Basis & Coordinate Sets & Decomposition & DA

Question 1

What is a Basis / Bases?

Answer 1

Set of Functions tat allo indexing every element in a set individually and uniquely as a linear combination of the basis

Question 2

What is the Expression for Basis?

Answer 2

∀v ∈ V : v = ∑{i} w{i} * e{i}

w{i} = Weights
e{i} = Functions

Question 3

What can any element in a set be expressed as?

Answer 3

Summation of the Linear Combination of w{i} * e{i}

Question 4

What is Dimension of a set?

Answer 4

No. of functions in the basis

Question 5

What is the Representation?

Answer 5

The tuples of values that used to point to the element in the set.

Question 6

How can a point is space be represented?

Answer 6

by a linear weighted sum of chosen coordiante basis functions

p = ∑{i} w{i} * e{i}

Question 7

For different tools and expression that simplify to ∑{i} w{i} * e{i} tell us?

Answer 7

There MUST be a coordinate basis

Question 8

Give the 3 well known coordinate basis?

Answer 8

• Canonical Basis of Euclidina Space R^n
• Orthogonal Unit Vectors in Physics i,j,k
• Argand Basis of Complex plan *C

Question 9

In simple terms what does the Coordinate Basis allow you to do?

Answer 9

Uniquely Pinpoint elements in a set

Can be scalar, but can be much more complex

Question 10

Do coordinate Basis have to be Normalised or Orthogonal?

Answer 10

NO

Question 11

What have Basis?

Answer 11

• ALL non-empty topological Sets
• ALL vector spaces (That have + & * encoded)

Question 12

What are NON-empty Topological sets guarantee?

Answer 12

At least one basis guaranteed

Note a set can have 0, many / infinite basis

Question 13

How do we switch between coordinates basis given we have 2 different representaion of one point?

Answer 13

• Get 2 points to make a system of equations and reslove the new sustem to get a new representation & bais
• JUST SIMULTANEOUS EQUATIONS:
  You use simultaneous equations to express one basis in terms of the other.That gives you the change-of-basis matrix.Then just multiply by the matrix to convert coordinates

Question 14

What is the Idea of Coordinate Systems in DA?

Answer 14

Find a coordinate basis to move/shift a point & return to the other basis to find the new basis.

ONLY WHEN IT IS DIFFICULT TO DO SO ON OG BASIS

Question 15

How can Sampling property of discrete impulse function be expressed?

Answer 15

x[n] = ∑{-∞,∞} x[k]δ[n-k]

• Expresses x[n] as a combo of the coordinates in delta t
• Signal maybe expressed as the value of the sample at the time at a certain positon. Do it for every postion to get a signal

GPT : is the discrete-time sifting property, expressing x[n] as a sum of scaled and shifted unit impulses — the fundamental “building blocks” of any discrete-time signal.

Question 16

WHat is x[n]

Answer 16

x[n] = Discrete signal (possible to express every sample as though it were a unit δ[n] * amplitude

Question 17

What is Dimension of Basis?

Answer 17

Min Number of descriptors that are required to uniquely identify any given element

Question 18

What is δ[n-k]?

Answer 18

Offset of δ[n] to select the section that we want

Question 19

What is a projection?

Answer 19

When a signal of n points can be represented as an impulse in a space with n dimensions, a point in it is called PROJECTION

Question 20

Processing

What is Processing?

Answer 20

• Internal operation
• Stays on the same plane
• When the function applies a shift to move a FULL signal to another position

Question 21

Analysis

What is Analysis?

Answer 21

• External Processes
• Moving Across Spaces
• SAME IDEA AS PROCESSING MOVING FULL SIGNALS

Question 22

What is Decomposition?

Answer 22

Algo to find Coordinate Basis of a space with some desired mathematical properties

Optionally with predefiend properties

JUST THE FINDING A BASIS WHERE IT IS EASIER TO CALCULATE THE SHIFT/OPERATION ON

Question 23

What is General Linear Model?

Answer 23

• Most Pervasive Model
• Expresses 95% of nature & the other 5% is just tranfoming
• Simple & Powerful

• I can encode GLM on the same basis as our SIGNALS
• When working with quadratics or bigger i can rename them so they can work with GLM

Question 24

What is the expression for colinear?

Answer 24

y =B1x1 + B0
AKA y = mx+c

Question 25

What is it when Linear Models are Not Colinear?

Answer 25

Not all point fit the model or fit on a straight line in that basis

Question 26

How do you Express Non Colinear Linear Models?

Answer 26

Fit all the data you have with error/uncertiaty **y =B1x1 + B0 + ε** ## Footnote Go from deterministic lines to stochastic lines

Question 27

How to solve Non Colinear Linear Models?

Answer 27

Use Least Squares

Question 28

What is the Expression for a Linear Model?

Answer 28

y = B1 * e1 + B0 * e0 + e # SINGLE y = Bn * en + ... + B1 * e1 + B0 * e0 *They both simpilfy to:* Σ{i} Bi * xi **THIS IS A BASIS** ## Footnote For both single and multivariable

Question 29

What is the Matric Form of GLM?

Answer 29

**Y** = **X** * B | Y = Matrix of Dependent Variables X = Matrix of Indepenedent Variables ## Footnote Use this formal for most of DA & expresses 95% of the universe

Question 30

What is the Matrix Form for Finding B?

