REVISED: Sunday, March 3, 2013
You will learn Matrix Multiplication.
1 2 15
1 2 3 15
4 5 6 42
mxn matrix nx1matrix m-dimensional
(m rows, (n-dimensioal vector
n columns) vector)
To get ci, multiply A's ith row with elements of vector b, and add them up.
The multiplication steps to create vector c are shown below:
c
(1*3) + (2*6) = 3 + 12 = 15
(1*3) + (2*6) = 3 + 12 = 15
(4*3) + (5*6) = 12 + 30 = 42
For this to work, the number of columns in Matrix A, has to match the number of rows in Matrix b.
B. Prediction Example
Our hypothesis is that we can predict the sales price of houses based on their square footage as follows:
hθ(x) = -20 + 0.35x
House sizes:
5678
9123
4567
S is our 3x2 house square footage matrix.
h is our 2x1 hypothesis matrix.
p is our 3x1 matrix product prediction of the sales price of houses resulting from the matrix multiplication of S*h.
S h p
1 5678 1967.30
1 9123 -20 3173.00
1 4567 0.35 1578.40
(-20 * 1) + (0.35 * 5678) = -20 + 1987.30 = 1967.30
(-20 * 1) + (0.35 * 9123) = -20 + 3193.00 = 3173.00
(-20 * 1) + (0.35 * 4567) = -20 + 1598.40 = 1578.40
As shown below we get the same answer using linear algebra.
hθ(5678) = -20 + (0.35 * 5678) = -20 + 1987.30 = 1967.30
hθ(9123) = -20 + (0.35 * 9123) = -20 + 3193.00 = 3173.00
hθ(4567) = -20 + (0.35 * 4567) = -20 + 1598.40 = 1578.40
C. Types
1. Supervised Learning
Supervised Learning is when you are given the right answer for each example in the data set.
2. Classification
Classification is when you have discrete-valued output.
3. Regression Problem
Predict real-valued output.
d. Notation
m = Number of training examples, rows.
(x,y) = One training example, one row.
(x(i), y(i)) = ith training example. The superscript refers to the index of the training set, the ith row.
x's = "input" variable/features.
y's = "output" variable / "target" variable.
h = Hypothesis; h is a function that maps from x's to y's.
The hypothesis is used to make predictions.
hθ (x) = θo + θ1x
shorthand h(x)
θo and θ1are parameters.
h is predicting y is a linear function of x.
Linear regression with one variable, x.
Univariate, one variable, linear regression.
I. MATRIX VECTOR MULTIPLICATION
A. Generic Example
As shown below the multiplication of a 3x2 Matrix A, by a 2x1 Vector b, results in a 3x1 Vector c.
A * b = cA. Generic Example
As shown below the multiplication of a 3x2 Matrix A, by a 2x1 Vector b, results in a 3x1 Vector c.
1 2 15
1 2 3 15
4 5 6 42
mxn matrix nx1matrix m-dimensional
(m rows, (n-dimensioal vector
n columns) vector)
To get ci, multiply A's ith row with elements of vector b, and add them up.
The multiplication steps to create vector c are shown below:
c
(1*3) + (2*6) = 3 + 12 = 15
(1*3) + (2*6) = 3 + 12 = 15
(4*3) + (5*6) = 12 + 30 = 42
For this to work, the number of columns in Matrix A, has to match the number of rows in Matrix b.
B. Prediction Example
Our hypothesis is that we can predict the sales price of houses based on their square footage as follows:
hθ(x) = -20 + 0.35x
House sizes:
5678
9123
4567
S is our 3x2 house square footage matrix.
h is our 2x1 hypothesis matrix.
p is our 3x1 matrix product prediction of the sales price of houses resulting from the matrix multiplication of S*h.
S h p
1 5678 1967.30
1 9123 -20 3173.00
1 4567 0.35 1578.40
(-20 * 1) + (0.35 * 5678) = -20 + 1987.30 = 1967.30
(-20 * 1) + (0.35 * 9123) = -20 + 3193.00 = 3173.00
(-20 * 1) + (0.35 * 4567) = -20 + 1598.40 = 1578.40
As shown below we get the same answer using linear algebra.
hθ(5678) = -20 + (0.35 * 5678) = -20 + 1987.30 = 1967.30
hθ(9123) = -20 + (0.35 * 9123) = -20 + 3193.00 = 3173.00
hθ(4567) = -20 + (0.35 * 4567) = -20 + 1598.40 = 1578.40
C. Types
1. Supervised Learning
Supervised Learning is when you are given the right answer for each example in the data set.
2. Classification
Classification is when you have discrete-valued output.
3. Regression Problem
Predict real-valued output.
d. Notation
m = Number of training examples, rows.
(x,y) = One training example, one row.
(x(i), y(i)) = ith training example. The superscript refers to the index of the training set, the ith row.
x's = "input" variable/features.
y's = "output" variable / "target" variable.
h = Hypothesis; h is a function that maps from x's to y's.
The hypothesis is used to make predictions.
hθ (x) = θo + θ1x
shorthand h(x)
θo and θ1are parameters.
h is predicting y is a linear function of x.
Linear regression with one variable, x.
Univariate, one variable, linear regression.
How to Link to My Home Page
It will appear on your website as:"Link to ELCRIC OTTO CIRCLE's Home Page"
No comments:
Post a Comment