DS-Lecture-3c-Covariance and Correlation.pdf
- 1. Data Science Instructor:Maham Naeem September 11, 2024 1 National University of Computer and Emerging Sciences, Lahore Covariance and Correlation Lecture No. 3c
- 2. Covariance Covariance measures the direction of the relationship between two variables. A positive covariance means that both variables tend to be high or low at the same time. A negative covariance means that when one variable is high, the other tends to be low
- 3. Covariance vs Correlation Coefficient Covariance measures the direction of a relationship between two variables Correlation measures the strength of that relationship. Both correlation and covariance are positive when the variables move in the same direction, and negative when they move in opposite directions. However, a correlation coefficient must always be ������� − � ��� + �, with the extreme values indicating a strong relationship.
- 4. Computing Covariance ��� �, � = �=1 � �� − � × �� − � � Average of variable A Average of variable B ��� �, � = �=1 � �� × �� � − � × � �������� �� = � � �=� � �� − � � �������� �� = � � �=� � �� � − �� Recall from Lecture 4: Note: Variance is special case of covariance where two attributes are identical (covariance with itself)
- 5. Computing Covariance ��� �, � = �=1 � �� × �� � − � × � � �������������� = 6 + 5 + 4 + 3 + 2 5 = 20 5 = 4 � ���ℎ���ℎ = 20 + 10 + 14 + 5 + 5 5 = 54 5 = 10.80 ��� ��������������, ���ℎ���ℎ = 6 × 20 + 5 × 10 + 4 × 14 + 3 × 5 + 2 × 5 5 − 4 × 10.80 ��� ��������������, ���ℎ���ℎ = 50.2 − 43.2 ��� ��������������, ���ℎ���ℎ = 7 What if the data is nominal?
- 6. Correlation Correlation is a statistical term describing the degree to which two variables move in coordination with one another. If the two variables move in the same direction, then those variables are said to have a positive correlation. If they move in opposite directions, then they have a negative correlation. The strength of the correlation is determined by the correlation coefficient, which varies between −� and +�. What if correlation is 0?
- 7. Correlation Between Two Variables Weak relationship have small correlation value… Strong relationship have large correlation value… Correlation is 1 if a straight line with positive slope can be drawn from the center of all data points
- 8. Correlation Between Two Variables
- 9. Correlation Between Two Variables
- 10. Correlation Between Two Variables But … How to Compute Correlation?
- 11. Computing Correlation X Y 1 2 2 4 3 7 4 9 5 12 6 14 X Y XY �� �� 1 2 2 4 3 7 4 9 5 12 6 14
- 12. Computing Correlation X Y 1 2 2 4 3 7 4 9 5 12 6 14 X Y XY �� �� 1 2 2 2 4 8 3 7 21 4 9 36 5 12 60 6 14 84
- 13. Computing Correlation X Y 1 2 2 4 3 7 4 9 5 12 6 14 X Y XY �� �� 1 2 2 1 4 2 4 8 4 16 3 7 21 9 49 4 9 36 16 81 5 12 60 25 144 6 14 84 36 196 21 48 211 91 490 � � �� �� ��
- 14. Computing Correlation X Y 1 2 2 4 3 7 4 9 5 12 6 14 21 48 211 91 490 � � �� �� �� � = � × ��� − �� × �� � × �� − �� � × � × ��� − �� � � = ���� − ���� ��� − ��� × ���� − ���� = ��� ��� × ��� = ��� ����� = �. ��� More Details: https://www.investopedia.com/ask/answers/032515/what-does-it-mean-if-correlation-coefficient-positive-negative-or-zero.asp This is “Pearson’s Correlation Coefficient”
- 15. Correlation of Multiple Variables IRIS Dataset: Again… Heatmap
- 16. Correlation Heatmaps: Complex Examples We can also create heatmaps for Similarity and Dissimilarity (Distance) between two objects/records… But … What is similarity and dissimilarity between two objects and how to measure it?
- 17. Correlation Interpretation More Details: https://statistics.laerd.com/statistical-guides/pearson-correlation-coefficient-statistical-guide.php
- 18. Correlation does NOT imply causality… if A and B are correlated, this does not necessarily imply that A causes B or that B causes A. For example, in analyzing a demographic database, we may find that attributes representing the number of hospitals and the number of car thefts in a region are correlated. This does not mean that one causes the other. Both are actually causally linked to a third attribute, namely, population.
- 19. What Correlation Tells Us? Some attributes are redundant Redundancy here means that an attribute (such as annual_revenue) may be redundant if it can be “derived” from other attribute or set of attributes. Inconsistencies in attribute (or dimension) naming can also cause redundancies in the resulting dataset. Some redundancies can be detected by correlation analysis.
- 20. Covariance vs Correlation Covariance : Direction (any value) Correlation: Direction and Strength (-1 to +1)
- 21. Correlation Analysis Correlation Analysis: Given two attributes, how strongly one attribute implies the other, based on the available data For Numerical Data: Correlation coefficient (or Pearson’s Correlation Coefficient) Covariance For Nominal Data: �2 (Chi-Square) test
- 22. Reading Chap 2 nd 3 (Han and Kamber)
- 23. Activity Calculate Correlation between these two variables X, Y X Y 2 10 4 9 6 8 8 7 10 6
- 24. Activity Calculate Covariance of the following two variables X, Y X Y 2.1 8 2.5 10 3.6 12 4 14