Publication date: 08/13/2020

Model Reports

Use the options in the Models submenu in the Functional Data Explorer red triangle menu to fit models to your data. See Models for the available models. Each time you fit a different type of model to the data, a model report appears. Each model report contains the following reports:

Model Controls

Model Selection

Diagnostic Plots

Function Summaries

Functional PCA

Model Controls

The Model Controls report enables you to define parameters of models to compare in the Model Selection report. The appearance of the Model Controls report depends on the type of model that is fit.

B-Spline and P-Spline Model Controls

When a B-Spline or P-Spline model is fit, you can specify the following parameters:

Number of Knots

Add, remove, or specify a range for the number of knots in each spline.

Note: The maximum number of knots allowed for B-Spline models is the maximum number of observations per function or the number of unique inputs. The maximum number of knots allowed for P-Spline models is two less than the number of unique inputs. If you specify a number larger than the maximum, a warning message appears.

Spline Degree

Add or remove spline degree fits from the Model Selection report.

Fourier Basis Model Controls

When a Fourier Basis model is fit, you can specify the following parameters:

Number of Fourier Pairs

Add, remove, or specify a range for the number of Fourier pairs to compare.

Period

Change the period of the function.

After you specify the model controls, click Go to view the updated models in the Model Selection report.

Tip: To specify the Model Controls prior to fitting a model, press Shift, click the Functional Data Explorer red triangle, and select the desired model. See Models.

Model Selection

The Model Selection report contains an overall prediction plot, a grid of individual prediction plots for each level of the ID variable, a solution path plot, and a table of fit statistics. The solution path plot shows a model selection criterion plotted over values of a model parameter. The Bayesian Information Criterion (BIC) is the default fitting criterion. See Model Report Options. For B-Spline and P-Spline models, there is a separate curve in the solution path for each spline degree plotted across the defined number of knots. For Fourier Basis models, the solution path is plotted across the number of Fourier pairs for a defined period. Use the Model Controls to change the solution path plot parameters.

The current solution is designated by the dotted vertical line in the solution path plot. By default, the slider is placed at the number of knots or Fourier pairs that corresponds to the model that has the smallest model selection criterion value. You can drag the slider at the top of the dotted vertical line to change the number of knots or Fourier pairs in the current model. Dragging the slider automatically updates the prediction plots in the Model Selection report, as well as the information in all other reports.

The Fit Statistics table contains a description of the current solution model. It also displays the -2 Log Likelihood, the values for the AICc, BIC, and GCV model fitting criterion, and a value for the response standard deviation, denoted as <Y, Output> Std Dev. The response standard deviation is defined as the residual sigma from the fitted model. When a P-Spline model is selected, the penalty parameter λ (Lambda) is also displayed.

The prediction plots show the raw data and prediction curves that correspond to the current model. For spline models, the default model selected is the degree of spline with the best fit. Click a specific spline in the solution path plot or the legend to change the current model selection. The curve in the overall prediction plot is a prediction of the mean curve. The curves in the individual prediction plots are prediction curves for each specific function. For B-Spline models, the overall prediction plot also displays the location of the knots. You can change the location of the knots by dragging the blue slider bars to different locations. To update the model reports according to the new knot locations, click the Update Models button. To reset the knots to their default locations, click the Reset Knots button.

Diagnostic Plots

The Diagnostic Plots report contains the Actual by Predicted plot and the Residual by Predicted plot. These plots help assess how well the current model fits the data. The Diagnostic Plots report is closed by default.

Function Summaries

Displays summaries from the Functional PCA for each level of the ID variable. The functional principal components associated with eigenvalues that explain more than 1% variation in the data are displayed by default. The mean, standard deviation, median, minimum, maximum, integrated difference, root integrated square error (RISE), and root integrated function square (RIFS) are also shown. The integrated difference and RISE summary values are used to determine how much the ID specific function differs from the overall mean function. The RIFS summary value is used for optimal curve fitting. See Function Summaries Details. The Function Summaries red triangle menu contains the following options:

Customize Function Summaries

Displays a window that enables you to select the number of FPCs and the summary statistics that are shown in the Function Summaries report. If the number of FPCs to be shown is specified, the Functional PCA report is also updated.

Tip: If you have multiple functional processes, you can customize all Function Summaries reports to show the same summary values by clicking Ctrl and selecting Customize Function Summaries.

Save Summaries

Saves the summary statistics specified in the Function Summaries report to a new data table. The name of the new data table describes the model fit. This data table contains formula columns for the eigenfunctions, mean function, prediction function, and conditional prediction function. In the data table, there is a profiler script that launches the prediction profilers for the prediction and conditional prediction formulas. These formulas are functions of the input variable, the ID variable, and the eigenfunctions.

Basis Function Coefficients

Displays the estimated basis function coefficients and their standard deviations. These are common across all levels of the ID variable and are fixed estimates in the mixed model framework. To view standard errors and confidence intervals for the coefficients, right-click in the table and select Columns.

Random Coefficients by Function

Displays the estimated random coefficients for each basis function and functional process combination. These are unique to each level of the ID variable and are random effects estimates in the mixed model framework.

Functional PCA

Functional principal components analysis (functional PCA) is performed on the fitted functional model. The Functional PCA report lists the eigenvalues that correspond to each functional principal component (FPC) in order from largest to smallest. The percent of variation accounted for by each FPC and the cumulative percent is listed and shown in a bar chart. There is a graph of the mean function as well as a graph for each component. The component graphs show the values of the eigenfunction. FPCs that explain more than 1% variation in the data are included in the report.

Note: The Functional PCA report is not shown if only a single function is modeled. Otherwise, if JMP is unable to perform Functional PCA, an error message appears in the Functional PCA report.

The following options are available in the Functional PCA red triangle menu:

Score Plot

Shows or hides a score plot of the FPC scores. Use the lists under Select Component to specify which FPCs are plotted on each axis of the Score Plot. If there is only one FPC, the FPC scores are plotted on the line y = x and the lists to change the components are not shown. Score plots are useful for detecting outliers. In the case of FPC scores, the Score Plot is useful for detecting levels of the ID variable that have outlier functions.

Tip: Place your cursor over a point in the score plot to view a prediction plot of the fitted curve for that level of the ID variable.

FPC Profiler

Shows or hides a profiler of the FPC scores. The FPC Profiler includes a column for the input variable and a column for each FPC score. For each target function that is specified, there are two additional profilers. One measures the difference from the target function, and the other measures the integrated error from the target function. For more information about FPC Profiler red triangle menu options, see Profiler in Profilers.

Customize Number of FPC’s

Specifies the number of FPC scores to show in the Functional PCA. Specifying the number of FPC scores in this option also updates the Function Summaries report.

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