Variations. ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra
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1 Variations ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra
2 Last Time Probability Density Functions Normal Distribution Expectation / Expectation of a function Independence Uncorrelated Chain Rule 2
3 Two Random Variables (and their relationships)
4 Several Random Variables Joint PDFs of Random Variables pp xx, yy Joint PDF of 2 random variables Vector form Define WW = XX YY, ww = xx yy Credit: #/media/file:multivariatenormal.png pp ww = pp xx, yy Joint PDF of 2 random variables (short version) 4
5 Several Random Variables Chain Rule for 2 Random Variables pp xx, yy = pp xx pp yy xx = pp yy pp xx yy If Random Variables are independent pp xx, yy = pp xx pp yy 5
6 Several Random Variables Moments of two random variables XX, YY Mean: E XX, E YY 2 nd Moment: E XX 2, EE YY 2 Variance of X: Var XX = E XX E XX 2 Variance of Y: Var YY = E XX E YY 2 Co-variance: Cov XX, YY = E XX E XX YY E YY = E XXXX E XX E YY Variance of sum: Var aaaa + bbbb = a 2 Var XX + b 2 Var YY + 2aaaa Cov XX, YY 6
7 Several Random Variables If XX, YY are uncorrelated Mean: E XX, E YY 2 nd Moment: E XX 2, EE YY 2 Variance of X: Var XX = E XX E XX 2 Variance of Y: Var YY = E XX E YY 2 Co-variance: Cov XX, YY = E XXXX E XX E YY = 0 Variance of sum: Var aaaa + bbbb = a 2 Var XX + b 2 Var YY Important Note: Independent implies uncorrelated, But uncorrelated does not imply independent 7
8 Multiple Random Variables (and their relationships)
9 Several Random Variables Joint PDFs of Random Time Series pp xx = pp xx 1, xx 2, xx 3,, xx NN Joint PDF of N random variables Credit: e:mv_indp.png Credit: le:multivariatenormal.png 9
10 Several Random Variables Chain Rule for Multi-variant Variables pp xx, yy, zz = pp xx pp yy, zz xx pp xx, yy, zz = pp xx pp yy xx pp zz xx, yy or many other permutations If Random Variables are independent pp xx, yy, zz = pp xx pp yy pp zz 10
11 Several Random Variables Expectations EE gg XX, YY, ZZ = gg xx, yy, zz pp xx, yy, zz dddd dddd dddd Generally very difficult to compute Question: Are the ways/models that simplify this problem? 11
12 Several Random Variables Moments of a random vector xx = XX 1 XX 2 XX NN TT Mean: E xx = EE XX 1 EE XX 2 EE XX NN TT Co-variance matrix: Cov xx = E xx E xx xx E xx TT = E xxxx TT E xx E xx TT Variance of sum: Var NN NN NN XX nn = Var xx TT 11 = Cov XX nn, XX mm = 11 TT Cov xx 11 nn=1 mm=1 nn=1 NN NN NN Var aa nn XX nn = Var xx TT aa = aa nn aa mm Cov XX nn, XX mm = aa TT Cov xx aa nn=1 mm=1 nn=1 12
13 Several Random Variables Uncorrelated random vector xx = XX 1 XX 2 XX NN TT Mean: E xx = EE XX 1 EE XX 2 EE XX NN TT Co-variance matrix: Cov xx = E xx E xx xx E xx TT = E xxxx TT E xx E xx TT = DD Diagonal matrix Variance of sum: Var NN NN XX nn = Var xx TT 11 = Var XX nn = 11 TT Cov xx 11 nn=1 nn=1 NN NN Var aa nn XX nn = Var xx TT aa = aa 2 nn Var XX nn = aa TT Cov xx aa nn=1 nn=1 Diagonal matrix 13
14 Several Random Variables Multi-Variate Normal (MVN) vector xx NN μμ, ΣΣ Gaussian Random Vector of Length NN Mean vector Covariance Matrix Probability Distribution Function: pp xx = 1 2ππ NN/2 ΣΣ exp 1 2 xx μμ TT ΣΣ 1 xx μμ Mean: Determinant Co-variance Matrix: EE xx = μμ = EE XX 1 EE XX 2 EE XX NN EE xx μμ xx μμ TT = ΣΣ i is row EE XX ii μμ ii XX jj μμ jj = Σ iiii j is column 14
