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In [[statistics]], the '''hat matrix''', '''''H''''', sometimes also called '''projection matrix''', maps the vector of [[observed value]]s to the vector of [[fitted value]]s. It describes the influence each observed value has on each fitted value.<ref name="Hoaglin1977">
{{Citation
| title = The Hat Matrix in Regression and ANOVA
| first1= David C. | last1= Hoaglin
|first2= Roy E. | last2=Welsch
|journal= [[The American Statistician]]
| volume=32 | month=February| year= 1978| pages=17&ndash;22
| doi = 10.2307/2683469 |issue=1| jstor = 2683469}}
</ref> The diagonal elements of the hat matrix are the [[leverage (statistics)|leverage]]s, which describe the influence each observed value has on the fitted value for that same observation.


If the vector of observed values is denoted by '''y''' and the vector of fitted values by '''ŷ''',
:<math>\hat{\mathbf{y}} = H \mathbf{y}.</math>
As '''ŷ''' is usually pronounced "y-hat", the hat matrix is so named as it "puts a [[circumflex|hat]] on '''y'''".


Suppose that we wish to solve a [[linear model]] using [[linear least squares (mathematics)|linear least squares]]. The model can be written as
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:<math>\mathbf{y} = X \boldsymbol \beta  + \boldsymbol \varepsilon,</math>
where ''X'' is a matrix of [[explanatory variable]]s (the [[design matrix]]), '''''β''''' is a vector of unknown parameters to be estimated, and '''''ε''''' is the error vector.
 
== Uncorrelated errors ==
For uncorrelated [[errors and residuals in statistics|errors]], the estimated parameters are
 
:<math>\hat{\boldsymbol \beta} = \left(X^\top X \right)^{-1} X^\top \mathbf{y},</math>
 
so the fitted values are
 
:<math>\hat{\mathbf{y}} = X \hat{\boldsymbol \beta} = X \left(X^\top X \right)^{-1} X^\top \mathbf{y}.</math>
 
Therefore the hat matrix is given by
 
:<math>H = X \left(X^\top X \right)^{-1} X^\top.</math>
 
In the language of [[linear algebra]], the hat matrix is the [[orthogonal projection]] onto the [[column space]] of the design matrix ''X''. (Note that <math>\left(X^\top X \right)^{-1} X^\top</math> is the [[Moore–Penrose_pseudoinverse#Full_rank|pseudoinverse of X]].)
 
The hat matrix corresponding to a [[linear model]] is [[symmetric matrix|symmetric]] and [[idempotent]], that is, ''H''<sup>2</sup> = ''H''. However, this is not always the case; in [[local regression|locally weighted scatterplot smoothing (LOESS)]], for example, the hat matrix is in general neither symmetric nor idempotent.
 
The formula for the vector of [[errors and residuals in statistics|residual]]s '''r''' can be expressed compactly using the hat matrix:
 
:<math>\mathbf{r} = \mathbf{y} - \mathbf{\hat{y}} = \mathbf{y} - H \mathbf{y} = (I - H) \mathbf{y}.</math>
 
The [[covariance matrix]] of the residuals is therefore, by [[error propagation]], equal to <math>\left(I-H \right)^\top \Sigma\left(I-H \right) </math>, where Σ is the covariance matrix of the errors (and by extension, the observations as well).  For the case of linear models with [[independent and identically distributed]] errors in which Σ = ''σ''<sup>2</sup>''I'', this reduces to (''I''&nbsp;&minus;&nbsp;''H'')''σ''<sup>2</sup>.<ref name="Hoaglin1977"/>
 
For [[linear models]], the [[trace (linear algebra)|trace]] of the hat matrix is equal to the [[rank (linear algebra)|rank]] of ''X'', which is the number of independent parameters of the linear model. For other models such as LOESS that are still linear in the observations '''y''', the hat matrix can be used to define the [[degrees of freedom (statistics)#Effective degrees of freedom|effective degrees of freedom]] of the model.
 
The hat matrix has a number of useful algebraic properties.<ref>Gans, P. (1992) ''Data Fitting in the Chemical Sciences,'', Wiley. ISBN 978-0-471-93412-7</ref><ref>Draper, N.R., Smith, H. (1998) ''Applied Regression Analysis'', Wiley. ISBN 0-471-17082-6 {{Please check ISBN|reason=Check digit (6) does not correspond to calculated figure.}}</ref> Practical applications of the hat matrix in regression analysis include [[Leverage (statistics)|leverage]] and [[Cook's distance]], which are concerned with identifying observations which have a large effect on the results of a regression.
 
