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{{Short description|Concept in statistics}}
{{For|the linear transformation|Projection (linear algebra)}}
In [[statistics]], the '''projection matrix''' <math>(\mathbf{P})</math>,<ref>{{cite book |first=Alexander |last=Basilevsky |title=Applied Matrix Algebra in the Statistical Sciences |publisher=Dover |year=2005 |isbn=0-486-44538-0 |pages=160–176 |url=https://books.google.com/books?id=ScssAwAAQBAJ&pg=PA160 }}</ref> sometimes also called the '''influence matrix'''<ref>{{cite web |title=Data Assimilation: Observation influence diagnostic of a data assimilation system |url=http://old.ecmwf.int/newsevents/training/lecture_notes/pdf_files/ASSIM/ObservationInfluence.pdf |archive-url=https://web.archive.org/web/20140903115021/http://old.ecmwf.int/newsevents/training/lecture_notes/pdf_files/ASSIM/ObservationInfluence.pdf |url-status=dead |archive-date=2014-09-03 }}</ref> or '''hat matrix''' <math>(\mathbf{H})</math>, maps the vector of [[response variable|response values]] (dependent variable values) to the vector of [[fitted value]]s (or predicted values). It describes the [[influence function (statistics)|influence]] each response value has on each fitted value.<ref name="Hoaglin1977" >{{Cite journal | title = The Hat Matrix in Regression and ANOVA
| first1= David C. | last1= Hoaglin |first2= Roy E. | last2=Welsch |journal= [[The American Statistician]] | volume=32 |date=February 1978| pages=17–22 | doi = 10.2307/2683469 |issue=1| jstor = 2683469 |url=http://dspace.mit.edu/bitstream/1721.1/1920/1/SWP-0901-02752210.pdf | hdl= 1721.1/1920 | hdl-access= free }}</ref><ref name = "Freedman09">{{cite book |author=
==
If the vector of [[Response variable|response values]] is denoted by <math>\mathbf{y}</math> and the vector of fitted values by <math>\mathbf{\hat{y}}</math>,
:<math>\mathbf{\hat{y}} = \mathbf{P} \mathbf{y}.</math>
As <math>\mathbf{\hat{y}}</math> is usually pronounced "y-hat", the projection matrix <math>\mathbf{P}</math> is also named ''hat matrix'' as it "puts a [[circumflex|hat]] on <math>\mathbf{y}</math>".
==Application for residuals== The formula for the vector of [[errors and residuals in statistics|residual]]s <math>\mathbf{r}</math> can also be expressed compactly using the projection matrix: :<math>\mathbf{r} = \mathbf{y} - \mathbf{\hat{y}} = \mathbf{y} - \mathbf{P} \mathbf{y} = \left( \mathbf{I} - \mathbf{P} \right) \mathbf{y}.</math>
where <math>\mathbf{I}</math> is the [[identity matrix]]. The matrix <math>\mathbf{M}
:<math>\mathbf{\Sigma}_\mathbf{r} = \left( \mathbf{I} - \mathbf{P} \right)^\textsf{T} \mathbf{\Sigma} \left( \mathbf{I}-\mathbf{P} \right)</math>,
where <math>\mathbf{\Sigma}</matH> is the [[covariance matrix]] of the error vector (and by extension, the response vector as well). For the case of linear models with [[independent and identically distributed]] errors in which <math>\mathbf{\Sigma} = \sigma^{2} \mathbf{I}</math>, this reduces to:<ref name="Hoaglin1977"/>
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From the figure, it is clear that the closest point from the vector <math>\mathbf{b}</math> onto the column space of <math>\mathbf{A}</math>, is <math>\mathbf{Ax}</math>, and is one where we can draw a line orthogonal to the column space of <math>\mathbf{A}</math>. A vector that is orthogonal to the column space of a matrix is in the nullspace of the matrix transpose, so
:<math>\mathbf{A}^\textsf{T}(\mathbf{b}-\mathbf{Ax}) = 0</math>.
From there, one rearranges, so
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\Rightarrow && \mathbf{A}^\textsf{T}\mathbf{b} &= \mathbf{A}^\textsf{T}\mathbf{Ax} \\
\Rightarrow && \mathbf{x} &= \left(\mathbf{A}^\textsf{T}\mathbf{A}\right)^{-1}\mathbf{A}^\textsf{T}\mathbf{b}
\end{align}</math>.
Therefore, since <math>\mathbf{
== Linear model ==
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Therefore, the projection matrix (and hat matrix) is given by
:<math>\mathbf{P}
=== Weighted and generalized least squares ===
{{further|Weighted least squares|Generalized least squares}}
The above may be generalized to the cases where the weights are not identical and/or the errors are correlated. Suppose that the [[covariance matrix]] of the errors is
: <math>
\hat{\mathbf\beta}_{\text{GLS}}= \left( \mathbf{X}^\textsf{T} \mathbf{\
</math>.
