# Piece-wise linear and non-linear regression in R

I have a question which is perhaps more a statistical query than one related to r directly, however it may be that I am just invoking an r package incorrectly so I will post the question here. I have the following dataset:

x<-c(1e-08, 1.1e-08, 1.2e-08, 1.3e-08, 1.4e-08, 1.6e-08, 1.7e-08, 1.9e-08, 2.1e-08, 2.3e-08, 2.6e-08, 2.8e-08, 3.1e-08, 3.5e-08, 4.2e-08, 4.7e-08, 5.2e-08, 5.8e-08, 6.4e-08, 7.1e-08, 7.9e-08, 8.8e-08, 9.8e-08, 1.1e-07, 1.23e-07, 1.38e-07, 1.55e-07, 1.76e-07, 1.98e-07, 2.26e-07, 2.58e-07, 2.95e-07, 3.25e-07, 3.75e-07, 4.25e-07, 4.75e-07, 5.4e-07, 6.15e-07, 6.75e-07, 7.5e-07, 9e-07, 1.15e-06, 1.45e-06, 1.8e-06, 2.25e-06, 2.75e-06, 3.25e-06, 3.75e-06, 4.5e-06, 5.75e-06, 7e-06, 8e-06, 9.25e-06, 1.125e-05, 1.375e-05, 1.625e-05, 1.875e-05, 2.25e-05, 2.75e-05, 3.1e-05) y2<-c(-0.169718017273307, 7.28508517630734, 71.6802510299446, 164.637259265704, 322.02901173786, 522.719633360006, 631.977073772459, 792.321270345847, 971.810607095548, 1132.27551798986, 1321.01923840546, 1445.33152600664, 1568.14204073109, 1724.30089942149, 1866.79717333592, 1960.12465709003, 2028.46548012508, 2103.16027631327, 2184.10965255236, 2297.53360080873, 2406.98288043262, 2502.95194879366, 2565.31085776325, 2542.7485752473, 2499.42610084412, 2257.31567571328, 2150.92120390084, 1998.13356362596, 1990.25434682546, 2101.21333152526, 2211.08405955931, 1335.27559108724, 381.326449703455, 430.9020598199, 291.370887491989, 219.580548355043, 238.708972427248, 175.583544448326, 106.057481792519, 59.8876372379487, 26.965143266819, 10.2965349811467, 5.07812046132922, 3.19125838983254, 0.788251933518549, 1.67980552001939, 1.97695007279929, 0.770663673279958, 0.209216903989619, 0.0117903221723813, 0.000974437796492681, 0.000668823762763647, 0.000545308757270207, 0.000490042305650751, 0.000468780182460397, 0.000322977916070751, 0.000195423690538495, 0.000175847622407421, 0.000135771259866332, 9.15607623591363e-05)

which when plot looks like this: Segmentation test http://i48.tinypic.com/25pltoy.png

I have then attempted to use the segmentation package to generate three linear regressions (solid black line) in three regions (10^⁻8--10^⁻7,10^⁻7--10^⁻6 and >10^-6) since I have a theoretical basis for finding different relationships in these different regions. Clearly however my attempt using the following code was unsuccessful:

lin.mod <- lm(y2~x) segmented.mod <- segmented(lin.mod, seg.Z = ~x, psi=c(0.0000001,0.000001))

Thus my first question- are there further parameters of the segmentation I can tweak other than the breakpoints? So far as I understand I have iterations set to maximum as default here.

My second question is: could I perhaps attempt a segmentation using the nls package? It looks as though the first two regions on the plot (10^⁻8--10^⁻7 and 10^-7--10^-6) are further from linear then the final section so perhaps a polynomial function would be better here?

As an example of a result I find acceptable I have annoted the original plot by hand: Annotated segmentation example http://i45.tinypic.com/zjb439.jpg.

Edit: The reason for using linear fits is the simplicity they provide, to my untrained eye it would require a fairly complex nonlinear function to regress the dataset as a single unit. One thought that had crossed my mind was to fit a lognormal model to the data as this may work given the skew along a log x-axis. I do not have enough competence in R to do this however as my knowledge only extends to fitdistr which so far as I understand would not work here.

Any help or guidance in a relevant direction would be most appreciated.

## Answers

If you are not satisfied with the segmented package, you can try the earth package with the **mars** algorithm. But here, I find that the result of the segmented model is very acceptable. see the R-Squared below.

lin.mod <- lm(y2~x) segmented.mod <- segmented(lin.mod, seg.Z = ~x, psi=c(0.0000001,0.000001)) summary(segmented.mod) Meaningful coefficients of the linear terms: Estimate Std. Error t value Pr(>|t|) (Intercept) -2.163e+02 1.143e+02 -1.893 0.0637 . x 4.743e+10 3.799e+09 12.485 <2e-16 *** U1.x -5.360e+10 3.824e+09 -14.017 NA U2.x 6.175e+09 4.414e+08 13.990 NA Residual standard error: 232.9 on 54 degrees of freedom Multiple R-Squared: 0.9468, Adjusted R-squared: 0.9419 Convergence attained in 5 iterations with relative change 3.593324e-14

You can check the result by plotting the model :

plot(segmented.mod)

To get the coefficient of the plots , you can do this:

intercept(segmented.mod) $x Est. intercept1 -216.30 intercept2 3061.00 intercept3 46.93 > slope(segmented.mod) $x Est. St.Err. t value CI(95%).l CI(95%).u slope1 4.743e+10 3.799e+09 12.4800 3.981e+10 5.504e+10 slope2 -6.177e+09 4.414e+08 -14.0000 -7.062e+09 -5.293e+09 slope3 -2.534e+06 5.396e+06 -0.4695 -1.335e+07 8.285e+06