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Re: linear regression with errors in both variables
*To*: mathgroup at smc.vnet.net
*Subject*: [mg96324] Re: linear regression with errors in both variables
*From*: dh <dh at metrohm.com>
*Date*: Wed, 11 Feb 2009 05:24:15 -0500 (EST)
*References*: <gmrmga$a1k$1@smc.vnet.net>
Hi Joerg,
this is an addendum to my former message for the case where the x and y
axis have different scales. The following example assumes that the error
in the y axis is 100 times that of the x axis. sx and sy denote the
standard deviations. a anb be define again the line: y==a+b x. res
denotes the weighted sum of squares of distances. The metric is taken
as: {{1/sx^2,0},{0,1/sy^2}}
d = Table[{RandomReal[], 100 RandomReal[]}, {4}];
sx = 1; sy = 100;
res[a_, b_] =
Module[{perp, met = {{1/sx^2, 0}, {0, 1/sy^2}}},
perp = {b sx^2, -sy^2};
perp = perp/Sqrt[perp.met.perp];
Plus @@ (perp.met.(# + {a/b, 0}) & /@ d)^2];
sol = Minimize[res[a, b], {a, b}]
fun[x_] = a + b x /. sol[[2]]
Plot[fun[x], {x, 0, 1}, Epilog -> {Red, Point[d]},
PlotRange -> ({0.95 Min @@ #, 1.05 Max @@ #} & /@ Transpose[d])]
hope this helps, Daniel
Joerg wrote:
> Hi,
>
> I want to test the hypothesis that my data
> follows a known simple linear relationship,
> y = a + bx. However, I have (known) measurements
> errors in both the y and the x values.
>
> I suppose just a simple linear regression
> does not do here.
>
> Any suggestions how do test this correctly?
>
> Thanks,
>
> joerg
>
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