1. 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.

2. A datum
(0)
(16)

3. Two data
(8)
(0)
(16)
Principal of independence of observation : perfectly opposed direction

4. Two data
(16,8)
(8)
(0, 0)
(0)
(16)

5. Two data
(16,8)
(0, 0)

6. Starting point: Zero
Ending point
(16,8)
Starting point
(0,0)

7. Starting point: Mean
Ending point
x = (x1, x2)
Starting point

8. Starting point: Mean
Starting point
(12, 12)
Ending point
x = (16, 8)

9. One group

10. Many groups

11. Degrees of freedom

12. We removed the effect of the mean We centralized the data
Starting point (mean)
(12, 12)
Ending point
x = (16, 8)
(0, 0)
= (4, -4)

13. We removed the effect of the mean (many groups)

14. We removed the effect of the mean (many groups)

15. We removed the effect of the mean (many groups)
What is the real dimensionality?

16. 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.

17. Variance

18. 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)

19. 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

20. What is the difference between these three arrows?
Therefore, the point (4,3) is at a distance of 5 from its starting point.
= sum of squares = variance×(n-1)
(4,3) 5

21. What is the difference between these three arrows?
What is the length of these three lines? 1 ? A) 1 1 1 C) 3 ? 1 2 ? 1 B)
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? 1

22. What is the difference between these three arrows?
We divide the length of the data set by its true dimensionality
= (quadratic) distance (from the mean) corrected by the (true) dimensionality of the data.

23. Normalization et standardization

24. 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)

25. Two groups or two variables

26. One group of three participants

27. Two groups of three participants

28. Two groups of three participants
They can be represented by a plane

29. Two groups of three participants
They can be represented by a plane

30. Two groups of three participants
They can be represented by a plane

31. Two groups of three participants
They can be represented by a plane
This is true whatever the number of participants

32. Correlation and regression

33. 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.

34. Relation between two vectors
If the angle reaches 90 degrees, then they share nothing in common.

35. Relation between two vectors
The cosine of the angle is the coefficient of correlation