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Correlation
Regression and Correlation
Facultad de Comercio, Turismo y Ciencias Sociales Jovellanos
Reg. and Corr.
Correlation
Main goals
Linear regression
Correlation
Reg. and Corr.
Goals Lin. regressionCorrelation
Main goals: regression and correlation
I (^) In practice, X and Y use to be: I ((((((((((
( Statistically independent I I ((((((((((^ Functionally dependentStatistically dependent. I (^) Main goals in the study of dependence between two quantitative variables: I (^) Find the best function f such that Y is closer to f (X ). I (^) Quantify to what extent f is adequate to express Y as a function of X.
Reg. and Corr.
Goals Lin. regressionCorrelation
Examples
I (^) First goal: find the best function
best function: straight line best function: straight line
f (x)
x best function: f I (^) Second goal: quantify the suitability of the best function
f (x)
x good fit^
bad fit perfect fit
Reg. and Corr.
Correlation
Regression line: example
I (^) X represents the temperature indoors I (^) Y represents the heart rate of a vertebrate
- We look for the (straight) line that best fits the data (regression).
- We measure the strength and the direction of relationship (correlation).
Reg. and Corr.
Correlation
Review: Equation of the line
b∗
a∗^ = tg(α)
α
Equation: y = a∗^ + x + b∗
I (^) The points on the straight line are all those pairs (x, y ) satisfying Equation y = a∗x + b∗. I (^) a∗: slope (pendiente) I (^) b∗: intercept (ordenada en el origen)
Reg. and Corr.
Goals Lin. regressionCorrelation
Minimum mean square error criterion
y = ax + b
b
ei
Mean square error associated to the line y = ax + b:
MSE(a, b) = e^2 =
∑n i=1 e
2 i n
∑n i=1[yi^ −^ (axi^ +^ b)]
2 n
Reg. and Corr.
Goals Lin. regressionCorrelation
Regression line of Y against X
y = ax + b
b
ei
I (^) It is the (straight) line y = a∗x + b∗that minimizes the mean square error. I (^) Equation of the regression line: I (^) a∗^ = s sxy 2 x I (^) b∗^ = y − a∗^ x = y − s sxy 2 x^ x Reg. and Corr.
Lin. regressionGoals Correlation
Interpolation-extrapolation
HEIGHT VS AGE (BABIES)
48
68
88
108
128
148
168
188
208
228
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 age
height
Regression estimations are not reliable in general to make predictions outside the range of the independent variable on which the regression line is based. Reg. and Corr.
