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Gautschi proposed a method for computing incomplete beta functions I x ( a , b ) using Miller’s algorithm with a three-term recurrence relation and showed a computation program in ALGOL. In this paper, first, Miller’s algorithm using the recurrence relation satisfied by f k ( x ) = I x ( a + k , b ) is described. Next, another solution that is first-order independent of f k ( x ) of the recurrence relation is given, and its general solution can be expressed as a linear sum of these. Using this general solution, an error analysis for the function I x ( a , b ) is performed for the first time. The relative error of the function values is then expressed in a new formula to a trend of the error behavior. Also, Miller’s algorithm with a normalizing sum is explained and its error analysis is performed. Since Miller’s algorithm requires a predefined number of iterations of the recurrence relation, it is necessary to repeat the computation of the recurrence relation with increasing number of iterations until the required accuracy will be met. Therefore, in this paper, we apply Deuflhard’s algorithm, which can automatically obtain the function value with the required accuracy. This algorithm requires far fewer iterations than Gautschi’s algorithm to obtain the same accuracy.

Charlier polynomials (CHPs) and their moments are commonly used in image processing due to their salient performance in the analysis of signals and their capability in signal representation. The major issue of CHPs is the numerical instability of coefficients for high-order polynomials. In this study, a new recurrence algorithm is proposed to generate CHPs for high-order polynomials. First, sufficient initial values are obtained mathematically. Second, the reduced form of the recurrence algorithm is determined. Finally, a new symmetry relation for CHPs is realized to reduce the number of recurrence times. The symmetry relation is applied to calculate $$ \\sim $$ ∼ 50% of the polynomial coefficients. The performance of the proposed recurrence algorithm is evaluated in terms of computational cost and reconstruction error. The evaluation involves a comparison with existing recurrence algorithms. Moreover, the maximum size that can be generated using the proposed recurrence algorithm is investigated and compared with those of existing recurrence algorithms. Comparison results; indicate that the proposed algorithm exhibits better performance because it can generate a polynomial 44 times faster than existing recurrence algorithms. In addition, the improvement of the proposed algorithm over the traditional recurrence algorithms in terms of maximum-generated size is between 19.25 and 42.85.

Tchebichef polynomials (TPs) play a crucial role in various fields of mathematics and applied sciences, including numerical analysis, image and signal processing, and computer vision. This is due to the unique properties of the TPs and their remarkable performance. Nowadays, the demand for high-quality images (2D signals) is increasing and is expected to continue growing. The processing of these signals requires the generation of accurate and fast polynomials. The existing algorithms generate the TPs sequentially, and this is considered as computationally costly for high-order and larger-sized polynomials. To this end, we present a new efficient solution to overcome the limitation of sequential algorithms. The presented algorithm uses the parallel processing paradigm to leverage the computation cost. This is performed by utilizing the multicore and multithreading features of a CPU. The implementation of multithreaded algorithms for computing TP coefficients segments the computations into sub-tasks. These sub-tasks are executed concurrently on several threads across the available cores. The performance of the multithreaded algorithm is evaluated on various TP sizes, which demonstrates a significant improvement in computation time. Furthermore, a selection for the appropriate number of threads for the proposed algorithm is introduced. The results reveal that the proposed algorithm enhances the computation performance to provide a quick, steady, and accurate computation of the TP coefficients, making it a practical solution for different applications.

This paper proposes efficient fixed-point and floating-point implementations for radix-10 square root in Xilinx FPGAs devices. The method implements digit recurrence with restoring algorithm, which supports the three decimal floating-point (DFP) types specified in the IEEE 754-2008 standard. The technique used for restoring is optimal and novel. The designs use new techniques based on the efficient utilization of dedicated resources in the programmable devices. Implementations were made in Xilinx 7-series devices. For fixed-point square root, they are capable of operating up to 212 MHz for p=7, 197 MHz for p=16, and 190 MHz for p=34. As for DFP square root, the operation frequency obtained is 194 MHz for p=7, 183 MHz for p=16, and 174 MHz for p=34. The proposed architecture achieves better computation times than related works.

RNA secondary structure folding kinetics is known to be important for the biological function of certain processes, such as the hok/sok system in E. coli. Although linear algebra provides an exact computational solution of secondary structure folding kinetics with respect to the Turner energy model for tiny (≈20 nt) RNA sequences, the folding kinetics for larger sequences can only be approximated by binning structures into macrostates in a coarse-grained model, or by repeatedly simulating secondary structure folding with either the Monte Carlo algorithm or the Gillespie algorithm. Here we investigate the relation between the Monte Carlo algorithm and the Gillespie algorithm. We prove that asymptotically, the expected time for a K-step trajectory of the Monte Carlo algorithm is equal to ⟨N⟩ times that of the Gillespie algorithm, where ⟨N⟩ denotes the Boltzmann expected network degree. If the network is regular (i.e. every node has the same degree), then the mean first passage time (MFPT) computed by the Monte Carlo algorithm is equal to MFPT computed by the Gillespie algorithm multiplied by ⟨N⟩; however, this is not true for non-regular networks. In particular, RNA secondary structure folding kinetics, as computed by the Monte Carlo algorithm, is not equal to the folding kinetics, as computed by the Gillespie algorithm, although the mean first passage times are roughly correlated. Simulation software for RNA secondary structure folding according to the Monte Carlo and Gillespie algorithms is publicly available, as is our software to compute the expected degree of the network of secondary structures of a given RNA sequence—see http://bioinformatics.bc.edu/clote/RNAexpNumNbors.

