Multivariate data are typically represented by a rectangular matrix (table) in which the rows are the objects (cases) and the columns are the variables (measurements). When there are many variables one often reduces the dimension by principal component analysis (PCA), which in its basic form is not robust to outliers. Much research has focused on handling rowwise outliers, that is, rows that deviate from the majority of the rows in the data (e.g., they might belong to a different population). In recent years also cellwise outliers are receiving attention. These are suspicious cells (entries) that can occur anywhere in the table. Even a relatively small proportion of outlying cells can contaminate over half the rows, which causes rowwise robust methods to break down. In this article, a new PCA method is constructed which combines the strengths of two existing robust methods to be robust against both cellwise and rowwise outliers. At the same time, the algorithm can cope with missing values. As of yet it is the only PCA method that can deal with all three problems simultaneously. Its name MacroPCA stands for PCA allowing for Missingness And Cellwise & Rowwise Outliers. Several simulations and real datasets illustrate its robustness. New residual maps are introduced, which help to determine which variables are responsible for the outlying behavior. The method is well-suited for online process control.

Dozens of missing epochs in the monthly gravity product of the satellite mission Gravity Recovery and Climate Experiment (GRACE) and its follow-on (GRACE-FO) mission greatly inhibit the complete analysis and full utilization of the data. Despite previous attempts to handle this problem, a general all-purpose gap-filling solution is still lacking. Here we propose a non-parametric, data-adaptive and easy-to-implement approach - composed of the Singular Spectrum Analysis (SSA) gap-filling technique, cross-validation, and spectral testing for significant components - to produce reasonable gap-filling results in the form of spherical harmonic coefficients (SHCs). We demonstrate that this approach is adept at inferring missing data from long-term and oscillatory changes extracted from available observations. A comparison in the spectral domain reveals that the gap-filling result resembles the product of GRACE missions below spherical harmonic degree 30 very well. As the degree increases above 30, the amplitude per degree of the gap-filling result decreases more rapidly than that of GRACE/GRACE-FO SHCs, showing effective suppression of noise. As a result, our approach can reduce noise in the oceans without sacrificing resolutions on land. The gap filling dataset is stored in the “SSA_filing/" folder. Each file represents a monthly result in the form of spherical harmonics. The data format follows the convention of the site ftp://isdcftp.gfz-potsdam.de/grace/. Low degree corrections (degree-1, C20, C30) have been made. The code to generate the dataset is located in the “code_share/“ folder, with an example for C30. The model-based Greenland mass balance result for data validation (results given in the paper) is provided in the "Greenland_SMB-D.txt” file.

We collected data from 36 published articles on PubMed [13–48] to train and validate our machine learning models. Some articles comprised more than one type of cartilage injury models or treatment condition. In total, 15 clinical trial conditions and 29 animal 66 model conditions (1 goat, 6 pigs, 2 dogs, 9 rabbits, 9 rats, and 2 mice) on osteochondral injury or osteoarthritis were included, where MSCs were transplanted to repair the cartilage tissue. We documented each case with specific treatment condition into an entry by considering the cell- and treatment target-related factors as input properties, including species, body weight, tissue source, cell number, cell concentration, defect area, defect depth, and type of cartilage damage. The therapeutic outcomes were considered as output properties, which were evaluated using integrated clinical and histological cartilage repair scores, including the international cartilage repair society (ICRS) scoring system, the O’Driscoll score, the Pineda score, the Mankin score, the osteoarthritis research society international (OARSI) scoring system, the international knee documentation committee (IKDC) score, the visual analog score (VAS) for pain, the knee injury and osteoarthritis outcome score (KOOS), the Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC), and Lyscholm score. In this study, these scores were linearly normalized to a number between 0 and 1, with 0 representing the worst damage or pain, and 1 representing the completely healthy tissue. The list of entries was combined together to form a database.

We have provided the details for the imputation algorithm in the subsection Handling missing data under Methods and a flowchart in Fig 2. Data imputation algorithm for the vector x was added in the manuscript for illustration. The pseudo-code for uncertainty calculation was shown in S1 Algorithm: A ensemble model to measure the ANN's prediction uncertainty. The original database gathered from the literature, and a ‘complete’ database with missing information filled from our neural network are also included, along with a sample neural network architecture file in Python.

Here we provide a Python notebook comprising a neural network that delivers the performance and results described in the manuscript. Documentation in the form of comments and installation guide is included in the Python notebook. This Python notebook along with the methods described in the manuscript provides sufficient details for other interested readers to either extend this script or write their own scripts and reproduce the results in the paper.