       Re: Linear Regression and Hat Matrix

• To: mathgroup at smc.vnet.net
• Subject: [mg44610] Re: Linear Regression and Hat Matrix
• From: Antti Penttilä@smc.vnet.net
• Date: Tue, 18 Nov 2003 06:41:39 -0500 (EST)
• Organization: University of Helsinki
• References: <bpa1m8\$19f\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Axel Kowald wrote:
> Hi everybody,
>
> I'm doing some linear regression and am a bit confused by Mathematicas
> output. I do:
> <<Statistics`LinearRegression`
> data = {{1, 0.1}, {2, 0.2}, {3, 0.3}, {4, 0.4}, {5, 0.5},
>         {6,0.6}, {7, 0.7}, {8, 0.8}, {9, 0.3}};
> Regress[data, {1, x}, x, RegressionReport -> {HatDiagonal}]
>
> and get the HatDiagonal:
> {HatDiagonal -> {0.377778,
>     0.261111, 0.177778, 0.127778, 0.111111, 0.127778, 0.177778,
> 0.261111, \
> 0.377778}}
>
>
> That's fine. But if I try to construct the Hat Matrix by hand I get
> something completely different. The hat matrix is defined as: X(X^T
> X)^-1 X^T, so I do:
>
> hat = data.Inverse[Transpose[data].data].Transpose[data]

Your formula is right, but the matrix 'data' is not the desing matrix of this
model. The design matrix is a matrix with all the explaining variables as
colums, and with intercept term included in the model, your design or X-matrix
is now:

1 1
1 2
1 3
1 4
X =  1 5
1 6
1 7
1 8
1 9

--
Antti Penttilä       Antti.I.Penttila at helsinki.removethis.fi

```

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