suppressPackageStartupMessages({
  library(tidyverse)
  library(scales)
  library(glue)
  library(datarium)
})

1 Linear Regression

Linear regression is a simplest form of machine learning allowing to build a linear model based on the availble data and use it to predict outcomes for the new (previously unseen) observations.

1.1 One dimensional linear regression

set.seed(42)

df <- tibble(x = seq(0,10,length.out = 100)) %>% 
  mutate(y = rnorm(n = 100, mean = 2, sd =0.5)   *   x +  rnorm(100, mean = 10, sd = 5))   # adding some noise 

df
df %>% 
  ggplot(aes(x, y))+
  geom_point(size  = 1)+
  geom_smooth(method = "lm", se = FALSE)+
  geom_line(aes(y  = 2*x + 10), color = "red")+
  theme_bw()+
  labs("One dimensional linear regression")

1.2 The math behind linear regression

How do we find the equation of the linear regression?

The definition:

Linear regression is a line of the form \(y = mx+b\) that minimizes the mean squared error of the prediction

For an observation \(x_i\), the prediction error is defined as

\[ e(i) = y_i-y_{i,pred} = y_i - (mx_i+b) \]

The squared error is:

\[ e^2(i) = (y_i-y_{i,pred})^2 = (y_i - (mx_i+b))^2 \] Mean squared error (MSE) across all \(N\) observations

\[ MSE = \frac{1}{N}\sum_{i=1}^{N}e^2(i) = \frac{1}{N}\sum_{i=1}^{N} (y_i-y_{i,pred})^2 = \sum_{i=1}^{N}(y_i - mx_i - b)^2 \] The linear regression coefficients are then found by finding the mimimum of MSE

\[ m,b \leftarrow \arg \min \sum_{i=1}^{N}(y_i - mx_i - b)^2 \]

Finding the solution requires some calculus, but the end result is given by

\[ m = \frac{\sum_{i=1}^{N}(x_i-\overline x)(y_i-\overline y)}{\sum_{i=1}^{N}(x_i-\overline x)^2} = \frac{Cov(x,y)}{Var(x)} = r_{xy}\frac{Var(y)}{Var(x)} \] \[ b = \overline y - m\overline x \]

Luckily it is very easy to find linear regression using R.

fit <- lm(y~x, data  = df)

summary(fit)

Call:
lm(formula = y ~ x, data = df)

Residuals:
     Min       1Q   Median       3Q      Max 
-11.8128  -3.8642   0.3551   2.8431  12.0895 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   8.8910     1.0201   8.716 7.35e-14 ***
x             2.1445     0.1762  12.168  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.139 on 98 degrees of freedom
Multiple R-squared:  0.6017,    Adjusted R-squared:  0.5977 
F-statistic: 148.1 on 1 and 98 DF,  p-value: < 2.2e-16
m <- round(fit$coefficients[2],2)
b <- round(fit$coefficients[1],2)

So from the noisy data we obtained the following line 2.14x+8.89

R-squared (Pearson’s correlation coefficient) is a simple way to determine the quality of the prediction. The closer R-squared is to 1 the better linear regression represents the data.

2 How to we predict the result for new observations?


y_pred <- predict(fit, tibble(x = 1:100))

tibble(x = 1:100,
       y = 2*x + 10,
       y_pred = y_pred) %>% 
  ggplot(aes(x, y))+
  geom_line(color = "red")+
  geom_point(aes(y  =y_pred), color = "blue")

NA
NA

When not to use linear regression :)

picture picture

3 Multidimensional linear regression

The idea is to use linear combination of multiple variables (predictors) to predict the outcome:

\[ \hat y = b + m_1x_1+m_2x_2+...+m_nx_n \] where \(b\) is the intercept (or bias), \(m_1,...,m_n\) are linear coefficients and \(x_1,...,x_n\) are predictors (or independent variables).

For this section we will be using marketing dataset from datarium package.

