The hard part is to decide

- What is the dependent variable
- The values that we want to predict
- The vertical axis

- What are the independent variables
- The values that we can control
- The horizontal axis

- What is the formula connecting them

Here we are using a linear model \[y=β_0 + β_1\cdot x\] In the example of last class, we have \[\text{time} = β_0 + β_1\cdot \text{volume}\]

\(β_0\) and \(β_1\) are the *coefficients* of the
model

**Which are the best \(β_0\)
and \(β_1\) values?**

We find the best coefficients by minimizing the squared error \[\begin{aligned}
β_0 &= \overline{𝐲} - β_1 \overline{𝐱}\\
β_1 &= \frac{\text{cov}(𝐱, 𝐲)}{\text{var}(𝐱)}
\end{aligned}\] Finding the best coefficients is called
*“fitting the model to the data”*

- Coils and rubber bands have a natural size
- If you apply a force to them, they
*expand* - What is the relation between the
*expansion*and the*force*?

Robert Hooke (1635–1703) was an English *natural philosopher*,
architect and *polymath*.

In 1660, Hooke discovered the law of elasticity which describes the
linear variation of *tension* with extension

**“The extension is proportional to the
force”**

**Natural philosophy** was the study of nature and the
physical universe that was dominant before the development of modern
science

**Polymath** (from Greek “having learned much”) is a
person whose expertise spans a significant number of different subject
areas

**Biologist.** Hooke used the microscope and was the
fists to use the term *cell* for describing biological
organisms.

The *essence* of the coil is:

- It has a natural length \(L\)
- If we change the length by \(x\),
it
*pulls*with a force \[\text{force}(x)= K \cdot (L-x)\]

n_balls | length | repetition |
---|---|---|

0 | 78.00 | 1 |

1 | 82.61 | 1 |

2 | 85.85 | 1 |

3 | 90.26 | 1 |

4 | 95.05 | 1 |

0 | 79.21 | 2 |

2 | 85.55 | 2 |

The table shows only part of the data.

Get all data at http://dry-lab.org/static/2017/rubber1.txt

Remember that *straight lines* can be represented by the
formula \[\text{length} = β_0 + β_1 \cdot
\text{n_balls}\] The coefficient \(β_0\) is the value where the line
*intercepts* the vertical axis

The coefficient \(β_1\) is *how
much* **length** goes up when
**n_marbles** increases. This is called *slope*

The formula from Hooke’s Law is “\(\text{force}=K\cdot(L-\text{length})\)”.

Since **force** is the weight of the balls, we can write
\[-m
g\cdot\text{n_balls}=K\cdot(L-\text{length})\] which can be
re-written as \[\text{length}=\underbrace{L}_{β_0}+\underbrace{\frac{m
g}{K}}_{β_1}\cdot\text{n_balls}\]

When there are no balls, the length of the coil is \(L\), in this case \[L = β_0\] If we assume that the mass of each ball is 20gr, we can find \(K\) as \[K = \frac{β_1}{0.020 \cdot 9.8}\]