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- Re: Peak of a probability distribution

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May 18, 2017 12:43 PM
(754 views)

Just to add on.

Mark's way is really the best way to get the peak value, but for a quick way to get an estimated value you can fit the curve then use the JMP Crosshairs tool found in the Tools menu or on your menu bar. Click the tool then put the crosshairs symbol on the spot you feel best represents the peak and a value will pop up. If you are using JMP 13 you will get a magnifier with the Crosshair tool that will help you get close.

HTH

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May 19, 2017 2:39 AM
(690 views)

Mark's method is neat. Here's an alternative that uses the function directly:

```
NamesDefaultToHere(1);
ClearLog();
// Make some representative data
dt = NewTable("Johnson Su", NewColumn("Data", Numeric, Continuous, Formula(Random Johnson Su( 5, 2, 1, 1 ))));
dt << addRows(50);
// Fit the data
dist = dt << Distribution(Continuous Distribution( Column( :Data ), Fit Distribution( Johnson Su ) ));
// Get the fitted parameters
params = Report(dist)[NumberColBox(3)] << get;
gamma = params[1]; // Shape
delta = params[2]; // Shape
theta = params[3]; // Location
sigma = params[4]; // Scale
// http://www.mathwave.com/articles/johnson_su_distribution.html
// f(x) = delta/(lambda * Sqrt(2*pi()) * Sqrt(z^2 + 1)) * exp( -1/2*(gamma + delta * ln(z + Sqrt(z^2+1)))^2 );
// where:
// z = (x - eta)/lambda;
// and:
// delta > 0 lambda > 0
jsu = Expr(delta/(lambda * Sqrt(2*pi()) * Sqrt(z^2 + 1)) * exp( -1/2*(gamma + delta * ln(z + Sqrt(z^2+1)))^2 ) );
SubstituteInto(jsu, Expr(z), Expr((x - eta)/lambda));
// Use the fitted parameter values
// Note the change in parameter names! 'lambda' -> 'sigma' and 'eta' -> 'theta'
SubstituteInto(jsu,
Expr(gamma), Eval(gamma),
Expr(delta), Eval(delta),
Expr(eta), Eval(theta),
Expr(lambda), Eval(sigma)
);
// Get upper and lower bounds for the optimisation (either side of the (assumed) single peak)
vals = Column(dt, "Data") << getValues;
low = Quantile(0.25, vals);
high = Quantile(0.75, vals);
// Do the maximization
Maximize(jsu, {x(low, high)}, << Tolerance(10^-10), << showDetails(true));
Print(x);
```

('By eye' it seems to work, but no testing of course).

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May 19, 2017 10:59 AM
(1531 views)

Solution

One of the things that I love about JMP is that it often gives you many ways to solve a problem. All of the previous suggestions for finding the mode of the fitted Johnson SU density for the data are great, but here is one more option that uses the Profiler.

Once you have used Distribution to fit the Johnson Su distribution to your data, you can choose* Save Density Formula* from the *Fitted Johnson Su* menu. This saves the Johnson Su Density formula to a column in your data table.

Next you can open the Profiler by choosing *Graph>>Profiler*. Cast your density formula as the *Y, Prediction Formula* and click OK. Next turn on the *Desirability Functions* from *Prediction Profiler* menu. After that you choose *Maximize Desirability* from the *Prediction Profiler* menu. This will find the mode (maximum) of your Johnson Su density for this data.

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May 22, 2017 1:59 AM
(622 views)