Math 412 - Topology with Applications

Highlights of Data Expedition: • Students explored daily observations of local climate data spanning the past 35 years. • Topological Data Analysis, or TDA for short, provides cutting-edge tools for studying the geometry of data in arbitrarily high dimensions. • Using TDA tools, students discovered intrinsic dynamical features of the data and learned how to quantify periodic phenomenon in a time-series. • Since nature invariably produces noisy data which rarely has exact periodicity, students also considered the theoretical basis of almost-periodicity and even invented and tested new mathematical definitions of almost-periodic functions. Summary The dataset we used for this data expedition comes from the Global Historical Climatology Network. “GHCN (Global Historical Climatology Network)-Daily is an integrated database of daily climate summaries from land surface stations across the globe.” Source: https://www.ncdc.noaa.gov/oa/climate/ghcn-daily/ We focused on the daily maximum and minimum temperatures from January 1, 1980 to April 1, 2015 collected from RDU International Airport. Through a guided series of exercises designed to be performed in Matlab, students explore these time-series, initially by direct visualization and basic statistical techniques. Then students are guided through a special sliding-window construction which transforms a time-series into a high-dimensional geometric curve. These high-dimensional curves can be visualized by projecting down to lower dimensions as in the figure below (Figure 1), however, our focus here was to use persistent homology to directly study the high-dimensional embedding. The shape of these curves has meaningful information but how one describes the “shape” of data depends on which scale the data is being considered. However, choosing the appropriate scale is rarely an obvious choice. Persistent homology overcomes this obstacle by allowing us to quantitatively study geometric features of the data across multiple-scales. Through this data expedition, students are introduced to numerically computing persistent homology using the rips collapse algorithm and interpreting the results. In the specific context of sliding-window constructions, 1-dimensional persistent homology can reveal the nature of periodic structure in the original data. I created a special technique to study how these high-dimensional sliding-window curves form loops in order to quantify the periodicity. Students are guided through this construction and learn how to visualize and interpret this information. Climate data is extremely complex (as anyone who has suffered from a bad weather prediction can attest) and numerous variables play a role in determining our daily weather and temperatures. This complexity coupled with imperfections of measuring devices results in very noisy data. This causes the annual seasonal periodicity to be far from exact. To this end, I have students explore existing theoretical notions of almost-periodicity and test it on the data. They find that some existing definitions are also inadequate in this context. Hence I challenged them to invent new mathematics by proposing and testing their own definition. These students rose to the challenge and suggested a number of creative definitions. While autocorrelation and spectral methods based on Fourier analysis are often used to explore periodicity, the construction here provides an alternative paradigm to quantify periodic structure in almost-periodic signals using tools from topological data analysis.

Assessment of Two Diabetes Point-of-Care Analyzers Measuring Hemoglobin A1c in the Peruvian Amazon

Aims: Measurement of glycated hemoglobin (HbA1c) is an important indicator of glucose control over time. Point-of-care (POC) devices allow for rapid and convenient measurement of HbA1c, greatly facilitating diabetes care. We assessed two POC analyzers in the Peruvian Amazon where laboratory-based HbA1c testing is not available.

Methods: Venous blood samples were collected from 203 individuals from six different Amazonian communities with a wide range of HbA1c, 4.4-9.0% (25-75 mmol/mol). The results of the Afinion AS100 and the DCA Vantage POC analyzers were compared to a central laboratory using the Premier Hb9210 high-performance liquid chromatography (HPLC) method. Imprecision was assessed by performing 14 successive tests of a single blood sample.

Results: The correlation coefficient r for POC and HPLC results was 0.92 for the Afinion and 0.93 for the DCA Vantage. The Afinion generated higher HbA1c results than the HPLC (mean difference = +0.56% [+6 mmol/mol]; p < 0.001), as did the DCA Vantage (mean difference = +0.32% [4 mmol/mol]). The bias observed between POC and HPLC did not vary by HbA1c level for the DCA Vantage (p = 0.190), but it did for the Afinion (p < 0.001). Imprecision results were: CV = 1.75% for the Afinion, CV = 4.01% for the DCA Vantage. Sensitivity was 100% for both devices, specificity was 48.3% for the Afinion and 85.1% for the DCA Vantage, positive predictive value (PPV) was 14.4% for the Afinion and 34.9% for the DCA Vantage, and negative predictive value (NPV) for both devices was 100%. The area under the receiver operating characteristic (ROC) curve was 0.966 for the Afinion and 0.982 for the DCA Vantage. Agreement between HPLC and POC in classifying diabetes and prediabetes status was slight for the Afinion (Kappa = 0.12) and significantly different (McNemar’s statistic = 89; p < 0.001), and moderate for the DCA Vantage (Kappa = 0.45) and significantly different (McNemar’s statistic = 28; p < 0.001).

Conclusions: Despite significant variation of HbA1c results between the Afinion and DCA Vantage analyzers compared to HPLC, we conclude that both analyzers should be considered in health clinics in the Peruvian Amazon for therapeutic adjustments if healthcare workers are aware of the differences relative to testing in a clinical laboratory. However, imprecision and bias were not low enough to recommend either device for screening purposes, and the local prevalence of anemia and malaria may interfere with diagnostic determinations for a substantial portion of the population.