From marketing help page: A data frame containing the impact of three advertising medias (youtube, facebook and newspaper) on sales. Data are the advertising budget in thousands of dollars along with the sales. The advertising experiment has been repeated 200 times.

data("marketing")

head(marketing)

First let’s explore the data by plotting the data

marketing %>% 
  pivot_longer(cols = c(youtube, facebook, newspaper), names_to = "media", values_to = "budget") %>% 
  ggplot(aes(x = budget,  y = sales, color = media)) +
  geom_point()+
  geom_smooth(se = FALSE, method = "lm")+
  facet_wrap(~media, scales = "free")+
  theme_bw()+
  labs(title = "Sales vs Advertisement Budget - Univariate Analysis",
       x = "Budget [$, thousands]", y = "Sales [$, millions]")

We can see a relatively good correlation between sales and youtube media, some correlation with facebook and not as much correlation with newspaper advertising.

The challenge is that all three media cotribute to sales together and even though we can plot them seperately, the effect is compound and it is difficlut to understand the impact of each media channel separately.

3.1 Building a multivariate regression model

Technically it is possible to you all data to fit our model we will only use 75% of the data for fitting the model and then use the remaining 25% for testing how the model performs. This is generally an accepted approach in training any machine learning model.

set.seed(42)

N <- nrow(marketing)

in_train <- sample(seq(1,N), size = N * 0.75)  # make sure N *0.75 is integer, otherwise need to round the result

train <- marketing %>% 
  slice(in_train)

test <- marketing %>% 
  slice(-in_train)

  
nrow(train)
[1] 150
nrow(test)
[1] 50
fit <- lm(sales ~ youtube + facebook + newspaper, data = train)

summary(fit)

Call:
lm(formula = sales ~ youtube + facebook + newspaper, data = train)

Residuals:
     Min       1Q   Median       3Q      Max 
-10.1290  -0.9734   0.4990   1.4681   3.2090 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 3.655491   0.458705   7.969 4.19e-13 ***
youtube     0.046252   0.001673  27.647  < 2e-16 ***
facebook    0.175388   0.010508  16.691  < 2e-16 ***
newspaper   0.001930   0.007118   0.271    0.787    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.121 on 146 degrees of freedom
Multiple R-squared:  0.886, Adjusted R-squared:  0.8836 
F-statistic: 378.1 on 3 and 146 DF,  p-value: < 2.2e-16

3.2 Interpreting the model

We can see a relatively good fit with R squared equal to 0.89. We can also see that youtube and facebook are relatively good predictors (three astersiks means good), while newspaper is a poor predictor.

Basically we can see that spending an additional 1000 dollars on facebook advertising leads to an increase in sales by approximately 175 thousand dollars. The youtube coefficient suggests that for every 1000 dollars increase in budget we can get an increase of 46 thousand dollars, on average.

Since the newspaper does not contribute much to the sales prediction, it can be removed from the analysis, as well as from the marketing strategy. The updated model will look like this

fit <- lm(sales ~ youtube + facebook, data = train)

summary(fit)

Call:
lm(formula = sales ~ youtube + facebook, data = train)

Residuals:
     Min       1Q   Median       3Q      Max 
-10.1963  -0.9979   0.4783   1.4798   3.2449 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 3.695331   0.433163   8.531 1.64e-14 ***
youtube     0.046268   0.001667  27.761  < 2e-16 ***
facebook    0.176391   0.009805  17.989  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.115 on 147 degrees of freedom
Multiple R-squared:  0.8859,    Adjusted R-squared:  0.8844 
F-statistic: 570.7 on 2 and 147 DF,  p-value: < 2.2e-16

\[ sales = 3.7 + 0.046 * youtube + 0.176 * facebook \] (here sales is expressed in millions and facebook/youtube is the advertising budget in thousands)

