Coefficient of determination

Summary

In statistics, the coefficient of determination, denoted R2 or r2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s). It is a statistic used in the context of statistical models whose main purpose is either the prediction of future outcomes or the testing of hypotheses, on the basis of other related information. It provides a measure of how well observed outcomes are replicated by the model, based on the proportion of total variation of outcomes explained by the model. There are several definitions of R2 that are only sometimes equivalent. One class of such cases includes that of simple linear regression where r2 is used instead of R2. When only an intercept is included, then r2 is simply the square of the sample correlation coefficient (i.e., r) between the observed outcomes and the observed predictor values. If additional regressors are included, R2 is the square of the coefficient of mul

Official source

https://en.wikipedia.org/wiki/Coefficient_of_determination

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Anaerobic digestion of organic waste to methane gas (CH4) is an attractive method to produce renewable energy. Due to the trend towards the reuse of waste and away from fossil energy sources, this technology saw a worldwide development in the recent years. A fundamental parameter in this domain is the Biochemical Methane Potential (BMP), which defines the amount of CH4 which can be produced out of a certain organic substrate. Such information is essential in order to plan and optimise anaerobic digestion plants, and to evaluate the reactor feed with a new substrate or a new substrate combination (co-digestion). BMPs are assessed in batch test, which can be regarded as the simulation of a full-scale biogas plant. Two experiments are required, the test in which the test substrate is digested on the inoculum, and the blank, in which the inoculum is digested alone. The part of the inoculum which can be digested is called endogenous substrate. Since the test experiment produced not only on the test substrate but only on the endogenous substrate, the CH4 production on the test substrate alone is obtained by the subtraction of the test production minus the blank production. The units of a BMP are generally indicated in litre of CH4 per gram of volatile solids (VS) test substrate [L/gVS]. Despite the importance and the wide application of this parameter, its exact determination remains a challenging issue and results are often not consistent in between (inter-) and within (intra-) laboratories. Experimental protocols exist but do not lead to a satisfactory BMP tests consistency neither. The aim of this study was to identify parameters affecting the outcome of BMP tests, based on the investigation of two different data sets. An inter-laboratory study of BMP tests providing the final BMP values of 327 experiments and information about 40 related experimental parameters, and a second data set containing the complete CH4 production curves of 136 BMP experiments provided by one single laboratory. The method consisted in graphical and statistical analysis (Mixed Effect Modelling), using [R] programming language and software environment. This study found out, that a significant part of the inter-laboratory BMP inconsistency can be explained by an imprecise assessment of the VS, which was not expected. As the VS of the substrate are directly implied in the computation of the BMP, the impact was significant and therefore the BMP were corrected regarding the VS imprecision. With these data, up to 70% of the inconsistency of BMPs could be explained by inter-laboratory effects. The statistical analysis led to the conclusion that the concentration of the endogenous substrate and the moisture content in the digestate would be the principal factors affecting the outcome of BMP tests. An effort was made to identify and correct errors contained in CH4 production curves, which turned out to be delicate for certain cases. Also indications regarding the precision and the reliability of BMPs were formulated. Further, the investigation of the CH4 production curves led to the development of a new method in order to compute the BMP result. This method was based on the fact, that a certain inoculum reaches always the same slope toward the end of the experiment, no matter what was digested before. The advantage of this method would be that the experiment end-point could be set clearly and that the concentration of the endogenous substrate would not have an impact on the outcome of a BMP test. According to the findings of this report, it was proposed to add the following requirements into experimental protocols for BMP tests: Tests, blanks and the analysis of VS should be carried out in triplicates and their standard deviation should be indicated for each of them, together with the BMP result If triplicates contain set-up errors, these experiments must be repeated, including the corresponding blank/test In the annex: An experimental protocol for the TS and VS analyses An indication of a required moisture content of the digestate (mechanism to investigate in detail first) An indication of the range of required VS concentrations or a maximal production rate for blank tests according to certain experimental conditions (mechanism to investigate in detail first). This report identified several parameters, which contribute to the inter-laboratory inconsistency of BMPs. These findings should be investigated further in order to prove and quantify their impact. An effort should be made to demonstrate the newly developed BMP-computation-method, which could eventually lead to more consistent results. The limitation of this study was, that a relatively low amount of data was available compared to their characteristics. Consequently, the findings could only be proven on a few examples and should therefore only be seen as evidences. Further, this led also to the risk of overfitting, since a relative high amount of parameters needed to be included into the statistical model.

