Scientific Computing Maastricht Science Program Week 4 Frans - - PowerPoint PPT Presentation
Scientific Computing Maastricht Science Program Week 4 Frans Oliehoek <frans.oliehoek@maastrichtuniversity.nl> Recap Matlab...! Supervised Learning find f that maps {x 1 (j) ,...,x D (j) } y (j) Interpolation f
Frans Oliehoek <frans.oliehoek@maastrichtuniversity.nl>
Matlab...! Supervised Learning
find f that maps {x1
(j),...,xD (j)} → y(j)
Interpolation
f goes through the data points
linear regression
lossy fit, minimizes 'vertical' SSE
Unsupervised Learning
PCA We just have data points {x1
(j),...,xD (j)}
x2 x1
u=(u1,u2)
Finding derivatives or primitives of a function f not always easy or possible....
no closed form solution exists the solution is a very complex expression that is hard to
we may not know f (as before!)
If we want to know the rate of change... E.g.:
fluid in a cylinder with a hole in the bottom, measured
High-speed camera images of animal movements,
(jumping in frogs and insects, suction feeding in fish, and the strikes of mantis shrimp)
determine speed and acceleration
Determine the vertical speed at t=0.25 what would you do?
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 frog height(t)
Determine the vertical speed at t=0.25...
a few options...
0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.35 0.36 0.36 0.37 0.37 0.38 0.38 0.39 0.39 frog height(t)
Determine the vertical speed at t=0.25...
a few options...
0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.35 0.36 0.36 0.37 0.37 0.38 0.38 0.39 0.39 frog height(t)
Determine the vertical speed at t=0.25...
a few options...
0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.35 0.36 0.36 0.37 0.37 0.38 0.38 0.39 0.39 frog height(t)
forward finite difference backward finite difference
Determine the vertical speed at t=0.25...
a few options...
0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.35 0.36 0.36 0.37 0.37 0.38 0.38 0.39 0.39 frog height(t)
forward finite difference backward finite difference
Determine the vertical speed at t=0.25...
a few options...
0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.35 0.36 0.36 0.37 0.37 0.38 0.38 0.39 0.39 frog height(t)
Centered finite difference
Integration: the reversed problem... Suppose we travel in a car with a broken odometer Speedometer is working...
maintain speeds, to figure out traveled distance
t v(t) km/h 80 30 120 65 128 120 122 728 120 733 798 20 836 20 941 70 970 120 1350 123 1404 90
enter highway ramp exit highway ramp traffic jam
maintain speeds, to figure out traveled distance
t v(t) km/h 80 30 120 65 128 120 122 728 120 733 798 20 836 20 941 70 970 120 1350 123 1404 90
enter highway ramp exit highway ramp traffic jam
200 400 600 800 1000 1200 1400 1600 20 40 60 80 100 120 140 v(t) km/h
maintain speeds, to figure out traveled distance
t v(t) km/h 80 30 120 65 128 120 122 728 120 733 798 20 836 20 941 70 970 120 1350 123 1404 90
enter highway ramp exit highway ramp traffic jam
200 400 600 800 1000 1200 1400 1600 20 40 60 80 100 120 140 v(t) km/h
Approximate the integral with a finite sum
integration interval
integration interval
integration interval
Approximation of the integral:
k=1 M
integration interval
integration interval
Approximation of the integral:
k=1 M
Finally: when faced with
Finally: when faced with