So I am exploring using a logistic regression model to predict the probability of a shot resulting in a goal. I have two predictors but for simplicity lets assume I have one predictor: distance from the goal. When doing some data exploration I decided to investigate the relationship between distance and the result of a goal. I did this graphical by splitting the data into equal size bins and then taking the mean of all the results (0 for a miss and 1 for a goal) within each bin. Then I plotted the average distance from goal for each bin vs the probability of scoring. I did this in python
#use the seaborn library to inspect the distribution of the shots by result (goal or no goal)
fig, axes = plt.subplots(1, 2,figsize=(11, 5))
#first we want to create bins to calc our probability
#pandas has a function qcut that evenly distibutes the data
#into n bins based on a desired column value
df['Goal']=df['Goal'].astype(int)
df['Distance_Bins'] = pd.qcut(df['Distance'],q=50)
#now we want to find the mean of the Goal column(our prob density) for each bin
#and the mean of the distance for each bin
dist_prob = df.groupby('Distance_Bins',as_index=False)['Goal'].mean()['Goal']
dist_mean = df.groupby('Distance_Bins',as_index=False)['Distance'].mean()['Distance']
dist_trend = sns.scatterplot(x=dist_mean,y=dist_prob,ax=axes[0])
dist_trend.set(xlabel="Avg. Distance of Bin",
ylabel="Probabilty of Goal",
title="Probability of Scoring Based on Distance")
Probability of Scoring Based on Distance
So my question is why would we go through the process of creating a logistic regression model when I could fit a curve to the plot in the image? Would that not provide a function that would predict a probability for a shot with distance x.
I guess the problem would be that we are reducing say 40,000 data point into 50 but I'm not entirely sure why this would be a problem for predict future shot. Could we increase the number of bins or would that just add variability? Is this a case of bias-variance trade off? Im just a little confused about why this would not be as good as a logistic model.
The binning method is a bit more finicky than the logistic regression since you need to try different types of plots to fit the curve (e.g. inverse relationship, log, square, etc.), while for logistic regression you only need to adjust the learning rate to see results.
If you are using one feature (your "Distance" predictor), I wouldn't see much difference between the binning method and the logistic regression. However, when you are using two or more features (I see "Distance" and "Angle" in the image you provided), how would you plan to combine the probabilities for each to make a final 0/1 classification? It can be tricky. For one, perhaps "Distance" is more useful a predictor than "Angle". However, logistic regression does that for you because it can adjust the weights.
Regarding your binning method, if you use fewer bins you might see more bias since the data may be more complicated than you think, but this is not that likely because your data looks quite simple at first glance. However, if you use more bins that would not significantly increase variance, assuming that you fit the curve without varying the order of the curve. If you change the order of the curve you fit, then yes, it will increase variance. However, your data seems like it is amenable to a very simple fit if you go with this method.
