Post-Regularization Confidence Bands for Ordinary Differential Equations
Authors: Xiaowu Dai, Lexin Li
JMLR 2024 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | We demonstrate the efficacy of the proposed method through both simulations and illustrations with two data applications. |
| Researcher Affiliation | Academia | Xiaowu Dai EMAIL Department of Statistics and Data Science and Department of Biostatistics University of California, Los Angeles, CA 90095-1554, USA Lexin Li EMAIL Department of Biostatistics and Epidemiology University of California, Berkeley, CA 94720-1776, USA |
| Pseudocode | Yes | Algorithm 1 Estimation and inference procedure for a given pair (j, k) {1, . . . , p}. 1: Initialization with the values for θjk = θjk l = 1, for k , l = 1, . . . , p, k = k, l = k , k. 2: Obtain the smoothing spline estimate bxj(t) from (8). 3: Estimate bθj0 from (9). 4: Solve (bαjk,t0, b Hjk) in (14) given bθj through (15) and (17). 5: Solve bθj in (18) given (bαjk,t0, b Hjk) through the Lasso penalized regression (18). 6: until the stopping criterion is met. 7: Construct the confidence band by Gaussian multiplier bootstrap from (20). |
| Open Source Code | No | The paper does not provide a direct link to a source-code repository or an explicit statement about releasing the code for the described methodology. It only mentions a CC-BY 4.0 license for the paper itself. |
| Open Datasets | Yes | The first example is a three-node enzyme regulatory system of a negative feedback loop with a buffering node (Ma et al., 2009, NFBLB). The second example is the classical Lotka-Volterra system, which consists of pairs of first-order nonlinear differential equations describing the dynamics of biological system in which predators and prey interact (Volterra, 1928). The data we analyze is the in silico benchmark gene expression data generated by Gene Net Weaver (GNW) using dynamical models of gene regulations and nonlinear ODEs (Schaffter et al., 2011). The data we analyze is an ECo G study of the brain during decision making (Saez et al., 2018). |
| Dataset Splits | Yes | In this example, we focus on the performance of recovery of the entire regulatory system through the proposed confidence band coupled with the Benjamini Hochberg (BH) procedure for multiple testing correction (Benjamini and Hochberg, 1995). Since the ODE equations in (25) only involve the linear and interaction terms, we use the first-order Matérn kernel for the step in (8), and use the linear kernel in (10). As such, the linear and kernel ODEs yield the same estimates. We continue to use the quadratic density for the local weight function Rh(t). We control the FDR at the level of 20%. ...by the squared root of the sum of predictive mean squared errors for Fj(xj(t)), j = 1, . . . , 10, at the unseen future time point t [100, 200], i.e., n P10 j=1 R 200 100 [ b Fj(bxj(t)) Fj(xj(t))]2dt o1/2 , where the integral is evaluated at 10000 evenly distributed time points in [100, 200]. |
| Hardware Specification | No | The paper does not provide specific details about the hardware used to run the experiments. It focuses on the methodology and simulation/data application results without mentioning CPU, GPU, or memory specifications. |
| Software Dependencies | No | The paper does not explicitly state the version numbers for any software dependencies or programming languages used for implementation. It mentions using a Matérn kernel but not the specific library or its version. |
| Experiment Setup | Yes | In our implementation, we use a first-order Matérn kernel for both steps in (8) and (10) of the collocation method, where KH1(x, x ) = (1 + sqrt(3) x x /ν) exp(sqrt(3) x x /ν), and ν is chosen by tenfold cross-validation. We have found the inference results are not overly sensitive to the choice of kernel functions here. Moreover, we use the quadratic density Rh(t) = (15/16) (1 t2/h2)21(|t| < h) for the local weight function, where the bandwidth h is chosen by tenfold cross-validation. We compute the quantile bcn(α) in (20) by bootstrap with 500 repetitions. We control the FDR at the level of 20%. |