Data Analysis II

Anthony E. ButterfieldCH EN 4903-1

"There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened.”

~ Douglas Adams, Hitchhiker's Guide to the Galaxy

Data Analysis II

• Review of Data Analysis I.• Hypothesis testing.– Types of errors.– Types of tests.– Student’s T-Test

• Fit lines of lines to data.

http://www.che.utah.edu/~geoff/writing/index.html

Quick Review of PDFs and CDFs• What is the probability of measuring a value

between -0.5 and 1.5 , with =0 and =1?• What is the probability of measuring a value

between -0.5 and 1.5 or between -2 and -1?

Hypothesis Testing• How do we know if one hypothesis is more likely true over

alternatives?

• Null Hypothesis (H0) – The hypothesis to be tested to determine if it is true (often that the data observed are the result of random chance).

• Alternative Hypothesis (Hi) – A hypothesis that may be found to be the more probable source of the observations if the null hypothesis is not (often that the observations are the result of more than chance, a real effect).

Possible Types of Error in Tests• Type I Error:– Rejecting a true hypothesis, a (significance level).

• Type II Error:– Accepting a false hypothesis, b (1-test’s power).

• Tradeoff between a and b.

Testing Alternatives, Tail Tests• One Tail (One-Sided) Test.– H0: = 0.

“Our new drug is no better than the old drug”H1: > 0.“Our new drug works better than the old one.”

– H0: = 0. “The catalytic converter is just as effective as it was when new.”H1: < 0.“The catalytic converter has fowled.”

• Two Tail (Two-sided) Test.– H0: = 0.

“Our liquid is a Newtonian fluid.”H1: ≠ 0. “Our liquid is a non-Newtonian fluid.”

Student’s T-Test

• T-distribution :

• Used for small data sets, where the standard deviation is unknown.

• As the degrees of freedom, v, goes to ∞, the t-distribution becomes the normal distribution.

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Student’s T-Test• Can use to determine the likelihood of two

means being the same.

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T Statistics Example• The test statistic puts the data in question into

a scale in which we can use the T-distribution.• Is a = b, or a ≠ b,

or a > b, or a < b?

T Statistics Example

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v = 38ab = 0.324

t = -1.53

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Student’s T-Test Example

• Two sets of data, 10 measurements each, with different variances and with means separated by an increasing value.

• Note the error.• What if we take 100

measurements?

Student’s T-Test for Our p Data

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• Use t statistic and the CDF to find probability.

• Two-tailed test (P 2).• Would need t=0.064 for

95% confidence.

Linear Fitting• How to best fit a straight line, Y=b+mx, to data?

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• Coefficient of Determination (R2):

• The closer R2 is to 1 the better the fit.

Linear Fit Quality

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Nonlinear Fits

• Linearized fits.– Prone to problems.

• Nonlinear fits.– Best for nonlinear

equations.– End up with n

nonlinear equations and n unknowns.

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Fitting Example

• Equation:• Linearized fit puts inordinate emphasis on

data taken at larger values of x, in this case.

( ) ( ) yy xbxay 3exp2exp

C.I. For Fitted Constants

• Method uses Student’s T-Test, residuals and Jacobian (Matrix of partial derivatives with respect to parameters for each data point).

• You may use a statistics program.• For example: Matlab • nlfit – get fit parameters, residuals, and Jacobian.• nlparci – find the CI for parameters.• nlpredici – find CI for predicted values.

• Open the functions, though, to see how they function (“>> open nlparci” and “>> help nlparci”).

C.I. For Fitted Constants, Example

• Put code for this example online, here. >> nlinfitex2Fit to equation: y = b1 + b2 * exp(-b3 * x) x data y data 0.000 3.022 0.222 2.002 0.444 1.644 0.667 1.241 0.889 0.888 1.111 1.052 1.333 1.043 1.556 1.104 1.778 1.055 2.000 0.800b1 was 1.0, and is estimated to be: 0.949577 ± 0.158716 (95% CL)b2 was 2.0, and is estimated to be: 2.073648 ± 0.317758 (95% CL)b3 was 3.0, and is estimated to be: 2.903019 ± 1.056934 (95% CL)

Data Analysis Conclusions• Data analysis is necessary to near any objective

use of measurements.• Must have a basic grasp on statistics.• All data and calculated values should come with

some confidence interval at some probability.• You can reject data under some circumstances,

but avoid them.• Use Student’s T-Test and fitting techniques to

judge if your data match theory.