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Advanced Linear Models for Data Science 2: Statistical Linear Models

Universidad Johns Hopkins

4.7 (30 calificaciones) | 7.2K estudiantes inscritos

Welcome to the Advanced Linear Models for Data Science Class 2: Statistical Linear Models. This class is an introduction to least squares from a linear algebraic and mathematical perspective. Before beginning the class make sure that you have the following: - A basic understanding of linear algebra and multivariate calculus. - A basic understanding of statistics and regression models. - At least a little familiarity with proof based mathematics. - Basic knowledge of the R programming language. After taking this course, students will have a firm foundation in a linear algebraic treatment of regression modeling. This will greatly augment applied data scientists' general understanding of regression models.

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4.7 (30 calificaciones)

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De la lección

Residuals

In this module we will revisit residuals and consider their distributional results. We also consider the so-called PRESS residuals and show how they can be calculated without re-fitting the model.

Residuals distributional results5:14

Code demonstration3:15

Leave one out residuals8:37

Press residuals14:36

Impartido por:

Brian Caffo, PhD
Professor, Biostatistics

Okay, so let's start creating some standardized residuals just to show

an example of doing it manually, and then with RS default built-in functions.

So if I do data(mtcars) for the motor train cars data set,

my y is miles per gallon.

In this case, my x is an intercept horsepower in weight, okay?

I define my n and my p, I create my hat matrix, and

then what I'm calling the residual matrix is just being

i minus the hat matrix, and then my e is my residmat times y.

So residmat is the matrix that is orthogonal to

the projection onto the space span by the columns of x.

Okay, and then my residual variance estimate is the sum of the residual

squared divided by n minus p.

And then my standardized residuals are just my residuals divided by

s divided by the square root of the diagonal of the residual matrix, okay?

And then, if I compare those, these are just exactly with what

you get if you take the output of your fitted lm object, okay?

And here, I'm just putting in my design matrix, which includes an intercept,

so I'm subtracting in r as default intercept.

And then the function, r standard, for standardized residuals,

is what returns the standardized residuals as we just calculated in.

You can see here, they're identical.

What you want to look for in standardized residuals, so

I would just want to do something like plot(rstd, by predict(.

And then, I should have assigned my output of my fitted object to a variable,

but I'm just going to refit it there.

And you usually,

Usually want the predictive values on the horizontal axis.

And then, you get your residuals versus, there we go, and

you get your residuals versus your fit of values plot.

And what you want to look for, when you look at these plots,

is any obvious patterns or a systematic relationships.

For example, variances increasing with the predictive value or things like that.

Often you usually want, You usually want to add a horizontal like at zero

because since we've included an intercept, our residuals have to sum to zero, so

they, at some level, have to be distributed above and below zero.

We're just doing our residual plot, only now,

we know that the interpretation of this minus one and

two here are in terms of residuals being minus one or one or

two standard deviations above or below the mean of the residuals.

The mean of the residuals, again, being zero in this case, okay?

So that's a pretty easy trick to get your residuals on a more interpretable scale.

Now, we're going to go through and talk about other kinds of standardized and

special kinds of residuals that take on a particularly nice form in linear models.

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