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mathematics for machine learning

my name is richard kirschner with the

simply learn team that's get certified

get ahead

we're going to cover mathematics for

machine learning so today's agenda is

going to cover data and its types then

we're going to dive into linear algebra

and its concepts

calculus statistics for machine learning

probability for machine learning

hands-on demos

and of course thrown in there in the

middle is going to be your matrixes and

a few other things to go along with all

this

data in its types

data denotes the individual pieces of

factual information collected from

various sources it is stored processed

and later used for analysis

and so we see here just a huge grouping

of information a lot of tech stuff money

dollar signs numbers

and then you have your performing

analytics to drive insights and

hopefully you have a nice share your

shareholders gather it at the meeting

and you're able to explain it in

something they can understand

so we talk about data types of data

we have in our types of data we have a

qualitative categorical

you think nominal or ordinal

and then you have your quantitative or

numerical which is discrete or

continuous

and let's look a little closer at those

data type vocabulary always people's

favorite is the vocabulary words okay

not mine

but let's dive into this what we mean by

nominal

nominal they are used to label various

uh label our variables without providing

any measurable value

country gender race hair color etc

it's something that you either mark true

or false this is a label it's on or off

either they have a red hat on or they do

not

so a lot of times when you're thinking

nominal data

labels

think of it as a true false kind of

setup and we look at ordinal this is

categorical data with a set order or a

scale to it

and you can think of salary range as a

great one

movie ratings etc you can see here the

salary rates if you have 10 000 to 20

000 number of employees earning that

rate is 150 20 000 to 30

100 and so forth some of the terms

you'll hear is bucket

this is where you have 10 different

buckets and you want to separate it into

something that makes sense into those 10

buckets

and so when we start talking about

ordinal a lot of times when you get down

to the brass bones again we're talking

true false

so if you're a member of the 10 to 20k

reigns

so forth those would each be either part

of that group or you're not but now

we're talking about buckets so we want

to count how many people are in that

bucket quantitative numerical data

falls into two classes discrete or

continuous

and so data with a final set of values

which can be categorized class strength

questions answered correctly and runs

hit and cricket a lot of times when you

see this you can think integer

and a very restricted integer i.e

you can only have 100 questions

on a test so you can it's very discreet

i only have 100 different values that it

can attain so think usually you're

talking about integers but within a very

small range they don't have an open in

or anything like that

so discrete is very solid

simple to count

set number

continuous on the other hand uh

continuous data can take any numerical

value within a range so water pressure

weight of a person etc usually we start

thinking about float values where they

can get phenomenally small and they're

in what they're worth

and there's a whole series of values

that falls right between discrete and

continuous

you can think of the stock market you

have dollar amounts it's still discrete

but it starts to get complicated enough

when you have like you know jump in the

stock market from

525.33 cents to

580.67

there's a lot of point values in there

it'd still be called discrete but you

start looking at it as almost continuous

because it does have such a variance in

it now uh we talked about no we did we

went over nominal and ordinal

almost true false charts and we looked

at quantitative and numerical data which

will start to get into numbers discrete

you can usually a lot of times discrete

will be put into it could be put into

true false but usually it's not uh so we

want to address this stuff and the first

thing you want to look at is the very

basic which is your algebra so we're

going to take a look at linear algebra

you can remember back when your

euclidean geometry we have a line well

let's go through this we have a linear

algebra is the domain of mathematics

concerning linear equations

and the representations in vector spaces

and through matrixes i told you we're

going to talk about matrix is

uh so a linear equation

is simply

2x plus 4y minus 3z equals 10. very

linear 10x plus 12.4 y equals z and now

you can actually solve these two

equations by combining them

and that's we're talking about a linear

equation

in the vectors we have a plus b equals c

now we're starting to look at a

direction

and these values usually think of an x y

z plot

so each one is a direction

and the actual

distance of like a triangle a b is c

and then your matrix can describe all

kinds of things

i find matrixes

confuse a lot of people

not because they're particularly

difficult but because of the magnitude

and the different things they're used

for and a matrix is a chart

or a you know think of a spreadsheet but

you have your rows and your columns

and you'll see here we have a times b

equals c

very important to know your counts

so depending on how the math is being

done what you're using it for making

sure you have the same rows and number

of columns or a single number there's

all kinds of things that play in that

that can make matrixes confusing

but really has a lot more to do with

what domain you're working in are you

adding in

multiple polynomials where you have like

a x squared plus b y plus you know you

start to see that it can be very

confusing versus a very straightforward

matrix

and let's just go a little deeper into

these because these are such primary

this is what we're here to talk about is

these different math uh mathematical

computations that come up

so we're looking at linear equations

let's dig deeper into that one an

equation having a maximum order of one

is called a linear equation

so it's linear because when you look at

this we have ax plus b equals c which is

a one variable

we have two variable a x plus b y equals

c a x plus b y plus z c z equals d

and so forth but all of these

are to the power of one you don't see x

squared you don't see x cubed so we're

talking about linear equations that's

what we're talking about in their

addition

if you have already dived into say

neural networks you should recognize

this ax plus by plus cz

setup plus the intercept

uh which is basically your your neural

network each node adding up all the

different inputs and we can drill down

into that most common formula is your y

equals mx plus c

so you have your y

equals the m which is your slope

your x value plus c

which is your

y-intercept you kind of labeled it wrong

here

threw me for a loop but the the c would

be your y-intercept so when you set x

equal to zero y equals c and that's

that's your y-intercept right there

uh and that's they just had a reverse

value of y when x equals zero equals the

y-intercept which is c and your slow

gradient line which is your m so you get

your y equals two x plus three

and there's lots of easy ways to compute

this this way this is why we always

start with the most basic one when we're

solving one of these problems and then

of course the

one of the most important takeaways is

the slope gradient of the line

so the slope is very important that m

value

in this case we went ahead and solved

this

if you have y equals 2x plus 3 you can

see how it has a nice line graph here on

the right

so matrixes a matrix refers to a

rectangular representation of an array

of numbers arranged in columns and rows

so we're talking m rows by n columns

here a11 is denotes the element of the

first row in the first column similarly

a12 and it's really pronounced a11 in

this particular setup so it's a row one

column one a 12 is a

row one column two

first row and second column and so on

and there's a lot of ways to denote this

i've seen these as like a capital letter

a smaller case a for the top row or i

mean you can see where they can go all

kinds of different directions as far as

the value

you just take a moment to realize

there's need to be some designation as

far as what row it's in and what column

it's in

and we have our basic operations we have

addition so when you think about

addition you have uh

two matrixes of two by two and you just

add each individual number in that

matrix and then when you get to the

bottom you have uh in this case the

solution is twelve 10 plus 2 is 12 5

plus 3 is 8 and so on and the same thing

with subtraction

now again your counting matrix is you

want to check your

dimensions of the matrix

the shape you'll see shape come up a lot

in programming so we're talking about

dimensions we're talking about the shape

if the two shapes are equal

this is what happens when you add them

together or subtract them

and we have multiplication when you look

at the multiplication you end up with a

very uh a slightly different setup going

now

if we look at our last one we're uh uh

we're like why

this always gets to me when we get to

matrixes they don't really say why you

multiply matrixes

you know my first thought is one times

two four times three but if you look at

this we get one times two plus four

times three

one times three plus four times five

uh six times two plus three times three

six times three plus three times five if

you're looking at these matrixes uh

think of this more as an equation

and so we have if you remember we went

back up here for our multiple line

equations let's just go back up a couple

slides where we were looking at

two variables so this is a two variable

equation a x plus b y equals c

and this is a way to make it very quick

to solve these variables and that's why

you have the matrix and that's why you

do

the multiplication the way they do

and this is the dot product of uh one

times two

plus four times three

one times three plus four times five

six times two plus three times three

six times three plus three times five

and it gives us a nice little

14 23 21 and 33 over here which then can

be used and reduced down to a sample

formula as far as solving the variables

as you have enough inputs

and then in matrix operations when

you're dealing with a lot of matrixes

now keep in mind multiplying matrixes is

different than finding the product of

two matrixes okay so we're talking about

multiplication we're talking about

solving uh for equations when you're

finding the product you are just finding

one times two keep that in mind because

that does come up i've had that come up

a number of times where i am altering

data and i get confused as to what i'm

doing with it

uh transpose flipping the matrix over is

diagonal comes up all the time where you

have you still have 12 but instead of it

being 12 8 it's now 12 14

8 21 you're just flipping the columns

and the rows

and then of course you can do an inverse

changing the signs of the values across

this main diagonal

and you can see here we have the inverse

a to the minus 1 and ends up with

instead of 12 8 14 12 is now minus 22

minus 12.

