This document discusses vector space interpretation of random variables. It begins by introducing vector spaces and their properties such as closure under addition and scalar multiplication. Random variables can be interpreted as elements of a vector space. Inner products, norms, orthogonality and projections are discussed in the context of both vector spaces and random variables. Interpreting expectations as inner products allows treating random variables as vectors in an inner product space.
De Morgan's Laws Proof and real world application.
De Morgan's Laws are transformational Rules for 2 Sets
1) Complement of the Union Equals the Intersection of the Complements
not (A or B) = not A and not B
2) Complement of the Intersection Equals the Union of the Complements
not (A and B) = not A or not B
Take 2 Sets A and B
Union = A U B ← Everything in A or B
Intersection = A ∩ B ← Everything in A and B
U = Universal Set (All possible elements in your defined universe)
Complement = A’ Everything not in A, but in the Universal Set
What is interpolation?
How to interpolate a polynomial through a given set of data?
General approach, Newton method, Lagrange method
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Interpolation
De Morgan's Laws Proof and real world application.
De Morgan's Laws are transformational Rules for 2 Sets
1) Complement of the Union Equals the Intersection of the Complements
not (A or B) = not A and not B
2) Complement of the Intersection Equals the Union of the Complements
not (A and B) = not A or not B
Take 2 Sets A and B
Union = A U B ← Everything in A or B
Intersection = A ∩ B ← Everything in A and B
U = Universal Set (All possible elements in your defined universe)
Complement = A’ Everything not in A, but in the Universal Set
What is interpolation?
How to interpolate a polynomial through a given set of data?
General approach, Newton method, Lagrange method
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Interpolation
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
very detailed illustration of Log of Odds, Logit/ logistic regression and their types from binary logit, ordered logit to multinomial logit and also with their assumptions.
Thanks, for your time, if you enjoyed this short article there are tons of topics in advanced analytics, data science, and machine learning available in my medium repo. https://medium.com/@bobrupakroy
Computer Oriented Numerical Analysis
What is interpolation?
Many times, data is given only at discrete points such as .
So, how then does one find the value of y at any other value of x ?
Well, a continuous function f(x) may be used to represent the data values with f(x) passing through the points (Figure 1). Then one can find the value of y at any other value of x .
This is called interpolation
Newton’s Divided Difference Formula:
To illustrate this method, linear and quadratic interpolation is presented first.
Then, the general form of Newton’s divided difference polynomial method is presented.
Numerical method-Picards,Taylor and Curve Fitting.Keshav Sahu
Here i have given some topics which is related to numerical method and computing.I covered picards method, Taylors series method, Curve fitting of method of least square and fitting a non leaner curve.
The little Oh (o) notation is a method of expressing the an upper bound on the growth rate of an algorithm’s
running time which may or may not be asymptotically tight therefore little oh(o) is also called a loose upper
bound we use little oh (o) notations to denote upper bound that is asymptotically not tight.
Course: Intro to Computer Science (Malmö Högskola):
knowledge representation and abstraction, decision making, generalization, data acquistion (abstraction), machine learning, similarity
another version of abstraction
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
very detailed illustration of Log of Odds, Logit/ logistic regression and their types from binary logit, ordered logit to multinomial logit and also with their assumptions.
Thanks, for your time, if you enjoyed this short article there are tons of topics in advanced analytics, data science, and machine learning available in my medium repo. https://medium.com/@bobrupakroy
Computer Oriented Numerical Analysis
What is interpolation?
Many times, data is given only at discrete points such as .
So, how then does one find the value of y at any other value of x ?
Well, a continuous function f(x) may be used to represent the data values with f(x) passing through the points (Figure 1). Then one can find the value of y at any other value of x .
This is called interpolation
Newton’s Divided Difference Formula:
To illustrate this method, linear and quadratic interpolation is presented first.
Then, the general form of Newton’s divided difference polynomial method is presented.
