# Positive coefficient regression in R

August 5, 2011
By

I’m currently working on a paper about simulating urban demands for electricity and gas at 5 minute resolution. To do this, I have a simple regression model that tries to explain observed consumption based on local population figures and simulated levels of activity demands (e.g. minutes spent at work, leisure, etc). The data set looks like this, where each of the letters is an activity code:

> head(data)
zone  pop     elec       gas  A    B   I    J  L M     O   R    S     W
1    0 7412 46221768 124613714  0    0   0    0  0 0     0   0    0     0
2    4 7428 37345875 100944002 60 3060 120 1020  0 0  4900 390  510 28635
3    7 7464 20914281  64109628  0 1155 510  255  0 0 11000 225 2475 29580
4   10 7412 46221768 124613714  0  680   0  390  0 0  5145   0    0  9300
5   14 7128 69233086  36611811  0 1335 105  210 60 0  5970   0 2520 14910
6   17 7608 40783190  59343776  0  150   0  150  0 0  1500   0    0   555

Created by Pretty R at inside-R.org

I then performed a basic regression using lm, removing the intercept (the “- 1″) as I want the population coefficient to serve a similar purpose for this analysis:

lm.elec <- lm(elec/365 ~ pop + A + B + I + J + L + M + O + R + S + W - 1, data)

Created by Pretty R at inside-R.org

Using Andrew Gelman’s helpful arm package, I can get a quick overview of the result as shown below. It doesn’t look too bad at first, but then I noticed all sorts of negative coefficients for the activity levels. This makes no sense: when individuals perform activities such as going to work, going to school, or shopping, we expect that their demand for electricity should go up, not down.

> display(lm.elec)
lm(formula = elec/365 ~ pop + A + B + I + J + L + M + O + R +
S + W - 1, data = data)
coef.est coef.se
pop    13.99     1.35
A     177.07    83.87
B     -63.90    27.80
I      -0.80    21.05
J      91.27    34.19
L   -1075.76   391.53
M      -5.57    73.50
O      55.47     9.90
R    -336.14   178.98
S      28.12     3.84
W      -8.19     3.16
---
n = 391, k = 11
residual sd = 159994.17, R-Squared = 0.65

Created by Pretty R at inside-R.org

Since a linear regression is essentially an optimization problem, my immediate thought was: can I just constrain the coefficient values so that they are all positive? This would mean that some activities might have no significant effect on consumption, but at least they couldn’t have a negative impact. And it turns out, yes, you can do this using the nnls package and function.

The nnls function is not quite as user-friendly as lm so the first thing you have to do is manually define your input variable matrix and output vector. For example:

A <- as.matrix(data[,c("pop","A","B","I","J","L","M","O","R","S","W")])
b.elec <- data\$elec/365

The analysis can then be run as:

nnls.elec <- nnls(A,b.elec)

Similarly, we can’t use predict to generate our results and have to manually perform the matrix multiplication. This can be done as shown below.

coef.elec <- coef(nnls.elec)
pred.elec.nnls <- as.vector(coef.elec%*%t(A))

This method also doesn’t give you an r2 value per se, but you can estimate it with the following dummy regression:

> lm.elec.dummy <- lm(b.elec~pred.elec.nnls - 1)
> display(lm.elec.dummy)
lm(formula = b.elec ~ pred.elec.nnls - 1)
coef.est coef.se
pred.elec.nnls 1.00     0.04
---
n = 391, k = 1
residual sd = 164861.13, R-Squared = 0.62

As you can see, the r2 value is slightly lower in this case than the standard lm model but in terms of interpreting the coefficients the result makes much more sense. This can be clearly seen in the following graph, where demands for electricity and gas rise during the day as expected, rather than sinking in the basic case.

Simulated profiles for electricity and gas, comparing lm and nnls regressions

So there you go. If you ever need to run a regression and ensure that all the coefficients are greater than or equal to zero, nnls is your friend.

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