vectors

vector just means we have

a value and a direction

and we have down four numbers here on

our vector

uh in mathematics a one dimensional

matrix is called a vector uh so

if you have your x plot and you have a

single value that values along the x

axis and it's a single dimension

if you have two dimensions you can think

about putting them on a graph you might

have x and you might have y and each

value denotes a direction and then of

course the actual distance is going to

be the hypothesis of that triangle and

you can do that with three dimensionals

x y and z

and you can do it all the way to nth

dimensions so when they talk about the k

means

for categorizing and how close data is

together they will compute that based on

the pythagorean theorem so you would

take

the square of each value add them all

together and find the square root and

that gives you a distance

as far as where that point is where that

vector exists or an actual point value

and then you can compare that point

value to another one it makes a very

easy comparison versus comparing 50 or

60 different numbers

and that brings us up to i gene vectors

and i gene values

hygiene vectors the vectors that don't

change their span while transformation

and i gene values the scalar values that

are associated to the vectors

conceptually

you can think of the vector as your

picture you have a picture it's uh

two dimensions x and y

and so when you do those two dimensions

and those two values or whatever that

value is

that is that point but the values change

when you skew it and so

if we take and we have a vector a

and that's a set value uh

b is um your is your you have a and b

which is your i gene vector two is the i

gene value so we're altering

all the values by two that means we're

maybe we're stretching it out one

direction making it tall uh if you're

doing picture editing

that's one of the places this comes in

but you can see when you're transforming

uh your different information how you

transform it is then your hygiene value

and you can see here vector after line

transit transition uh we have 3a a is

the hygiene vector 3 is the aging value

so a doesn't change that's whatever we

started with that's your original

picture and 3

is skewing it one direction and maybe

a b is being skewed another direction

and so you have a nice tilted picture

because you altered it by those by the

hygiene values

so let's go ahead and pull up a demo on

linear algebra and to do this i'm going

to go through my trusted anaconda into

my jupiter notebook

and we'll create a new

notebook called linear algebra

since we are working in python we're

going to use our numpy i always import

that as np or numpy array probably the

most popular

module for doing matrixes and things in

given that this is part of a series i'm

not going to go too much into numpy we

are going to go ahead and create two

different variables a for a numpy array

10 15 and b 29

we'll go ahead and run this and you can

see there's our two arrays 10 15 29 and

i went and added a space there in

between

so it's easier to read

and since it's the last line we don't

have to put the print statement on it

unless you want we can simply but we can

simply do a plus b so when i run this uh

we have 10 15 29 and we get 30 24 which

is what you expect 10 plus 20 15 plus 9

you could almost look at this addition

as being

just adding up the columns on here

coming down and if we wanted to do it a

different way we could also do a dot t

plus b dot t

remember that t flips them and so if we

do that we now get them uh we now have

30 24 going the other way

we could also do something kind of fun

there's a lot of different ways to do

this

as far as a plus b i can also do a plus

b

dot t

and you're going to see that that will

come out the same the 30 24 whether i

transpose a and b or transpose them both

at the end

and likewise we can very easily subtract

two vectors i can go a minus b

and we run that and we get minus 10 6

now remember this is the last line in

this particular section that's right not

to put the print around it

and just like we did before

we can transpose either the individual

or we can transpose the main setup and

then we get a minus 10 6 going the other

way

now we didn't mention this in our notes

but you can also do a scalar

multiplication

and just put down the scalar so you can

remember that

uh what we're talking about here is i

have

this

array here u

and if i go a

times u

we'll take the value 2 we'll multiply it

by every value in here so 2 times 30 is

60 2 times 15

and just like we did before

this happens a lot because when you're

doing matrixes you do need to flip them

you get 60 30 coming this way

so in numpy uh we have what they call

dot product

and uh with this this is in a two

dimensional vectors it is equivalent of

two matrix multiplication remember we

were talking about matrix multiplication

uh where it is the

well let's walk through it

we'll go ahead and start by defining two

numpy arrays we'll have uh 10 20 25 6 or

our u and our v uh and then we're going

to go ahead and do

if we take

the values

and if you remember correctly

an array like this would be 10 times 25

plus 20 times 6.

we'll go ahead and

print that

there we go

and then we'll go ahead and do the np

dot dot

of u comma

v

and we'll find when we do this we go and

run this

we're going to get 370

370. so this is a strain multiplication

where they use it to solve

linear algebra when you have multiple

numbers going across and so this could

be very complicated we could have a

whole string of different variables

going in here but for this we get a nice

value for our dot multiplication

and we did

addition earlier which is just your

basic addition

and of course the matrix you can get

very complicated on these or

in this case we'll go ahead and do

let's create two

complex matrixes

this one is a matrix of

you know 12 10 4 6 4 31. we'll just

print out a so you can see what that

looks like here's print

a

we print a out you can see that we have

a

2 by 3 layer matrix for a

and we can also put together

always kind of fun when you're playing

with print values

we could do something like this we could

go in here

there we go

we could print a we have it end with

equals a

run

and this kind of gives it a nice look

here's your matrix that's all this is

comma n means it just tags it on the end

that's all that is doing on there

and then we can simply add in what is a

plus b and you should already guess

because this is the same as what we did

before there's no difference

we do a simple vector addition we have

12 plus 2 is 14 10 plus 8 is 18 and so

on

and just like we did the matrix addition

we can also do a minus b

and do our matrix subtraction

and we look at this we have what 12

minus 2 is 10 10 minus 8 um where are we

oh there we go eight minus uh

confusing what i'm looking at i should

have reprinted out the original numbers

uh but we can see here 12 minus 2 is of

course 10 10 minus 8 is 2

4 minus 46 is minus 42 and so forth so

same as a subtraction as before we just

call it matrix subtraction it's

identical

now if you remember up here we had

scalar addition we're adding just one

number

to a matrix you can also do scalar

multiplication

and so simply if you have a single value

a and you have b which is your array we

can also do a times b when we run that

you can see here we have 2 times 4 is 8

5 times 4 is 20 and so forth you're just

multiplying the 4 across each one of

these values

and this is an interesting one that

comes up a little bit of a brain teaser

is matrix and vector multiplication

and so when we're looking at this

we are

just doing regular arrays it doesn't

necessarily have to be a numpy array we

have a

which has our

array of arrays and b which is a single

array and so we can from here

do the dot

a b

and this is going to return two values

and the first value is that is you could

say it's like

we're doing the

this array b array

first with a and then with a second one

and so it splits it up so you have a

matrix of vector multiplication and you

can mix and match

when you get into really complicated uh

back end stuff this becomes more common

because you're now you've got layers

upon layers of data and so you'll end up

with a matrix and a set of bolt uh

vector matrices do you want to multiply

now keep in mind

that if you're doing data science a lot

of times you're not looking at this this

is what's going on behind the scenes so

if you're in the scikit looking at sk

learn where you're doing linear

regression models

this is some of the math that's hidden

behind the scenes that's going on

other times you might find yourself

having to do part of this and manipulate

the data around so it fits right and

then you go back in and you run it

through the psi kit and if we can do

up here where we did a

matrix and vector multiplication we can

also do matrix two matrix multiplication

and if we run this where we have the two

matrixes

you can see we have a very complicated

array that of course comes out on there

for our dot

and just to reiterate it we have our

transpose a matrix which is your dot t

and so if we create a matrix a and we do

uh transpose it you can see how it flips

it from

5 10 15 20 25 30 to 5 15 25

10 20 30

rows and columns

and certainly with the math uh this

comes up a lot um it also comes up a lot

with xy plotting when you put in the pi

plot you have one format where they're

looking at pairs and numbers and then

they want all of x's and all y's so you

know the transpose is an important tool

both for your math and for plotting and

all kinds of things

another tool that we didn't discuss uh

is your identity matrix

uh and this one is more definition

the identity matrix

we have here one where we just did two

so it comes down as one zero zero one

one zero zero zero one zero it creates a

diagonal of one and what that is is when

you're doing your identities you could

be comparing

all your different features to the

different features and how they

correlate

and of course when you have feature one

compared to feature one to itself it is

always

one

where

usually it's between zero one depending

on how well correlates so when we're

talking about identity matrix that's

what we're talking about right here is

that you create this preset matrix and

then you might adjust these numbers

depending on what you're working with

and what the domain is

and then another thing we can do

to kind of wrap this up we'll hit you

with the most complicated

piece of this

puzzle here is an inverse

a matrix

and let's just go ahead and put the um

it's a lengthy description

let's go and put the description this is

straight out of the uh

the website for

numpy so given a square matrix a here's

our square matrix a which is 2 1 0 0 1 0

1 2 1. keep in mind 3 by 3 is square

it's got to be equal it's going to

return the matrix a inverse satisfying

dot a

a inverse so here's our matrix

multiplication

and then of course it equals the dot

yeah a inverse of a

with an identity shape of

a dot shaped zero this is just reshaping

the identity

that's a little complicated there uh so

we're going to have our here's our array

we'll go ahead and run this

and you can see what we end up with

is we end up with uh an array 0.5 minus

0.5 and so forth with our 2 1 1 going

down 2 1 0 0 1 0 1 2 1.