Numerical method-Picards,Taylor and Curve Fitting.Keshav Sahu
Here i have given some topics which is related to numerical method and computing.I covered picards method, Taylors series method, Curve fitting of method of least square and fitting a non leaner curve.
The little Oh (o) notation is a method of expressing the an upper bound on the growth rate of an algorithm’s
running time which may or may not be asymptotically tight therefore little oh(o) is also called a loose upper
bound we use little oh (o) notations to denote upper bound that is asymptotically not tight.
Course: Intro to Computer Science (Malmö Högskola):
knowledge representation and abstraction, decision making, generalization, data acquistion (abstraction), machine learning, similarity
another version of abstraction
Vector Space & Sub Space Presentation
Presented By: Sufian Mehmood Soomro
Department: (BS) Computer Science
Course Title: Linear Algebra
Shah Abdul Latif University Ghotki Campus
Math for Intelligent Systems - 01 Linear Algebra 01 Vector SpacesAndres Mendez-Vazquez
These are the initial notes for a class I am preparing for this summer in the Mathematics of Intelligent Systems. we will start with the vectors spaces, their basis and dimensions. The, we will look at one the basic applications the linear regression.
I am Fabian H. I am a Calculus Homework Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics, Deakin University, Australia. I have been helping students with their homework for the past 6 years. I solve homework related to Calculus.
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Linear algebra power of abstraction - LearnDay@Xoxzo #5Xoxzo Inc.
LearnDay@Xoxzo is a monthly online seminar initiated by the Xoxzo team. We will have speakers from the team or guest speakers which will talk for 20 minutes each, on a subject of their choosing.
Linear algebra power of abstraction by Akira.
XOXZO Learn day
2018/12/21
======================
We have recorded sessions of our previous LearnDay here: https://www.youtube.com/channel/UCiV-bQprArQxKBSzaKY1vQg
For updates and news on our future LearnDays, follow us on Twitter (https://twitter.com/xoxzocom/) or sign up for our Exchange Newsletter (https://info.xoxzo.com/en/exchange-mailing-list/)
Opendatabay - Open Data Marketplace.pptxOpendatabay
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Leverage these privacy-preserving datasets for training and testing AI models without compromising sensitive information. Opendatabay prioritizes transparency by providing detailed metadata, provenance information, and usage guidelines for each dataset, ensuring users have a comprehensive understanding of the data they're working with. By leveraging a powerful combination of distributed ledger technology and rigorous third-party audits Opendatabay ensures the authenticity and reliability of every dataset. Security is at the core of Opendatabay. Marketplace implements stringent security measures, including encryption, access controls, and regular vulnerability assessments, to safeguard your data and protect your privacy.
Techniques to optimize the pagerank algorithm usually fall in two categories. One is to try reducing the work per iteration, and the other is to try reducing the number of iterations. These goals are often at odds with one another. Skipping computation on vertices which have already converged has the potential to save iteration time. Skipping in-identical vertices, with the same in-links, helps reduce duplicate computations and thus could help reduce iteration time. Road networks often have chains which can be short-circuited before pagerank computation to improve performance. Final ranks of chain nodes can be easily calculated. This could reduce both the iteration time, and the number of iterations. If a graph has no dangling nodes, pagerank of each strongly connected component can be computed in topological order. This could help reduce the iteration time, no. of iterations, and also enable multi-iteration concurrency in pagerank computation. The combination of all of the above methods is the STICD algorithm. [sticd] For dynamic graphs, unchanged components whose ranks are unaffected can be skipped altogether.
Explore our comprehensive data analysis project presentation on predicting product ad campaign performance. Learn how data-driven insights can optimize your marketing strategies and enhance campaign effectiveness. Perfect for professionals and students looking to understand the power of data analysis in advertising. for more details visit: https://bostoninstituteofanalytics.org/data-science-and-artificial-intelligence/
1. VECTOR SPACE INTERPRETATION OF RANDOM VARIABLES
Interpretation of random variables as elements of a vector space helps in understanding many
operations involving random variables. We start with an introduction to the concepts of the
vector space. We will discuss the principles of the minimum mean-square error estimation and
the linear minimum mean-square error estimation of a signal from noise and the vector-space
interpretation of the later.
