Regression. Petter Mostad 2005.10.10. Some problems you might want to look at. Given the annual number of cancers of a certain type, over a few decades, make a prediction for the future, with uncertainty.
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We would like to study connection between x and y!
Fit a line!
The sum of the squares of
the ”errors” minimized
Least squares method!
where and all sums are done for i=1,...,n.
This is two equations with two unknowns, and the solution of these give the answer.
Some grasshoppers make sound by rubbing their wings against each other. There is a connection between the temperature and the frequency of the movements, unique for each species. Here are some data for Nemobius fasciatus fasciatus:
If you measure 18 movements per sec, what is estim. temperature?
Data from Pierce, GW. The Songs of Insects. Cambridge, Mass.: Harvard University Press, 1949, pp. 12-21
Answer: Estimated temperature
y against x
Regression of x against y
(even in uncentered case)
SST = SSR + SSE
where are independent, normally distributed, with equal variance
where estimates variance of b
(log(x2), log(y2)), ...,(log(xn),log(yn))
Note: In this model, the curve goes through (0,0)
(x11, x12, x13, y1), (x21, x22, x23, y2), ...