Answer 30

**B = ((X^T * X)^ -1) * X^T * Y** ## Footnote Solves Deterministic & Stochastic Models Always gets one answer Can't tell if solution is good mathematically only YOU can determine that

Question 31

What is the Derivation to find B?

Answer 31

XB = Y => (X^T X)B = X^T Y # * *X^T guarnatees prodcut is square and therefore we can calculate inverse* => ((X^T X)^ -1)(X^T X)B = X^T Y ((X^T X)^ -1) => IB = X^T Y ((X^T X)^ -1) => B = X^T Y ((X^T X)^ -1)

Question 32

Whats the rules with Multiplying Matrixes?

Answer 32

Only possible if one matrix has n rows and the other matrix has n columns

Question 33

What is Fourier Analysis?

Answer 33

Method of Operating on signals that is very good at spectral & Filtering ## Footnote Another form of defining a Coordinate Space

Question 34

What does Fourier Analysis do?

Answer 34

Decomposes a signal in the time domain f(t) into its components frequencies g(wi) | No matter complexity can always conver it into sine & cosine

Question 35

What are the 2 ways to Express Fourier Analysis?

Answer 35

- **Complex Exponentials:** f(t) =Σ{-N,N} cn * e ^ (jwtn) - **Sinusoidals:** - f(t) = Σ{0,N}(an * cos(jwtn) + bn * sin(jwtn)) ## Footnote Both related to Euler's Model => e^(jθ) = cos(θ) + jsin(θ) **Just weighted Summation Formulas**

Question 36

What is the Idea of Fourier like the process?

Answer 36

Taking the Signal (the Wave graph), making it a point on a ne basis then moving the point another place and seeing the change on the orginal basis ## Footnote All that matters is ur shifting or moving the point around

Question 37

What is Low Pass Filter?

Answer 37

Makes high component 0 and leaves low as it is. ## Footnote All that matters is ur shifting or moving the point around

Question 38

What is High Pass Filter?

Answer 38

Makes Low Component Cooridnate 0 and Leaves the high one the same. ## Footnote All that matters is ur shifting or moving the point around

Question 39

What is Wavelets?

Answer 39

Approximate a Complex function using superpositioning of similar functions Number of possible wavlets families are latge , therefore can capture a wide range of signal features. Each Wavlet is a wave; Amp at 0 , increase and decrease back to 0

Question 40

What is the Formula for wavelets?

Answer 40

f(t) = Σ{k * n} ck,n Ψ(t; k ,n) ## Footnote Similar to Sinusoidals but compresses time Shifted and scaled by mother wavelet Ψ Function represents Coordinate basis & coeffient represents signal ck,n

Question 41

What is Principal Component Anlaysis & Classical Multidimensional Scaling?

Answer 41

Tool used to find a coordinate basis of a data set such that it aligns wiht MAX Var in datase, in each dimension. Maths based on eigen-decompostion of the correlation matrix ## Footnote PCA 7 cMDS are mathematical duality & thhey yield the same solution

Question 42

Formula for PCA?

Answer 42

p' = Σ{i} (eig i) ^T * (P - μP) ## Footnote eig i => Function (P - μP) => Weight μP => Mean of all data

Question 43

Steps for PCA?

Answer 43

1) Mean Correct Data: *subtract mean from each point* X' = X - E[X] => E[X'] = 0 2) Normalise Data *Define Matrix* H{k * M} = (1 / (sqrt(m-1))) * X'^T Therefore Cov x',x = (X' X^T) /( m-1) = H^T H

Question 44

In PCA What forms the New Basis?

Answer 44

The EigenVectors

Question 45

What does PCA do?

Answer 45

Rotates the Axix/Data: Can do Low pass, high pass, & bandpass & pand rejection filtering

Question 46

What do all Decompostion Follow at is core?

Answer 46

p = **∑{i} w{i} * e{i}**

Question 47

What is A manifold?

Answer 47

Manifolds are sets that can be mapped from one projection onto another - Set with a countable base - Homeomorphic to flat spaces *Hold locally a certain property anywhere* - Family of Topological Spaces - Root in low level maths - Expensive an Very powerful that it often hard to think of practical problems that cannot be seene as a specific case

Question 48

What does Homeomorphic Mean?

Answer 48

- Holds locally a certain property everywhere - Every point **P** in **M** there is an open neighbourhood **U** of **P** & homeomorphism **f: U => V** that maps set **U** to open set **V**. ## Footnote Don't have to be euclidean Cocneputally the object is flat in the local space

Question 49

What is Manifold Embedding

Answer 49

Projection of a manifold onto another, commonly to achieve dimensionality reduction #

Question 50

What is Manifold Equation?

Answer 50

p = Σ{i} wi^(u * p) * ei^(-u * p) (u * p) => the neighbourhood of p (Having a local coordinate basis at each point p)

Question 51

Genreal Remarks/ NOTES:

Answer 51

- Need to use non-linear model use it - Need to generalise then do, if not dont - Know when to stop with mathematical knowledge

Question 52

Explain Info in an inclosed system?

Answer 52

- Info cannot be created or destoryed in an inclosed system - More u manipulate data the entropy grows (*Move points where i want without abusing our power *)