Name `sport`

is an abbreviation for Sequential Pairwise Online Rating Techniques. Package contains functions calculating ratings for two-player or multi-player matchups. Methods included in package are able to estimate ratings (players strengths) and their evolution in time, also able to predict output of challenge. Algorithms are based on Bayesian Approximation Method, and they don’t involve any matrix inversions nor likelihood estimation. `sport`

incorporates glicko algorithm, glicko2, bayesian Bradley-Terry and dynamic logistic regression. Parameters are updated sequentially, and computation doesn’t require any additional RAM to make estimation feasible. Additionally, package is written in `c++`

what makes computations even faster.

Before start, it’s recommended to read theoretical foundations of algorithms in other `sport`

vignette “The theory of the online update algorithms”.

Package can be installed from CRAN or from github.

Package contains actual data from Speedway Grand-Prix. There are two data.frames:

`gpheats`

- results SGP heats. Column`rank`

is a numeric version of column`position`

- rider position in race.`gpsquads`

- summarized results of the events, with sum of point and final position.

```
## 'data.frame': 1002 obs. of 11 variables:
## $ id : num 1 1 1 1 2 2 2 2 3 3 ...
## $ season : int 1995 1995 1995 1995 1995 1995 1995 1995 1995 1995 ...
## $ date : POSIXct, format: "1995-05-20 19:00:00" "1995-05-20 19:00:00" ...
## $ round : int 1 1 1 1 1 1 1 1 1 1 ...
## $ name : chr "Speedway Grand Prix of Poland" "Speedway Grand Prix of Poland" "Speedway Grand Prix of Poland" "Speedway Grand Prix of Poland" ...
## $ heat : int 1 1 1 1 2 2 2 2 3 3 ...
## $ field : int 1 2 3 4 1 2 3 4 1 2 ...
## $ rider : chr "Tomasz GOLLOB" "Gary HAVELOCK" "Chris LOUIS" "Tony RICKARDSSON" ...
## $ points : int 2 0 3 1 3 0 1 2 0 2 ...
## $ position: chr "2" "4" "1" "3" ...
## $ rank : num 2 4 1 3 1 4 3 2 4 2 ...
```

Data used in `sport`

package must be in so called long format. Typically `data.frame`

contains at least `id`

, `name`

of the player and `rank`

, with one row for one player within specific match. Package allows for any number of players within event and allows ties also.

In all methods, output variable needs to be expressed as a rank/position in event. Don’t mix up rank output with typical 1-win, 0-lost. In `sport`

package output for two player game should be coded as 1=winner 2=looser. Below example of two matches with 4 players each.

```
## id rider rank
## 1 1 Tomasz GOLLOB 2
## 2 1 Gary HAVELOCK 4
## 3 1 Chris LOUIS 1
## 4 1 Tony RICKARDSSON 3
## 5 2 Sam ERMOLENKO 1
## 6 2 Jan STAECHMANN 4
## 7 2 Tommy KNUDSEN 3
## 8 2 Henrik GUSTAFSSON 2
```

To compute ratings using each algorithms one has to specify formula. - RHS of the formula have to be specified with `player(player)`

term or `player(player | team)`

when players competes in team match. `player(...)`

is a term function which helps identify column with `player`

names and/or `team`

names. - LHS of the formula should contain `rank`

term which points to column where results (ranks) are stored and `id`

(optional). RHS should rather be specified by `rank | id`

to split matches - if `id`

is missing all data will be computed under same event id.

```
glicko <- glicko_run(formula = rank | id ~ player(rider), data = data)
glicko2 <- glicko2_run(formula = rank | id ~ player(rider), data = data)
bbt <- bbt_run(formula = rank | id ~ player(rider), data = data)
dbl <- dbl_run(formula = rank | id ~ player(rider), data = data)
print(glicko)
```

```
##
## Call: rank | id ~ player(rider)
##
## Number of unique pairs: 1500
##
## Accuracy of the model: 0.63
##
## True probabilities and Accuracy in predicted intervals:
## Interval Model probability True probability Accuracy n
## 1: [0,0.1] 0.066 0.196 0.804 92
## 2: (0.1,0.2] 0.152 0.305 0.695 243
## 3: (0.2,0.3] 0.251 0.294 0.706 299
## 4: (0.3,0.4] 0.350 0.424 0.575 416
## 5: (0.4,0.5] 0.454 0.448 0.549 481
## 6: (0.5,0.6] 0.553 0.560 0.556 419
## 7: (0.6,0.7] 0.650 0.576 0.575 416
## 8: (0.7,0.8] 0.749 0.706 0.706 299
## 9: (0.8,0.9] 0.848 0.695 0.695 243
## 10: (0.9,1] 0.934 0.804 0.804 92
```