getting into a little deep on the math

understanding

when you need this is probably really is

is what's really important when you're

doing data science

versus

handwriting this out and looking up the

math and handwriting all the pieces out

you do need to know about the linear

algorithm inverse of a

so if it comes up you can easily pull it

up or at least remember where to look it

up we took a look at the algebra side of

it let's go ahead and take a look at the

calculus side of what's going on here

with the machine learning so calculus oh

my goodness and differential equations

you got to throw that in there because

that's all part of the

bag of tricks especially when you're

doing large neural networks but also

comes up in many other areas the good

news is most of it's already done for

you in the back end uh so when it comes

up you really do need to understand from

the data science not data analytics data

analytics means you're digging deep into

actually solving these math equations

and a neural network is just a giant

differential equation

so we talk about calculus

we're going to go ahead

and understand it by talking about cars

versus time and speed

so helps to calculate the spontaneous

rate of change

so suppose we plot a graph of the speed

of a car with respect to time

so as you can see here going down the

highway probably merged into the highway

from an on-ramp so i had to accelerate

so my speed went way up

uh stuck in traffic merged into the

traffic traffic opens up and i

accelerate again up to the speed limit

and uh maybe peter's off up there so you

can look at this as as

the speed versus time i'm getting faster

and faster because i'm continually

accelerating and if i hit the brakes you

go the other way

so the rate of change of speed with

respect of time is nothing but

acceleration how fast are we

accelerating

the acceleration is the area between the

star point of x and the end point of

delta x

so we can calculate a simple

if you had x and delta x we could put a

line there and that slope of the line is

our acceleration

now that's pretty easy when you're doing

linear algebra but i don't want to know

it just for that line in those two

points i want to know what across

the whole of what i'm working with

that's where we get into calculus

so we talk about the distance between x

and delta x it has to be the smallest

possible near to zero in order to

approximate the acceleration

uh so the idea is instead of i mean if

you ever did took a basic calculus class

they would draw bars down here and you

would divide this area up

let's go back up the screen you divide

this area of this time period up into

maybe 10 sections and you'd use that and

you could calculate the acceleration

between each one of those 10 sections

kind of thing

and then we just keep making that space

smaller and smaller until delta x is

almost

infinitesimally small

and so we get a function of a

equals a limit as h goes to 0 of a

function of a plus h minus a function of

a over h and that is you're

computing the slope of the line

we're just computing that slope and

they're smaller and smaller and smaller

samples

and that's what calculus is calculus is

the integral you can see down here we

have our nice uh integral sign looks

like a giant s

and that's what that means is that we've

taken this down to as small as we can

for that sampling

so we're talking about calculus we're

finding the area under the slope is the

main process in the integration

similar small intervals are made of the

smallest possible length of x plus delta

x where delta x approaches almost an

infinitesimally small space

and then it helps to find the overall

acceleration by summing up all the links

together

so we're summing up all the

accelerations from the beginning to the

end

and so here's our integral we sum of a

of x times d of x

equals a plus c

uh that is our basic calculus here

so when we talk about multivariate

calculus

multivariate calculus deals with

functions that have multiple variables

and you can see here we start getting

into some very complicated equations um

changing w over change of time

equals change of w over change of z

the differential of z to dx differential

of x to dt it gets pretty complicated uh

and it really translates into the

multivariate integration using double

integrals and so you have the the sum of

the sum of f of x of y of d of a equals

the sum from c to d and a to b of f of x

y d x d y equals

uh the sum of a to b sum of c to d of f

x y d y d x

understanding the very specifics of

everything going on in here and actually

doing the math is usually calculus 1

calculus 2 and differential equations so

you're talking about three full length

courses to dig into

and solve these math equations

what we want to take from here is we're

talking about calculus

we're talking about summing of all these

different slopes and so we're still

solving a linear

expression we're still solving

y equals m x plus b

but we're doing this for infinitesimally

small x's and then we want to sum them

up that's what this integral sign means

the sum of a of x d of x equals a plus c

and when you see these very complicated

multivariate differentiation using the

chain rule

when we come in here and we have the

change of w to the change of t equals

the change of w dz

uh and so forth

that's what's going on here that's what

these means we're basically looking for

the area under the curve which really

comes to

how is the change changing speeds going

up how is that changing and then you end

up with a multiple layer so if i have

three layers of neural networks how is

the third layer changing based on the

second layer changing which is based on

the first layer changing and you get the

picture here that now we have a very

complicated

multivariate integration

with integrals

the good news is we can solve this

mathematically and that's what we do

when you do neural networks and reverse

propagation

so the nice thing is that you don't have

to solve this on paper unless you're a

data analysis and you're working on the

back end of integrating these formulas

and building the script to actually

build them so we talk about applications

of calculus it provides us the tools to

build an accurate predictive model

so it's really behind the scenes we want

to guess at what the change of the

change of the change is

that's a little goofy i know i just

threw that out there it's kind of a meta

term but if you can guess how things are

going to change then you can guess what

the new numbers are

multivariate calculus explains the

change in our target variable in

relation to the rate of change in the

input variables

so there's our multiple variables going

in there

if one variable is changing how does it

affect the other variable

and then in gradient descent calculus is

used to find the local and global maxima

and this is really big

we're actually going to have a whole

section here on gradient descent because

it is

really i mean i talked about neural

networks and how you can see how the

different layers go in there but

gradient descent is one of the most key

things for trying to guess the best

answer to something

so let's take a look at the code behind

gradient descent

and uh before we open up the code let's

just do real quick

uh gradient descent

let's say we have a curve like this and

most common

is that this is going to represent your

error oops

error there we go error

hard to read there and i want to make

the error as low as possible

and so what i'm looking at it is i want

to find this line here

which is the

minimum value so we're looking for the

minimum and it does that by uh sampling

there

and then based on this it guesses it

might be someplace here and it goes hey

this is still going down

it goes here and then goes back over

here and then goes a little bit closer

and it's just playing a high low until

it gets to that spot that bottom spot

and so we want to minimize the error in

on the flip note you could also want to

be maximizing something you want to get

the best output of it uh that's simply

uh minus the value

so if you're looking for where the peak

is

this is the same as a negative

for where the valley is looking for that

valley

that's all that is and this is a way of

finding it so

the cool thing is all the heavy lifting

is done i actually

ended up putting together one of these a

while back is when i didn't know about

sidekick and it was just starting

boy it's a long while back

and uh

is playing high low how do you play high

low not get stuck in the valleys uh

figure out these curves and things like

that well you do that and the back end

is all the calculus and differential

equations to calculate this out

the good news is you don't have to do

those

so instead we're going to put together

the code and let's go ahead

and see what we can do with that

so uh guys in the back put together a

nice little piece of code here which is

kind of fun

uh some things we're gonna note and this

is this is really important stuff

because when you start doing your data

science and digging into your machine

learning models

you're going to find

these things are stumbling blocks

the first one is current x where do we

start at

keep in mind

your

model that you're working with is very

generic so whatever you use to minimize

it the first question is where do we

start

and we started at this because the

algorithm starts at x equals three

so we arbitrarily picked five

learning rate is uh how many bars to

skip going one way or the other i'm in

fact i'm going to separate that a little

bit because these two are really

important

um if we're dealing with something like

this where we're talking about

well here's our here's the function

we're going to use our

gradient of our function

2 times x plus 5 keep it simple so

that's a function we're going to work

with so if i'm dealing with increments

of 1000.1 is going to be a very long

time and if i'm dealing with increments

of 0.001

0.1 is going to skip over my answer so i

won't get a very good answer

and then we look at precision this tells

us when to stop the algorithm so again

very specific to what you're working on

if you're working with money

and

you don't convert it into a float value

you might be dealing with 0.01 which is

a penny that might be your precision

you're working with

and then of course the previous step

size max iterations we want something to

cut out at a certain point usually

that's built into a lot of minimization

functions

and then here's our actual uh formula

we're going to be working with

and then we come in we go while previous

step size is greater than precision and

it is less than max and max iters

say that 10 times fast

um

we're just saying if it's uh if we're if

we're still greater than our precision

level we still got to keep digging

deeper

and then we also don't want to go past a

thou or whatever this is a million or 10

000

running that's actually pretty high we

almost never do max iterations more than

like 100 or 200

rare occasions you might go up to four

or 500 if it's depending on the problem

you're working with

uh so we have our previous equals our

current that way we can track time wise

the current now equals the current minus

the rate times the formula of our

previous x

so now we've generated our new version

previous step size equals the absolute

current previous

so we're looking for the change in x

errors equals iterations plus one that's

so we know to stop if we get too far

and then we're just going to print the

local minimum occurs at

x on here and if we go ahead and run

this

you can see right here it gets down to

this point and it says hey

local minimum is minus

3.3222 for this particular series we

created this is created off of our

formula here lambda x2 times x plus five

now

when i'm running this stuff you'll see

this come up a lot

in

with the sk learn kit and one of the

nice reasons of breaking this down the

way we did

is i could go over those top pieces

those top pieces are everything when you

start looking at these minimization tool

kits in built-in code and so from

we'll just do it's actually

docs

dot

scipy.org

and we're looking at

the scikit

there we go optimize minimize

you can only minimize one value

you have the function that's going in

this function can be very complicated

so we used a very simple function up

here

it could be

there's all kinds of things that could

be on there and there's a number of

methods to solve this as far as how they

shrink down

uh and your x naught there's your

there's your start value so your

function your start value

there's all kinds of things that come in

here that you can look at which we're

not going to

optimization automatically creates

constraints bounds

some of this it does automatically but

you really the big thing i want to point

out here is you need to have a starting

point you want to start with something

that you already know is mostly the

answer

if you don't then it's going to have a

heck of a time trying to calculate it

out

or you can write your own little script

that does this and does a high low

guessing and tries to find the max value

that brings us to statistics what this

is kind of all about is figuring things

out a lot of vocabulary and statistics