15 Several Random Variables Multi-Variate Normal (MVN) vector xx NN μμ, ΣΣ Gaussian Random Vector of Length NN Mean: Co-variance Matrix: EE xx = μμ = EE XX 1 EE XX 2 EE XX NN EE xx μμ xx μμ TT = ΣΣ EE XX ii μμ ii XX jj μμ jj = Σ iiii i is row j is column 15
16 Several Random Variables Multi-Variate Normal (MVN) vector xx NN μμ, ΣΣ Gaussian Random Vector of Length NN Bivariate Case EE XX = μμ = μμ 1 μμ 2 EE XX μμ XX μμ TT = ΣΣ = σσ 1 2 σσ 1 σσ 2 σσ 1 σσ 2 σσ 2 2 Question: How can we interpret these values? 16
17 Several Random Variables Example: Multi-Variate Normal (MVN) vector xx = XX 1 XX 2 XX 3 XX 4 ~ NN μμ, ΣΣ Consider μμ = , ΣΣ = What is the pdf of pp xx 1? 17
18 Several Random Variables Example: Multi-Variate Normal (MVN) vector xx = XX 1 XX 2 XX 3 XX 4 ~ NN μμ, ΣΣ Consider μμ = , ΣΣ = What is the pdf of pp xx 1, xx 3? 18
19 Several Random Variables Example: Multi-Variate Normal (MVN) vector xx = XX 1 XX 2 XX 3 XX 4 ~ NN μμ, ΣΣ Consider μμ = , ΣΣ = Determine EE XX 2? 2 19
20 Several Random Variables Example: Multi-Variate Normal (MVN) vector xx = XX 1 XX 2 XX 3 XX 4 ~ NN μμ, ΣΣ Consider μμ = , ΣΣ = Determine EE XX 2 + XX 3? 5 20
21 Several Random Variables Example: Multi-Variate Normal (MVN) vector xx = XX 1 XX 2 XX 3 XX 4 ~ NN μμ, ΣΣ Consider μμ = , ΣΣ = Determine EE XX 2 2 2? 6 21
22 Several Random Variables Example: Multi-Variate Normal (MVN) vector xx = XX 1 XX 2 XX 3 XX 4 ~ NN μμ, ΣΣ Consider μμ = , ΣΣ = Determine EE XX 2 2 XX 3 3? 7 22
23 Several Random Variables Example: Multi-Variate Normal (MVN) vector xx = XX 1 XX 2 XX 3 XX 4 ~ NN μμ, ΣΣ Consider μμ = , ΣΣ = Determine EE XX 2 XX 3? EE XX 2 EE XX 2 XX 3 EE XX 3 = EE XX 2 XX 3 EE XX 2 EE XX 3 EE XX 2 XX 3 = CCCCCC XX 2, XX 3 + EE XX 2 EE XX 3 = = 13 23
24 Several Random Variables Example: Multi-Variate Normal (MVN) vector xx = XX 1 XX 2 XX 3 XX 4 ~ NN μμ, ΣΣ Consider μμ = , ΣΣ = Determine EE xx μμ xx μμ TT? EE xx μμ xx μμ TT = ΣΣ 24
25 Several Random Variables Example: Multi-Variate Normal (MVN) vector xx = XX 1 XX 2 XX 3 XX 4 ~ NN μμ, ΣΣ Consider μμ = , ΣΣ = Determine EE xx μμ TT xx μμ? EE xx μμ TT xx μμ = EE XX 1 μμ 1 2 XX 2 μμ 2 2 XX 3 μμ 3 2 XX 4 μμ 4 2 =
26 Several Random Variables Example: Multi-Variate Normal (MVN) vector xx = XX 1 XX 2 XX 3 XX 4 ~ NN μμ, ΣΣ Consider μμ = , ΣΣ = Determine EE xx TT xx? EE xx μμ TT xx μμ = EE XX 1 2 XX 2 2 XX 3 2 XX 4 2 = vvvvvv XX 1 vvvvvv XX 2 + vvvvvv XX 3 vvvvvv XX 4 EE XX 1 2 EE XX 2 2 EE XX 3 2 EE XX 4 2 =
27 Several Random Variables Example: Multi-Variate Normal (MVN) vector xx = XX 1 XX 2 XX 3 XX 4 ~ NN μμ, ΣΣ Consider μμ = , ΣΣ = Determine EE xxxx TT? EE xx μμ xx μμ TT = EE xxxx TT μμμμ TT EE xxxx TT = ΣΣ + μμμμ TT = =
28 Several Random Variables Example: Multivariate Exponential Distribution Let XX 1, XX 2, XX 3,, XX NN be iid (independent and identically distributed) random variables such that XX 1, XX 2, XX 3,, XX NN : eeeeeeeeeeeeeeeeeeeeee λλ pp xx; λλ = λλee λλλλ for xx 0 0 for xx < 0 Compute the joint PDF pp xx 1, xx 2,, xx NN ; λλ Credit: 3/previews/html/gkde2test_05.png 28
29 Several Random Variables Example: Multivariate Exponential Distribution Let XX 1, XX 2, XX 3,, XX NN be iid (independent and identically distributed) random variables such that XX 1, XX 2, XX 3,, XX NN : eeeeeeeeeeeeeeeeeeeeee λλ pp xx; λλ = λλee λλλλ for xx 0 0 for xx < 0 Compute the joint PDF pp xx 1, xx 2,, xx NN ; λλ pp xx 1, xx 2,, xx NN ; λλ = pp xx 1 ; λλ pp xx 2 ; λλ pp xx 3 ; λλ pp xx NN ; λλ pp xx 1, xx 2,, xx NN ; λλ = NN nn=1 pp xx nn ; λλ pp xx 1, xx 2,, xx NN ; λλ = λλ NN ee λλ xx 1+xx 2 +xx 3 + +xx NN pp xx 1, xx 2,, xx NN ; λλ = λλ NN ee λλ nn=1 NN xx nn pp xx; λλ = λλ NN ee λλ xxtt 11 Vector of all ones 29
