## Linear and Nonlinear Models in R

This post will talk about building linear and non-linear models of trip rates in R.  If you haven't read the first part of this series, please do so, partly because this builds on it.

# Simple Linear Models

Simple linear models are, well, simple in R.  An example of a fairly easy linear model with two factors is:

`inTab.hbsh`

`This creates a simple linear home-based-shopping trip generation model based on workers and household size.  Once the estimation completes (it should take less than a second), the summary should show the following data:`

`> summary(hbsh.lm.W_H)`

`Call:`
`lm(formula = N ~ Workers4 + HHSize6, data = hbsh)`

`Residuals:`
`    Min      1Q  Median      3Q     Max `
`-2.2434 -1.1896 -0.2749  0.7251 11.2946 `

`Coefficients:`
`            Estimate Std. Error t value Pr(>|t|)    `
`(Intercept)  1.79064    0.10409  17.203  < 2e-16 ***`
`Workers4    -0.02690    0.05848  -0.460    0.646    `
`HHSize6      0.24213    0.04365   5.547 3.58e-08 ***`
`---`
`Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 `

`Residual standard error: 1.649 on 1196 degrees of freedom`
`Multiple R-squared: 0.03228,    Adjusted R-squared: 0.03066 `
`F-statistic: 19.95 on 2 and 1196 DF,  p-value: 3.008e-09`

`What all this means is:`

`Trips = -0.0269*workers+0.24213*HHSize+1.79064`

`The important things to note on this is that the intercept is very significant (that's bad) and the R2 is 0.03066 (that's horrible).  There's more here, but it's more details.`

```Non-Linear Least Squares
```

`When doing a non-linear model, the nls function is the way to go.  The two lines below create a trips data frame, and then run a non-linear least-squares model estimation on it (note that the first line is long and wraps to the second line).`

`trips=3),`
`start=c(a=1,b=1),trace=true)`

`The second line does the actual non-linear least-squares estimation.  The input formula is T=a*e^(HHSize+b).  In this type of model, starting values for a and b have to be given to the model.`

`The summary of this model is a little different:`

`> summary( trips.hbo.nls.at3p)`

`Formula: T.HBO ~ a * log(HHSize6 + b)`

`Parameters:`
`  Estimate Std. Error t value Pr(>|t|)    `
`a   1.8672     0.1692  11.034  < 2e-16 ***`
`b   1.2366     0.2905   4.257 2.58e-05 ***`
`---`
`Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 `

`Residual standard error: 2.095 on 402 degrees of freedom`

`Number of iterations to convergence: 4 `
`Achieved convergence tolerance: 1.476e-07 `

`It doesn't perform R2 on this because it can't directly.  However, we can because we know the actual values and the model predicts the values.  So, one thing that can be done is a plot:`

`> plot(c(0,10),c(0,10),type='l',xlab='Observed Trips',ylab='Predicted Trips')`
`> points(subset(trips,AreaType>=3)\$T.HBO,fitted(trips.hbo.nls.at3p),col='red')`

`The resulting graph looks like this.  Not particularly good, but there is also no scale as to the frequency along the 45° line.`

`R2 is still a good measure here.  There's probably an easier way to do this, but this way is pretty simple.  `

`testTable=3)\$T.HBO,fitted(trips.hbo.nls.at3p)))`
`cor(testTable\$X1,testTable\$X2)`

`Since I didn't correct the column names when I created the data frame, R used X1 and X2, as evidenced by checking the summary of testTable:`

`> summary(testTable)`
`       X1               X2       `
` Min.   : 0.000   Min.   :1.503  `
` 1st Qu.: 1.000   1st Qu.:1.503  `
` Median : 2.000   Median :2.193  `
` Mean   : 2.072   Mean   :2.070  `
` 3rd Qu.: 3.000   3rd Qu.:2.193  `
` Max.   :23.000   Max.   :3.696  `

`So the R2 value is pretty bad...`

`> cor(testTable\$X1,testTable\$X2)`
`[1] 0.2755101`

`It's better than some of the others, after all, this is semirandom human behavior.`

`That's it for now.  My next post will be... MORE R!`

`Also, I have a quick shout-out to Jeremy Raw at FHWA for help via email related to this.  He helped me through some issues via email, and parts of his email helped parts of this post.`