uh so statistics well i guess it's all

relative it's definitely not an edel

class

so a bunch of stuff going on statistics

statistics concerns with the collection

organization analysis interpretation

and presentation of data

that is a mouthful

so we have from end to end

where does it come from is it valid what

does it mean how do we organize it um

how do we analyze it and then you gotta

take those analysis and interpret it

into something that people can use kind

of reduce it to

understandable

and nowadays you have to be able to

present it if you can't present it then

no one else is going to understand what

the heck you did

so we look at the terminologies

there is a lot of terminologies

depending on what domain you're working

in

so clearly if you're working in

a domain that deals with

viruses and t cells and and

how does you know where does that come

from you're studying the different

people that you can have a population

if you are working with um

mechanical gear

you know a little bit different if

you're looking for the wobbling

statistics uh to know when to replace a

rotor on a machine or something like

that

that can be a big deal you know we have

these huge fans that turn

in our sewage processing systems and so

those fans they start to wobble and hum

and do different things that the sensors

pick up at one point do you replace them

instead of waiting for it to break in

which case it costs a lot of money

instead of replacing a bushing you're

replacing the whole fan unit

an interesting project that came up for

our city a while back

so population all objects are

measurements whose properties are being

observed

so that's your population all the

objects it's easy to see it with people

because we have our population in large

but in the case of the sewer fans we're

talking about having the fan units

that's the population of fans that we're

working with

you have a parameter a matrix that is

used to represent a population or

characteristic

you have your sample a subset of the

population studied you don't want to do

them all because then you don't have a

if you come up with a conclusion for

everyone you don't have a way of testing

it so you take a sample

sometimes you don't have a choice you

can only take a sample of what's going

on you can't study the whole population

and a variable a metric of interest for

each person or object in a population

types of sampling

we have a probabilistic approach

selecting samples from a larger

population using a method based on the

theory of probability

and we'll go into a little bit more

deeper on these we have random

systematic stratified and then you have

a non-probabilistic approach selecting

samples based on the subjective judgment

of the researcher rather than random

selection

it has to do with convenience trying to

reach a quota

or snowball

and they're very biased that's one of

the reasons you'll see this big stamp on

that says biased so you gotta be very

careful on that

so probabilistic sampling uh when we

talk about a random sampling we select

random size samples from each group or

category so we it's as random as you can

get we talk about systematic sampling

we're selecting random size samples from

each group or category with a fixed

periodic interval

uh so we kind of split it up this would

be like a time set up or different

categories

and you might ask your question what is

a category or a group

if you look at i'm going to go back a

window let's say we're studying

economics of different of an area

we know pretty much that based on their

culture where they came from

they might need to be separated and so

uh and when i say separated i don't mean

separated from their

place where they live i mean as far as

the analysis we want to look at the

different groups and make sure they're

all represented

so if we had like an eighty percent uh

of a group that is uh say hispanic and

or indian and also in that same area we

have 20 20 percent who are

let's call our expatriates they left

america and they're nice and

your caucasian group we might want to

sample a group that is representative of

both uh so we're talking about

stratified sampling and we're talking

about groups those are the groups we're

talking about and it brings us to

stratified sampling selecting

approximately equal size samples from

each group or category

this way we can actually separate the

categories and give us an insight into

the different cultures and how that

might affect them in that area

so you can see these are very very

different kind of

depends on what you're working with

as far as your data and what you're

studying

and so we can see here just a little bit

more we'd have selecting 25 employees

from a company of 250 employees randomly

don't care anything about them what

groups are in which office they're in

nothing

and we might be selecting one employee

from every 50 unique employees and a

company of 250 employees

and then we have selecting one employee

from every branch in the company office

so we have all the different branches

there's our group or categories by the

branch and the category could depend on

what you're studying so it has a lot of

variation on there

you see this kind of grouping and

categorizing is also used to generate a

lot of misinformation

so if you only study one group and you

say this is what it is

then everybody assumes that's what it is

for everybody and so you've got to be

very careful of that and it's very

unethical thing to kind of do

so types of statistics uh we talk about

statistics

we're going to talk about descriptive

and inferential statistics

there are so many different terms and

statistics to break it up uh so we so

we're talking about a particular

setup

so we're talking about descriptive and

inferential uh statistics

the base of the word describe

is pretty solid you're describing the

data what does it look like with

inferential statistics we're going to

take that from the small population to a

large population so if you're working

with a drug company you might look at

the data and say these people were

helped by this drug

they did

80 percent better

as far as their health or 80 percent

better survival rate than the people

who did not have the drug so we can

infer that that drug will work in the

greater populace and will help people so

that's where you get your inferential so

we are

predicting how it's going to affect the

greater population

so descriptive statistics it is used to

describe the basic features of data and

form the basis of quantitative analysis

of data

so we have a measure of central

tendencies we have your mean median and

mode

and then we have a measure of spread

like your range your interquartile range

your variance and your standard

deviation

and we're going to look at all these a

little deeper here in a second

but one of them you can think of is

how it the data difference

differences you know what's the max min

range all that stuff is your spread and

anything that's just a single number is

usually your central uh tendencies

measure of central tendencies

so we talk about the mean it is the

average of the set of values considered

what is the average outcome of whatever

is going on

and then your median

separates the higher half and the lower

half of data

so where's the center point of all your

different data points so your mean might

have some a couple really big numbers

that skew it

so that the average is much higher than

if you took those outliers out where the

median would by separating the high from

the low might give you a much lower

number you might look at and say oh

that's that's odd why is the average so

much higher than the median well it's

because you have some outliers or why is

it so much lower

and then the mode is the most frequent

appearing value

this is really interesting if you're

studying economics and how people are

doing you might find that the most

common

income like in the us was

1.24 000 a year

where the average was closer to 80 000

and it's like wow what a difference well

there's some people have a lot of money

and so that skews that way up so the

average person is not making that kind

of money and then you look at the median

income and you're like well the median

income is a little bit closer to the

average so it does create a very

interesting way of looking at the data

again these are all uh central

tendencies single numbers you can look

at for the whole spread of the data

and we look at the measure of central

tendencies the mean is the average marks

of a students in a classroom so here we

have the mean sum of the marks of the

students total number of students and as

we talked about the median

we have

0 through 10 and we take half the

numbers and put them on one side of the

line half the numbers on the other side

of the line uh we end up with five in

the middle and then the mode what mark

was scored by most of the students in a

test

in a simple case where most people

scored like an 82 percent and got

certain problems wrong easy to figure

out uh not so easy when you have

different areas where like you have like

the um oh let's go back to economy a

little bit more difficult to calculate

if you have a large group that scores

that makes 30 000

and a slightly bigger group that makes

26 000 so what do you put down for the

mode uh certainly there's a number of

ways to calculate that and there's

actually a different variations

depending on what you're doing

so now we're looking at a measure of

spread uh range what's the difference

between the highest and the lowest value

first thing you want to look at you know

it's uh we had everybody in the test

scored between 60 and 100 so we got 100

or maybe 60 to 90 it was so hard a lot

of people could not get a hundred

percent

you have your

inter-quartile range quartiles divide a

rank ordered data set into four equal

parts

very common thing to do as part of all

the basic packages whether you're

working in

data frames with pandas whether you're

working in scala whether you're working

in r

you'll see this come up where they have

range your min your max and then it'll

have your interquartile range how does

it look like in each quarter of data

variance measures how far each number in

the set is from the mean and therefore

from every other number in the set

so you have like how much turbulence is

going on in this

data and then the standard deviation

it is to measure the variance or the

dispersion of a set of values from the

mean

and you'll usually see uh if i'm doing a

graph i might have the value graphed

and then based on the the error i might

graph graph the standard deviation in

the error on the graph as a background

so you can see how far off it is

uh so standard deviation is used a lot

so measurement of spread uh marks of a

student out of 100 we have here from 50

to 63 or 50 to 90.

so the range maximum marks minimum marks

we have 90 to 45 and the spread of that

is 45 90 minus 45. and then we have the

interquartile range

using the same marks over there you can

see here where the median is

and then there's the first quarter the

second quarter and the third quarter

based on splitting it apart by those

values

and to understand the variance and

standard deviation we first need to find

out the mean uh so here's our our you

know calculating the average there we

end up at approximately 66 for the

average and then we look at that the

variance once we know the means we can

do equals the marks minus the mean

squared

y is a squared

because one you want to make sure it's

you don't have like if you if you're

putting all this stuff together you end

up with an error as far as one's

negative one's positive one's a little

higher one's a little lower

so you always see

the squared value and over the total

observations

and so the standard deviation equals the

square root of the variance which is

approximately 16.

and if you were looking at

a predictable model you would be looking

at the deviation based on the error how

much error does it have

that's again

really important to know if your if your

prediction is predicting something

what's the chance of it being way off or

just a little bit off

now that we've looked at the

tools as far as some of the basics for

doing your statistics and we're talking

about

let's go ahead and pull up a little demo

and show you what that looks like in

python code

so you can get some little hands on here

for that let's go back into our jupiter

notebook and python now almost all of

this you can do in numpy last time we

worked in numpy this time we're going to

go ahead and use pandas

and if you remember from pandas on here

this is basically a data frame rows

columns let's just go ahead and do a

print

df.head

and run that

and you can see we have the name jane

michael william rosie hannah sat in

their salaries on here and of course

instead of having to do all those hand

calculations and add everything together

and divide by the total

we can do something very simple on this

like

use the command mean in pandas and so if

i go ahead and do this print df

pick our column salary because we want

to find the means of that calorie

we want to find the means of that column

and we go and print this out and you can

see that the

average income on here is 71 000.