30 Several Random Variables Example: Multivariate Exponential Distribution Let XX 1, XX 2, XX 3,, XX NN be iid (independent and identically distributed) random variables such that XX 1, XX 2, XX 3,, XX NN : eeeeeeeeeeeeeeeeeeeeee λλ pp xx; λλ = λλee λλλλ for xx 0 0 for xx < 0 Compute EE XX TT XX, where XX = XX 1, XX 2, XX 3,, XX NN TT 30
31 Several Random Variables Example: Multivariate Exponential Distribution Let XX 1, XX 2, XX 3,, XX NN be iid (independent and identically distributed) random variables such that XX 1, XX 2, XX 3,, XX NN : eeeeeeeeeeeeeeeeeeeeee λλ pp xx; λλ = λλee λλλλ for xx 0 0 for xx < 0 Compute EE XX TT XX, where XX = XX 1, XX 2, XX 3,, XX NN TT Due to independence: EE XX TT XX = EE XX EE XX EE XX NN 2 = 2NN λλ 2 31
32 Linear Models
33 Linear Models Linear Systems In this class, we will work often with linear systems (or linear models of systems) Question: What is a linear system? How is it defined? Why do we want linear systems? 33
34 Linear Models Linear Systems In this class, we will work often with linear systems (or linear models of systems) Question: What is a linear system? How is it defined? Why do we want linear systems? Must satisfy superposition H αα xx 1 nn + ββ xx 2 nn = αα H xx 1 nn + ββ H xx 2 [nn] αα, ββ R xx 1 nn, xx 2 nn R NN 34
35 Linear Models The Standard Linear Model yy nn = H xx nn + ww nn yy nn is an output H is a linear system xx nn is an input ww nn is noise 35
36 Linear Models The Standard Linear Model yy = HHHH + ww yy is an output HH is a linear system (also known as the observation matrix or channel) xx is an input ww is noise 36
37 Linear Models The Standard Linear Model xx = HHHH + ww xx is an output (Kay s notation) HH is a linear system (also known as the observation matrix or channel) θθ is an input (Kay s notation) ww is noise 37
38 Linear Models The Standard Linear Model xx = HHθθ + ww xx is an output (Kay s notation) HH is a linear system (also known as the observation matrix or channel) θθ is an input (Kay s notation) ww is noise Question: If ww ~ NN 0, II, how is xx distributed? 38
39 Linear Models The Standard Linear Model xx = HHθθ + ww xx is an output (Kay s notation) HH is a linear system (also known as the observation matrix or channel) θθ is an input (Kay s notation) ww is noise Question: If ww ~ NN 0, II, how is xx distributed? xx ~ NN HHHH, II 39
40 Linear Models The Standard Linear Model xx = HHθθ + ww xx is an output (Kay s notation) HH is a linear system (also known as the observation matrix or channel) θθ is an input (Kay s notation) ww is noise Question: If θθ ~ NN 0, II, ww = 00, how is xx distributed? 40
41 Linear Models The Standard Linear Model xx = HHθθ + ww xx is an output (Kay s notation) HH is a linear system (also known as the observation matrix or channel) θθ is an input (Kay s notation) ww is noise Question: If θθ ~ NN 0, II, ww = 00, how is xx distributed? xx ~ NN 00, HH TT HH Real Complex xx ~ NN 00, HH HH HH 41
42 Linear Models The Standard Linear Model xx = HHθθ + ww xx is an output (Kay s notation) HH is a linear system (also known as the observation matrix or channel) θθ is an input (Kay s notation) ww is noise Question: If θθ ~ NN 0, II, ww = 00, how is xx distributed? Covariance matrix II EE xx = EE HHθθ = HH EE θθ = 00 EE xx EE xx xx EE xx TT = EE xxxx TT = EE HHθθθθ TT HH TT = HH EE θθθθ TT HH TT = HHHH TT 42
