Regression

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Regression
For a given experimental data set (input,
output), find a mathematical relationship between
the input and output.
WHY?
Use the mathematical relationship to predict
output for a given input.
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In linear regression we are after linear
relationships that can be expressed as
Think of control systems you use every
day! Dial on the stove -- gives varying heat
output Volume dial on radio --- changes radio
volume Trigger on variable speed drill -- causes
different rotational speeds
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Question? - How would you devise a system to
predict output - element temperature, drill
speed, radio volume, etc. based on control
setting?
Empirical approach keep all other parameters
constant temperature, supply voltages,
etc. vary the input parameter dial setting or
trigger depression
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record the associated output parameter levels
element temperature or chuck speed determine the
mathematical relationship output as a function
of input parameters use the mathematical
relationship to predict
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Output for a known input parameter
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How do you get the equation that best describes
the relationship between the input (control) and
output data?
Calibration examine the output for a range of
inputs range of inputs must cover the entire
range of normal operation apply a curve fit
(will assume linear although other curve shapes
may be used)
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Goal Given a set of measured responses (yi) for
a range of inputs (xi) define a and b such that
the sum of squared errors between measured and
predicted responses is minimized.
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Method Least Squares Find the error between
the measured value of the output, yi and the
predicted output, Square the error ..
WHY? Sum the squared errors Minimize the sum of
the squared error
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represents the vertical distance between each
point and the prediction line. The goal is to
minimize the sum of these distances squared!
Differentiate SSE with respect to b and a and set
each derivative equal to zero.
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Quality of Curve Fit
Simply given a set of data, we may assume a
linear relationship between the independent and
dependent variables, but how do we know if we
have a good fit?
A common technique to indicate quality of fit is
to state the sample correlation coefficient, r or
the sample coefficient of determination, r2,
where r2 1.0 for a perfect fit and equals 0
when there is no correlation.
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A simple way to demonstrate regression equations
and correlation 10 volunteers to provide
height and weight Step through the regression
equations.
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To regress weight on height w weight, h
height, N 10 Calculate averages w and
h Generate columns for Sw, Sh, Swh, Sw2,
Sh2 Calculate slope (b) and intercept (a), and
correlation coefficient (r2)
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The regression equation can be used to predict
weight given height for unknowns selected from
the class.