Curve_Fit_Lin_Reg_PLS

# Curve_Fit_Lin_Reg_PLS - Department of Mechanical...

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Department of Mechanical Engineering Curve Fitting Linear Regression ES 140 Section 5 Fall 2006

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Department of Mechanical Engineering What is curve fitting? A statistical or analytical representation of experimental or calculated data. It is different from interpolation because all the points don’t need to fall on the curve—it just needs to be a “good” fit.
Department of Mechanical Engineering Why do we need to know? To mathematically describe the relationship between two variables The curve equation provides a valuable tool for making calculations

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Department of Mechanical Engineering Variable Relationships Data obtained through experimentation are called “empirical.” Relationships among data points may be complicated. We can try to fit them into a “closed form” equation —there are many ways that this can be done. Generally, we use one of the following four categories of equation: Linear Exponential Power Periodic
Department of Mechanical Engineering Linear Regression When experimental data plot as a straight line on rectangular grid paper, the equation of the line may be written as: y = mx + b where m = the slope of the line (constant) b = a constant (the y intercept, i.e. the value of y when x = 0). By using two different points on the line, the constants can be determined.

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Department of Mechanical Engineering Example – Linear Regression For example, if we know that the points (x 1 ,y 1 )=(12,8) and (x 2 ,y 2 )=(24,13) lie on the line, substituting into: y = mx + b we get: 8 = (m X 12) + b and 13 = (m X 24) + b Solving simultaneously yields: m=0.42 the slope of the line, b=3.0 the y-intercept Our equation is: y = 0.42x + 3 Sketch this curve.
Department of Mechanical Engineering Power Curves When the data do not form a straight line, you must determine which type of curve the line most closely approximates. Example: A solid object is dropped from a tall building, and the values are recorded and graphed in the next slide.

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Department of Mechanical Engineering Example – Power Curve
Department of Mechanical Engineering Power Curve Calculations We can try representing the data by the following equation: y = bx m (from physics we know that y=½gt 2 where g=acceleration due to gravity) Mathematically, this equation can be put into linear form by taking the logarithm of both sides: log y = m log x +log b (log base 10 or ln base e) This relationship indicates that if the logs of both y and x were recorded and the results plotted on rectangular paper, the line would be straight as shown on the following slide

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Department of Mechanical Engineering Example – Log-Log Power Curve
Department of Mechanical Engineering Exponential Curves Suppose the data do not produce a nearly straight line on either rectangular coordinate paper or on log- log paper.

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Department of Mechanical Engineering Example – Exponential Curves Plotting the data on semilog graph paper produces a reasonably straight line. The equation of this type
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Curve_Fit_Lin_Reg_PLS - Department of Mechanical...

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