Vector space
Consider a set V with elements called vectors and the set of real numbers . The elements of
will be called scalars. A vector will be denoted by bold-face character like v. Suppose two
operations called vector addition and scalar multiplication respectively on V.
V is called a vector space if the following properties are satisfied.
(1) ( V ,+) is a commutative group. Thus (V ,+) satisfies the following properties.
(i) Closure property: For any pair of elements ,v,w V there exists a unique
element .(v + w) V
(ii) Associativity: Vector addition is associative: v +(w +z) = (v +w)+z for
any three vectors v,w,z V
(iii) Existence of the zero vector: There is a vector 0 V such that
v +0 = 0+ v v for any .v V
(iv) Existence of the additive inverse: For any v V there is a vector v V such
that
.v +(-v) = 0 = (-v)+ v
(v) Commutativity: For any ,v,w V .v + w = w + v
(2) For any element v V and any .r the scalar multiplication r v V
This the scalar multiplication has the following properties for any , .r s and any
,v,w V
2. (1) Associativity: ( ) ( ) for , andr s rs r s v v v V
(2) Distributivity with respect to vector addition: r r r(v + w) = v + w
(3) Distributivity with respect to sacalar addition:
r s r s( )v = v + v
(4) Unity scalar: 1v = v
Example 1:
Let S be an arbitrary set and V be the set of all functions from S to . Suppose :f S and
:g S denote two functions and s S .and a . Then by definition,
( )( ) ( ) ( )f g s f s g s and
( )( ) ( )af s af s .
Therefore, addition of functions and scalar multiplication of a function are function in V. It is
easy to verify that addition of functions and the multiplication of a function by scalar satisfies
the properties of vector space. Particularly, the zero function is a function that maps all
elements in S to the real number 0. Thus,
0( ) 0,s s S
The random variables defined on a probability space ( , , )S P are functions on the sample space.
Therefore, the set of random variables forms a vector space with respect to the addition of
random variables and scalar multiplication of a random variable by a real number.
Subspace
Suppose W is a non-empty subset of V. W is called a subspace of V if W is a vector space
with respect to the vector addition and scalar multiplication defined on V.
For a non-empty subset W is a subspace of , it is sufficient that W is closed under the vector
addition and the scalar multiplication of V. Thus the sufficient conditions are:
(1) , , and
(2) , ,r r
v w W (v + w) W
v W v W
Linear Independence and Basis
Consider a subset of n vectors 1 2 nB = {b ,b ,...,b }
If 1 2 .... nc c c 1 2 nb b b 0 implies that
1 2 ... 0,nc c c then n1 2b ,b ,...,b are called linearly independen (LI)t.
3. The subset 1 2 nB = {b ,b ,...,b } of n LI vectors is called a basis if each v V can be expressed
as a linear combination of elements of B. The number of elements in B is called the dimension
of V. Thus B = {i, j,k} is a basis of 3
.
Norm of a vector
Suppose v is a vector in a vector space V defined over . The norm, v is a scalar such that
andV r v,w
1. 0
2. 0 only when
3.
4. ( Triangle Inequality )
r r
v
v v 0
v v
v w v w
A vector space V where norm is defined is called an normed vector space. For example,
following are valid norms of 1 2[ ... ] n
nv v v v :
(i) 2 2 2
1 2 ... nv v v v
(ii) 1 2max( , ,..., )nv v vv
Inner Product
If vand w are real vectors in a vector space V defined over , the inner product v,w is a
scalar such that andV r v,w,z
2
1.
2. 0, where is a induced by the inner product
3.
