PACE Documentation

Documentation for PACE

For large data set, it may be helpful to save computating time by performing prebinning before the analysis. The input option numBins is designed for this purpose. The data that fall in each bin will be averaged and the mid point of the bin becomes the corresponding time point (or predictor coordinate). After binning, initially irregular data with the input parameter regular = 0 will be analyzed as regular data with missing values, that is, the parameter "regular" will be reset to 1. For completely balance data, that is, if input data has the parameter setting regular = 2, after binning, regular remains 2.

When numBins = 0, no binning will be performed.
When numBins ≥ 10, the data will be prebinned with number of bins chosen as numBins.
When numBins = [], default binning will be implemented:

     If regular = 1 or 2, m = max of n_i, if regular = 0, m = median of n_i.
     No binning occurs if n ≤ 5000 and m ≤ 400 or m ≤ 20.
     If n ≤ 5000 and m > 400, bin with numBins = 400.
     If n > 5000, compute
          m* = max(20, (5000-n)*19/2250+400)
          if m > m*, bin with numBins = m*,
          if m ≤ m*, no binning occurs

2. For dense regular functional data with number of observed time points much larger than sample size, one may set regular = 3 for a cross-sectional estimate of mean and covariance function, which is much faster. Note that this method assumes no measurement error.

3. Alternative way of estimating eigenvalues for sparse functional data: option ls_fit = 1 (default is 0) regresses raw covariances on eigenfunctions with least square algorithm. This method reduces bias of eigenvalues in a reciprocal sense.

4. How is the number of principal components selected in PACE? (pdf)

5. Improvement of the estimation of the FPC scores. (pdf)

6. Quasi R², denoted as Q, from FPCreg.m output.

          - If the response Y is a scalar:

                  Q = 1-sum((Y_i-Yhat_i)²)/sum((Y_i-mean(Y))²)

              - If Y is a function,

                  Q = 1-sum((Y_i-Yhat_i)'*(Y_i- Yhat_i)/n_i)/sum((Y_i-mean(Y))'*(Y_i-mean(Y))/n_i)

7. AIC_R from FPCreg.m: use the regression AIC criterion to choose the number of principal components. This method chooses the components of X based on the linear relationship between X and Y, rather than on properties of X itself.

Selected important help files: