``` R
### start with assigning lm.fit to the model:
lm.fit = lm(XVAL~YVAL, data = datavar)
#1.  lm.fit provides “basic” information *such as* Coefficients
lm.fit
```
Below is an example of the type of output information you can get calling lm.fit (from Pg. 111 of Introduction to Statistical Learning with Applications in R).


lm.fit: Output Basic Statistical Information in R 

- predict() returns y-hat (estimate) based on your training model. 
- This method produces the confidence and prediction intervals for the prediction of YVAL given a value of the XVAL. In other words, for each training set, we have corresponding x and y values. 
- Now we have a new data set and we are passing a vector with one column from that new data and we want to predict the prediction/confidence intervals for that new data value based on the training data model.
```R
predict(lm.fit, data.frame(XVAL=(c(new_data_row_value1, new_data_row_value2), interval = “”)
#Example: 
predict(lm.fit, data.frame(XVAL=(c(5, 10,15)), interval = “prediction”)
	#Above, data.frame is included as an optional data frame that looks for variables to predict. Without it, fitted values are used.
	#interval = “confidence” , “none”, “prediction” are all options
        ```

predict(): Predict New Data from the (Simple) Linear Model in R

R Tutorial Series: Simple Linear Regression. John M. Quick. (2009, november 26). Retrieved from [www.r-bloggers.com/r-tutorial-series-simple-linear-regression/](https://www.r-bloggers.com/r-tutorial-series-simple-linear-regression/) 

Linear Regression in R

In R, lm() is a function that can take many parameters. Common parameters include formula, which describes a model, and data, which contains a dataset. 
 

In simple linear regression in R, a formula parameter is written YVAR~XVAR, where XVAR is the predictor variable and YVAR is the response variable. 

A simple linear regression equation with data and formula parameters may look like:
``` R
lm(formula = YVAR~XVAR, data = datavar)
```


University of Michigan - Ann Arbor

Simple Linear Regression predicts a model for the relationship between a predictor matrix (independent variable) $X$ and the response (dependent variable) $Y$.
$$Y=\beta _1 X+\beta _0+\epsilon$$
The coefficients $\beta _1$ and $\beta _0$ are called slope and intercept, respectively. The estimated values for these coefficients are shown as $\hat{\beta _1}$ and $\hat{\beta _0}$.
$$\hat{Y}=\hat{\beta _1} X+\hat{\beta _0}$$

Simple Linear Regression

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

Simple linear regression commonly uses a least squares approach, where the linear model attempts to minimize the residual distance between the actual points and the predicted line. Consequently, its residual is defined as:
$$e_i = y_i -\hat{y}_i$$

where $y_i$ represents the actual ith response value and $\hat{y}_i$ represents the ith response value of the linear model. 

Least Squares Approach

```python
#imports 
import numpy as np 
from sklearn.linear_model import LinearRegression

#make up data 
array_x = np.array([5, 10, 15, 20, 25]).reshape((-1, 1))
#calling reshape makes sure that the array is 2D 
array_y = np.array([5, 3, 15, 23, 6])

#create a model 
model_xy = LinearRegression()

#using model 
mode.fit(array_x, array_y)

#getting results 
rsq_val = model_xy(array_x, array_y)
intercept = model_xy.intercept_
slope = model_xy.coef_

#predicting a response 
y_pred = model_xy.predict(x)

```

Python: Simple Linear Regression 

Simple Linear Regression in R: lm()

Link: $\href{http://www.sthda.com/english/articles/40-regression-analysis/165-linear-regression-essentials-in-r/}{sthda}$
The image references a simple linear regression line, where the 'e' represents the residual error, 'b0' represents the y-intercept, and 'b1' represents the change in Y (response value) for one unit increase in X.

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