The Difference Between a Value and The Mean of a Dataset

QUIZ #03 (30 points) NAME:____________________ (uppercase) 1. (20 points) Terms, Definitions, and (possible) Values A F 0(only) [+, -] B G +(positive) [+, 0, -] C Choices -(negative) D [-, 0] E [0, +] Assume that 𝐱 = [𝒙𝟏 , 𝒙𝟐 , ⋯ , 𝒙𝒏 ]𝑻 , where 𝒙𝒊 ∈ ℝ (i.e., 𝒙𝒊 is a scalar.); thus 𝐱 ∈ ℝ𝑛 (i.e., 𝐱 is an n-dimensional (column) vector.). For each of (1) – (12) in the table below, choose all that apply from the choices (A – T) in the first table. Term (01) (arithmetic) mean (02) median (03) mode (04) difference (05) distance (06) deviation (07) variance (08) standard deviation (09) skewness (10) kurtosis Definition (Provide a definition in a short sentence or in a formula.) Possible values (One from A – G) 2. (10 points) (Vector) Norms Similarly, WLOG, assume that the two column vectors (say, 𝐱 = [𝒙𝟏 , 𝒙𝟐 , ⋯ , 𝒙𝒏 ]𝑻 and 𝐲 = [𝒚𝟏 , 𝒚𝟐 , ⋯ , 𝒚𝒏 ]𝑻). We start by asking the following questions: (1) How to compare two scalars (say, 𝒙𝒊 and 𝒙𝒋 )? (2) How to get a difference between 𝒙𝒊 and 𝒙𝒋 (or between 𝒙𝒋 and 𝒙𝒊 )? (3) How to define a length from a difference between 𝒙𝒊 and 𝒙𝒋 (or between 𝒙𝒋 and 𝒙𝒊 )? (4) Can you the same questions and answers be directly applicable to x and y? (5) What are the properties of a vector? In other words, how do you define a vector? (6) How to calculate the direction of a vector? (7) How to calculate the magnitude of a vector? (8) What is the inner product of two vectors? (9) What is the outer product of two vectors? (10) What is the cross product of two vectors? (11) What is the dot product of two vectors? (12) What do you mean by being “orthogonal” between two vectors? (13) What is the norm of a vector? (14) What is the normalization of a vector? (15) What do you mean by being “orthonormal”? Term vector norm Definition p-norm 1. ‖x‖ ≥ 0 (non-negativity) and ‖x‖ = 0, iff 𝑥𝑖 = 0. (x is a zero vector.) 2. ‖αx‖ = |𝛼|‖x‖, for ∀𝛼 ∈ ℝ. (scaling) 3. ‖x + y‖ ≤ ‖x‖ + ‖y‖. (triangle inequality) ‖⋅‖: ℝ𝑛 → ℝ 𝑛 Properties (or related contents) 1 𝑝 ‖x‖𝑝 = (∑|𝑥𝑖 |𝑝 ) ℓ𝑝 -norm 𝑖=1 ℓ1-norm L1 norm Taxicab norm Manhattan norm ℓ2 -norm L2 norm Euclidean norm ℓ -norm supremum norm, Chebyshev norm, uniform norm, infinity norm, maximum norm p-norm normalization visualization of norms – – Relationship among norms – –