00. Introduction

Mathematical Statistical and Computational Foundations for Data Scientists

©2018 Raazesh Sainudiin. Attribution 4.0 International (CC BY 4.0)

  1. Introduction
  2. What is SageMath and why are we using it?
  3. Interaction - learning/teaching style
  4. What can you expect to get out of this course?


This is a course over three full-day workshops (3 hp) on May 11, 18 and 25 2018.

Prerequisites: current proficiency in high-school level mathematics (pre-calculus, geometry and algebra with some programming experience beyond Excel).

Target Audience: any MSc or PhD student at UU who wants to understand the mathematical statistical foundations in the data scientist’s computational toolbox. The approach will use formal mathematical communication of concepts starting from sets and logic, but with concomitant development of computer programming skills to algorithmically construct and implement the concepts. Topics will include: Sets, Maps, Functions, Modular Arithmetic, Axiomatic Probability, Conditional probability, Pseudo-random constructive understanding of random variables and structures including graphs, Statistics, Likelihood Principle, Bayes Rule, Decisions (parametric and non-parametric) including tests and estimators, Markov chains and their pseudorandom constructions, etc.

This course is designed to help you take the first steps along the mathematical-statistical-computational path of being a data scientist.

What is SageMath and why are we using it?

We will be using Sage or SageMath for our hands-on work in this course. Sage is a free open-source mathematics software system licensed under the GPL. Sage can be used to study mathematics and statistics, including algebra, calculus, elementary to very advanced number theory, cryptography, commutative algebra, group theory, combinatorics, graph theory, exact linear algebra, optimization, interactive data visualization, randomized or Monte Carlo algorithms, scientific and statistical computing and much more. It combines various software packages into an integrative learning, teaching and research experience that is well suited for novice as well as professional researchers.

Sage is a set of software libraries built on top of Python, a widely used general purpose programming language. Sage greatly enhance Python's already mathematically friendly nature. It is one of the languages used at Google, US National Aeronautic and Space Administration (NASA), US Jet Propulsion Laboratory (JPL), Industrial Light and Magic, YouTube, and other leading entities in industry and public sectors. Scientists, engineers, and mathematicians often find it well suited for their work. Obtain a more thorough rationale for Sage from Why Sage? and Success Stories, Testimonials and News Articles. Jump start your motivation by taking a Sage Feature Tour right now!

Interaction - learning/teaching style

This is an interactive jupyter notebook with SageMath interpreter and interactive means...


We will embed relevant videos in the notebook, such as those from The Khan Academy or open MOOCs from google, facebook, academia, etc.


We will formally present mathematical and statistical concepts in the Notebook using Latex as follows:

$$ \sum_{i=1}^5 i = 1+2+3+4+5=15, \qquad \prod_{i=3}^6 i = 3 \times 4 \times 5 \times 6 = 360 $$$$ \binom{n}{k}:= \frac{n!}{k!(n-k)!}, \qquad \lim_{x \to \infty}\exp{(-x)} = 0 $$$$ \{\alpha, \beta, \gamma, \delta, \epsilon, \zeta, \mu,\theta, \vartheta, \phi, \varphi, \omega, \sigma, \varsigma,\Gamma, \Delta, \Theta, \Phi, \Omega\}, \qquad \forall x \in X, \quad \exists y \leq \epsilon, \ldots $$

Interactive Visualizations

We will use interactive visualisations to convey concepts when possible. See the Taylor approximation below for a given order.

In [8]:
x0  = 0
f   = sin(x)*e^(-x)
p   = plot(f,-1,5, thickness=2)
dot = point((x0,f(x=x0)),pointsize=80,rgbcolor=(1,0,0))
def _(order=[1..12]):
    ft = f.taylor(x,x0,order)
    pt = plot(ft,-1, 5, color='green', thickness=2)
    show(dot + p + pt, ymin = -.5, ymax = 1, figsize=[6,3])

Lab-Lecture Style of Teaching-Learning

We will write computer programs within code cells in the Notebook right after we learn the mathematical and statistical concepts.

Thus, there is a significant overlap between traditional lectures and labs in this course -- in fact these interactions are lab-lectures.

Live Data Explorations and Modeling

Let us visualize the CO2 data, fetched from US NOAA, and do a simple linear regression.

In [26]:
# Author: Marshall Hampton 
import urllib2 as U
import scipy.stats as Stat
from IPython.display import HTML
co2data = U.urlopen('ftp://ftp.cmdl.noaa.gov/ccg/co2/trends/co2_mm_mlo.txt').readlines()
datalines = []
for a_line in co2data:
    if a_line.find('Creation:') != -1:
        cdate = a_line
    if a_line[0] != '#':
        temp = a_line.replace('\n','').split(' ')
        temp = [float(q) for q in temp if q != '']
trdf = RealField(16)
def mauna_loa_co2(start_date = slider(1958,2010,1,1958), end_date = slider(1958, 2010,1,2009)):
    htmls1 = '<h3>CO2 monthly averages at Mauna Loa (interpolated), from NOAA/ESRL data</h3>'
    htmls2 = '<h4>'+cdate+'</h4>'
    sel_data = [[q[2],q[4]] for q in datalines if start_date < q[2] < end_date]
    c_max = max([q[1] for q in sel_data])
    c_min = min([q[1] for q in sel_data])
    slope, intercept, r, ttprob, stderr = Stat.linregress(sel_data)
    pretty_print(html(htmls1+htmls2+'<h4>Linear regression slope: ' + str(trdf(slope))
                      + ' ppm/year; correlation coefficient: ' + str(trdf(r)) + '</h4>'))
    show(list_plot(sel_data, plotjoined=True, rgbcolor=(1,0,0)) 
                   + plot(slope*x+intercept,start_date,end_date), 
                      xmin = start_date, ymin = c_min-2, axes = True, xmax = end_date, ymax = c_max+3, 
                      frame = False, figsize=[8,3])

We will use publicly available resources generously!

In [2]:
def showURL(url, ht=500):
    """Return an IFrame of the url to show in notebook with height ht"""
    from IPython.display import IFrame
    return IFrame(url, width='95%', height=ht) 

What can you expect to get out of this course?

While teaching SDS-2.2: Scalable Data Science from Atlantis, a fast-paced industrially aligned course in data science to research students at Uppsala University from various Departments in the Technical and Natural Sciences, I realized that the students have a significant variance in their mathemtical, statistical and computational backgrounds.

Most of the students of that course were able to learn and apply the methods and interpret the outputs of the models and methods on datasets. However, only those with a background in probability and statistics as well as computing were able to understand the models well enough to adapth them for the problem and dataset at hand - a crucial distinguishing skillset of a data scientist.

This course is nearly reverse-engineered from my experience in SDS-2.2 with the goal of making the mathematical, statistical and computational foundations reasonably strong for a data scientist who is fairly rusty on these interweaving foundations.

We will take a "middle path" to reach as many course participants (MSc students, PhD students, postdoctoral researchers, industrial employees), given their varied backgrounds across:

  • pure mathemtics,
  • computational biology,
  • industrial researchers and developers from several sectors,
  • Masters students from cultural geography, IT,
  • etc.

What is Data Science?

We will steer clear of academic/philosophical discussions on "what is data science?" and focus instead on the core skillset in mathematics, statistics and computing that is expected in a typical data science job today.

In [3]:
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