and let's just go ahead and do this

we'll go ahead and put in

means

and if we're going to do that we also

might want to find the median

and the median is

very similar

except it actually is just median we're

used to means in average it's kind of

interesting that those are the use of

two different words

uh there can be in some computation

slight differences but for the most part

the means is the average uh and then the

median

oops let's put a

median here

do you have salary that way it displays

a little better we can see the median is

54

000 so the halfway mark is significantly

below the average why because we have

somebody in here makes 189 000. darn you

rosie for throwing off our numbers

but that's something you'd want to

notice this is this is the difference

between these is huge and so is what is

the meaning behind that when you're

studying a populace and looking at

the different data coming in and of

course we also want to find out hey

what's the most

common

income that people make

in this little tiny sample and so we'll

go ahead and do the mode and you can see

here with the mode

it's at 50 000.

so this is this is very telling that

most people are making 50 000

the middle point is at 54 000. so half

the people are making more than that

what that tells me is that if the most

common income is weight is below the

median

then

there's a few there's a skill there's a

lot of high salaries going up but

there's some really low salaries in

there and so this trend which is very

common in statistics when you're

analyzing

the economy in different people's income

is pretty common and the bigger

difference between these is also

very important when we're studying

statistics

and when you hear someone just say hey

the average income was you might start

asking questions at that point why

aren't you talking about the median

income why aren't you talking about the

mode the most common income what are you

hiding

and if you're doing these analysis you

should be looking at these saying hey

why are these discrepancies why are

these so different and of course with

any analysis it's important to find out

the minimum

and the maximum so we'll go ahead it's

just simply uh

dot min

it'll pull up your minimum and then dot

max pulls up the maximum

pretty straightforward on as far as um

translating it and knowing which you

know put the your lowest value which

your highest value is here

um which you'll use to generate like a

spread later on and real quick on no

mode uh note that it puts mode zero like

i said there's a couple different ways

you can compute the mode

although the standard one is pretty good

we can of course do the range

which is your max minus your min so now

we have a range of 149 000 between the

upper end and the lower end and you

might want to be looking up the

individual values on all of these but it

turns out there is a describe

feature in pandas

and so in pandas we can actually do df

salary describe and if we do this you

can see we have that there's seven uh

setups here's our mean

um our standard deviation which we

didn't compute yet which would just be a

dot std

and you gotta be a little careful

because when it computes it it looks for

axes and things like that

we have our minimum value and here's our

quartiles

our maximum value and then of course the

name salary

so these are the these are the basic

statistics you can pull them up and like

just describe

this is a dictionary so i could actually

do something like

in here i could actually go uh count

and run

and now it just prints the count

so because this is a dictionary you can

pull any one of these values out of here

it's kind of a quick and dirty way to

pull all the different information and

then split it up and depending on what

you need

now if i just walked in and gave you

this information

in a meeting

at some point you would just kind of

fall asleep that's what i would do

anyway

um

so we want to go ahead and see about

graphing it here and we'll go ahead and

put it into a histogram and plot that

graph on it

of the salaries and let's just go ahead

and put that in here so

we do our map plot inline remember

that's a jupiter's notebook thing

a lot of the new version of the matte

plot library does it automatically

but just in case i always put it in

there import matplot library pi plot is

plt that's my plotting

and then we have our data frame i don't

i guess i really don't need to respell

the data frame

maybe we could just remind ourselves

what's in it so we'll go ahead and just

print

df

that way we still have it

and then we have our salary df salary

salary.plot history title salary

distribution color gray

uh plot ax v line salary the mean value

so we're going to take the mean value

color violet

line style dash this is just all making

it pretty

uh what color dashed line line width of

2 that kind of thing

and the median and let's go ahead and

run this just so you can see what we're

talking about

and so up here we are taking on our plot

um so here's the data here's our our

data frame print it out so you can see

it with the salaries we'll look at the

salary distribution and just look at

this the way the salary is distributed

you have our

in this case we did

let's see we had red for the median

we have violet

for our average or mean

and you can just see how it really

i mean here's our outlier here's our

person who makes a lot of money here's

the

average and here's the median

and so as you look at this you can say

wow

based on the average it really doesn't

tell you much about what people are

really taking home all it does is tell

you

how much money is in this you know what

the average salary is

so

some of the things you want to take away

in addition to this is that it's very

easy to plot

an ax v line

these are these up and down lines for

your markers

and as you just display the data i mean

you can add all kinds of things to this

and get really complicated keeping it

simple is pretty straightforward i look

at this and i can see we have a major

outlier out here we can definitely do a

histogram and stuff like that

but you know pictures worth a thousand

words

what you really want to make sure you

take away is that we can do a basic

describe

which pulls all this information out and

we can print any of the individual

information from the describe

because this is a dictionary

and so if we want to go ahead and look

up

the mean value we can also do describe

mean so if you're doing a lot of

statistics

being able to

doesn't have the print on there so it's

only going to print the last one which

happens to be the mean

you can very easily reference any one of

these and then you can also if you're

doing something a little bit more

complicated and you don't need just the

basics you can come through and pull any

one of the individual

references from the from the pandas on

here so now we've had a chance to

describe our data

let's get into inferential statistics

inferential statistics allows you to

make predictions or inferences from data

and you can see here we have a nice

little picture movie ratings and

if we took this group of people and said

hey how many people like the movie

dislike it can't say and then you ask

just a random person who comes out of

the movie who hasn't been in this study

you can infer that 55 percent chance of

saying liked

35 chance of saying disliked or a 10 or

11 chance of can't say

so that's real basics of what we're

talking about is you're going to infer

that the next person is going to follow

these statistics

so let's look at point estimation

it is a process of finding an

approximate value for a population's

parameter like mean

or average from random samples of the

population let's take an example of

testing vaccines for covid19

vaccines and flu bugs all that it's a

pretty big thing of how do you test

these out and make sure they're going to

work on the populace

a group of people are chosen from the

population medical trials are performed

results are generalized for the whole

population so here's a protected there's

our small group up here where we've

selected them we run medical trials on

them and then the results work for the

population

nice diagram with the arrows going back

and forth and the very scary coveted

virus in the middle of one

and let's take a look at the

applications of inferential statistics

very central is what they call

hypotheses testing

and the confidence interval which go

with that and then as we get into

probability we get into our binomial

theorem our normal distribution in

central limit theorem

hypothesis testing hypothesis testing is

used to measure the plausibility of a

hypothesis

assumption by using sample data

now

when we talk about theorem's

theory

hypothesis uh keep in mind that if you

are in a philosophy class

theory

is the same as hypothesis where theorem

is a scientific uh statement that is

something that has been proven

although it is always up for debate

because in science we always want to

make sure things are up to debate so

hypothesis is the same as a

philosophical class calling a theory

where theory in science is not the same

