The quality of tree approximation from AUC bounds

Formulation of the Tree Approximation Problem as a Detection Problem and Relation between the AUC and Information Theory Divergences

We conduct a study of graphical models and discuss the quality of model selection approximation by formulating the problem as a detection problem and examine the area under the curve (AUC). We are specifically looking at the model selection problem for jointly Gaussian random vectors. For Gaussian distributions, this problem simplifies to the covariance selection problem which is widely discussed in literature by Dempster [1]. In this paper, we discuss graphical models such as the pth order Markov chain and the pth order star network interpretation which also have junction tree graphical representations and give the definition for the correlation approximation matrix (CAM) which contains all information about the model selection problem. We compute the model covariance matrix as well as the KL divergence between the original distribution and the approximated model distribution. We conduct some simulations which show that the quality of the selected model increases as the model order, p, increases.

The following detection problem is studied, in which there are M sequences of samples out of which one outlier sequence needs to be detected. Each typical sequence contains n independent and identically distributed (i.i.d.) continuous observations from a known distribution π, and the outlier sequence contains n i.i.d. observations from an outlier distribution μ, which is distinct from n, but otherwise unknown. A universal test based on Kullback-Leibler (KL) divergence is built to approximate the maximum likelihood test, with known π and unknown μ. A KL divergence estimator based on data-dependent partitions is employed, and is shown to converge to its true value exponentially fast when the density ratio satisfies 0 <; Kl ≤ dμ/dπ ≤ K2, where K1 and K2 are positive constants. The performance of such a KL divergence estimator further implies that the outlier detection test is exponentially consistent. The detection performance of the KL divergence based test is compared with that of a recently introduced test for this problem based on the machine learning approach of maximum mean discrepancy (MMD). Regimes in which the KL divergence based test is better than the MMD based test are identified.

In assessing the performance of a diagnostic test, the widely used classification technique is the Receiver Operating Characteristic (ROC) Curve. The Binormal model is commonly used when the test scores in the diseased and healthy populations follow Normal Distribution. It is possible that in real applications the two distributions are different but having a continuous density function. In this paper we considered a model in which healthy and diseased populations follow half normal and exponential distributions respectively, hence named it as the Hybrid ROC (HROC) Curve. The properties and Area under the curve (AUC) expressions were derived. Further, to measure the distance between the defined distributions, a popular divergence measure namely Kullback Leibler Divergence (KLD) has been used. Simulation studies were conducted to study the functional behavior of Hybrid ROC curve and to show the importance of KLD in classification.

BACKGROUND/AIMS:The Lymph node staging system provides significant information for laryngeal squamous cell carcinoma (LSCC) prognosis.Additional parameters are suggested in order to improve the prognostic capacity of lymph node staging.This study aimed to investigate the prognostic value of different lymph node staging methods. MATERIALS AND METHODS:The long-term survival data and pathological features of fifty-two patients with LSCC were obtained retrospectively.The effects of metastatic lymph node count (MLNC), metastatic lymph node ratio (MLNR), and the logarithmic ratio of positive lymph nodes (LODDS) on disease-free survival (DFS), disease-specific survival, and overall survival (OS) were analyzed.Significant cut-off values for MLNR and LODDS were calculated using receiver operating characteristic analysis.In addition, Kaplan-Meier survival analysis with log-rank was used for comparisons of nodal disease-related study groups. RESULTS:Cancer recurrence was similar between the groups by T-stage (0.963), N-stage (0.935), MLNR groups (0.297), and LODSS groups (0.244).However, the recurrence rate was significantly lower in tumors with a severe lymphoid response (0.004) and with a total dissected number of lymph nodes 18 (0.037).Total lymph node count (0.303), total MLNC (0.768), MNLR (0.656), and LODDS (0.356) values were similar in those patients with and those without cancer recurrence (p>0.05).No significant cut-off value was detected for either DFS or OS for MLNR or LODDS values (p=0.672,area under the curve (AUC): 0.672, 95% confidence interval (CI): 0.365-0.706;p=0.352,AUC: 0.578, 95% CI: 0.411-0.746;p=0.450,AUC: 0.615, 95% CI: 0.222-1; p=0.450,AUC: 0.615, 95% CI: 0.230-0.999,respectively).CONCLUSION: MLNR and LODDS were significant in improving the prognostic value of TNM staging in LSSC.