3.3 Testing the accuracy of the model on the test data

pred <- predict(fit, test)

df <- test %>% 
  mutate(predicted = pred,
         pred_error = predicted/sales - 1) 

summary(df)
    youtube          facebook       newspaper          sales         predicted        pred_error      
 Min.   :  4.92   Min.   : 2.40   Min.   :  2.64   Min.   : 3.84   Min.   : 6.378   Min.   :-0.27067  
 1st Qu.: 80.55   1st Qu.:11.25   1st Qu.: 16.47   1st Qu.:12.63   1st Qu.:11.462   1st Qu.:-0.08124  
 Median :168.30   Median :25.08   Median : 31.20   Median :14.64   Median :15.468   Median :-0.01436  
 Mean   :165.75   Mean   :27.78   Mean   : 36.76   Mean   :16.44   Mean   :16.264   Mean   : 0.00790  
 3rd Qu.:259.14   3rd Qu.:43.89   3rd Qu.: 48.03   3rd Qu.:19.68   3rd Qu.:19.818   3rd Qu.: 0.04947  
 Max.   :340.32   Max.   :59.52   Max.   :136.80   Max.   :30.60   Max.   :28.331   Max.   : 0.66102  
df %>% 
  ggplot(aes(x = sales, y = pred_error))+
  geom_point()+
  scale_y_continuous(labels = scales::percent, breaks = pretty_breaks(6))+
  theme_bw()

---
title: "Linear Regression"
author: "Alexander Lifshitz"
date: '`r format(Sys.time(), "%B %d, %Y %H:%M:%S", tz="America/Los_Angeles",usetz=TRUE)`' 
output:
  html_notebook:
    code_folding: hide
    number_sections: yes
    theme: cerulean
    toc: yes
---

```{r}
suppressPackageStartupMessages({
  library(tidyverse)
  library(scales)
  library(glue)
  library(datarium)
})
```

# Linear Regression

Linear regression is a simplest form of machine learning allowing to build a linear model based on the availble data and use it to predict outcomes for the new (previously unseen) observations. 


## One dimensional linear regression

```{r}
set.seed(42)

df <- tibble(x = seq(0,10,length.out = 100)) %>% 
  mutate(y = rnorm(n = 100, mean = 2, sd =0.5)   *   x +  rnorm(100, mean = 10, sd = 5))   # adding some noise 

df
```


```{r}
df %>% 
  ggplot(aes(x, y))+
  geom_point(size  = 1)+
  geom_smooth(method = "lm", se = FALSE)+
  geom_line(aes(y  = 2*x + 10), color = "red")+
  theme_bw()+
  labs("One dimensional linear regression")

```

## The math behind linear regression

How do we find the equation of the linear regression?  

*The definition:*

Linear regression is a line of the form $y = mx+b$ that minimizes the mean squared error of the prediction

For an observation $x_i$, the prediction error is defined as 

$$
e(i) = y_i-y_{i,pred} = y_i - (mx_i+b)
$$


The squared error is:

$$
e^2(i) = (y_i-y_{i,pred})^2 = (y_i - (mx_i+b))^2
$$
Mean squared error (MSE) across all $N$ observations

$$
MSE = \frac{1}{N}\sum_{i=1}^{N}e^2(i) = \frac{1}{N}\sum_{i=1}^{N} (y_i-y_{i,pred})^2 = \sum_{i=1}^{N}(y_i - mx_i - b)^2
$$
The linear regression coefficients are then found by finding the mimimum of MSE 

$$
m,b \leftarrow \arg \min \sum_{i=1}^{N}(y_i - mx_i - b)^2
$$

Finding the solution requires some calculus, but the end result is given by

$$
m  = \frac{\sum_{i=1}^{N}(x_i-\overline x)(y_i-\overline y)}{\sum_{i=1}^{N}(x_i-\overline x)^2} = \frac{Cov(x,y)}{Var(x)} = r_{xy}\frac{Var(y)}{Var(x)} 
$$
$$
b = \overline y - m\overline x
$$


Luckily it is very easy to find linear regression using R.


```{r}
fit <- lm(y~x, data  = df)

summary(fit)
```

```{r}
m <- round(fit$coefficients[2],2)
b <- round(fit$coefficients[1],2)
```

So from the noisy data we obtained the following line `r glue("{m}x+{b}")`



`R-squared` (Pearson's correlation coefficient) is a simple way to determine the quality of the prediction. The closer R-squared is to 1 the better linear regression represents the data.


