Lecture 12: Simple Linear Regression

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# Lecture 12: Simple Linear Regression

## Introducing (revisiting) a less-simple model

### Data 100 Spring 2021

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# What is a *model*?

> A useful simplification of reality.

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## The modeling process: 3 steps

### I. Choose a model

- Constant model `$\hat{y} = \theta$`
  - `$\hat{y}$` is a prediction, `$\theta$` is a parameter

### II. Choose an objective function

- Loss functions
  - Squared loss: `$L_2(y, \hat{y}) = (y - \hat{y})^2$`
  - Absolute loss: `$L_1(y, \hat{y}) = |y - \hat{y}|$`

### III. Fit the model by optimizing your objective function

- Analytical approach
  - Calculus

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## Modeling tip rate

.right-column[
<img src="lec-12-1_files/figure-html/unnamed-chunk-3-1.png" width="504" />
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.left-column[
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class: animated, fadeIn
## Modeling tip rate

.right-column[
<img src="lec-12-1_files/figure-html/unnamed-chunk-4-1.png" width="504" />
]

.left-column[
]

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class: animated, fadeIn
## Modeling tip rate

.right-column[
<img src="lec-12-1_files/figure-html/unnamed-chunk-5-1.png" width="504" />
]

.left-column[
**Constant Model**
with L2 loss

`$$\hat{y} = .157$$`
]

---
class: animated, fadeIn
## Modeling tip rate

.right-column[
<img src="lec-12-1_files/figure-html/unnamed-chunk-6-1.png" width="504" />
]

.left-column[
**Constant Model**
with L2 loss

`$$\hat{y} = .157$$`
]

---
class: animated, fadeIn
## Modeling tip rate

.right-column[
<img src="lec-12-1_files/figure-html/unnamed-chunk-7-1.png" width="504" />
]

.left-column[
**Constant Model**
with L2 loss

`$$\hat{y} = .157$$`
]

---
class: animated, fadeIn
## Modeling tip rate

.right-column[
<img src="lec-12-1_files/figure-html/unnamed-chunk-8-1.png" width="504" />
]

.left-column[
**Constant Model**
with L2 loss

`$$\hat{y} = .157$$`
]

---
class: animated, fadeIn
## Modeling tip rate

.right-column[
<img src="lec-12-1_files/figure-html/unnamed-chunk-9-1.png" width="504" />
]

.left-column[
**Constant Model**
with L2 loss

`$$\hat{y} = .157 \\
MSE = .00211$$`
]

---
class: animated, fadeIn
## Modeling tip rate

.right-column[
<img src="lec-12-1_files/figure-html/unnamed-chunk-10-1.png" width="504" />
]

.left-column[
**Constant Model**
with L2 loss

`$$\hat{y} = .157 \\
MSE = .00211$$`

*Can we improve the prediction?*
]

---
## Modeling tip rate

.right-column[
![](lec-12-1_files/figure-html/unnamed-chunk-11-1.gif)
]

.left-column[
**Constant Model**
with L2 loss

`$$\hat{y} = .157 \\
MSE = .00211$$`

*Can we improve the prediction?*
]

---
class: animated, fadeIn
## Modeling tip rate

.right-column[
<img src="lec-12-1_files/figure-html/unnamed-chunk-12-1.png" width="504" />
]

.left-column[
**Constant Model**
with L2 loss

`$$\hat{y} = .157 \\
MSE = .00211$$`

*Can we improve the prediction?*
]

---
class: animated, fadeIn
## Modeling tip rate

.right-column[
<img src="lec-12-1_files/figure-html/unnamed-chunk-13-1.png" width="504" />
]

.left-column[
**Constant Model**
with L2 loss

`$$\hat{y} = .157 \\
MSE = .00211$$`

*Can we improve the prediction?*

**Linear Model**
with ? loss

`$$\hat{y} = \theta_0 + \theta_1 x$$`
]

---
class: animated, fadeIn
## Modeling tip rate

.right-column[
<img src="lec-12-1_files/figure-html/unnamed-chunk-14-1.png" width="504" />
]

.left-column[
**Constant Model**
with L2 loss

`$$\hat{y} = .157 \\
MSE = .00211$$`

*Can we improve the prediction?*

**Linear Model**
with ? loss

`$$\hat{y} = \theta_0 + \theta_1 x$$`

*Did it help?*

]

---
class: animated, fadeIn
## Modeling tip rate

.right-column[
<img src="lec-12-1_files/figure-html/unnamed-chunk-15-1.png" width="504" />
]

.left-column[
**Constant Model**
with L2 loss

`$$\hat{y} = .157 \\
MSE = .00211$$`

*Can we improve the prediction?*

**Linear Model**
with ? loss

`$$\hat{y} = \theta_0 + \theta_1 x$$`

*Did it help?*
]

---
class: animated, fadeIn
## Modeling tip rate

.right-column[
<img src="lec-12-1_files/figure-html/unnamed-chunk-16-1.png" width="504" />
]

.left-column[
**Constant Model**
with L2 loss

`$$\hat{y} = .157 \\
MSE = .00211$$`

*Can we improve the prediction?*

**Linear Model**
with ? loss

`$$\hat{y} = \theta_0 + \theta_1 x$$`

*Did it help?*
]

---
## Comparing Models

.pull-left[
### Constant Model

`$$\hat{y} = \theta$$`

**Useful**
  - Desc: average tip
  - Pred: `$MSE = .0021$`
  
**Simple**
  - 1 parameter
  
<img src="lec-12-1_files/figure-html/unnamed-chunk-18-1.png" width="144" style="display: block; margin: auto;" />

]

.pull-right[
### Simple Linear Model

`$$\hat{y} = \theta_0 + \theta_1 x$$`

**Useful**
  - Desc: negative assoc.
  - Pred: `$MSE = ?$`
  
**Simple**
  - 2 parameters
  
<img src="lec-12-1_files/figure-html/unnamed-chunk-19-1.png" width="144" style="display: block; margin: auto;" />

]