV. (

**a**) V

_{leader}and ΔV when staying STR (blue) and transitioning from STR to DIV (red). (

**b**) V

_{leader}and ΔV when staying DIV (green) and transitioning from DIV to STR (black). A crossing point of the error bar shows the mean value, and the width of the error bar shows the standard deviation. The median of V

_{leader}is 14 m/s. (

**c**) The probability of DIV for V

_{leader}when transitioning from STR (red) and DIV (green). For V

_{leader}, in the grid with 0.5 m/s in (

**a**) and (

**b**), the probability of transitioning to a divided state was calculated. (

**d**) The probability of DIV for ΔV when transitioning from STR (red) to DIV (green). For ΔV, in the grid of 0.5 m/s in (

**a**) and (

**b**), the probability of transitioning to a divided state was calculated.

From | Comparison | V_{Leader} | V_{main} | ΔV | ΔT | ||||
---|---|---|---|---|---|---|---|---|---|

Welch F Value (df) | Steel–Dwass t Value (df) | Welch F Value (df) | Steel–Dwass t Value | Welch F Value (df) | Steel–Dwass t Value | Welch F Value (df) | Steel–Dwass t Value | ||

Stretched state (STR) | STR→DEN STR→STR | 3.644 * (2, 25.1) | 1.361 (59) | 1.351 (2, 22.1) | - | 8.143 ** (2, 21.7) | 0.929 | 14.929 *** (2, 26.3) | 2.443 |

STR→DEN STR→DIV | 2.426 * (24) | - | 4.337 *** | 3.859 *** | |||||

STR→STR STR→DIV | 0.906 (53) | - | 4.019 *** | 2.223 | |||||

Divided state (DIV) | DIV→STR DIV→DIV | 0.079 (2, 13.7) | - (55) | 3.010 (2, 13.8) | - | 5.045 * (2, 14.6) | 2.560 * | 4.829 (2, 14.0) | 1.718 |

DIV→STR DIV→ESC | - (16) | - | 2.252 | 2.344 * | |||||

DIV→DIV DIV→ESC | - (55) | - | 0.810 | 2.758 * |

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**MDPI and ACS Style**

Okumura, F.; Yokoyama, K.; Yamamoto, Y.
Descriptive Parameters and Its Hysteresis of the Group Separation and Recombination in Bicycle Points Races: Leader’s Velocity and Speed Difference between Leader and Main Group. *Appl. Sci.* **2023**, *13*, 1315.
https://doi.org/10.3390/app13031315

**AMA Style**

Okumura F, Yokoyama K, Yamamoto Y.
Descriptive Parameters and Its Hysteresis of the Group Separation and Recombination in Bicycle Points Races: Leader’s Velocity and Speed Difference between Leader and Main Group. *Applied Sciences*. 2023; 13(3):1315.
https://doi.org/10.3390/app13031315

**Chicago/Turabian Style**

Okumura, Fumihiro, Keiko Yokoyama, and Yuji Yamamoto.
2023. "Descriptive Parameters and Its Hysteresis of the Group Separation and Recombination in Bicycle Points Races: Leader’s Velocity and Speed Difference between Leader and Main Group" *Applied Sciences* 13, no. 3: 1315.
https://doi.org/10.3390/app13031315