Objects returned by `<method>_run`

are of class `rating`

and have their own `print`

and `summary`

which provides simple overview. `print.sport`

shows

condensed informations about model performance like accuracy and consistency of model predictions with observed probabilities. More precise overview are

given by `summary`

by showing ratings, ratings deviations and comparing model win probabilities with observed.

```
## $formula
## rank | id ~ player(rider)
##
## $method
## [1] "dbl"
##
## $`Overall Accuracy`
## [1] 0.635
##
## $`Number of pairs`
## [1] 3000
##
## $r
## rider r rd
## 1: rider=Tomasz GOLLOB 0.523 0.073
## 2: rider=Gary HAVELOCK 0.865 0.116
## 3: rider=Chris LOUIS 0.355 0.048
## 4: rider=Tony RICKARDSSON 1.167 0.048
## 5: rider=Sam ERMOLENKO 0.243 0.049
## 6: rider=Jan STAECHMANN -1.769 0.292
## 7: rider=Tommy KNUDSEN 0.855 0.122
## 8: rider=Henrik GUSTAFSSON 0.957 0.048
## 9: rider=Mikael KARLSSON -1.464 0.292
## 10: rider=Hans NIELSEN 1.522 0.053
## 11: rider=Andy SMITH -0.946 0.068
## 12: rider=Mark LORAM -0.082 0.048
## 13: rider=Greg HANCOCK 1.079 0.049
## 14: rider=Marvyn COX -1.011 0.054
## 15: rider=Dariusz ŚLEDŹ 0.103 0.774
## 16: rider=Craig BOYCE -0.330 0.059
## 17: rider=Billy HAMILL 1.235 0.054
## 18: rider=Peter KARLSSON 0.600 0.175
## 19: rider=Franz LEITNER -0.597 0.735
## 20: rider=Gerd RISS 0.002 0.540
## 21: rider=Josh LARSEN -2.481 0.735
## 22: rider=Lars GUNNESTAD -0.480 0.735
## 23: rider=Jason CRUMP -0.167 0.264
## 24: rider=Leigh ADAMS -0.333 0.358
## 25: rider=Joe SCREEN -0.155 0.264
## 26: rider=Stefano ALFONSO -1.733 0.735
## rider r rd
```

To visualize top n ratings with their 95% confidence interval one can use dedicated `plot.rating`

function. For `dbl`

method top coefficients are presented which doesn’t have to be player specific (ratings). It’s also possible to examine ratings evolution in time, by specifying `players`

argument.

Except dedicated `print`

,`summary`

and `plot`

there is possibility to extract more detailed information for analyses. `rating`

object contains following elements:

`## [1] "final_r" "final_rd" "r" "pairs"`

`rating$final_r`

and`rating$final_rd`

contains the last estimate of the ratings and ratings deviations. For`glicko2`

there is also`rating$final_sigma`

.`r`

contains`data.table`

with prior ratings estimations from first event to the last. Number of rows in`r`

equals number of rows in input data.`pairs`

pairwise combinations of players in analyzed events with prior probability and result of a challenge.

```
## id rider r rd
## 1: 250 Peter KARLSSON 1597.472 37.17764
## 2: 250 Tomasz GOLLOB 1552.346 32.34887
## 3: 250 Billy HAMILL 1697.257 30.04788
## 4: 251 Craig BOYCE 1477.183 30.23765
## 5: 251 Hans NIELSEN 1778.792 34.01788
## 6: 251 Chris LOUIS 1579.143 28.47306
```

```
## id rider opponent Y P
## 1: 251 Craig BOYCE Hans NIELSEN 0 0.1520817
## 2: 251 Craig BOYCE Chris LOUIS 1 0.3584955
## 3: 251 Hans NIELSEN Craig BOYCE 1 0.8479183
## 4: 251 Hans NIELSEN Chris LOUIS 1 0.7573203
## 5: 251 Chris LOUIS Craig BOYCE 0 0.6415045
## 6: 251 Chris LOUIS Hans NIELSEN 0 0.2426797
```

Examples presented in package overview might be sufficient in most cases, but sometimes it is necessary to adjust algorithms to fit data better. One characteristic of the online update algorithms is that variance of the parameters drops quickly to zero. Especially, when the number of events for the player is big ($n_i>100 $), after hundreds iterations rating parameters are very difficult to change, and output probabilities use to be extreme. To avoid these mistakes some additional controls should be applied, which is explained in this section with easy to learn examples.

In all methods formula must contain `rank | id ~ player(player)`

elements, to correctly specify the model.

