Jean - ESSENTIAL MATH FOR DATA SCIENCE: take control of your data with fundamental calculus, linear... algebra, probability, and statistics
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Master the math needed to excel in data science and machine learning. If youre a data scientist who lacks a math or scientific background or a developer who wants to add data domains to your skillset, this is your book. Author Hadrien Jean provides you with a foundation in math for data science, machine learning, and deep learning.
Through the course of this book, youll learn how to use mathematical notation to understand new developments in the field, communicate with your peers, and solve problems in mathematical form. Youll also understand whats under the hood of the algorithms youre using.
Learn how to:
- Use Python and Jupyter notebooks to plot data, represent equations, and visualize space transformations
- Read and write math notation to communicate ideas in data science and machine learning
- Perform descriptive statistics and preliminary observation on a dataset
- Manipulate vectors, matrices, and tensors to use machine learning and deep learning libraries such as TensorFlow or Keras
- Explore reasons behind a broken model and be prepared to tune and fix it
- Choose the right tool or algorithm for the right data problem
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by Hadrien Jean
Copyright 2019 OReilly Media. All rights reserved.
Printed in the United States of America.
Published by OReilly Media, Inc. , 1005 Gravenstein Highway North, Sebastopol, CA 95472.
OReilly books may be purchased for educational, business, or sales promotional use. Online editions are also available for most titles ( http://oreilly.com ). For more information, contact our corporate/institutional sales department: 800-998-9938 or corporate@oreilly.com .
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- Illustrator: Rebecca Demarest
- August 2020: First Edition
- 2019-10-10: First Release
See http://oreilly.com/catalog/errata.csp?isbn=9781098115562 for release details.
The OReilly logo is a registered trademark of OReilly Media, Inc. Essential Math for Data Science, the cover image, and related trade dress are trademarks of OReilly Media, Inc.
The views expressed in this work are those of the author(s), and do not represent the publishers views. While the publisher and the author(s) have used good faith efforts to ensure that the information and instructions contained in this work are accurate, the publisher and the author(s) disclaim all responsibility for errors or omissions, including without ...
In this first Chapter, you will learn about basic mathematical buildingblocks. We will start with variables and then use them to constructequations. Finally, we will see that equations can express functionalrelations.
In algebra, variables are letters (like a,b, x) representing numerical quantities. Wegenerally use a single letter with an italic typeface as variable names(such as x).
This allows us to formulate general rules that work for any values. Itis also useful to express relationships (through equations for instance)or to deal with unknowns (and eventually find their values).
Symbols representing unchanging numerical quantities (like for instance) are called mathematical constants orjust constants.
In computer programming, a variable is a reference to a piece of storedinformation. Usually, it is like a pointer to a specific location wherewe can access the information. Its name is used to reference the storedvalue.
In mathematics, the concept is similar. We want to be able to writegeneral rules that cover various situations. To do that we need a way ofrepresenting objects that can take various values.
Lets start with an example. Well work on a dataset giving the globaltemperatures averaged from thousands of meteorological stations aroundthe world. Here is a plot representing temperature anomalies (anomaliesare deviations from a reference):
# See https://github.com/jjrennie/GHCNpydata = pd.read_csv("data/ghcn-m-v1.csv")data1 = data.replace(-9999, np.NaN)data_month = data1.groupby(['year', 'month']).mean().reset_index()data_month_mean = data_month.iloc[:, 2:].mean(axis=1)data_year = data1.groupby(['year']).mean().reset_index()data_year_mean = data_year.iloc[:, 2:].mean(axis=1)plt.figure()plt.plot(np.arange(data_year_mean.shape[0]) + 1880, data_year_mean / 100)plt.xlabel('Date (Year)')plt.ylabel('Temperature anomalies (degree Celsius)')Text(0, 0.5, 'Temperature anomalies (degree Celsius)')image::images/output_12_1.png
We have two variables here, the date and the temperature. Lets call thetemperature variable T. We can see that, for example, in1953 T=-0.32 while in 2003 T=0.19. Thisshows that the value associated with the variable T ischanging.
We can also consider the date as a variable. It changes as well.
Variables are very powerful because they can express ideas using objectsthat are not fixed. In our example, we can take the physical concept oftemperature, assign it a variable with a short name (T)for convenience, and characterize it mathematically. For instance, wecan say that T increases over time. If the exact relationbetween time and temperature was known we could have written an equationsynthetizing this relation (see next chapter on equations andinequalities).
Sometimes, you might encounter the name unknown to talk aboutvariables. Like variables, unknowns are letters standing for a valuethat we dont know. It is useful to create equations using values thatwe dont know. One goal can be to solve the equation and find theunknown value.
You can also encounter the terms dependent and independent variables.Dependent variables are the variable of interest, which depends on theother variables called independent variables. For instance, in thedataset expressing the temperatures as a function of time, we areinterested in the evolution of the temperatures. This is the dependentvariable. Independent variables are all the other variables that canaffect the dependent variable. When we look at the temperatures as afunction of time, we want to see the effect of time on the temperatureand not the opposite. The date is, in this case, an independentvariable. The dependent variable is usually plotted on the y-axis in acartesian plane (See ).
We can use variables to construct expressions like equations orinequalities. Equations can be used to find value of an unknown and toexpress relations between entities.
Equations express relations between elements through equality. Forinstance, we can have:
Or using variables:
which is the equation of a line defined by the parametersa and b.
Equations can use single variable, like in:
In this case, solving the equation means asking the question whatvalue of x gives 4 when this is multiply by 2?.
It is possible to write real-world problems as equations. The syntheticand expressive notation helps to solve problems.
Example
Sonja found a great recipe book, but it uses the metric system. Sheneeds to convert the quantities from grams to cups. She looked on theInternet and saw that 1 cup of flour corresponds to 120 grams of flour.From the recipe, she needs 200 grams of flour, so how many cups shouldshe use? Lets formulate this problem under mathematical notation. Shewants to know how many cups (the unknown, lets say x)multiplied by the weight of flour of each cup (120 grams) gives 200grams.
This is an equation. It means: what number gives 200 when we multiplyit by 120?. 120 is the weight in grams of 1 cup of flour. 200 is theweight in grams that we want to get. But what is x?x is the unknown: the numerical value that we dont knowin our problem. Solving the equation means finding the value ofx that satisfy the expression. This is the number of cupsSonja needs to get 200 grams of flour.
So, x120 corresponds to the total weight of flourSonja will use. Since one cup of flour corresponds to 120 grams shemultiplies the number of cups by the weight of one cup to get the totalweight. For instance, if x=1, she uses 1 cup of flour,thus 120 grams of flour but thats not enough. If x=2, sheuses 2 cups of flour: 1202=240. Thats too much.It looks like x is between 1 and 2.
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