Transcript review

Vectors geometry: Playing with arrows
• How using a vector (arrow) we can represent concepts of
– Mean, variance (standard deviation), normalization and
standardization.
• How using two vectors we can represent concepts of
– Correlation and regression.
A datum
(0)
(16)
Two data
(8)
(0)
Principal of independence of
observation : perfectly opposed
direction
(16)
Two data
(8)
(16,8)
(0,(0)
0)
(16)
Two data
(16,8)
(0, 0)
Starting point: Zero
Ending point
(16,8)
Starting point
(0,0)
Starting point: Mean
Ending point
x = (x1, x2)
Starting point
(x, x)
Starting point: Mean
Starting point
(12, 12)
Ending point
x = (16, 8)
One group
Many groups
Degrees of freedom
We removed the effect of the mean
We centralized the data
Starting point (mean)
(12, 12)
(0, 0)
Ending point
x = (16, 8)
x  x = (4, -4)
We removed the effect of the mean
(many groups)
We removed the effect of the mean
(many groups)
We removed the effect of the mean
(many groups)
What is the real
dimensionality?
We removed the effect of the man
• If we have two data, we will get one dimension.
• If we have three data, we will get two dimensions
.
.
.
• If we have n data, we will get n-1 dimensions.
 In other words, degrees of freedom represent the true dimensionality of
the data.
Variance
What is the difference between these three
vectors (composed of two data each) ?
 Length (distance)
 The higher the variability, the longer the length
will be.
(-0.5, 0,5)
(1.5, -1.5) (2.5, -2.5)
What is the difference between these three
arrows?
How do we measure the length (distance)?
Pythagoras
Hypotenuse of a triangle
? = (4^2+3^2) = 25 = 5
(4,3)
5?
3
4
What is the difference between these three
arrows?
Therefore, the point (4,3) is at a distance of 5 from
its starting point.
n
52 

( xi  x ) 2 = sum of squares = variance×(n-1)
i 1
(4,3)
5
What is the difference between these three
arrows?
What is the length of these three lines?
1?
A)
1
1
1
2?
B)
C)
3
?
1
1
1
 The dimensionality inflates the variability.
In order to a have a measure that can take
into account the dimensionality, what do we
need to do?
What is the difference between these three
arrows?
•We divide the length of the data set by its true dimensionality
n
Variance 
2
(
x

x
)
 i
i 1
n-1
= (quadratic) distance (from the mean)
corrected by the (true) dimensionality of
the data.
n
Standard deviation 
2
(
x

x
)
 i
i 1
n-1
Normalization et standardization
Normalization vs Standardization
•
To normalize is equivalent as to bring a given vector x (arrow) centered (mean = 0) to a
length of 1..
•
Normalization: z = x  by its length
Sz2 = 1
•
Standardization: zx = x  SD
Szx2 = n-1
=> zx = z*(n-1)
Two groups or two variables
One group of three participants
Two groups of three participants
Two groups of three participants
• They can be
represented by a
plane
Two groups of three participants
• They can be
represented by a
plane
Two groups of three participants
• They can be
represented by a
plane
Two groups of three participants
• They can be
represented by a
plane
• This is true
whatever the
number of
participants
Correlation and regression
Relation between two vectors
•
•
If two groups (u and v) have the same data, then the two vectors are superposed on
each other.
As the angle between them increases, the direction changes.
Relation between two vectors
•
If the angle reaches 90 degrees, then they share nothing in
common.
Relation between two vectors
•
The cosine of the angle is the coefficient of correlation
b 2  c 2  a1
 u  xu v  xv 
cos  

2
2
2bc
 u  xu   v  xv 
 u  x v  x 
u
cos  
v
n 1
 u  x 
 v  x 
n 1
n 1
2
u
2
v

cov uv
 ruv
su s v