The hard part is to decide
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”
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:
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 \[K = \frac{β_1}{0.020 \cdot 9.8}\]