Lm 11

Summary

Introduction to Modeling: Takeaways

by Dataquest Labs, Inc. - All rights reserved © 2020

Syntax

Visualizing bivariate relationships

• Generating scatterplots to visualize bivariate relationships:

  ggplot(data = uber_trips, aes(x = distance, y = cost)) + geom_point()

• Visualizing a linear regression line:

  ggplot(data = uber_trips, aes(x = distance, y = cost)) + geom_point() + geom_smooth(method = "lm", se = FALSE)

Analyzing the residuals

• Calculating mean abolute error (MAE):

  MAE <- mean(abs(df$residuals))

Notation

General form of a predictive model:

$Y = f(X) + \epsilon$

• In this context, $X$ represents a set of inputs and $Y$ represents a set of outputs. The random error term $\epsilon$ is independent of $X$ and has a mean of approximately zero. In reality, the error term is unknown, so we can represent our estimate of $Y$ as a function of $X$ as:

$\hat{Y} = \hat{f}(X)$

• We can omit the error term because it averages to zero. The "hat" symbol indicates an estimate.

Bivariate linear model:

$\hat{y} = \hat{\beta}_0 + \hat{\beta}_1x$

• Here $\hat{y}$ indicates a prediction of $Y$ assuming $X = x$. Intercept ($\hat{\beta}_0$) refers to the value on the y-axis where the value on the x-axis is equal to zero. Slope ($\hat{\beta}_1$)is the change in $\hat{y}$ for every single unit change in $x$.

Mean absolute error (MAE):

$MAE = \frac{1}{n}\ \displaystyle\sum_{i = 1}^{n}|y_i - \hat{y}_i$|

• $y_i$ = observed output value

• $\hat{y_i}$ = predicted output value

• $\displaystyle\sum_{i = 1}^{n}$ = the sum of...

• $|y_i - \hat{y}_i$| = the absolute value of each residual

• $\frac{1}{n}\$ = divide the above by the total number of observations (to return the average value)

Concepts

• An input, or input variable is also sometimes referred to as a predictor, independent variable, feature, attribute, descriptor, or simply variable*.

• An output or output variable is also known as a dependent variable, outcome, response variable, response, target, or class.

• Prediction: If our primary purpose of building a model is to generate an accurate prediction, we aren't too concerned if the function of the input to predict the response is a "black box." Rather than understanding $f$, our primary concern is that our model gives us accurate output predictions for each input.

• Inference: When motivated by inference, we may or may not be interested in generating predictions for our response. Instead, we wish to understand $f$ and how the response variable is affected by changes in the input variable.

• Error: The accuracy of our prediction for an output depends on two types of error: reducible error, and irreducible error. Even though the "E" in mean absolute error stands for error, it does not refer to the epsilon error described here.

• Reducible error can be minimized by selecting a statistical method that provides a good estimate of the response. In linear regression, one key to minimizing reducible error is to select the input variable that provides the most accurate estimate of the output vatiable.

• Irreducible error is out of our control. Unmeasurable variation contributes to error. Another term for this is random noise.

• Residual: The difference between the observed value and the model’s prediction.

• Summary measures: Statisticians have developed various summary measurements (mean absolute error, for example) that can take the residuals from our model and transform them into a single value that represents the predictive ability of our model.

• Overfitting: When a model finds patterns in the training data that are not present in the unseen data.

Resources

• An Introduction to Statistical Learning with Applications in R by James et al.
• Applied Predictive Modeling by Max Kuhn and Kjell Johnson.

Takeaways by Dataquest Labs, Inc. - All rights reserved © 2020