pseudotime TDA

4CE longitudinal lab values

https://github.com/covidclinical/Phase2.1TDAPseudotimeRPackage

CRP enrichment

Topological Data Analysis (TDA): mine “connected” disease states via data shape
Pseudo Time: infer progression among states to reconstruct disease progression trajectories

Challenges

Missing data
- Not at random
- Slow imputation
Aggregate result from different sites
- Consistency?
- Parameter tuning?
- Train/test site
Assumptions
- Number of trajectory clusters
- All patients start out at a similar state

4CE longitudinal lab values

Does the missingness pattern reflect the healthcare dynamic? (doctor's worry)

Step 0: Drop variable with high missing rate (> 80%: troponin_high, procalcitonin, fibrinogen, troponin_nomal)

Step 1: Naively impute missing data points of each variable using functional PCA {fdapace}

Step 2: Drop rows with the most (originally) missing values, record the proportion of rows dropped for each patient (pdrop)

Step 3: Put NAs back in the CRP variable where it was missing.

Step 4: Train CRP on Leukocytes, Albumin and pdrop (mixed effect model, XGBoost, Amelia II) with available data

Step 5: Use the fitted model to predict the missing CRP values.

Step 6: Repeat Steps 3–5 separately for each variable that has missing data (Leukocytes and Albumin).

CRP, Albumin, Leukocytes

for each cycle:

nRMSD(a) = \sqrt{\frac{\sum{_{p,i}}I_{p,a,i}\left(\frac{X_{p,a,i} - Y_{p,a,i}}{max(Y_{p,a}) - min(Y_{p,a})}\right)^2}{\sum_{p,i}I_{p,a,i}}}

patient \(p\)

lab \(a\)

time index \(i\)

mask one extra value per lab per patient

x_n = \frac{x-min(x)}{max(x) - min(x)}

\(f(t_i)\) have a joint Gaussian distribution

P(f(t))

locality constraint

closer time points have more similar measurement values

cov(f(t_1), f(t_2)) = \alpha e^{-(t_1-t_2)^2/l}

Step 1: extract separate univariate time series for each patient and variable

Step 2: GPfit: MLE over \(\alpha\) and \(l\)

Step 3: infer values at unobserved time points