# How to we predict the result for new observations?

```{r}

y_pred <- predict(fit, tibble(x = 1:100))

tibble(x = 1:100,
       y = 2*x + 10,
       y_pred = y_pred) %>% 
  ggplot(aes(x, y))+
  geom_line(color = "red")+
  geom_point(aes(y  =y_pred), color = "blue")


```


When not to use linear regression :)

![picture](https://imgs.xkcd.com/comics/linear_regression.png)
![picture](https://imgs.xkcd.com/comics/extrapolating.png)

# Multidimensional linear regression

The idea is to use linear combination of multiple variables (predictors) to predict the outcome:

$$
\hat y = b + m_1x_1+m_2x_2+...+m_nx_n
$$
where $b$ is the intercept (or bias), $m_1,...,m_n$ are linear coefficients and $x_1,...,x_n$ are predictors (or independent variables). 

For this section we will be using `marketing` dataset from `datarium` package.

From `marketing` help page: A data frame containing the impact of three advertising medias (youtube, facebook and newspaper) on sales. Data are the advertising budget in thousands of dollars along with the sales. The advertising experiment has been repeated 200 times.

```{r}
data("marketing")

head(marketing)
```



First let's explore the data by plotting the data

```{r, fig.width=12, fig.height=6}
marketing %>% 
  pivot_longer(cols = c(youtube, facebook, newspaper), names_to = "media", values_to = "budget") %>% 
  ggplot(aes(x = budget,  y = sales, color = media)) +
  geom_point()+
  geom_smooth(se = FALSE, method = "lm")+
  facet_wrap(~media, scales = "free")+
  theme_bw()+
  labs(title = "Sales vs Advertisement Budget - Univariate Analysis",
       x = "Budget [$, thousands]", y = "Sales [$, millions]")
```


We can see a relatively good correlation between sales and youtube media, some correlation with facebook and not as much correlation with newspaper advertising. 

The challenge is that all three media cotribute to sales together and even though we can plot them seperately, the effect is compound and it is difficlut to understand the impact of each media channel separately. 



## Building a multivariate regression model

Technically it is possible to you all data to fit our model we will only use 75% of the data for fitting the model and then use the remaining 25% for testing how the model performs. This is generally an accepted approach in training any machine learning model.

```{r}
set.seed(42)

N <- nrow(marketing)

in_train <- sample(seq(1,N), size = N * 0.75)  # make sure N *0.75 is integer, otherwise need to round the result

train <- marketing %>% 
  slice(in_train)

test <- marketing %>% 
  slice(-in_train)

  
```

```{r}
nrow(train)
nrow(test)
```

```{r}
fit <- lm(sales ~ youtube + facebook + newspaper, data = train)

summary(fit)
```

## Interpreting the model

We can see a relatively good fit with R squared equal to `r round(summary(fit)$r.squared,2)`.
We can also see that youtube and facebook are relatively good predictors (three astersiks means good), while newspaper is a poor predictor.

Basically we can see that spending an additional 1000 dollars on facebook advertising leads to an increase in sales by approximately 175 thousand dollars. The youtube coefficient suggests that for every 1000 dollars increase in budget we can get an increase of 46 thousand dollars, on average.

Since the newspaper does not contribute much to the sales prediction, it can be removed from the analysis, as well as from the marketing strategy.
The updated model will look like this

```{r}
fit <- lm(sales ~ youtube + facebook, data = train)

summary(fit)
```


$$
sales = 3.7 + 0.046 * youtube + 0.176 * facebook
$$
(here sales is expressed in millions and facebook/youtube is the advertising budget in thousands)


## Testing the accuracy of the model on the test data

```{r}
pred <- predict(fit, test)

df <- test %>% 
  mutate(predicted = pred,
         pred_error = predicted/sales - 1) 

summary(df)
```

```{r}
df %>% 
  ggplot(aes(x = sales, y = pred_error))+
  geom_point()+
  scale_y_continuous(labels = scales::percent, breaks = pretty_breaks(6))+
  theme_bw()
```

