31  Constructing derivatives

\[ \newcommand{\dnorm}{\text{dnorm}} \newcommand{\pnorm}{\text{pnorm}} \newcommand{\recip}{\text{recip}} \]

Reading questions

Reading question 31.1 Consider two explanatory variables, represented by vectors \(\vec{x}\) and \(\vec{y}\). Suppose each vectors is defined by a column (“variable”) of a data frame with 25 rows.

  1. What is the dimension of the embedding space?
2       3       25       50      

question id: projection-123-1

A set of vectors (such as \(\vec{x}\) and \(\vec{y}\)) define a subspace, which you can think of as a “slice” through the embedding space.

  1. How do we describe the “size” of the subspace?

The length of the longest of the vectors in the set.

The length of the shortest of the vectors in the set.

A dimension: the number of vectors in the set.

A dimension: it’s the same as the embedding space.

question id: projection-123-2

Suppose another column is pulled out from the same data frame as \(\vec{x}\) and \(\vec{y}\). We’ll call this column \(\vec{b}\).

  1. What is dimension of the space in which \(\vec{b}\) is embedded?
1       2       3       23       25      

question id: projection-123-3

By no means does \(\vec{b}\) have to live in the subspace defined by \(\vec{x}\) and \(\vec{y}\). But we can always re-write \(\vec{b}\) as the sum of two vectors:

  1. A vector that lives in the x-y subspace.
  2. A vector that lives entirely outside of the x-y subspace.
  1. What is the dimension of the space that we get by excluding the x-y subspace from the embedding space?
1       2       3       23       25      

question id: projection-123-4

  1. The component of \(\vec{b}\) that lives in the x-y subspace is called what?
The explanatory variable       The response variable       The target vector       The fitted model vector \(\hat{b}\)       The residual vector.      

question id: projection-123-5

  1. In the context of the projection problem, what is \(\vec{b}\) called?
The explanatory variable       The response variable       The target vector       The fitted model vector \(\hat{b}\)       The residual vector.      

question id: projection-123-6

  1. In the context of the model fitting problem, what is \(\vec{b}\) called?
The explanatory variable       The response variable       The target vector       The fitted model vector \(\hat{b}\)       The residual vector.      

question id: projection-123-7

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