4.
norm
,
r , r
v, w w, v
v, v v v
v w z v, z w, z
v w v, w
A vector space V where an inner product is defined is called an inner product space. Following
are examples of inner product spaces:
(i) n
with the inner product v, w =
1 1 2 2 1 2 1 2... , [ ... ] , [ ... ]n n
n n n nv w v w v w v v v w w w v w v w =
(ii) The space 2
( )L of square-integrable real functions with the inner product
1 2 1 2, ( ) ( )f f f x f x dx
2
1 2, ( )f f L
4. Cauchy Schwarz Inequality
For any two vectors vand w belonging to an inner product space V ,
v, w w v
Let us consider
2
2 22
then , 0
, 2 , , 0
2 , 0
c c c
c c c
c c
z v w z v w v w
v v v w w w
v v w w
The left-hand side of the last inequality is a quadratic expression in variable c For the above
quadratic expression to be non-negative, the discriminant must be non-positive.. So
2 22
2 22
2 22
| 2 , | 4 0
| , | 0
,| |
| , |
v w v w
v w v w
wv v w
v w v w
The equality holds when
0 0c
c
z v w
v w
Hilbert space
Note that the inner product induces a norm that measures the size of a vector. Thus, v w is a
measure of distance between the vectors ,v,w V We can define the convergence of a
sequence of vectors in terms of this norm.
Consider a sequence of vectors , 1,2,...,n nv The sequence is said to converge to a limit
vector v if corresponding to every 0 , we can find a positive integer N such that
for .n N nv v The sequence of vectors , 1,2,...,n nv is said to be a
Cauchy sequence if
,
lim 0 .m
n m
nv v
In analysis, we may require that the limit of a Cauchy sequence of vectors is also a member of
the inner product space. Such an inner prduct space where every Cauchy sequence of vectors is
convergent is known as a Hilbert space.
Orthogonal vectors
Two vectors vand w belonging to an inner product space V are called orthogonal if
0 v, w
Orthogonal vectors are independent and a set of n orthogonal vectors 1 2 nB = {b ,b ,...,b } forms a
basis of the n-dimensional vector space.
5. Orthogonal projection
It is one of the important concepts in linear algebra widely used in random signal processing.
Suppose W is a subspace of an inner product space V. Then the subset
{ | , 0 }v v
W V v,w w W
is called the orthogonal complement of W.
Any vector v in a Hilbert space V can be expressed as
1v = w + w
where 1,
w W w W . In such a decomposition, w is called the orthogonal projection of v
on W and represents closest approximation of v by a vector in W in the following sense
min
u W
v w v u
We omit the proof of this result. The result can be geometrically illustrated as follows:
Gram-Schimdt orthogonalisation
Joint Expectation as an inner product
Interpreting the random variables andX Y as vectors, the joint expectation EXY satisfies the
properties of the inner product of two vectors. and .X Y Thus
< > XYX,Y = E
We can also define the norm of the random variable X by
2 2
< > XX X, X = E
Similarly, two random variables andX Y are orthogonal if < > XYX,Y = E = 0
We can easily verify that EXY satisfies the axioms of inner product.
The norm of a random variable X is given by
2 2
X = EX
u
1w
v
v
w
v
W
v
6. For two n dimensional random vectors
1
2
n
X
X
X
X and
1
2
,
n
Y
Y
Y
Y the inner product is
1
n
i i
i
< >= E EX Y
X, Y X Y
The norm of an n-dimensional random vector X is given by
2 2
1
n
i
i
< >= E EX
X X, X X X
Orthogonal Random Variables and Orthogonal Random Vectors
Two vectors vand w are called orthogonal if 0 v, w
Two random variables YX and are called orthogonal if 0.EXY
Similarly two n-dimension random vectors andX Y are called orthogonal if
1
0
n
i i
i
E EX Y
X Y
Just like the independent random variables and the uncorrelated random variables, the
orthogonal random variables form an important class of random variables.
If YX and are uncorrelated, then
( )( ) 0
( ) is orthogonal to ( )
X Y
X Y
E X Y
X Y
If each of YX and is of zero-mean
( , )Cov X Y EXY
In this case, 0 ( ) 0.EXY Cov XY