Penalized Multi-band Learning
We use a synthetic dataset to implement the method. ‘lis3’ is a data list consisting of three matrices, each giving activity data for one individual. Each column of the matrix is one-day observation, and here the physical activity (PA) is measured every one minute, so the matrix is 1440 by ‘nday’. ‘lis3’ is also a named list, the name of which is the individual ID.
To convert the data to the required format as the input of the function for penalized multi-band learning, we use function ‘form’:
## # A tibble: 13 x 3
## ID ID_Nday activity
## <dbl> <int> <list>
## 1 1 1 <dbl [1,440]>
## 2 1 2 <dbl [1,440]>
## 3 1 3 <dbl [1,440]>
## 4 1 4 <dbl [1,440]>
## 5 3 1 <dbl [1,440]>
## 6 3 2 <dbl [1,440]>
## 7 3 3 <dbl [1,440]>
## 8 15 1 <dbl [1,440]>
## 9 15 2 <dbl [1,440]>
## 10 15 3 <dbl [1,440]>
## 11 15 4 <dbl [1,440]>
## 12 15 5 <dbl [1,440]>
## 13 15 6 <dbl [1,440]>The resulting dataset ‘pa3’ is in tbl_df format. It has 13 observations for 3 individuals, and the variables are “ID”, “ID_Nday” (the ith day observation for an individual), and activity. The activity variable is an embedded list with each element consisting of a vector of one-day observation.
Further, ‘pa3’ serves as part of the input for the function bandSelect. It uses combined L1/L2 penalty to select important harmonics sequentially based on Fast Fourier Transform (FFT) results. \(X_k\) is the vector of the \(k\)th harmonic FFT results. \(\lambda\) is the tuning parameter for penalty, and it ranges from 0 (no penalty) to 2max(\(||X_k||^2\)). \(\theta_i\)’s are the estimated coefficents for different harmonics given a specified penalty.
alpha: controls the balance between L1 and L2 penalties. Because here we want to pick a few important harmonics, we set \(\alpha\)=1 to use L1 penalty only.
Nlength: In the analysis we need to use individual observations of equal lengths (equal number of day observations), because different lengths of inputs into FFT will result in different sets of harmonics and expect different magnitudes of signals (i.e. for certain harmonics, the signals are expected to be stronger in observations of longer time). Therefore, we use the ‘Nlength’ argument to specify the length of observation for an individual to be entered into the analysis. Here we choose 3 days, namely 1440*3 observations per individual. We then show the top 3 important harmonics in the population (‘Ntop=3’). Because the sample data only have 3 individuals, we do not perform 5-fold cross-validation (‘cross=FALSE’).
plot=TRUE: We want to see the effect of the first a few harmonics compared to other harmonics and all harmonics, so we plot Mean Squared Error (MSE) against the number of nonzero \(\theta\)’s as \(\lambda\) decreases. Note that it only plots the points whether the number of nonzero \(\theta\)’s increases; between plotted points, the values of the already nonzero \(\theta\)’s increase.
data(pa3)
re <- bandSelect(df=pa3,Nlength=1440*3,Nlambda=100,alpha=1,Ntop=3,
cross=FALSE,Ncross=NULL,plot=TRUE)## With a threshold of 4320 observations per individual,## 3 individuals are used for penalized multi-band learning.
The top three frequencies are 1440, 720, 480, corresponding to 1-day, 1/2-day, and 1/3-day periodicities. The convolution of the three frequencies can give two-peak day activity patterns, with one activity peak in the morning, one peak in the afternoon and evening, and interims of one short nap in the afternoon and one long sleep at night. The output ‘xscore’ and ‘xprop’ gives FFT results (signal magnitudes and proportions standardized by individuals respectively) in 3 by 2159 matrices, with row standing for 3 individuals and column standing for 2159 frequencies. We may want to see the top frequencies with mean signal proportions:
freq <- data.frame(Frequency=re$freq,Proportion=colMeans(re$xprop))
rownames(freq) <- NULL
head(freq[order(freq$Proportion,decreasing = TRUE),],5)| Frequency | Proportion |
|---|---|
| 1440 | 0.0107955 |
| 720 | 0.0086717 |
| 480 | 0.0082258 |
| 360 | 0.0059348 |
| 2160 | 0.0045538 |
As is shown in the table, the top three proportions are quite dominant.
