Who am I?

Daniel Kastinen

• Living in Kiruna since 2017
• Finished PhD Nov. 2022
  From Meteors to Space Safety: Dynamical Models and Radar Measurements of Space Objects
• Hired as permanent staff scientist February 2023
• Task: Conduct research on meteors and space objects!

My focus: connecting the small and big pictures

• Radar data analysis & signal processing
• Meteoroid and small body dynamical simulations
• Statistical methods and solving inverse problems
• Meteor phenomena and meteoroid stream formation modelling
• (now also) Optical data processing and hardware control
• Space object observations, orbit determination, sensor simulations

This is a WHOLE lot of software development

And so this course was born

• Next time we will cover some motivation
• Now let us get programming!
• Please open up https://danielk.developer.irf.se/phd-course-software-development/ And go to "Handouts > 0. Development environment"

Lets get programming!

Calculating $\pi$ with randomness

• Probability is
  $\mathbb{P}(\bar{x} \in A) = \int_{A} f_X(\bar{x})\mathrm{d}\bar{x}$
  where $\bar{x} \sim X$
• Uniform 2d random distribution on $S = [-1, 1] \times [-1, 1]$
  $f_\mathcal{U}(x, y) = \mathbf{1}_S(x, y) \frac{1}{\text{Area}} = \mathbf{1}_S(x, y)\frac{1}{4}$
• What is $\mathbb{P}(\sqrt{x^2 + y^2} \leq 1)$?
• $= \int_{\text{Circle}} \mathbf{1}_S(x, y)\frac{1}{4} \mathrm{d}\bar{x} = \frac{1}{4} A_{\text{Circle}} = \frac{\pi}{4}$

$$\mathbb{P}(\sqrt{x^2 + y^2} \leq 1) = \frac{\pi}{4}$$

Calculating $\pi$ with randomness

$$\mathbb{P}(\sqrt{x^2 + y^2} \leq 1) = \frac{\pi}{4}$$

$$ \lim_{\# \mapsto \infty} 4\frac{\# \text{number of points in circle}}{\# \text{number of points}} \mapsto \pi $$

                    $ pip install numpy
                

Calculating $\pi$ with randomness

                    $ pip install numpy
                

                    import numpy as np

samples = 100
x = np.random.rand(samples)*2 - 1
y = np.random.rand(samples)*2 - 1

r = np.sqrt(x**2 + y**2)
pi = 4*np.sum(r < 1)/samples

print(f"{samples=} gives {pi=} (error {np.pi - pi})")
                    
                

                    $ python hello.py
                

Try samples = 1_000_000

Homework