Publication date: 11/10/2021

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:

• Random Coefficients by Function

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.

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. The knots must be non-zero integers.

Note: The maximum number of knots allowed for B-Spline models is one less than 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.

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.

The Model Selection report contains an overall prediction plot, a grid of individual prediction plots, a solution path plot, and a table of fit statistics. The grid of individual prediction plots has the same layout and controls as the grid of individual plots in the Data Processing report. At most, there are twenty plots shown at a time. There are drop-down menus and arrows that enable you to view different groups of individual prediction plots. 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 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 Selection option 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. If there is a validation set, the predicted curves are not shown for functions that are in the validation set. 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.

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.

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. There is also a checkbox, Save Graphs, that determines whether a graph for each function is included in the data table produced by the Save Summaries option.

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. There is also a column that contains the image of a graph of the raw data and the specified model fit for each 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.

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.

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 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.

You can perform model selection in the Functional PCA report to refine the selected number of functional principal components. There is a solution path plot that shows the Bayesian Information Criterion (BIC) plotted versus the number of FPCs. The current number of FPCs is designated by the dotted vertical line in the solution path plot. It is possible that models with different numbers of FPCs might have similar fits. Therefore, the solution path plot provides zones, which are intervals of values of the BIC statistic. There is a green zone and a yellow zone. The green zone contains values in the interval of the minimum BIC to the minimum BIC plus four and the yellow zone contains values in the interval of the minimum BIC plus four to the minimum BIC plus 10. By default, the model with the smallest number of FPCs within the green zone is selected. You can drag the slider at the top of the vertical line to change the number of FPCs. Dragging the slider automatically updates the other information in the Functional PCA report.

Note: The zones may appear small on your plot. Zoom in on the y-axis to better visualize the zones.

When Direct Functional PCA is performed, there is also an overall prediction plot and a grid of individual prediction plots. The grid of individual prediction plots has the same layout and controls as the grid of individual plots in the Data Processing report. At most, there are twenty plots shown at a time and there are drop-down menus and arrows that enable you to view different groups of individual prediction plots. Updating the number of FPCs automatically updates the prediction plots as well.

The prediction plots show the raw data and prediction curves that correspond to the current model. If there is a validation set, the predicted curves are not shown for functions that are in the validation set. The curve in the overall prediction plot is a prediction of the mean curve, given the specified number of FPCs. The curves in the individual prediction plots are prediction curves for each specific function, given the specified number of FPCs.

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:

Diagnostic Plots

Shows or hides the Actual by Predicted and the Residual by Predicted plots. Use these plots help assess how well the model fits the data, given the selected number of functional principal components.

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. If you select a point in the score plot, the FPC Profiler is set to the scores for that function.

Tip: Hover 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.

Tip: Use the Reset button to reset all of the FPC scores to 0 in the profiler.

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. To view the mean model, set the number of FPC’s to 0.

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