Polynomial regression is a type of regression analysis in which the relationship between the independent variable x and the dependent variable y is modeled as an nth-degree polynomial. In R, you can perform polynomial regression using the `lm()`

function, which fits a linear model.

Here’s an example of how to perform polynomial regression in R:

Suppose we have the following data:

```
x <- c(1, 2, 3, 4, 5)
y <- c(2, 6, 9, 10, 12)
```

We can fit a second-degree polynomial regression model using the `lm()`

function as follows:

```
model <- lm(y ~ poly(x, 2, raw=TRUE))
```

In this case, `poly(x, 2, raw=TRUE)`

creates a matrix of the predictors, where the columns are `x`

raised to the power of 0, 1, and 2 (i.e., the intercept, x, and x^2). The `raw=TRUE`

argument specifies that the predictors should not be standardized.

We can then use the `summary()`

function to obtain the model summary:

```
summary(model)
```

This will output a summary of the model, including the coefficients, standard errors, t-values, and p-values for each predictor.

We can also use the `predict()`

function to make predictions based on the model:

```
new_x <- seq(1, 5, length.out=100)
new_y <- predict(model, newdata=data.frame(x=new_x))
```

This will generate 100 new values of `x`

and use the model to predict the corresponding values of `y`

.

Finally, we can use the `ggplot2`

package to visualize the data and the fitted model:

```
library(ggplot2)
df <- data.frame(x, y, new_x, new_y)
ggplot(df, aes(x, y)) +
geom_point() +
geom_line(aes(x=new_x, y=new_y), color="blue")
```

This will create a scatter plot of the data points, overlaid with a blue line representing the fitted model.

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