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: <math>
\mathbf{H} = \mathbf{X}\left( \mathbf{X}^\textsf{T} \mathbf{\
</math>
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The projection matrix has a number of useful algebraic properties.<ref>{{cite book |last=Gans |first=P. |year=1992 |title=Data Fitting in the Chemical Sciences |url=https://archive.org/details/datafittinginche0000gans |url-access=registration |publisher=Wiley |isbn=0-471-93412-7 }}</ref><ref>{{cite book |last=Draper |first=N. R. |last2=Smith |first2=H. |year=1998 |title=Applied Regression Analysis |publisher=Wiley |isbn=0-471-17082-8 }}</ref> In the language of [[linear algebra]], the projection matrix is the [[orthogonal projection]] onto the [[column space]] of the design matrix <math>\mathbf{X}</math>.<ref name = "Freedman09" /> (Note that <math>\left( \mathbf{X}^\textsf{T} \mathbf{X} \right)^{-1} \mathbf{X}^\textsf{T}</math> is the [[Moore–Penrose pseudoinverse#Full rank|pseudoinverse of X]].) Some facts of the projection matrix in this setting are summarized as follows:<ref name = "Freedman09" />
* <math>\mathbf{u} = (\mathbf{I} - \mathbf{P})\mathbf{y},</math> and <math>\mathbf{u} = \mathbf{y} - \mathbf{P} \mathbf{y} \perp \mathbf{X}.</math>
* <math>\mathbf{P}</math> is symmetric, and so is <math>\mathbf{M}
* <math>\mathbf{P}</math> is idempotent: <math>\mathbf{P}^2 = \mathbf{P}</math>, and so is <math>\mathbf{M}</math>.
* If <math>\mathbf{X}</math> is an {{nowrap|''n'' × ''r''}} matrix with <math>\operatorname{rank}(\mathbf{X}) = r</math>, then <math>\operatorname{rank}(\mathbf{P}) = r</math>
* The [[eigenvalue]]s of <math>\mathbf{P}</math> consist of ''r'' ones and {{nowrap|''n'' − ''r''}} zeros, while the eigenvalues of <math>\mathbf{M}</math> consist of {{nowrap|''n'' − ''r''}} ones and ''r'' zeros.<ref>{{cite book |first=Takeshi |last=Amemiya |title=Advanced Econometrics |___location=Cambridge |publisher=Harvard University Press |year=1985 |isbn=0-674-00560-0 |pages=[https://archive.org/details/advancedeconomet00amem/page/460 460]–461 |url=https://archive.org/details/advancedeconomet00amem |url-access=registration }}</ref>
* <math>\mathbf{X}</math> is invariant under <math>\mathbf{P}</math> : <math>\mathbf{P X} = \mathbf{X},</math> hence <math>\left( \mathbf{I} - \mathbf{P} \right) \mathbf{X} = \mathbf{0}</math>.
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== Blockwise formula ==
Suppose the design matrix <math>\mathbf{X}</math> can be decomposed by columns as <math>\mathbf{X} =
Define the hat or projection operator as <math>\mathbf{P}[\mathbf{X
Then the projection matrix can be decomposed as follows:<ref>{{cite book|last1=Rao|first1=C. Radhakrishna|last2=Toutenburg|first2=Helge|author3=Shalabh|first4=Christian|last4=Heumann|title=Linear Models and Generalizations|url=https://archive.org/details/linearmodelsgene00raop|url-access=limited|year=2008|publisher=Springer|___location=Berlin|isbn=978-3-540-74226-5|
:<math> \mathbf{P}[\mathbf{X}] = \mathbf{P}[\mathbf{A}] + \mathbf{P}\big[\mathbf{M}[\mathbf{A}] \mathbf{B}\big], </math>
where, e.g., <math>\mathbf{P}[\mathbf{A
There are a number of applications of such a decomposition. In the classical application <math>\mathbf{A}</math> is a column of all ones, which allows one to analyze the effects of adding an intercept term to a regression. Another use is in the [[fixed effects model]], where <math>\mathbf{A}</math> is a large [[sparse matrix]] of the dummy variables for the fixed effect terms. One can use this partition to compute the hat matrix of <math>\mathbf{X
==History==
▲where, e.g., <math>P\{A\} = A \left(A^\textsf{T} A \right)^{-1} A^\textsf{T}</math> and <math>M\{A\} = I - P\{A\}</math>.
The hat matrix was introduced by John Wilder in 1972. An article by Hoaglin, D.C. and Welsch, R.E. (1978) gives the properties of the matrix and also many examples of its application.
▲There are a number of applications of such a decomposition. In the classical application <math>A</math> is a column of all ones, which allows one to analyze the effects of adding an intercept term to a regression. Another use is in the [[fixed effects model]], where <math>A</math> is a large [[sparse matrix]] of the dummy variables for the fixed effect terms. One can use this partition to compute the hat matrix of <math>X </math> without explicitly forming the matrix <math>X</math>, which might be too large to fit into computer memory.
== See also ==
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[[Category:Regression analysis]]
[[Category:Matrices (mathematics)]]
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