Deep learning algorithms have been successfully used in medical image classification. In the next stage, the technology of acquiring explainable knowledge from medical images is highly desired. Here we show that deep learning algorithm enables automated acquisition of explainable features from diagnostic annotation-free histopathology images. We compare the prediction accuracy of prostate cancer recurrence using our algorithm-generated features with that of diagnosis by expert pathologists using established criteria on 13,188 whole-mount pathology images consisting of over 86 billion image patches. Our method not only reveals findings established by humans but also features that have not been recognized, showing higher accuracy than human in prognostic prediction. Combining both our algorithm-generated features and human-established criteria predicts the recurrence more accurately than using either method alone. We confirm robustness of our method using external validation datasets including 2276 pathology images. This study opens up fields of machine learning analysis for discovering uncharted knowledge. Technologies for acquiring explainable features from medical images need further development. Here, the authors report a deep learning based automated acquisition of explainable features from pathology images, and show a higher accuracy of their method as compared to pathologist based diagnosis of prostate cancer recurrence.

The aim of this study is to explore the feasibility of using machine learning (ML) technology to predict postoperative recurrence risk among stage IV colorectal cancer patients. Four basic ML algorithms were used for prediction—logistic regression, decision tree, GradientBoosting and lightGBM. The research samples were randomly divided into a training group and a testing group at a ratio of 8:2. 999 patients with stage 4 colorectal cancer were included in this study. In the training group, the GradientBoosting model’s AUC value was the highest, at 0.881. The Logistic model’s AUC value was the lowest, at 0.734. The GradientBoosting model had the highest F1_score (0.912). In the test group, the AUC Logistic model had the lowest AUC value (0.692). The GradientBoosting model’s AUC value was 0.734, which can still predict cancer progress. However, the gbm model had the highest AUC value (0.761), and the gbm model had the highest F1_score (0.974). The GradientBoosting model and the gbm model performed better than the other two algorithms. The weight matrix diagram of the GradientBoosting algorithm shows that chemotherapy, age, LogCEA, CEA and anesthesia time were the five most influential risk factors for tumor recurrence. The four machine learning algorithms can each predict the risk of tumor recurrence in patients with stage IV colorectal cancer after surgery. Among them, GradientBoosting and gbm performed best. Moreover, the GradientBoosting weight matrix shows that the five most influential variables accounting for postoperative tumor recurrence are chemotherapy, age, LogCEA, CEA and anesthesia time.

Whole-genome sequencing (WGS) brings comprehensive insights to cancer genome interpretation. To explore the clinical value of WGS, we sequenced 254 triple-negative breast cancers (TNBCs) for which associated treatment and outcome data were collected between 2010 and 2015 via the population-based Sweden Cancerome Analysis Network–Breast (SCAN-B) project (ClinicalTrials.gov ID:NCT02306096). Applying the HRDetect mutational-signature-based algorithm to classify tumors, 59% were predicted to have homologous-recombination-repair deficiency (HRDetect-high): 67% explained by germline/somatic mutations of BRCA1/BRCA2, BRCA1 promoter hypermethylation, RAD51C hypermethylation or biallelic loss of PALB2. A novel mechanism of BRCA1 abrogation was discovered via germline SINE-VNTR-Alu retrotransposition. HRDetect provided independent prognostic information, with HRDetect-high patients having better outcome on adjuvant chemotherapy for invasive disease-free survival (hazard ratio (HR) = 0.42; 95% confidence interval (CI) = 0.2–0.87) and distant relapse-free interval (HR = 0.31, CI = 0.13–0.76) compared to HRDetect-low, regardless of whether a genetic/epigenetic cause was identified. HRDetect-intermediate, some possessing potentially targetable biological abnormalities, had the poorest outcomes. HRDetect-low cancers also had inadequate outcomes: ~4.7% were mismatch-repair-deficient (another targetable defect, not typically sought) and they were enriched for (but not restricted to) PIK3CA/AKT1 pathway abnormalities. New treatment options need to be considered for now-discernible HRDetect-intermediate and HRDetect-low categories. This population-based study advocates for WGS of TNBC to better inform trial stratification and improve clinical decision-making.

Purpose A subset of skull base meningiomas (SBM) may show early progression/recurrence (P/R) as a result of incomplete resection. The purpose of this study is the implementation of MR radiomics to predict P/R in SBM. Methods From October 2006 to December 2017, 60 patients diagnosed with pathologically confirmed SBM (WHO grade I, 56; grade II, 3; grade III, 1) were included in this study. Preoperative MRI including T2WI, diffusion-weighted imaging (DWI), and contrast-enhanced T1WI were analyzed. On each imaging modality, 13 histogram parameters and 20 textural gray level co-occurrence matrix (GLCM) features were extracted. Random forest algorithms were utilized to evaluate the importance of these parameters, and the most significant three parameters were selected to build a decision tree for prediction of P/R in SBM. Furthermore, ADC values obtained from manually placed ROI in tumor were also used to predict P/R in SBM for comparison. Results Gross-total resection (Simpson Grades I–III) was performed in 33 (33/60, 55%) patients, and 27 patients received subtotal resection. Twenty-one patients had P/R (21/60, 35%) after a postoperative follow-up period of at least 12 months. The three most significant parameters included in the final radiomics model were T1 max probability, T1 cluster shade, and ADC correlation. In the radiomics model, the accuracy for prediction of P/R was 90%; by comparison, the accuracy was 83% using ADC values measured from manually placed tumor ROI. Conclusions The results show that the radiomics approach in preoperative MRI offer objective and valuable clinical information for treatment planning in SBM.

TAILORx established the role of the 21-gene predictor of genetic risk in ascertaining treatment for women with hormone-receptor–positive, human epidermal growth factor receptor 2–negative breast cancer. Clinical risk factors provided additional prognostic information for women with intermediate genetic risk.