This study aimed to quantitatively assess [68Ga]Ga-PSMA-11 uptake in pathological lesions and normal organs in prostate cancer using the total-body [68Ga]Ga-PSMA-11 PET/CT and to characterize the dynamic metabolic heterogeneity of prostate cancer. Dynamic total-body [68Ga]Ga-PSMA-11 PET/CT scans were performed on ten prostate cancer patients. Manual delineation of volume-of-interests (VOIs) was performed on multiple normal organs displaying high [68Ga]Ga-PSMA-11 uptake, as well as pathological lesions. Time-to-activity curves (TACs) were generated, and the four compartment models including one-tissue compartmental model (1T1k), reversible one-tissue compartmental model (1T2k), irreversible two-tissue compartment model (2T3k) and reversible two-tissue compartmental model (2T4k) were fitted to each tissue TAC. Various rate constants, including K1 (forward transport rate from plasma to the reversible compartment), k2 (reverse transport rate from the reversible compartment to plasma), k3 (tracer binding on the PSMA-receptor and its internalization), k4 (the externalization rate of the tracer) and Ki (net influx rate), were obtained. The selection of the optimal model for describing the uptake of both lesions and normal organs was determined using the Akaike information criteria (AIC). Receiver operating characteristic (ROC) curve analysis was performed to determine the cut-off values for differentiating physiological and pathological [68Ga]Ga-PSMA-11 uptake. Both 1T1k and 1T2k models showed relatively high AIC values compared to the 2T3k and 2T4k models in both pathological lesions and normal organs. The kinetic behavior of pathological lesions was better described by the 2T3k model compared to the 2T4k model, while the normal organs were better described by the 2T4k model. Significant variations in kinetic metrics, such as K1, k2, and k3, and Ki, were observed among normal organs with high [68Ga]Ga-PSMA-11 uptake and pathological lesions. The high Ki value in normal organs was primarily determined by elevated K1 and low k3, rather than k2. Conversely, the high Ki value in pathological lesions, ranking second to the kidney and similar to salivary glands and spleen, was predominantly determined by the highest k3 value. Notably, k3 exhibited the highest performance in distinguishing between physiological and pathological [68Ga]Ga-PSMA-11 uptake, with an area under the curve (AUC) of 0.844 (95% CI, 0.773-0.915), sensitivity of 82.9%, and specificity of 74.1%. The k3 values showed better performance than SUVmean (AUC, 0.659), SUVmax (AUC, 0.637), and other kinetic parameter including K1 (AUC, 0.604), k2 (AUC, 0.634), and Ki (AUC, 0.651). Significant discrepancies in kinetic metrics were detected between pathological lesions and normal organs, despite their shared high uptake of [68Ga]Ga-PSMA-11. Notably, the k3 value exhibits a noteworthy capability to distinguish between pathological lesions and normal organs with elevated [68Ga]Ga-PSMA-11 uptake. This discovery implies that k3 holds promise as a prospective imaging biomarker for distinguishing between pathologic and non-specific [68Ga]Ga-PSMA-11 uptake in patients with prostate cancer.

In gastric cancer, the classification of lymph node status is still a controversial prognostic factor. Recent studies have proposed a new prognostic factor (metastatic lymph node ratio: MLR) for gastric cancer patients who undergo curative resection. The present study tested the hypothesis that MLR was better than the current pN staging system by analyzing the correlation between MLR and the International Union Against Cancer/American Joint Committee on Cancer (UICC/AJCC) staging system, by analyzing the correlation between MLR and 5-year overall survival (OS), by comparing area under the curve (AUC), and by performing univariate and multivariate analyses for OS. Of 409 patients who were diagnosed with gastric adenocarcinoma between January 2003 and December 2006, 370 patients underwent curative resection and were included in this study. The prognostic significance of the number of metastatic lymph nodes and the metastatic lymph node ratio were compared in AUC and univariate and multivariate Cox regression analyses. MLR was significantly correlated with the depth of invasion and the number of lymph node metastases (p < 0.001). Increasing MLR also was statistically correlated with a lower 5-year OS rate (p < 0.001). The AUC of MLR and the number of lymph node metastases were not significantly different (p = 0.825). MLR was an independent prognostic factor on multivariate analysis, but the number of metastatic lymph nodes was not. MLR can be a prognostic factor in patients who undergo radical resection for gastric cancer and can overcome the limitations of existing prognostic factors.

BackgroundA combination of plasma phospho‐tau (p‐tau), amyloid beta (Aβ)‐PET, brain magnetic resonance imaging, cognitive function tests, and other biomarkers might predict future cognitive decline. This study aimed to investigate the efficacy of combining these biomarkers in predicting future cognitive stage transitions within 3 years.MethodAll participants were from the validation cohort of the Korean Brain Aging Study for the Early Diagnosis and Prediction of Alzheimer’s Disease (KBASE‐V) study. Among the participants in the KBASE‐V study, 49 mild cognitive impairment (MCI) and 113 cognitively unimpaired (CU) participants with Aβ‐PET and brain imaging data were analyzed. Main outcome was cognitive stage transitions within 3 years. The best combination model that predicts cognitive stage transition was considered by significant change of area under the curve (AUC) and Akaike information criterion value.ResultWithin 3 years, 19.1% of the participants showed a cognitive stage transition. Older age, increased plasma p‐tau, Aβ‐PET positivity, and decreased semantic fluency were independently associated with cognitive stage transitions. Plasma p‐tau181 (AUC = 0.673) and the Centiloid scale (AUC = 0.769) alone predicted cognitive stage transition well and correlated with each other significantly. Combining age, p‐tau181, the Centiloid scale, semantic fluency, and hippocampal volume produced better accuracy (AUC = 0.879). In MCI participants, the model combining age, p‐tau181, and the Centiloid scale demonstrated the highest predictive value (AUC = 0.963).ConclusionPlasma p‐tau181 and Centiloid scale, in combination with other biomarkers, might have high predictive value in predicting future cognitive stage transition in non‐dementia patients.

