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In the field of machine learning, we’re often interested in building models using a set of predictor variables and a response variable. Our goal is to build a model that can effectively use the predictor variables to predict the value of the response variable.

Given a set of *p* total predictor variables, there are many models that we could potentially build. One method that we can use to pick the *best* model is known as **best subset selection** and it works as follows:

**1.** Let M_{0} denote the null model, which contains no predictor variables.

**2.** For k = 1, 2, … p:

- Fit all
_{p}C_{k}models that contain exactly*k*predictors. - Pick the best among these
_{p}C_{k}models and call it M_{k}. Define “best” as the model with the highest R^{2}or equivalently the lowest RSS.

**3.** Select a single best model from among M_{0}…M_{p} using cross-validation prediction error, Cp, BIC, AIC, or adjusted R^{2}.

Note that for a set of *p* predictor variables, there are 2^{p} possible models.

**Example of Best Subset Selection**

Suppose we have a dataset with p = 3 predictor variables and one response variable, y. To perform best subset selection with this dataset, we would fit the following 2^{p} = 2^{3} = 8 models:

- A model with no predictors
- A model with predictor x
_{1} - A model with predictor x
_{2} - A model with predictor x
_{3} - A model with predictors x
_{1}, x_{2} - A model with predictors x
_{1}, x_{3} - A model with predictors x
_{2}, x_{3} - A model with predictors x
_{1}, x_{2}, x_{3}

Next, we’d choose the model with the highest R^{2} among each set of models with *k* predictors. For example, we might end up choosing:

- A model with no predictors
- A model with predictor x
_{2} - A model with predictors x
_{1}, x_{2} - A model with predictors x
_{1}, x_{2}, x_{3}

Next, we’d perform cross-validation and choose the best model to be the one that results in the lowest prediction error, Cp, BIC, AIC, or adjusted R^{2}.

For example, we might end up choosing the following model as the “best” model because it produced the lowest cross-validated prediction error:

- A model with predictors x
_{1}, x_{2}

**Criteria for Choosing the “Best” Model**

The last step of best subset selection involves choosing the model with the lowest prediction error, lowest Cp, lowest BIC, lowest AIC, or highest adjusted R^{2}.

Here are the formulas used to calculate each of these metrics:

**Cp:** (RSS+2dσ̂) / n

**AIC: **(RSS+2dσ̂^{2}) / (nσ̂^{2})

**BIC: **(RSS+log(n)dσ̂^{2}) / n

**Adjusted R ^{2}:** 1 – ( (RSS/(n-d-1)) / (TSS / (n-1)) )

where:

**d:**The number of predictors**n:**Total observations**σ̂:**Estimate of the variance of the error associate with each response measurement in a regression model**RSS:**Residual sum of squares of the regression model**TSS:**Total sum of squares of the regression model

**Pros & Cons of Best Subset Selection**

Best subset selection offers the following **pros:**

- It’s a straightforward approach to understand and interpret.
- It allows us to identify the best possible model since we consider all combinations of predictor variables.

However, this method comes with the following **cons:**

- It can be computationally intense. For a set of
*p*predictor variables, there are 2^{p}possible models. For example, with 10 predictor variables there are 2^{10}= 1,000 possible models to consider. - Because it considers such a large number of models, it could potentially find a model that performs well on training data but not on future data. This could result in overfitting.

**Conclusion**

While best subset selection is straightforward to implement and understand, it can be unfeasible if you’re working with a dataset that has a large number of predictors and it could potentially lead to overfitting.

An alternative to this method is known as stepwise selection, which is more computationally efficient.