Difference between revisions of "Principal Component Analysis"

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{{ClassSidebar
{{ClassSidebarYouTube
| date = 16, 23, 30 September 2011
| date = 16, 23, 30 September 2011
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}}__NOTOC__
}}__NOTOC__
== Class 2 ==
== Class 2 (16 September 2011) ==


<pdfreflow>   
 
class_date        = 16 September 2011 [1.65 Mb]
[[Image:Nuvola_mimetypes_pdf.png|20px|link=Media:Lvm-class-2.pdf]] [[Media:Lvm-class-2.pdf|Download the class slides]] (PDF)
button_label      = Create my projector slides!
show_page_layout  = 1
show_frame_option = 1
pdf_file          = lvm-class-2.pdf
</pdfreflow>
or you may [[Media:Lvm-class-2.pdf|download the class slides]] directly.




* Download these 3 CSV files and bring them on your computer:
* Download these 3 CSV files and bring them on your computer:
** Peas dataset: http://datasets.connectmv.com/info/peas
** Peas dataset: http://openmv.net/info/peas
** Food texture dataset: http://datasets.connectmv.com/info/food-texture
** Food texture dataset: http://openmv.net/info/food-texture
** Food consumption dataset: http://datasets.connectmv.com/info/food-consumption
** Food consumption dataset: http://openmv.net/info/food-consumption


=== Background reading ===
=== Background reading ===
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** matrix multiplication
** matrix multiplication
** that matrix multiplication of a vector by a matrix is a transformation from one coordinate system to another (we will review this in class)
** that matrix multiplication of a vector by a matrix is a transformation from one coordinate system to another (we will review this in class)
** [http://en.wikipedia.org/wiki/Linear_combination linear combinations] (read the first section of that website: we will review this in class)
** [https://en.wikipedia.org/wiki/Linear_combination linear combinations] (read the first section of that website: we will review this in class)
** the dot product of 2 vectors, and that they are related by the cosine of the angle between them (see the [http://en.wikipedia.org/wiki/Dot_product geometric interpretation section])
** the dot product of 2 vectors, and that they are related by the cosine of the angle between them (see the [http://en.wikipedia.org/wiki/Dot_product geometric interpretation section])


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== Class 3 ==
== Class 3 (23, 30 September 2011) ==


<pdfreflow>   
[[Image:Nuvola_mimetypes_pdf.png|20px|link=Media:Lvm-class-3.pdf]] [[Media:Lvm-class-3.pdf|Download the class slides]] (PDF)
class_date        = 23, 30 September 2011 [580 Kb]
button_label      = Create my projector slides!
show_page_layout  = 1
show_frame_option = 1
pdf_file          = lvm-class-3.pdf
</pdfreflow>
or you may [[Media:Lvm-class-3.pdf|download the class slides]] directly.




===Background reading ===
===Background reading ===


* [http://stats4eng.connectmv.com/wiki/Least_squares_modelling Least squares]:
* Least squares:
** what is the objective function of least squares
** what is the objective function of least squares
** how to calculate the regression coefficient \(b\) for \(y =  bx + e\) where \(x\) and \(y\) are centered vectors
** how to calculate the regression coefficient \(b\) for \(y =  bx + e\) where \(x\) and \(y\) are centered vectors
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* Some optimization theory:
* Some optimization theory:
** How an optimization problem is written with equality constraints
** How an optimization problem is written with equality constraints
** The [http://en.wikipedia.org/wiki/Lagrange_multiplier Lagrange multiplier principle] for solving simple, equality constrained optimization problems.
** The [https://en.wikipedia.org/wiki/Lagrange_multiplier Lagrange multiplier principle] for solving simple, equality constrained optimization problems.




===Background reading ===
===Background reading ===
* Reading on [http://literature.connectmv.com/item/12/cross-validatory-estimation-of-the-number-of-components-in-factor-and-principal-components-models cross validation]
* Reading on [http://literature.connectmv.com/item/12/cross-validatory-estimation-of-the-number-of-components-in-factor-and-principal-components-models cross validation]

Latest revision as of 14:04, 17 September 2018

Class date(s): 16, 23, 30 September 2011
Video material (part 1)
Download video: Link (plays in Google Chrome) [290 Mb]


Video material(part 2)
Download video: Link (plays in Google Chrome) [306 Mb]


Video material (part 3)
Download video: Link (plays in Google Chrome) [294 Mb]


Video material (part 4)
Download video: Link (plays in Google Chrome) [152 Mb]


Video material (part 5)
Download video: Link (plays in Google Chrome) [276 Mb]


Video material (part 6)
Download video: Link (plays in Google Chrome) [333 Mb]


Video material (part 7)
Download video: Link (plays in Google Chrome) [198 Mb]


Video material (part 8)
Download video: Link (plays in Google Chrome) [180 Mb]

Class 2 (16 September 2011)

Nuvola mimetypes pdf.png Download the class slides (PDF)


Background reading

  • Reading for class 2
  • Linear algebra topics you should be familiar with before class 2:
    • matrix multiplication
    • that matrix multiplication of a vector by a matrix is a transformation from one coordinate system to another (we will review this in class)
    • linear combinations (read the first section of that website: we will review this in class)
    • the dot product of 2 vectors, and that they are related by the cosine of the angle between them (see the geometric interpretation section)

This illustration should help better explain what I trying to get across in class 2B

  • \(p_1\) and \(p_2\) are the unit vectors for components 1 and 2.
  • \( \mathbf{x}_i \) is a row of data from matrix \( \mathbf{X}\).
  • \(\hat{\mathbf{x}}_{i,1} = t_{i,1}p_1\) = the best prediction of \( \mathbf{x}_i \) using only the first component.
  • \(\hat{\mathbf{x}}_{i,2} = t_{i,2}p_2\) = the improvement we add after the first component to better predict \( \mathbf{x}_i \).
  • \(\hat{\mathbf{x}}_{i} = \hat{\mathbf{x}}_{i,1} + \hat{\mathbf{x}}_{i,2} \) = is the total prediction of \( \mathbf{x}_i \) using 2 components and is the open blue point lying on the plane defined by \(p_1\) and \(p_2\). Notice that this is just the vector summation of \( \hat{\mathbf{x}}_{i,1}\) and \( \hat{\mathbf{x}}_{i,2}\).
  • \(\mathbf{e}_{i,2} \) = is the prediction error vector because the prediction \(\hat{\mathbf{x}}_{i} \) is not exact: the data point \( \mathbf{x}_i \) lies above the plane defined by \(p_1\) and \(p_2\). This \(e_{i,2} \) is the residual distance after using 2 components.
  • \( \mathbf{x}_i = \hat{\mathbf{x}}_{i} + \mathbf{e}_{i,2} \) is also a vector summation and shows how \( \mathbf{x}_i \) is broken down into two parts: \(\hat{\mathbf{x}}_{i} \) is a vector on the plane, while \( \mathbf{e}_{i,2} \) is the vector perpendicular to the plane.

Geometric-interpretation-of-PCA-xhat-residuals.png


Class 3 (23, 30 September 2011)

Nuvola mimetypes pdf.png Download the class slides (PDF)


Background reading

  • Least squares:
    • what is the objective function of least squares
    • how to calculate the regression coefficient \(b\) for \(y = bx + e\) where \(x\) and \(y\) are centered vectors
    • understand that the residuals in least squares are orthogonal to \(x\)
  • Some optimization theory:
    • How an optimization problem is written with equality constraints
    • The Lagrange multiplier principle for solving simple, equality constrained optimization problems.


Background reading