theory in science says this has been

well proven gravity is a theory uh so if

you want to debate the theory of gravity

try jumping up and down if you want to

have a theory about why the economy is

collapsing in your area

that is a philosophical debate

very important i've heard people mix

those up and it is a pet peeve of mine

when we talk about hypotheses testing

the steps involved in hypotheses testing

is first we formulate a hypothesis

we figure out the right test to test our

hypothesis we execute the test and we

make a decision and so when you're

talking about hypothesis you're usually

trying to disprove it if you can't

disprove it

and it works for all the facts then you

might call that a theorem at some point

so in a use case uh let's consider an

example we have four students were given

a task to clean a room every day

sounds like working with my kids they

decided to distribute the job of

cleaning the room among themselves they

did so by making four chits which has

their names on it and the name that gets

picked up has to do the cleaning for

that day

rob took the opportunity to make chits

and wrote everyone's name on it so

here's our four people nick rob emlia

imlia and summer

now rick emilia and summer are asking us

to decide whether rob has done some

mischief in preparing the chits i.e

whether rob has written his name on one

of the chit

for that we will find out the

probability of rob getting the cleaning

job on first day second day third day

and so on till 12 days

the probability of rob getting the job

decreases every day i.e his turn never

comes up then definitely he has done

some mischief while making the chits

so the probability of rob not doing work

on day one is uh three out of four

there's a 0.75 chance that he didn't do

work

uh two days three fourths times three

fourths equals point five six

three days you have three fourths three

fourths three fourths which equals 0.42

when you get to day 12 it's 0.032 which

is less than 0.05

remember this .05 that comes up a lot

when we're talking about

certain values when we're looking at

statistics

rob is cheating as he wasn't chosen for

12 consecutive days that's a very high

probability when on day 12 he still

hasn't gotten the job cleaning the room

so we come up to our important important

terminologies

we have null hypothesis

a general statement that states that

there is no relationship between two

measured phenomena or no association

among the groups

alternative hypothesis

contrary to the null hypothesis it

states whenever something is happening a

new theory is preferred instead of an

old one and so the two hypotheses go

hand in hand uh so your null this is

always interesting in in talking about

data science and the math behind it

it's about proving that the things have

no correlation

null hypothesis says these two have zero

relation to each other where the

alternative hypothesis says hey we found

a relation this is what it is

we have p-value the p-value is a

probability of finding the observed or

more extreme results when the null

hypothesis of a study question is true

and the t value it is simply the

calculated difference represented in

units of standard error the greater the

magnitude of t the greater the evidence

against the null hypothesis and you can

look at the t values being specific to

the test you're doing

where the p value is derived from your t

value and you're looking for what they

call the five percent or the 0.05

showing that it has a high correlation

so

digging in deeper let's assume that a

new drug is developed with the goal of

lowering the blood pressure more than

the existing drug

and this is a good one because the null

value here isn't that you don't have any

drug the null value here is that it's

better than the existing drug

the new drug doesn't lower the blood

pressure more than the existing drug

now if we get that

that says our null hypothesis is correct

there is no correlation and the new drug

is not doing its job the alternative

hypothesis the new drug does

significantly lower the blood pressure

more than the existing drug uh yay we

got a new drug out there and that's our

alternative hypothesis or the h1 or ha

and we look at the p-value results from

the evidence like medical trial showing

positive results which will reject the

null hypothesis

and again they're looking for

a 0.05 or 5 percent and the t value

comparing all the positive test results

and finding means of different samples

in order to test hypothesis

so this is specific to the test how what

percentage of increase did they have

and this leads us to the confidence

intervals

a confidence interval is a range of

values we are sure our true values of

observations lie in

let's say you asked a dog owner around

you and asked them how many cans of food

do you buy for your per year for your

dog

through calculations you got to know

that the on an average around 95 percent

of the people bought around 200 to 300

cans of food hence we can say that we

have a confidence interval of two 300

where 95 percent of our values lie in

that sprint data spread

and this the graph really helps a lot so

you can start seeing what you're looking

at here we have the 95 percent you have

your peak in this case it's a normal

distribution so you have a nice bell

curve equal on both sides it's not

asymmetrical and 95 of all the values

lie within a very small range and then

you have your outliers the 2.5 percent

going each way

so we touched upon hypothesis uh we're

going to move into probability so you

have your hypothesis once you've

generated your hypothesis we want to

know the probability of something

occurring probability is a measure of

the likelihood of an event to occur any

event can be predicted with total

certainty

and can only be predicted as a

likelihood of its occurrence so any

event cannot be predicted with total

certainty can only be predicted as a

likelihood of its occurrence

score prediction how good you're going

to do in whatever

sport you're in weather prediction stock

prediction

if you've studied physics and chaos

theory even the location of the chair

you're sitting on has a probability that

it might move three feet over

granted that probability is one in like

uh i think we calculated as under one in

trillions upon trillions so it's

the better the probability the more

likely it's going to happen there are

some things that have such a low

probability

that we don't see them so we talk about

a random variable a random variable is a

variable whose possible values are

numerical outcomes of a random phenomena

so uh we have the coin tossed how many

heads will occur in the series of 20

coin flips probably you know the on

average they're 10 but you really can't

know because it's very random how many

times a red ball is picked from a bag of

balls if there's equal number of red

balls and blue balls and green balls in

there how many times the sum of digits

on two dice

results are five each

so you know there's how often you're

gonna roll two fives on your paradigms

so in a use case uh let's consider the

example of rolling two dice we have a

random variable outcome equals y you can

take values two three four five six

seven eight nine ten eleven twelve

so we have a random variable and a

combination of dice

and instead of looking at how many times

both dice for roll five let's go ahead

and look at uh total sum of five and you

have in as far as your random variables

you can have a one four equals five four

one two three three two

so four of those rolls can be four if

you look at all the different options

you have four of those random rolls can

be a five

and if we look at the total number

which happens to be 36 different options

you can see that we have four out of 36

chance every time you roll the dice that

you're gonna roll a total of five you

can have an outcome of five

and uh we'll look a little deeper as to

what that means but you could think of

that at what point if someone never

rolls a five or they always roll a five

can you say hey that person's probably

cheating

we'll look a little closer at the math

behind that but let's just consider this

is one of the cases is rolling two dice

and gambling

there's also binomial distribution is

the probability of getting success or

failure as an outcome in an experiment

or trial that is repeated multiple times

and the key is is by meaning two

binomial so passing or failing an exam

winning or losing a game and getting

either head or tails so if you ever see

binomial distribution it's based on a

true false kind of setup you win or lose

let's consider a

use case and let's consider the game of

football between two clubs

barcelona and dortmund the teams will

have to play a total of four matches and

we have to find out the chances of

barcelona winning the series so we look

at the total games we're looking at five

different games or matches

let's say that the winning chance for

barcelona is 75 percent or 0.75

that means that each game they have a 75

chance that they're going to win that

game and losing chances are 25 percent

or 0.25 clearly 0.75 plus 0.25 equals 1.

