In this short post I want to quickly demonstrate how the most basic neural network (no hidden layer) gives us the same results as the linear model.

First we need data

data ( swiss )
str ( swiss )

```
## 'data.frame': 47 obs. of 6 variables:
## $ Fertility : num 80.2 83.1 92.5 85.8 76.9 76.1 83.8 92.4 82.4 82.9 ...
## $ Agriculture : num 17 45.1 39.7 36.5 43.5 35.3 70.2 67.8 53.3 45.2 ...
## $ Examination : int 15 6 5 12 17 9 16 14 12 16 ...
## $ Education : int 12 9 5 7 15 7 7 8 7 13 ...
## $ Catholic : num 9.96 84.84 93.4 33.77 5.16 ...
## $ Infant.Mortality: num 22.2 22.2 20.2 20.3 20.6 26.6 23.6 24.9 21 24.4 ...
```

We can now specify a model.

m 1 <- formula ( Fertility ~ Agriculture + Examination + Education + Catholic + Infant.Mortality )

Let’s start by estimating a linear model.

lm1 <- lm ( formula = m 1 , data = swiss )
summary ( lm1 )

```
##
## Call:
## lm(formula = m1, data = swiss)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.2743 -5.2617 0.5032 4.1198 15.3213
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 66.91518 10.70604 6.250 1.91e-07 ***
## Agriculture -0.17211 0.07030 -2.448 0.01873 *
## Examination -0.25801 0.25388 -1.016 0.31546
## Education -0.87094 0.18303 -4.758 2.43e-05 ***
## Catholic 0.10412 0.03526 2.953 0.00519 **
## Infant.Mortality 1.07705 0.38172 2.822 0.00734 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.165 on 41 degrees of freedom
## Multiple R-squared: 0.7067, Adjusted R-squared: 0.671
## F-statistic: 19.76 on 5 and 41 DF, p-value: 5.594e-10
```

Now lets use the built in `nnet`

package to estimate a neural network without a hidden layer.

library ( nnet )
nn1 <- nnet ( formula = m 1 , data = swiss , size = 0 , skip = TRUE , linout = TRUE )

```
## # weights: 6
## initial value 66124.628241
## iter 10 value 2105.042930
## iter 10 value 2105.042930
## iter 10 value 2105.042930
## final value 2105.042930
## converged
```

summary ( nn1 )

```
## a 5-0-1 network with 6 weights
## options were - skip-layer connections linear output units
## b->o i1->o i2->o i3->o i4->o i5->o
## 66.92 -0.17 -0.26 -0.87 0.10 1.08
```

Let look at the results next to each other.

lm1
summary ( nn1 )

```
##
## Call:
## lm(formula = m1, data = swiss)
##
## Coefficients:
## (Intercept) Agriculture Examination
## 66.9152 -0.1721 -0.2580
## Education Catholic Infant.Mortality
## -0.8709 0.1041 1.0770
##
## a 5-0-1 network with 6 weights
## options were - skip-layer connections linear output units
## b->o i1->o i2->o i3->o i4->o i5->o
## 66.92 -0.17 -0.26 -0.87 0.10 1.08
```

We can also estimate this model using the `neuralnet`

package (this package needs to be installed first using `install.packages("neuralnet")`

). This package allows us to nicely visualise the results.

library ( neuralnet )

`## Loading required package: grid`

`## Loading required package: MASS`

nn1a <- neuralnet ( formula = m 1 , data = swiss , hidden = 0 , linear.output = TRUE )
plot ( nn1a , rep = 'best' )

Of course it only becomes interesting if we include a hidden layer.

nn2a <- neuralnet ( formula = m 1 , data = swiss , hidden = 2 )
plot ( nn2a , rep = 'best' )