This is the 4th post of blog post series ‘Data Science: The Complete Reference’, this post covers these topics related to data science introduction.
- What is Multivariate Calculus?
- Why Multivariate Calculus is important in Data Science?
- How Multivariate Calculus is applied in Data Science?
What is Multivariate Calculus?
Multivariate Calculus (also known as multivariable calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables, rather than just one. ~Wikipedia
Calculus is a set of tools for analyzing the relationship between functions and their inputs. In Multivariate Calculus we can take a function with multiple inputs and determine the influence of each of them separately.
Multivariable Calculus | Khan Academy
Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance…
Why Multivariate Calculus is important in Data Science?
In data science, we try to find the inputs which enable a function to best match the data. The slope or descent describes the rate of change off the output with respect to an input. Determining the influence of each input on the output is also one of the critical tasks. All this requires solid understanding of Multivariate Calculus.
Why is multivariable calculus important for data science?
Answer (1 of 2): The most important thing you will learn in multivariable calculus for data science is the gradient…
How Multivariate Calculus is applied in Data Science?
First, lets cover core concepts of Calculus:
An equation will be a function if, for any x in the domain of the equation, the equation will yield exactly one value of y when we evaluate the equation at a specific x.
y = f(x)
Calculus I - Functions
In this section we will cover function notation/evaluation, determining the domain and range of a function and function…
The derivative of f(x) with respect to x is the function f′(x) and is defined as:
Calculus I - The Definition of the Derivative
In this section we define the derivative, give various notations for the derivative and work a few problems…
If the two functions f(x) and g(x) are differentiable (i.e. the derivative exist) then the product is differentiable and:
Calculus I - Product and Quotient Rule
In this section we will give two of the more important formulas for differentiating functions. We will discuss the…
Suppose that we have two functions f(x) and g(x) and they are both differentiable.
- If we define F(x)=(f∘g)(x) then the derivative of F(x) is:
- If we have y=f(u) and u=g(x) then the derivative of y is:
Calculus I - Chain Rule
In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule. With the…
If F(x) is any anti-derivative of f(x) then the most general anti-derivative of f(x) is called an indefinite integral and denoted:
c is any constant
Calculus I - Indefinite Integrals
In this section we will start off the chapter with the definition and properties of indefinite integrals. We will not…
Given a function f(x) that is continuous on the interval [a,b] we divide the interval into n subintervals of equal width (Δx) and from each interval choose a point, x∗i. Then the definite integral of f(x) from a to b is:
Calculus I - Definition of the Definite Integral
In this section we will formally define the definite integral, give many of its properties and discuss a couple of…
A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).
Calculus III - Partial Derivatives
In this section we will the idea of partial derivatives. We will give the formal definition of the partial derivative…
Now, lets look at the core concepts of Multivariate Calculus & how those relate to Data Science:
Gradients show changes along a particular variable or set of variables as a function “move” in space, and they are very easy to compute within optimization frameworks. The algorithms find minima/maxima of these functions and base optimization on these estimates.
Why is the gradient so commonly referenced in machine learning? What is it the gradient of? What's…
Answer (1 of 4): One minimizes the so called "loss" function, which is a measure of how far off the model is in…
The Jacobian of a set of functions is a matrix of partial derivatives of the functions. If you have just one function instead of a set of function, the Jacobian is the gradient of the function.
What is the Jacobian, how does it work, and what is an intuitive explanation of the Jacobian and a…
Answer (1 of 8): Change of basis: You're close with both your pictures. I prefer the first one. The perspective we like…
Hessian is, in some crude sense, rate of change of rate of change of the function. In regards to optimization, Hessian being positive definite, negative definite or indefinite tells you about whether the optima in question is a maxima, minima or a saddle point.
What is the Hessian matrix? What is it used for and for what reason?
Answer (1 of 14): Hessian is, in some crude sense, rate of change of rate of change of the function. For example, if…
Multivariate Chain Rule
Suppose that z=f(x,y), where x and y themselves depend on one or more variables. Multivariable Chain Rules allow us to differentiate z with respect to any of the variables involved:
Multivariable chain rule, simple version
The chain rule for derivatives can be extended to higher dimensions. Here we see what that looks like in the relatively…
A function approximation problem asks us to select a function among a well-defined class that closely matches (“approximates”) a target function.
Statistical and connectionist approaches to machine learning are related to function approximation methods in mathematics.
By function approximation, we describe a surface that separates the objects into different regions. The simplest…
Common techniques include the Taylor series and the Fourier series approximations. Recall that given enough terms, a Taylor series can approximate any function to a certain level of precision about a given point, while a Fourier series can approximate any periodic function.
Taylor and Fourier series are the same
An interesting relationship between the coefficients can be obtained from this observation. Recall that the…
A power series is any series that can be written in the form where a and cn are numbers. The cn’s are often called the coefficients of the series. The first thing to notice about a power series is that it is a function of x.
In data science, Power Series can be used to give you some indication of the size of the error, that results from using these approximations.
Calculus II - Power Series
In this section we will give the definition of the power series as well as the definition of the radius of convergence…
Linearisation is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest.
Linearization - Wikipedia
In mathematics, linearization is finding the linear approximation to a function at a given point. The linear…
A Multivariate Taylor series is an idea used in data science and other kinds of higher-level mathematics. It is a series that is used to create an estimate (guess) of what a function looks like with multiple inputs.
How do I derive a Taylor series for multivariable functions?
Answer (1 of 2): Begin with the definition of a Taylor series for a single variable, which states that for small enough…
Multivariate Taylor is the Power Series to its more general multivariate form.