so that accounts for 100 of the game

probability for getting k wins in in

matches is calculated

and we we're talking like so if you have

five games and you want to know if i

play

how many wins in those five games should

i get what's a percentage on those and

the probability for getting k wins and

in matches is calculated by p x equals k

equals nc k p to the k q to the n minus

k

here p is the probability of success and

q is the probability of failure and so

we can do total games of n equals 5

where k equals 0 one two three four five

p which is the chance of winning is

point seven five q the chance of losing

equals one minus p

which equals one minus point o seven

five which equals point two five the

probability that barcelona will lose all

of the matches can then just plug in the

numbers

and we end up with a .0009765625

so very small chance they're going to

lose all their matches

and we can plug in the value for two

matches probability that barcelona will

win at least two matches is 0.0878 and

of course we can go on to the

probability that barcelona will win

three matches the 0.26 and course four

matches and so on and it's always nice

to take this information

and let's find the accumulated discrete

probabilities for each of the outcomes

where barcelona has won three or more

matches x equals three x equals four x

equals five

and we end up with the p equals point

two six four plus point three nine five

plus two three seven which equals point

eight nine

in reality

the probability of barcelona winning the

series is much higher than 0.75

and it's always nice to

put out a nice graph so you can actually

see the number of wins to the

probability and how that pans out with

our binomial case

continuing in our important terminology

location the location of the center of

the graph depends on the mean

value and this is some very important

things so much of the data we look at

and when you start looking at

probabilities almost always has a

normalized look like the graph in the

middle

but you do have left skewed where the

data is skewed off to the left and you

have more stuff happening off to the

left and you have right skewed data

and so when this comes up and these

probabilities come up where they're

skewed it's really important to take a

closer look at that

mostly you end up with a normalized set

of data but you got to also be aware

that sometimes it's a skewed data

and then the height height of the slope

inversely depends upon the standard

deviation

so you can see down here the standard

deviation is really large it kind of

squishes it out and if the standard

deviation is small then most of your

data is going to hit right there in the

middle you can have a nice peak

and so being aware of this that you

might have a probability that fits

certain data but it has a lot of

outliers so you're if you have a really

high standard deviation

if you're doing stock market analysis

this means your predictions are probably

not going to make you much money

where if you have a very small deviation

you might be right on target and set to

become a millionaire

which leads us to the z-score z-score

tells you how far from the mean a data

point is it is measured in terms of

standard deviations from the mean around

68 percent of the results are found

between one standard deviation

around 95 percent of the results are

found between two standard deviations

and you read the symbols of course they

love to throw some greek letters in

there we have mu minus two sigma

mu is just a quick way it's a kind of

funky u it just means the mean

uh and then the sigma is the standard

deviation and that's the o with the

little arrow off to the right or the

little

wagy tail going up the o with it with

the line on it uh so mu minus two sigma

is your

uh 95 percent of the results are found

between two standard deviations

central limit theorem

this goes back to the skew if you

remember we were looking at the skew

values on this previous slide have left

skewed normalized and right skewed when

we're talking about it being skewed or

not skewed the distribution of the

sample means will be approximately

normally distributed evenly distributed

not skewed

if you take large random samples from

the population with the mean mu

and the standard deviation sigma with

replacement

and you can see here

of course we have our

mu minus two sigma and the spread down

here the mean the median and the mode

and so you're talking about very large

populations

these numbers should come together and

you shouldn't have a skewed value if you

do that's a flag that something's wrong

that's why this is so important to be

aware of what's going on with your data

where your samples are coming from

and the math behind it

and if we're going to do all this we got

to jump into conditional probability

the conditional probability of an event

a is a probability that the event will

occur given the knowledge that an event

to be has already occurred

and you'll see this as bayes theorem

b-a-y-e-s base

and this is red

i mean you have these funky looking

little

p

brackets a b

this is the probability of a being true

while b is already true

and you have the probability of b being

true when a is already true so p

b of a

probability of a being true divided by

the probability of b being true

and we talk about bayes theorem which

occurred back in the 1800s when he

discovered this this is such an

important formula and it's really it's

not if you actually do the math you

could just kind of do

x y equals j k and then you divide them

out and you're going to see the same

math but it works with probabilities

which makes it really nice

and so if you have a set you might have

uh eight or nine different studies going

on in different areas different people

have done the studies they brought them

together

if we look at today's covet virus the

virus spread

certainly the studies done in china

versus the studies the way they're done

in the us

that data is different in each of those

studies but if you can find a place

where it overlaps where they're studying

the same thing together you can then

compute the changes that you need to

make in one study to make them equal

and this is also true if you have a

study of

one group and you want to find out more

about it so this formula is very

powerful and it really has to do with

the data collection part of the math and

data science and understanding where

your data is coming from and how you're

going to combine different studies in

different groups

and we're going to go into a use case

let's find out the chance of a person

getting lung disease due to smoking

this is kind of interesting the way they

word this

let's say that according to medical

report provided by the hospital states

that around 10 percent of all patients

they treated suffered lung disease

so we have kind of a generic medical

report

they further found out

by a survey that 15 percent of the

patients that visit them smoke

so we have 10 percent that are lung

disease and

15 of the patients smoke

and finally five percent of the people

continued smoke even when they had lung

disease uh not the brightest choice um

but you know it is an addiction so it

can be really difficult to kick and so

we can look at the probability of a uh

prior probability of 10 people having

lung disease

and then probability b probability that

a patient smokes is 15 percent

uh and the probability of b

if b then a the probability of a patient

smokes even though they have lung

disease is five percent

and probability of a is b probability

that the patient will have lung disease

if they smoke and then when you put the

formulas together you get a nice

solution here you get the probability of

a of b probability that the patient will

have lung disease if they smoke

and you can just plug the numbers right

in and we get a 3.33 percent chance

hence there is a 3.33 chance that a

person who smokes will get a lung

disease

so we're going to pull up a little

python code i'm always my favorite roll

up the sleeves

keep in mind we're going to be doing

this um kind of like the back end way

so that you can see what's going on and

then later on we're going to create

we'll get into another demo which shows

you some of the tools are already

pre-built for this

let's start by creating a set so we're

going to create a set with curly braces

this means that our set has

only unique values so you have a list

you have your tuples which can never

change and then you have

in this case the the set so four seven

you can't create a four seven comma four

it'll delete the four out so it's only

unique values

and if you use dictionaries

quick reminder this should look familiar

because it is a dictionary we have a

value and that value is assigned to or

that key is assigned to a value

so you could have a key value set up as

a dictionary so it's like a dictionary

without the value it's just the keys and

they all have to be unique

and if we run this we have a

set of four seven

we can also take a list a regular

setup and i'm going to go ahead and just

throw in another number in here four

and run it uh and you can see here if i

take my list one two three four four

and i convert it to a set and here it is

my set from list equals set my list

the result is one two three four so it

just deletes that last four right out of

there

and with the sets you can also go in

there and

print here is my set my set

uh three is in the set and then if you

do three in my set

that's going to be a logic function

uh and one in my set six is not in the

set and so forth if we run this

we get three is in the set true one is

in the set false because three five

seven is another one six is in the set

six is not in the set so not in my set

you can also use this with the list we

could have just used three five seven

and it would have

the same response on there is three and

usually do if three is in but three in

my set is still works on just a regular

list

then we'll go ahead and do a little

iteration we're going to do kind of the

dice one remember

one two three four five six and so we're

going to bring in the iteration tool and

import product as product

and i'll show you what that means in

just a second so we have our two dice we

have dice a

and it's going to be a set of values

they can only have one value for each

one that's why they put it in a set and

if you remember from range it is up to

seven so this is going to be one two

three four five six it will not include

the seven and the same thing for our

dice b

and then we're gonna do is we're gonna

create a list which is the product

of a and b so what's a plus b

and if we go ahead and run this it'll

print that out and you'll see

in this case when they say product

because it's an iteration tool

we're talking about creating a tuple of

the two

so we've now created a tuple of all

possible outcomes of the dice where dice

a is one two three one to six and dice b

is one to six and you can see one to one

one to two one to three and so forth

you remember we had a slide on this

earlier where we talked about

the different all the different outcomes

of a dice

we can play around with this a little

bit we can do in dice equals two

dice faces one two three four five six

uh another way of doing what we did

before and then we can create an event

space where we have a set which is the

product of the dice faces

repeat equals indice and we'll go ahead

and just run this

and you can see here it just again puts

it through all the different possible

variables we can have

and then if we wanted to take the same

set on here and print them all out like

we had before

we can just go through for outcome and

event space outcome and equals

so the event space is creating

a sequence and as you can see here when

we print it out it stacks them versus

going through and putting them in a nice

line

and we'll go ahead and do something

let's go print

since we have the end printing with a

comma that just means it's just gonna

it's not gonna hit the return going down

to the next line

and we'll go ahead and do the length

of our event space that'll be an

important variable we're going to want

to know in a minute

and of course if i get carried away with

my typing of length uh we'll print it

twice and it'll give me an error

so we have 36 different possible

variations here

and we might want to calculate something

like

what about the multiple of three what if

we want to have

uh the probability of the multiple three

in our setup

and so we can put together the code for

the outcome and event space of x y

equals outcome

if x plus y

remainder 3 so we're going to divide by

3 and look at the remainder and it

equals 0

then it's a favorable outcome we're

going to pop that outcome on the end

there

and we'll turn it into a set so the

favor outcome equals a set

not necessary

because we know it's not going to be

repeating itself but just in case we'll

go ahead and do that

and if we want to print out the outcome

we can go ahead and see what that looks

like and you can see here these are all

multiples of three uh one plus two is

three five plus four is nine which

divided by three is three and so forth

and just like we looked up the length of

the one before let's go ahead and print

the length

of our

f outcome

so we can see what that looks like

there we go

and of course i did forget to add the

print in the middle because we're

looping through and putting an end on

the on the setup on there so we're going

to put the print in there and if i run

this you can see

we end up with 12. so we have 36 total

options

we have 12 that are multiple that add up

to a multiple of three

and we can easily conver compute the

probability of this

by simply taking the length

of our favorable outcome of the length

of the event space

if we print it out let me put that in

there

probability

last line so we just type it in we end

up with the 0.3333 chance

that's roughly a third

and we want to make this look nice so

let's go ahead and put in another line

there the probability of getting the sum

which is a multiple of 3 is

0.3333

we can compute the same thing for five

dice

and if we do this for five dice and go

and run it uh you can see we just have a

huge amount of choices

so just goes on and on down here and we

can look at

the

length of the event space

and we have over 7776

choices that's a lot of choices

and if we want to ask the question like

we did above uh

what is the sum where the sum is a

multiple of five but not a multiple of

three

we can go through all of these different

options and then

you can see here

d1 d2 d3 d4 d5 equals the outcome

and if you add these all together and

the

division by five does not have a

remainder of zero

but the remainder is also of a division

by three is not equal to zero

so the multiple of five is equal to zero

but the multiple three is not we can

just append that on here and then we can

look at that uh favorable outcome

we'll go ahead and set that and we'll

just take a look at this what's our

length

of our favorable outcome

it's always good to see what we're

working with and so we have 904 out of

70

6 and then of course we can just do a

simple division to get the probability

on here what's the probability that

we're going to roll

a multiple of 5 when you add them

together

but not a multiple of three

and so we're just going to divide those

