Overall Least Squares Problem 1
Task 1 – On paper, write down each of the component terms that make up 𝑦, as well as their associated
parameters that need to be identified for each section.
𝜃 𝑥 + 𝜃2 ,
𝑦={ 1
𝜃3 𝑥 + 𝜃4 ,
𝑥 ≤ 𝑥𝐵
𝑥 > 𝑥𝐵
Task 2 – What is the constraint required to ensure that the model is continuous?
𝑦𝐵 = 𝜃1 𝑥𝐵 + 𝜃2 = 𝜃3 𝑥𝐵 + 𝜃4
∴ 𝜃4 = 𝑦𝐵 − 𝜃3 𝑥𝐵
Task 3 – Solve for the parameters 𝜽 and write out the full equation for 𝑦.
6.932 × 104
𝜽 = [ −0.0526 4 ]
1.568 × 10
80.52
Task 4 – Plot the least-squares error to the segmentation index.
Task 5 –Plot the identified model versus measured data. Is it a good fit?
Task 6 –When does the yielding point occur (in terms of stress/strain)?
(𝑥𝐵 , 𝑦𝐵 ) = (0.0015 m/m, 104 MPa)
Overall Least Squares Problem 2
Task 1 – On paper, write down each of the component terms that make up 𝑦, as well as their associated
parameters that need to be identified for each section. (The model should be continuous already).
𝑦={
𝜃1 𝑥 + 𝜃2 ,
𝜃1 𝑥 + 𝜃2 + 𝜃3 √𝑥 − 𝑥𝐵 ,
𝑥 ≤ 𝑥𝐵
𝑥 > 𝑥𝐵
Task 2 – Solve for the parameters 𝜽 and write out the full equation for 𝑦.
0.1762
𝜽 = [ 2.224 ]
4.915
Task 3 – Plot the least-squares error to the segmentation index.
Note: The plot’s y-axis is logarithmic.
Task 4, 5 – Plot the identified model versus measured data. Is it a good fit? Complete the other side of
the “pitchfork” by flipping the sign of the identified parameter that is proportional the square-root
term. Plot this new function.
Overall Least Squares Problem 3
Task 1 – On paper, write down each of the component terms that make up 𝑦, as well as their associated
parameters that need to be identified for each section.
1
𝜃 + 𝜃2 ,
𝑦 = { 1𝑥
𝜃3 𝑥 + 𝜃4 ,
𝑥 ≤ 𝑥𝐵
𝑥 > 𝑥𝐵
Task 2 – What is the constraint required to ensure that the model is continuous?
𝑦𝐵 = 𝜃1
1
+ 𝜃2 = 𝜃3 𝑥𝐵 + 𝜃4
𝑥𝐵
∴ 𝜃4 = 𝑦𝐵 − 𝜃3 𝑥𝐵
Task 3 – Solve for the parameters 𝜽 and write out the full equation for 𝑦.
1.975
2.005
𝜽=[
]
−1.341
5.333
Task 4 – Plot the least-squares error to the segmentation index.
Task 5 –Plot the identified model versus measured data.