`rank`

denotes column with output (order).`id`

denotes event id, within which update is computed.`player(...)`

function helps to identify column in which names of the players are stored.`player(...)`

can be specified in two ways:`player(player)`

if results of the event are observed per player.`player(player | team)`

when players competes within teams, and results are observed per team. This option is not available in`dbl_run`

which requires only formula for player matchups.

other variables - available only in

`dbl_run`

, which allows to specify other factors in model.

`r`

and `rd`

Main functionality which is common between all algorithms is to specify prior `r`

and `rd`

. Both parameters can be set by creating named vectors. Let’s suppose we have 4 players `c("A","B","C","D")`

competing in an event, and we have players prior `r`

and `rd`

estimates. It’s important to have `r`

and `rd`

names corresponding with levels of `name`

variable. One can run algorithm, to obtain new estimates.

We can also run models re-using previously estimated parameters from `model$final_r`

and `model$final_rd`

in the future when new data appear.

```
glicko_run(
formula = rank | id ~ player(rider),
data = gpheats[17:20, ],
r = model$final_r,
rd = model$final_rd
)$final_r
```

```
## Tomasz GOLLOB Gary HAVELOCK Chris LOUIS Tony RICKARDSSON
## 1696.809 1200.487 1799.513 1400.162
## Sam ERMOLENKO Jan STAECHMANN Tommy KNUDSEN Henrik GUSTAFSSON
## 1940.042 1200.487 1400.162 1599.838
## Mikael KARLSSON Hans NIELSEN Andy SMITH Mark LORAM
## 1200.487 1599.838 1455.702 1799.513
## Greg HANCOCK Marvyn COX Dariusz ŚLEDŹ Craig BOYCE
## 1599.838 1200.487 1400.162 1508.129
```

`weight`

All algorithms have a weight argument which increases or decreases update size. Higher weight increasing impact of corresponding event. Effect of the weight on update size can be expressed directly by following formula - \(\small R_i^{'} \leftarrow R_i \pm \omega_i * \Omega_i\). To specify weight \(\omega_i\) one needs to create additional column in input data, and pass the name of the column to `weight`

argument. For example weight could depend on importance of competition. In speedway Grand-Prix last three heats determine event winner, thus they weight more.

```
##
## Attaching package: 'dplyr'
```

```
## The following objects are masked from 'package:stats':
##
## filter, lag
```

```
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union
```

`kappa`

In situation when player plays games very frequently, `rd`

can quickly decrease to zero, making further changes limited. Setting `kappa`

(single value) avoids rating deviation decrease to be lower than specified fraction of `rd`

. In other words final `rd`

can’t be lower than initial `RD`

times `kappa`

\[\small RD' \geq RD * kappa\]

```
bbt1 <- bbt_run(formula = rank | id ~ player(rider),
data = data,
kappa = 0.99) # RD decreases at most 1%
bbt2 <- bbt_run(formula = rank | id ~ player(rider),
data = data,
kappa = 0.8) # RD decreases at most 20%
all(bbt1$final_rd >= bbt2$final_rd)
```

`## [1] TRUE`

`lambda`

In some cases player ratings tend to be more uncertain. If scientist have prior knowledge about higher risk of event or uncertainty of specific player performance, then one might create another column with relevant values and pass the column name to `lambda`

argument.

In above examples players competes as individuals, and each is ranked at the finish line. There are sports where players, competes in teams, and results are reported per team. `sport`

is able to compute player ratings, and requires only changing formula from `player(player)`

to `player(player | team)`

. `data.frame`

should always be a long format, with one player for each row. Ratings are updated according to their contribution in team efforts. `share`

argument can be added optionally if scientist have some knowledge about players contribution in match (eg. minutes spent on the field from all possible minutes).

```
glicko2 <- glicko2_run(
data = data.frame(
id = c(1, 1, 1, 1),
team = c("A", "A", "B", "B"),
player = c("a", "b", "c", "d"),
rank_team = c(1, 1, 2, 2),
share = c(0.4, 0.6, 0.5, 0.5)
),
formula = rank_team | id ~ player(player | team),
share = "share"
)
glicko2$final_r
```

```
## a b c d
## 1583.660 1625.489 1394.845 1394.845
```

Output object contains the same elements as normal, with one difference - `pairs`

contains probability and output per team, and `r`

contains prior ratings per individuals.

```
## id team opponent Y P
## 1: 1 A B 1 0.5
## 2: 1 B A 0 0.5
```

```
## id team player r rd sigma
## 1: 1 A a 1500 350 0.05
## 2: 1 A b 1500 350 0.05
## 3: 1 B c 1500 350 0.05
## 4: 1 B d 1500 350 0.05
```