two numbers and you can see here we get

point one one six two five five or

eleven point six two percent

and so you can really have a nice visual

that this is not really complicated math

right here on probabilities

it's just how many options do you have

and how many of those are you possibly

going to be able to come up with with

the solution you're looking for

and this leads us to a confusion matrix

a confusion matrix is a table which is

used to describe the performance of a

classification model on a set of test

data for which the true values are known

and so you'll see in the left we have

the predicted and the actual

and we have a negative uh false negative

positive true positive

and then we have false positive and true

negative and you can think of this as

your predicted model what does that mean

that means if you divided your data and

you use two-third of this to create the

model

you might then test it against an actual

case for the last third to see how well

it comes out how many times was it

true positive versus a

false positive it gave a false positive

response

and you can imagine in medical

situations this is a pretty big deal you

don't want to give a false positive so

you might adjust your model accordingly

so you don't have a false positive say

with a covavirus test it'd be better to

have a false negative and they go back

and get re-tested than to have 30 false

positives where then the test is pretty

much invalid

so in a use case like cancer prediction

let's consider an example where a cancer

prediction model is put to the test for

its accuracy and precision

actual result of a person's medical

report is compared with the prediction

made by the machine learning model

and so you can see here here's our

actual predicted whether they have

cancer or not you know cancer a big one

you don't want to have a

false positive i mean a false negative

in other words you don't want to have it

tell you that you don't have cancer when

you do so that would be something you'd

really be looking for in this particular

domain you don't want a false negative

and this is again you know you've

created a model you have hundreds of

people or thousands of pieces of data

that come in

there's a real famous case study where

they have the imagery and all the

measurements they take and there's about

36 different measurements they take

and then if you run the a basic model

you want to know just how accurate it is

how many negative results do you have

that are either telling people they have

cancer that don't or telling people that

don't have cancer that they do and then

we can take these numbers and we can

feed them into our accuracy our

precision and our recall

so accuracy precision and recall

accuracy metric to measure how

accurately the results are predicted

and this is your

total

true where you got the right results you

add them together the true positive the

true negative

over all the results so what percentage

of them were accurate versus what were

wrong

we talked about precision is a metric to

measure how many of the correctly

predicted cases are actually turned out

to be positive

uh so we have a precision on

true positive

again if you're talking about like uh

covid testing with the viruses uh you

really want this to be a high number you

want this true

that to be the center point where you

might have the opposite if you're

dealing with a cancer where you want no

false negatives

so this is your metric on here precision

is your test positive

true positive plus

false positive

and then your recall how many of the

actual positive cases we were able to

predict quickly with our model

so test positive is the test positive

plus the false negative on there and

we'll want to go ahead and do a demo on

the naive bayes classifier before i get

too far into a naive bayes classifier

because we're going to pull it from the

sk learn or the site kit

let's go ahead kind of an interesting

page here for classifiers when you go

into the sk learn kit there's a lot of

ways to do classification and i'll just

zoom up in here so you can see some of

the titles

there's everything from the nearest

neighbor linear

but we're going to be focusing on the

naive bayes over here

and this is just a sample data set that

they put together and you can see how

some of these have a very different

output

the naive bayes remember is set up as

probably the most simplified uh

calculator or set of predictions out

there

and so what we've been talking about

with the true false and stuff like that

where there's a

and then a belief that there is a

independent assumption between the

features where the features are very

assumed to have some kind of connection

uh then we can go ahead and use that for

the prediction and so that's what we're

using is a naive bayes classifier versus

many of the other classifiers that are

out there

for this we're going to use the social

network ads it's a little data set on

here

and let me go and just open that up the

file

here we go it has user id gender age

estimated salary uh purchased

and so we have you can see the user id

male 19

estimated salary 19 000

and purchased zero so it's either gonna

make a purchase or not

so look at that last one zero one we

should be thinking of binomials we

should be thinking of a simple naive

bayes classifier kind of setup

so if we close this out we're going to

go ahead and import our numpy as np

we're going to nice to have a good

visual of our data so we'll put in our

matplot library here's our pandas our

data frame

and then we're going to go ahead and

import the data set and the data set's

going to be we're going to read it from

the social network ads.csv then we're

going to print the head just so you can

see it again

even though i showed you it in the file

and x equals the data set i location

two three values and y is going to be

the four

column four let me just run this it's a

little easier to go over that

you can see right here we're going to be

looking at

0 1 2 as age

and estimated salary so 2 3

and that's what i location just means

that we're

looking at the number versus a regular

location a regular location you'd

actually say age and estimated salary

and then column four is did they make a

purchase they purchased something

so those are the three columns we're

going to be looking at when we do this

and we've gone ahead and imported these

and

imported the data so now our data set is

all set with this information in it

and we'll need to go ahead and split the

data up so we need our from the sk learn

model selection we can import train test

split

this does a nice job we can set the

random state so randomly picks the data

and we're just going to take uh 25 of

it's going to go into the test our x

test and our y test

and the 75 will go to x train and y

train

that way once we

create our model we can then have data

to see just how accurate or how well it

has performed with our prediction

the next step in pre-processing our data

is to go ahead and do feature scaling

now a lot of this is start to look

familiar if you've done a number of the

other modules and setup you should start

noticing that we

bring in our data we take a look at what

we're working with

we go ahead and split it up into

training and testing

in this case we're going to go ahead and

scale it scale it means we're putting it

between a value of minus 1 and 1

or or someplace in the middle ground

there

this way if you have any huge set you

don't have this huge um

setup if we go back up to here where

salary the salary is

20 000 versus age 35.

well there's a good chance with a lot of

the back end math that 20 000 will skew

the results and the estimated salary

will have a higher impact than the age

instead of balancing them out and

letting the calculations weigh them

properly

and finally we get to actually create

our naive bayes model

um and then we're going to go ahead and

import the gaussian naive bayes

and the gaussian is is the most basic

one that's what we're looking at now it

turns out though if you go to the sk

learn

kit

they have a number of different ones you

can pull in there there's a

bernoulli i know i've never used that

one categorical

complement and here's our gaussian

so there's a number of different options

you can look at

gaussian when you come to the naive

bayes is the most commonly used

so we're talking about the naive bayes

that's usually what people are talking

about when they when they're pulling

this in

and one of the nice things about the

gaussian if you go to their website um

to sk learn the naive bayes gaussian

there's a lot of cool features one of

them is you can do partial fit on here

that means if you have a huge amount of

data you don't have to process it all to

want you once you can batch it into the

gaussian nb model and there's many other

different things you can do with it as

far as fitting the data and how you

manipulate it

we're just doing the basics so we're

going to go ahead and create our

classifier we're going to equal the

gaussian in b

and then we're going to do a fit we're

going to fit our training data and our

training solution so x train y train

and we'll go ahead and run this uh it's

going to tell us that it ran the code

right there

and now we have our trained classifier

model

so the next step is we need to go ahead

and run a prediction we're going to do

our y predict equals the

classifier.predict

x test

so here we fit the data and now we're

going to go ahead and predict

and now we get to our confusion matrix

so from

the sk learn matrix metrics you can

import your confusion matrix

just as saves you from doing all the

simple math that does it all for you

and then we'll go ahead and create our

confusion metrics with the y test and

the y predict so we have our actual

and we have our predicted value

and you can see from here this is the

chart we looked at here's predicted so

true positive false positive

false negative true negative

and if we go ahead and run this there we

have it 65 3 7 25

and in this particular prediction we had

65 or predicted the truth as far as a

purchase they're going to make a

purchase

and we guessed three wrong

and then we had 25 we predicted would

not purchase and seven of them did so

there's our our confusion matrix

at this point if you were with your

shareholders or a board meeting

you would start to hear some snoozing if

they were looking at the numbers and you

say hey here's my confusion

matrix so let's go ahead and visualize

the results

we're going to pull from the matplot

library colors import listed color map

and this is actually my machine is going

to throw an error because this is being

because of the way the setup is i have a

newer version on here than when they put

together the demo

and we need our x set and our y set

which is our x train and y train

and then we'll create our x1 x2

and we'll put that into a grid

uh and we set our x set minimum stop and

our x at max stop

and if you come all the way over here

we're going to step .01 this is going to

give us a nice line is what that's doing

and we're going to plot the contour

plot the x limit plot the y limit

and put the scatter plot in there let's

go ahead and run this

to be honest when i'm doing these graphs

there's so many different ways to do

that there's so many different ways to

put this code together

to show you what we're doing it's a lot

easier to pull up the graph and then go

back up and explain it

so the first thing we want to note here

when we're looking at the data

is this is the training set

and so we have those who didn't make a

purchase we've drawn a nice area for

that

that's defined by the naive bayes setup

and then we have those who did make a

purchase the green and you can see that

some of the green drops fall into the

red area and some of the red dots fall

into the green

so even our training set isn't going to

be a hundred percent uh we couldn't do

that

and so we're looking at our different

data coming down

uh we can kind of arrange our x1 x2 so

we have a nice plot going on and we're

going to create the

contour

that's that nice line that's drawn down

the middle on here with the red green

that's where that's what this is doing

right here with the reshape and notice

that we had to

uh

do the dot t if you remember from numpy

um if you did the numpy module

you end up with pairs you know x x1 x2

x1 x2 next row and so forth you have to

flip it so it's all one row you have all

your x1s and all your x2s

so this is what we're kind of looking

for right here on this setup

and then the scatter plot is of course

your scattered data across there we're

just going through all the points that

puts these nice little dots on to our

setup on here and we have our estimated

salary and our h and then of course the

dots are did they make a purchase or not

and just a quick note this is kind of

funny you can see up here where it says

x set y set equals x train y train which

seems kind of a little weird to do

this is because this is probably

originally a definition

so it was its own module that could be

called over and over again

and which is really a good way to do it

because the next thing we're going to do

is do the exact same thing

but we're going to visualize the test

set results

that way we can see what happened with

our test group our 25 percent

and you can see down here we have the

test set

and it

if you look at the two

graphs next to each other this one

obviously has

75 percent of the data so it's going to

show a lot more

this is only 25 of the data you can see

that there's a number that are kind of

on the edge as to whether they could

guess by age and income they're going to

make a purchase or not

but that said it still is pretty clear

it's pretty good as far as how much the

estimate is and how good it does

now

graphs are really effective

for showing people what's going on but

you also need to have the numbers and so

we're going to do from sklearn we're

going to import metrics

and then we're going to print our

metrics classification port from the y

test and the y predict

and you can see here we have precision

precision of zeros is 90 there's our

recall

0.96 we have an f1 score and a support

and we have our precision the recall on

getting it right

and then we can do our accuracy the

macro average and the weighted average

so you can see it pulls in

pretty good as far as

how accurate it is

you could say it's going to be about 90

percent is going to guess correctly

that it that they're not going to

purchase and we had an 89 chance that

they are going to purchase

um and then the other numbers as you get

down

have a little bit different meaning but

it's pretty straightforward on here

here's our accuracy and here's our micro

average and the weighted average and

everything else you might need and if

you forgot the exact definition of

accuracy it is the

true positive true negative over all of

the different setups

precision is your true positive overall

positives true and false

and recall is a true positive over true

positive plus false negative

and we can just real quick flip back

there

so you can see those numbers on here

here's our precision here's our recall

and here's our accuracy on this

thank you for joining us for mathematics

for machine learning my name is richard

kirschner with the simply learn team

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