43 Linear Models The Standard Linear Model xx = HHθθ + ww xx is an output (Kay s notation) HH is a linear system (also known as the observation matrix or channel) θθ is an input (Kay s notation) ww is noise Question: If θθ ~ NN μμ, ΣΣ, ww = 00, how is xx distributed? 43
44 Linear Models The Standard Linear Model xx = HHθθ + ww xx is an output (Kay s notation) HH is a linear system (also known as the observation matrix or channel) θθ is an input (Kay s notation) ww is noise Question: If θθ ~ NN μμ, ΣΣ, ww = 00, how is xx distributed? xx ~ NN HHHH, HH ΣΣHH TT Real Complex xx ~ NN HHμμ, HHΣΣHH HH 44
45 Vector Space, Subspaces, and Dimensions
46 Fields A field We will discuss these with vector spaces A field FF is a set of scalars with two operations: addition and multiplication such that The elements of FF have closure under addition and multiplication. Associativity of addition and multiplication Commutativity of addition and multiplication Existence of additive and multiplicative identity elements Existence of additive inverses and multiplicative inverses Distributivity of multiplication over addition ECE 6534, Chapter 2 46
47 Fields A field We will be largely working with the following field R (real numbers) C (complex numbers) N (natural numbers) Z (integer numbers) Q (rational numbers) ECE 6534, Chapter 2 47
48 Vector Spaces Vector spaces A vector space over a field of scalars C (or R) is a set of vectors, VV, together with operations of vector addition and scalar multiplication. For any xx, yy, zz in VV and αα, ββ in C (or R), these operations must satisfy the following properties: 1. Commutativity over addition: xx + yy = yy + xx 2. Associativity over addition/scalar multiplication: xx + yy + zz = xx + yy + zz and αααα xx = αα ββββ 3. Distributivity: αα xx + yy = αααα + αααα and αα + ββ xx = αααα + ββββ 4. Additive Identity: There exists an element 0 in V such that xx + 0 = 0 + xx = xx 5. Additive Inverse: There exists a unique element -xx in V such that xx + xx = xx + xx = Multiplicative Identity: For every xx in VV, 1 xx = xx 48
49 Examples of vector spaces Vector space example: R 2, C 2 : two-dimensional space C 2 = xx = xx 1 xx 2 xx 1 C, xx 2 C 49
50 Examples of vector spaces Vector space example: R 2, C 2 : two-dimensional space Examples: R2, + 2jj 4 2 C 2, αα ββ C2 if αα, ββ C
51 Examples of vector spaces Vector space example: R 3, C 3 : three-dimensional space xx 1 C 3 = xx = xx 2 xx 1 C, xx 2 C, xx 3 C xx 3 51
52 Examples of vector spaces Vector space example: R 3, C 3 : two-dimensional space Examples: R 3, 1 + jj 1/2 2jj C 3, αα ββ γγ C 2 if αα, ββ, γγ C αα ββ γγ 52
53 Examples of vector spaces Vector space example: R NN, C NN : N-dimensional space C NN = xx = xx 0 xx 1 TT xx nn C, nn 0,1,, NN Example: aa 1 aa 2 C NN if aa 1, aa 2, aa 3 C aa NN 53
54 Examples of vector spaces Vector space example: Space of complex-valued sequence of N C N = xx = xx nn xx nn C, nn N 54
55 Examples of vector spaces Vector space example: Space of complex-valued functions of R C R = xx = xx tt xx(tt) C, tt R 55
56 Examples of vector spaces Vector space example: R MM NN, C MM NN : MN-dimensional space Example: aa 1,1 aa 1,2 aa 1,NN aa 2,1 aa 2,2 aa 2,NN aa MM,1 aa MM,2 aa MM,NN C MM NN if aa mm,nn C mm, nn 56
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