Coefficients: (20 not defined because of singularities)

It is the model matrix which is singular, *not* the variable.  You are 
trying to fit a collinear model.

Use alias() to see what is going on.

Hello,

I am trying to run a linear regression analysis on my data set. For some 
reason most variables are removed due to singularities.

My linear regression looks this way (I am using only partial data, which 
is selected by flags):

fm<-lm(log(cplex6.time..sec..[flags]) ~ cplex6.cities[flags] + 
log(1/features.meanOver.frust[flags]) + 
log(1/features.meanOver.minDist[flags]) +
[...]
avg..steps.to.loc..Opt..norm..[flags] + NN.List.opt..tour.max.[flags])

As I am using inversion and logarithms I set all data to positiv values, 
before running lm():

cplex6.time..sec..[cplex6.time..sec..<=0.00001]=0.00001
features.meanOver.frust[features.meanOver.frust<=0.00001]=0.00001
features.meanOver.minDist[features.meanOver.minDist<=0.00001]=0.00001
[...]
features.varOver.varDist[features.varOver.varDist<=0.00001]=0.00001

Retrieving the summary of fm, I get the message, that some coefficients 
have been removed.
No, that they are nor defined, as it says.
[...]
Coefficients: (20 not defined because of singularities)
                                                Estimate Std. Error t 
value
(Intercept)                                      87.2162    44.1148   
1.977
log(1/features.meanOver.frust[flags])            -2.5298     0.1515 
-16.702
log(1/features.meanOver.minDist[flags])         154.7170    11.3917  
13.582
log(1/features.meanOver.quant25Dist[flags])    -943.4625    71.3505 
-13.223
log(1/features.meanOver.quart1SpanDist[flags])  776.1049    60.0571  
12.923
log(1/features.meanOver.spanDist[flags])         -9.8069     0.1400 
-70.038
log(1/features.meanOver.varDist[flags])         -11.3211     0.6715 
-16.859
log(1/features.quant25Over.minDist[flags])      -46.9655     3.1438 
-14.939
avg..steps.to.loc..Opt..norm..[flags]             0.8324     1.0919   
0.762
                                               Pr(>|t|)
(Intercept)                                      0.0511 .
log(1/features.meanOver.frust[flags])            <2e-16 ***
log(1/features.meanOver.minDist[flags])          <2e-16 ***
log(1/features.meanOver.quant25Dist[flags])      <2e-16 ***
log(1/features.meanOver.quart1SpanDist[flags])   <2e-16 ***
log(1/features.meanOver.spanDist[flags])         <2e-16 ***
log(1/features.meanOver.varDist[flags])          <2e-16 ***
log(1/features.quant25Over.minDist[flags])       <2e-16 ***
avg..steps.to.loc..Opt..norm..[flags]            0.4478
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
[...]

The summary of one of the removed coefficients looks like this:
That's the summary of the variable, not the coefficient.
summary(features.spanOver.quart1SpanDist[flags])
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
0.05584 0.05797 0.06366 0.06311 0.06674 0.07290
summary(log(1/features.spanOver.quart1SpanDist[flags]))
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
  2.619   2.707   2.754   2.767   2.848   2.885

The summary of a coefficient that was kept looks this way:

summary(features.quant25Over.minDist[flags])
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
0.001030 0.001030 0.001030 0.001032 0.001030 0.001040
summary(log(1/features.quant25Over.minDist[flags]))
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
  6.869   6.878   6.878   6.877   6.878   6.878

So, I don't see the difference. Why has the first coefficient been 
removed and the second one kept?
Please help me.

I'm using R 1.6.2 on a Linux x86 machine.

Greetings,
Thomas Fischer

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Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
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Oxford OX1 3TG, UK                Fax:  +44 1865 272595

Coefficients: (20 not defined because of singularities)

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