Part 2: Analysis of Relationship Between Two Variables
Part 2: Analysis of Relationship Between Two Variables
Linear Regression Linear correlation Significance Tests Multiple regression
ESS210B Prof. Jin-Yi Yu
Linear Regression
Y=aX+b
Dependent Variable
Independent Variable
? To find the relationship between Y and X which yields values of Y with the least error.
ESS210B Prof. Jin-Yi Yu
Predictor and Predictand
In meteorology, we want to use a variable x to predict another variable y. In this case, the independent variable x is called the "predictor". The dependent variable y is called the "predictand"
Y = a + b X
the dependent variable the predictand
the independent variable the predictor
ESS210B Prof. Jin-Yi Yu
Linear Regression
We have N paired data point (xi, yi) that we want to approximate their relationship with a linear regression:
The errors produced by this linear approximation can be estimated as:
a0 = intercept a1 = slope (b)
The least square linear fit chooses coefficients a and b to produce a minimum value of the error Q.
ESS210B Prof. Jin-Yi Yu
Least Square Fit
Coefficients a and b are chosen such that the error Q is minimum:
This leads to:
covariance between x and y
Solve the above equations, we get the linear regression coefficients:
b=
where
variance of x
ESS210B Prof. Jin-Yi Yu
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