# colf" In colf: Constrained Optimization on Linear Function

## colf

This is a package dedicated to performing a least squares constrained optimization on a linear objective function. The functions minimize the same objective function as lm, applying a constraint on the beta parameters:

$$S(\beta) = \sum_{i=1}^m \vert y_i - \sum_{j=1}^nX_{ij}\beta_j \vert^2 = \Vert y - X\beta\Vert^2$$

And

$$\hat{\beta} = arg_\beta min \ S(\beta)$$ under the constraints:

$$lower \le \hat{\beta} \le upper$$

The idea behind the package is to give the users a way to perform a constrained "linear regression" in an easy and intuitive way. The functions require a formula in the same syntax and format as lm which is a style most R users are familiar with.

So far the package includes two functions in order to perform the constrained optimization:

• colf_nls - uses the port algorithm which comes from the stats::nls function.
• colf_nlxb - uses Nash's variant of Marquardt nonlinear least squares solution which comes from the nlsr::nlxb function.

You can find more details about the two algorithms if you have a look at ?nls and ?nlxb respectively.

## colf_nls

Now we will see how we can easily use the port algorithm to perform a constrained optimization. As you will see we are using colf_nls in the same way we would use lm with the addition of upper and lower bounds for our parameter estimates.

We will use the mtcars data set for a demonstration. Let's load the package and use mtcars to run a constrained least squares optimization model.

In the model below we use 4 variables to model mpg which means we will have 5 parameter estimates (don't forget the Intercept). Parameters are prefixed with param_ in the model's output. We set the lower bounds of those 4 parameter estimates to -2 and the upper bounds to 2 (obviously they do not need to be the same). Ideally, starting values should be provided. If omitted a cheap guess will be made, which is basically setting all starting values to 1. If the staring values do not fall within the boundaries defined by lower and upper then an error will be returned and you would need to manually change the starting values via the start argument.

#### Usage

library(colf)
mymod <- colf_nls(mpg ~ cyl + disp + hp + qsec, mtcars, lower = rep(-2, 5), upper = rep(2, 5))
mymod


As you can see all 5 parameter estimates fall within the defined boundaries. The above provided formula includes the Intercept. In the output, X.Intercept is a variable set to 1 and param_X.Intercept is the estimated intercept.

If starting values do not fall within the boundaries an error will be returned. As said previously if not provided they will be set to 1.

{R, error = TRUE} colf_nls(mpg ~ cyl + disp + hp + qsec, mtcars, lower = rep(-2, 5), upper = rep(0.5, 5))

So, then they need to be set by the user:

{R}
colf_nls(mpg ~ cyl + disp + hp + qsec, mtcars, lower = rep(-2, 5), upper = rep(0.5, 5),
start = rep(0, 5))


#### Alternative ways to define the formula

As with lm, colf_nls accepts the same kind of formula syntax:

#no intercept
colf_nls(mpg ~ 0 + hp + cyl, mtcars)

colf_nls(mpg ~ ., mtcars)

colf_nls(mpg ~ I(hp + cyl), mtcars)

colf_nls(mpg ~ (hp + cyl + disp)^3, mtcars)

colf_nls(mpg ~ hp:cyl, mtcars)

colf_nls(mpg ~ hp * cyl, mtcars)


Notice that when the above versions are used, the parameter names are created with the use of make.names in order to be syntactically valid (otherwise the optimizers fail). This is why you see an 'X.' in front of the intercept or too many dots in the names.

#### Predict and rest of Methods

colf provides a number of methods for colf objects:

• predict - uses parameter estimates to predict on a new data set
• coef - retrieve the coefficients
• resid - retrieve the residuals
• print - print the model
• summary - view a summary of the model
• fitted - retrieve the fitted values

In order to use the parameter estimates to make predictions on a new data set you need to remember two really important checks:

• The new data set needs to contain exactly the same column names as the original one
• The new data set's columns should have exactly the same column classes

If any of the two is not valid, predict will fail.

set.seed(10)
newdata <- data.frame(hp = mtcars$hp, cyl = mtcars$cyl, disp = mtcars$disp, qsec = mtcars$qsec)
predict(mymod, newdata)


But if I change any of the names or classes predict will fail

{R, error = TRUE}

# change column name

newdata2 <- newdata names(newdata2) <- 'col1' predict(mymod, newdata2)

# change column class

newdata2 <- newdata newdata2$cyl <- as.character(newdata2$cyl)
predict(mymod, newdata2)

The rest of the colf_nls methods are demonstrated below:

You need to be careful when using summary because it returns p-values. By default nls and
nlxb both return p-values for the coefficients, which were naturally passed on to colf. When
running an unconstrained regression the p-values show us how likely it is for the estimate to be
zero. In constrained regression though this may not even hold if you think that a restriction (and
actually a common one) is to force the coefficients to be positive. In such a case the hypothesis
test does not hold at all since we have restricted the coefficients to be positive. In constrained
regression other assumptions that we make in unconstrained regression do not hold either (like
the coefficients' distribution) so the use and interpretation of the p-values can be problematic
when we set lower and/or upper.

{R}
summary(mymod)

coef(mymod)
print(mymod)
resid(mymod)
fitted(mymod)


## colf_nlxb

colf_nlxb can be used in the exact same way as colf_nls. All aspects / features discussed about colf_nls do stand for colf_nlxb as well. Only the underlying algorithm changes.

#### Usage

mymod <- colf_nlxb(mpg ~ cyl + disp + hp + qsec, mtcars, lower = rep(-2, 5), upper = rep(2, 5))
mymod


Setting lower, upper and starting values:

{R, error = TRUE}

# start values are outside boundaries

colf_nlxb(mpg ~ cyl + disp + hp + qsec, mtcars, lower = rep(-2, 5), upper = rep(0.5, 5))

{R}
#so they need to be provided
colf_nlxb(mpg ~ cyl + disp + hp + qsec, mtcars, lower = rep(-5, 5), upper = rep(.5, 5),
start = rep(0, 5))


#### Alternative ways to use formula, similar to lm:

#no intercept
colf_nlxb(mpg ~ 0 + hp + cyl, mtcars)
colf_nlxb(mpg ~ ., mtcars)
colf_nlxb(mpg ~ I(hp + cyl), mtcars)
colf_nlxb(mpg ~ (hp + cyl + disp)^3, mtcars)
colf_nlxb(mpg ~ hp:cyl, mtcars)
colf_nlxb(mpg ~ hp * cyl, mtcars)


#### Predict and rest of the methods

set.seed(10)
newdata <- data.frame(hp = mtcars$hp, cyl = mtcars$cyl, disp = mtcars$disp, qsec = mtcars$qsec)
predict(mymod, newdata)


As with colf_nls, in colf_nlxb keeping names and classes the same is vital:

{R, error = TRUE}

# change column name

newdata2 <- newdata names(newdata2) <- 'col1' predict(mymod, newdata2)

# change column class

newdata2 <- newdata newdata2$cyl <- as.character(newdata2$cyl)
predict(mymod, newdata2)

Rest of methods provided:

Please make sure you read the section about the interpretation of the p-values at colf_nls when
running a constrained regression. The same principles described there hold for colf_nlxb.

{R}
summary(mymod)

coef(mymod)
print(mymod)
resid(mymod)
fitted(mymod)


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colf documentation built on May 2, 2019, 8:23 a.m.