==Correlated errors==
The above may be generalized to the case of correlated errors.  Suppose that the [[covariance matrix]] of the errors is Σ. Then since
 
:<math> \hat{\boldsymbol{\beta}} = \left(X^\top \Sigma^{-1} X \right)^{-1} X^\top \Sigma^{-1}\,\mathbf{y}, </math>
 
the hat matrix is thus
 
:<math> H = X \left(X^\top \Sigma^{-1} X\right)^{-1} X^\top \Sigma^{-1}, \, </math>
 
and again it may be seen that ''H''<sup>2</sup> =&nbsp;''H''
 
==Blockwise formula==
 
Suppose the design matrix <math>C</math> can be decomposed by columns as <math>C = [A, B]</math>.  
Define the Hat operator as <math>H(X) = X \left(X^\top X \right)^{-1} X^\top</math>. Similarly, define the residual operator as <math>M(X) = I - H(X)</math>.
Then the Hat matrix of <math>C</math> can be decomposed as follows:
 
<math>
H(C) = H(A) + H(M(A) B)
</math>
<ref>{{cite book|last=Rao|first=C. Radhakrishna|title=Linear Models and Generalizations|year=2008|publisher=Springer|location=Berlin|isbn=978-3-540-74226-5|pages=323|edition=3rd|coauthors=Toutenburg, Shalabh, Heumann}}</ref>
 
There are a number of applications of such a partitioning. The classical application has <math>A</math> 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>C</math> without explicitly forming the matrix <math>C</math>, which might be too large to fit into computer memory.
 
== See also ==
*[[Moore–Penrose pseudoinverse]]
*[[Studentized residuals]]
*[[Degrees of freedom (statistics)#Effective degrees of freedom|Effective degrees of freedom]]
*[[Idempotent matrix]]
 
== References ==
<references />
 
[[Category:Statistical terminology]]
[[Category:Regression analysis]]
[[Category:Matrices]]

Latest revision as of 15:06, 28 October 2014


The individual trauma arrangement approach is just not a simple a single. There are so many things you have to watch out for to obtain a decent resolution. As an alternative to keeping at night, stick to some easy assistance to remain on the top of things. This post will help you get acquainted with the personal injury procedure, what you should confirm and how to locate a excellent lawyer.

If you think maybe you could have grounds for any injury scenario, visit a medical doctor right away. It's crucial that your information be recorded at the earliest opportunity. It will likely be used as beneficial proof in the future, but by the time you might have the day in court, the accidents could possibly have cured and you will definitely have zero evidence of them.

When picking your own trauma lawyer, make sure that you are going for a regulation business office which has a complete office committed to personal injury. At the same time, it can be frequent for regulation office buildings to go after several several types of cases, you should select a lawyer who's knowledgeable inside your market.

You may get an agreement should you have had any preexisting situation just before your incident. Be truthful relating to your previous personal injuries along with your attorney. He requirements to head off being sightless sided in court.

Obtain a lawyer who concentrates on accidental injuries cases. Legislation includes a huge place, so most lawyers choose to target a unique authorized area. A real property or separation and divorce legal professional is going to be sick-loaded to handle your own injury situation successfully. If you currently have a reliable lawyer who focuses on an alternative industry, inquire further for a referral to some trusted colleague who does function in accidental injury.

By no means let anyone who is in fault to your trauma encourage you that you need to take a pay out without having getting in contact with a lawyer. The amount they have may appear reasonable, however they is not any actual way to make sure of that particular with out getting in touch with somebody that understands exactly how much your problems are worth.

Examine accidental injury legal representatives. There are plenty of injury attorneys around however a few of them are sharks. Speak to your nearby Greater Organization Bureau and also check online for customer reviews. Generally though, the best way to find a very good accidental injury attorney is thru word of mouth.

Bring every one of the needed documents in your original evaluation because this is crucial to your accidental injuries legal professional to produce an excellent case. Make copies of each one of these documents too, as it is advisable to have 1 in case. This places you in the greatest placement to group on top of your legal representative and earn.

How fast does an attorney go back to you? The answer to this question need to play a vital role in which lawyer or attorney you select for your personal injury circumstance. Anyone who simply leaves you hanging for days on finish is no good. To understand just how long it will take after employed, flourish the length of time it had taken to allow them to initial speak to you instances a few.

Really the only folks you must go over an individual injuries circumstance with are your insurance professional, your spouse along with your lawyer. If the insurance adjuster telephone calls you to definitely record a statement about how exactly the accident took place, give them the volume of your legal professional and let them know that it is who needs to be called when they have any additional queries.

In the event you create a preliminary appointment by having an lawyer or attorney, ask about their charges advance. You may not want any shocks later on. Some attorneys works on the contingency grounds for personal injuries. Which means you simply will not will need money beforehand, along with the legal professional becomes paid for only should you at the same time.

Go with a legal professional car wreck lawyer in greenville (http://www.youtube.com) who conversations a lot less and listens much more. You'll realize that an attorney who discussions your ear off is a who also talks excessive at trial, which is often on the hindrance of your respective scenario. A lawyer who listens to what you must say will be able to use your evidence much better at trial run.

Take things slow car wreck lawyer in greenville (http://www.youtube.com) and take good care of your self. Usually do not allow caring for your circumstance find more significant than your own personal overall health. What is important you should do when affected by a private injury is rest and ensure you are recovery like you should be.

Choose a accidental injury attorney that'll work towards a contingency cost. This simply means the individual covers all expenses related to the way it is and stay compensated at the end via a percentage of any settlement. You may need to shop around to find the right one here, as some legal professionals can be more comfortable with working your situation similar to this as opposed to others.

It by no means is painful to get knowledgeable about the numerous stuff that were talked about in this post. The point that you've needed to put up with an individual damage has already been too much! Use the easy specifics and tips from this article to actually will certainly get to a just pay out.

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