---
title: "Ad Analysis"
author: "Gustavo Cepparo and Milica Cudina"
output:
  pdf_document: default
---

First, we download the data set and take a look at it. 

```{r}
data<-read.csv("salesadv.csv",header=TRUE)
attach(data)
#View(data)
plot(data[,-4],
     col="blue", pch=20)
```

Just for laughs, let's fit a multiple linear regression. 

```{r}
lm.fit.m=lm(sales ~ adv + adv1, data=data)
summary(lm.fit.m)
lm.fit.mi=lm(sales ~ adv*adv1, data=data)
summary(lm.fit.mi)
```
We are going to focus on just the fit with `adv` as a single predictor. So, let's look at the simple linear regression. 

```{r}
plot(adv, sales, 
     pch=20, col="blue",
     main="Dependence of sales on advertising",
     xlab="Advertising",
     ylab="Sales")
lm.fit<-lm(sales ~ adv)
abline(lm.fit, col="red", lwd=2)
summary(lm.fit)
```

Since there appears to be a difference in the "signal" in the vicinity of $50$ in `adv`, it's time to consider linear splines. First, we import the library `splines`.

```{r}
library(splines)
```

Then, we do create a fit with `degree=1` - meaning that it's a linear spline. Note, that we specify a single knot at $50$. 

```{r}
lm.fit.ls<-lm(sales ~ bs(adv,knots=c(50),degree=1),data=data)
summary(lm.fit.ls)
```

The $R^2$ is better. What about the visuals? 

```{r}
adv.mesh=seq(from=min(adv),to=max(adv), by=0.5)
predictions=predict(lm.fit.ls,newdata=list(adv=adv.mesh),se=T)

plot(adv, sales, 
     pch=20, col="lightblue",
     main="Dependence of sales on advertising",
     xlab="Advertising",
     ylab="Sales")

lines(adv.mesh,predictions$fit,col="red", lwd=2)
lines(adv.mesh,predictions$fit+2*predictions$se, col="blue")
lines(adv.mesh,predictions$fit-2*predictions$se, col="blue")
```

Of course, we might be interested in the basis functions used in this implementation. 


```{r}
## returns a basis matrix
b <- bs(adv.mesh, degree = 1, knots = c(50))  
#b

#the functions in the basis
b1 <- b[, 1]  
b2 <- b[, 2]  
plot(adv.mesh, b1, type = "l")
plot(adv.mesh, b2, type = "l")

#the coefficients from the summary of `lm`
(betas=summary(lm.fit.ls)$coeff[,1])

plot(adv, sales, 
     pch=20, col="lightblue",
     main="Dependence of sales on advertising",
     xlab="Advertising",
     ylab="Sales")

lines(adv.mesh,betas[1]+betas[2]*b1+betas[3]*b2,col="red", lwd=2)
```