# How to Predict the Position of Runners in a Race

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There are many different approaches to predict the winner of a race. The race can be any distance and the runners can be dogs, horses and humans. Also, apart from trying to predict the winner, it may be possible to answer other questions like the probability of a runner being on the podium (top three positions) and so on.

Personally, in this kind of problems, I prefer to approach them with Monte Carlo simulation instead of trying to build Machine Learning models. Let ‘s describe the Monte Carlo approach.

## The Data

Let’s say that we want to predict the probability of each runner winning a race of 100 meters. For our model we want to get the past racing times of the runners during the last period of time, let’s say last 1 to 2 years provided that we have a sufficient number of races. Then we need to calculate the **mean **and the **standard deviation **of each runner. Notice that it makes sense to use an exponential moving average for the mean and maybe for standard deviation so that to give more weight to the most recent observations. Also, a good technique is to remove the worst time of each racer.

You can easily get the exponential moving average with pandas. Let’s show how we can do that. Assume that our data frame has the **NAME **of the runner and the **TIME** order by **DATE**. Our logic is to get the rolling EWM and then to keep the **last** for each runner

import pandas as pd import numpy as np # convert it to data df['DATE'] = pd.to_datetime(df.DATE) # sort by date df.sort_values('DATE', inplace=True) df['mean_tmp']=df.groupby('NAME')['TIME'].transform(lambda x: x.ewm(alpha=0.30).mean()) df['std_tmp']=df.groupby('NAME')['TIME'].transform(lambda x: x.ewm(alpha=0.30).std()) # remove the NAN in Std df.dropna(subset=['std_tmp'], inplace=True) # get the most recent observation of the EWM runners= df.groupby('NAME')[['mean_tmp', 'std_tmp']].last() runners.reset_index(inplace=True) runners.columns = ['NAME', 'mean', 'std'] runners

Assume that we come up with the following mean and standard deviation for the 8 runners.

runner = pd.DataFrame({'NAME':["A","B","C","D","E","F","G","H"], 'mean': [13.11, 13.17, 12.99, 12.96, 13.25, 13.00, 13.40, 13.29], 'std': [0.15, 0.15, 0.17, 0.20, 0.14, 0.16, 0.17, 0.2]})

## Make the Predictions

**Find the Probability of each Runner to Win**

Let’s get the probability of each runner to win by running a Monte Carlo Simulation by approximating the normal distribution with the corresponding parameters.

# number of simulations np.random.seed(5) # number of simulations sims = 1000 runner['monte_carlo'] = runner.apply(lambda x:np.random.normal(x['mean'], x['std'], sims), axis=1)

Once we simulated the data, we can get the probability of each runner to win.

# Probability to finish in top x positions top_x = 1 tmp_probs = pd.DataFrame((pd.DataFrame(list(runner['monte_carlo']),index=runner.NAME).rank()<=top_x).sum(axis=1)/sims) tmp_probs.reset_index(inplace=True) tmp_probs.columns=['NAME', 'Probability']

As we can see, **the runner D was 34.8% probability to win** and he is the favorite!

**Find the Probability of each Runner to Win a Medal**

Similarly, we can estimate the probability of each runner to be on the podium, i.e. in the top 3 positions.

# Probability to finish in top x positions top_x = 3 # in top three positions tmp_probs = pd.DataFrame((pd.DataFrame(list(runner['monte_carlo']),index=runner.NAME).rank()<=top_x).sum(axis=1)/sims) tmp_probs.reset_index(inplace=True) tmp_probs.columns=['NAME', 'Probability']

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