BackgroundChronic atrophic gastritis (CAG) severity risk assessment remains challenging due to limitations of current static histologic/endoscopic tools. We evaluated the leucine-rich repeat-containing G-protein coupled receptor 5 (LGR5)/trefoil factor 2 (TFF2) ratio — quantifying gastric stem cell niche imbalance— as a dynamic biomarker for high-risk CAG (OLGA/OLGIM III-IV).MethodsIn a retrospective cohort (n = 60 CAG patients), we constructed four prediction models (Baseline: demographics; Clinical: +H. pylori/family history; Endoscopic; Full: +LGR5/TFF2 ratio) using Firth penalized regression. Model performance was assessed via area under the curve (AUC), Akaike (AIC), and Bayesian (BIC) information criteria, and likelihood ratio tests. Restricted cubic splines and subgroup analyses validated threshold effects.ResultsThe LGR5/TFF2 ratio > 1.066 (AUC = 0.93) identified high-risk CAG with 48.5-fold increased odds (95% CI: 4.07–578.51, P = 0.002), outperforming endoscopic/histologic models (ΔAUC = + 0.15). A nonlinear dose-response relationship revealed sharp risk escalation beyond this threshold. The high-ratio group (n = 20) exhibited 60.00% prevalence of prior H. pylori eradication, Wnt activation (β-catenin↑32.3%, CyclinD1↑90%, P < 0.001), and stronger endoscopic correlation (ρ = 0.42 for IM nodules) than traditional factors (ρ = 0.28 for H. pylori).ConclusionThe LGR5/TFF2 ratio is the first biomarker to molecularly define CAG severity risk by quantifying stem cell niche imbalance (LGR5↑/TFF2↓), achieving superior accuracy (AUC = 0.93) and identifying high-risk cases missed by current methods. Its integration into clinical practice could transform risk-stratified surveillance paradigms.

To determine the ability of frequency doubling technology (FDT) and scanning laser polarimetry with variable corneal compensation (GDx-VCC) to detect glaucoma when used individually and in combination. One hundred ten normal and 114 glaucomatous subjects were tested with FDT C-20-5 screening protocol and the GDx-VCC. The discriminating ability was tested for each device individually and for both devices combined using GDx-NFI, GDx-TSNIT, number of missed points of FDT, and normal or abnormal FDT. Measures of discrimination included sensitivity, specificity, area under the curve (AUC), Akaike's information criterion (AIC), and prediction confidence interval lengths. For detecting glaucoma regardless of severity, the multivariable model resulting from the combination of GDx-TSNIT, number of abnormal points on FDT (NAP-FDT), and the interaction GDx-TSNIT×NAP-FDT (AIC: 88.28, AUC: 0.959, sensitivity: 94.6%, specificity: 89.5%) outperformed the best single-variable model provided by GDx-NFI (AIC: 120.88, AUC: 0.914, sensitivity: 87.8%, specificity: 84.2%). The multivariable model combining GDx-TSNIT, NAP-FDT, and interaction GDx-TSNIT×NAP-FDT consistently provided better discriminating abilities for detecting early, moderate, and severe glaucoma than the best single-variable models. The multivariable model including GDx-TSNIT, NAP-FDT, and the interaction GDx-TSNIT×NAP-FDT provides the best glaucoma prediction compared with all other multivariable and univariable models. Combining the FDT C-20-5 screening protocol and GDx-VCC improves glaucoma detection compared with using GDx or FDT alone.

The objective of the chapter is to offer intuitive understandings for statistical learning from an infomation geometric point of view. We introduce a wide class of information divergence indeces that express quantatitavely the degree of departure between any two probability density functions on a data space. In general the information divergence leads to a statistical method by minimization based on empirical data information. We introduce that the information divergence associates with a Riemannian metric and a pair of congugate linear connections for a family of probability density functions. The most familiar example is the Kullback-Leibler divergence, which leads to the maximum likelihood method and associates with the information metric and the pair of the exponential and mixture connections. In the class of statistical methods led to by minimum divergence we discuss statistical properties focusing on robsutness. As applicatuons to statistical learning we discuss the minimum divergence methods for the principal component analysis and independent component analysis, and we also discuss applications to statistical pattern recognition. U -divergence, U -model, U -loss function, robustness, information geometry.

