• Multiple Linear Regression is similar to simple linear regression but it can have multiple variables which then contribute to the output.
• The format for multiple linear regression is \(y = b_0 + b_1x_1 +b_2x_2 + ... + b_nx_n\).
• \(b_0\) is the y-intercept and all the other b’s are the “weights” assigned to that variable. \(x_1\) to \(x_n\) are the column values.

• The first step for multiple linear regression is exploring the data including head, summary, str etc.
• After exploring the data and figuring out the information, data visualizations can help to know more about the data.

• I explored the rest of the data and those two variables seemed to have the most impact on the price; however, since color is a categorical value I will label encode it so it works with multiple linear regression.
• After the data can be used to build the model, I will graph it with plotly. Although more than two variables can be used to predict the price, I will only use two to create a plot in 3 dimensions.

# Encode the data
diamonds$color_encoded <- as.numeric(factor(diamonds$color))
# Build the model
mlr <- lm(price ~ carat + color_encoded, data = diamonds)

# Predictions
diamonds$predicted_price <- predict(mlr)

A summary of the model is available at the end of the presentation.

There are multiple ways to improve this model for example: - We can increase the amount of variables without a problem to make it more accurate; however, this has to be done with caution since it is prone to over fitting. - We can check for patterns in the errors to see if it has some kind of bias in some variables, and we can modify it to adjust for these. - Remove outliers in the modifying the data part, we can use standard deviation to remove \(|x| > 3\) since outliers can affect models in a negative way.

## 
## Call:
## lm(formula = price ~ carat + color_encoded, data = diamonds)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -18831.5   -806.6    -82.2    621.1  12030.6 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   -1576.860     16.684  -94.52   <2e-16 ***
## carat          8013.954     14.204  564.19   <2e-16 ***
## color_encoded  -246.228      3.958  -62.21   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1496 on 53937 degrees of freedom
## Multiple R-squared:  0.8594, Adjusted R-squared:  0.8594 
## F-statistic: 1.649e+05 on 2 and 53937 DF,  p-value: < 2.2e-16