Graphical models provide a broad framework for probabilistic inference, with application to such diverse areas as speech recognition (Hidden Markov Models), medical diagnosis (Belief networks) and artificial intelligence (Boltzmann Machines). However, the computing time is typically exponential in the number of nodes in the graph. We present a general framework for a class of approximating models, based on the Kullback-Leibler divergence between an approximating graph and the original graph. We concentrate here on undirected approximations of both intractable directed and undirected graphical models. Simulation results on a small benchmark problem suggest that this method compares favourably against others previously reported in the literature.

This paper investigates and discusses the use of information divergence, through the widely used Kullback–Leibler (KL) divergence, under the multivariate (generalized) $$\gamma $$-order normal distribution ($$\gamma $$-GND). The behavior of the KL divergence, as far as its symmetricity is concerned, is studied by calculating the divergence of $$\gamma $$-GND over the Student’s multivariate t-distribution and vice versa. Certain special cases are also given and discussed. Furthermore, three symmetrized forms of the KL divergence, i.e., the Jeffreys distance, the geometric-KL as well as the harmonic-KL distances, are computed between two members of the $$\gamma $$-GND family, while the corresponding differences between those information distances are also discussed.

Markov-Switching Model Selection Using Kullback-Leibler Divergence

Introduction: Axillary lymph node dissection (ALND) has no apparent oncologic benefit for breast cancer patients with clinically negative axillae and 1-2 positive sentinel lymph nodes so omission has been considered appropriate for this patient cohort. For patients with clinically positive nodes who are not treated with neoadjuvant chemotherapy (NAC), ALND is routinely recommended. Yet, at ALND, up to 40-57% of patients are found to have ≤ 3 positive nodes (pN1). No accurate method exists for identifying patients with minimal nodal disease who can potentially be spared the morbidity of ALND. DNA methylation (DNAm) has emerged as a potential tool for creating different classifiers to stratify cancer patients using artificial intelligence methods, such as machine learning. This study employed machine learning algorithms to create DNAm-based classifiers that efficiently stratify patients with clinically positive nodes to pN1 vs &amp;gt;pN1 disease. Methodology: We used publically available data from The Cancer Genome Atlas (TCGA) of breast cancer patients (n=1,006). We curated the cohort to select patients with hormone receptor-positive and HER2 negative invasive ductal carcinoma who were not treated with NAC. We only included patients who had undergone an ALND and had available DNAm (HM450 BeadChip array) data (n=58). Two Random Forest methodologies were used to obtain different combinations of DNAm features with high accuracy in stratifying pN1 and &amp;gt;pN1 patients. The most efficient combination of CpG sites was selected according to the Area Under the Curve (AUC) of each combination. Results: First, the differentially methylated sites (DMS, n=218) between pN1 and &amp;gt;pN1 patients were computed. Using all DMS displayed a good predictive potential for the selection of pN1 patients (AUC=0.88). Three different panels were obtained, containing 20 (P20), 15 (P15), and 13 (P13) genomic regions per panel. All three signatures displayed an excellent performance stratifying pN1 from &amp;gt;pN1 patients (AUCP20=0.98, AUCP15=0.98, and AUCP13=0.97). Furthermore, these epigenetic signatures showed a higher discriminative potential than clinical and pathological variables, such as the location of the tumor, tumor size, age, and tumor grade. Discussion: Due to the lack of data, ALND is still recommended for patients with clinically node positive disease. However, over half of patients with clinically positive nodes have minimal (pN1) nodal disease at surgery and could theoretically avoid ALND. We have designed three artificial intelligence DNAm-based panels (P20, P15, and P13) which efficiently stratify pN1 from &amp;gt;pN1 patients. These classifiers are more accurate than clinical and pathological variables for selecting patients with a low number of positive nodes. Furthermore, our method employs a small number of genomic regions (13 to 20), facilitating the use of PCR-based assays, such as quantitative Methylation Specific PCR (qMSP), which decreases cost compared to high-throughput technology, and increases availability and accessibility of this predictive tool. Citation Format: Maggie L. DiNome, Miguel Ensenyat-Mendez, Javier I.J. Orozco, Dennis Rünger, Jennifer L. Baker, Joanne B. Weidhaas, Diego M. Marzese. Machine learning-based DNA methylation classifiers to predict pathologic nodal stage in breast cancer patients with clinically node positive disease [abstract]. In: Proceedings of the 2021 San Antonio Breast Cancer Symposium; 2021 Dec 7-10; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2022;82(4 Suppl):Abstract nr P1-01-17.