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On the relationship between total ozone and atmospheric dynamics and chemistry at mid-latitudes - Part 2: The effects of the El Nino/Southern Oscillation, volcanic eruptions and contributions of atmospheric dynamics and chemistry to long-term total ozone changes

Anthony Christopher Davison, Linda Frossard

We present the first spatial analysis of "fingerprints" of the El Nino/Southern Oscillation (ENSO) and atmospheric aerosol load after major volcanic eruptions (El Chichon and Mt. Pinatubo) in extreme low and high (termed ELOs and EHOs, respectively) and mean values of total ozone for the northern and southern mid-latitudes (defined as the region between 30 degrees and 60 degrees north and south, respectively). Significant influence on ozone extremes was found for the warm ENSO phase in both hemispheres during spring, especially towards low latitudes, indicating the enhanced ozone transport from the tropics to the extra-tropics. Further, the results confirm findings of recent work on the connection between the ENSO phase and the strength and extent of the southern ozone "collar". For the volcanic eruptions the analysis confirms findings of earlier studies for the northern mid-latitudes and gives new insights for the Southern Hemisphere. The results provide evidence that the negative effect of the eruption of El Chichon might be partly compensated by a strong warm ENSO phase in 1982-1983 at southern mid-latitudes. The strong west-east gradient in the coefficient estimates for the Mt. Pinatubo eruption and the analysis of the relationship between the AAO and ENSO phase, the extent and the position of the southern ozone "collar" and the polar vortex structure provide clear evidence for a dynamical " masking" of the volcanic signal at southern mid-latitudes. The paper also analyses the contribution of atmospheric dynamics and chemistry to long-term total ozone changes. Here, quite heterogeneous results have been found on spatial scales. In general the results show that EESC and the 11-yr solar cycle can be identified as major contributors to long-term ozone changes. However, a strong contribution of dynamical features (El Nino/Southern Oscillation (ENSO), North Atlantic Oscillation (NAO), Antarctic Oscillation (AAO), Quasi-Biennial Oscillation (QBO)) to ozone variability and trends is found at a regional level. For the QBO (at 30 and 50 hPa), strong influence on total ozone variability and trends is found over large parts of the northern and southern mid-latitudes, especially towards equatorial latitudes. Strong influence of ENSO is found over the Northern and Southern Pacific, Central Europe and central southern mid-latitudes. For the NAO, strong influence on column ozone is found over Labrador/Greenland, the Eastern United States, the Euro-Atlantic Sector, and Central Europe. For the NAO's southern counterpart, the AAO, strong influence on ozone variability and long-term changes is found at lower southern mid-latitudes, including the southern parts of South America and the Antarctic Peninsula, and central southern mid-latitudes.

Copernicus Publications2013

Neuroimaging of human auditory cortex

Naghmeh Ghazaleh

Modern neuroimaging technologies provide us with non-invasive methods for study- ing structure and function, and plasticity of the human auditory cortex. Here we have acquired high spatial resolution functional magnetic resonance imaging (fMRI) data of the auditory cortex in patients with unilateral hearing loss and tinnitus. Compared to healthy controls, we found evidence of over-representation and hyperactivity in parts of cortical tonotopic map that correspond to low frequencies sounds, irrespective of the hearing loss and tinnitus range, which in most cases affected higher frequencies. These findings suggest that hearing loss has a destabilizing effect on the tonotopic organization in primary auditory cortex that is not restricted to the corresponding frequency range of hearing loss and tinnitus. One of the challenges in tinnitus studies is the high relevance of the hyperacusis con- dition in these patients. The similarity and co-occurrence of these two disorders con- tributes to the debate about the specificity of findings associated with tinnitus alone. For classifying tinnitus patients with and without hyperacusis, we thus implemented partial least square correlation (PLSC) analysis that builds upon the cross-covariance information between audiogram and fMRI data. This method provides us with a multi modal measure to successfully detect the hyperacusis condition in tinnitus patients with 80% cross-validation accuracy, for the first time. The PLSC driven biomarker, also yields the brain area playing a major role in the condition. There is a vast heterogeneity in the anatomy of auditory cortex across subjects, therefore individualized delineation the exact border between primary and secondary auditory cortex is a demanding research topic. Here we established a model of ex- trinsic connectivity of auditory cortex with the two other regions of the brain that have discriminative connectivity with primary and secondary auditory cortices. The resting state fMRI connectivity features are used for segmentation and are performed exploiting two methods as dynamical causal modeling (DCM) of effective connectivity with 74% accuracy and functional connectivity with 71% accuracy. This data-driven segmentation method, not only facilitate the definition of auditory cortex subregions for fundamental research in the normal brain, but also yields a useful tool for following the modification in the functional anatomy of this region as a result of hearing disorders like tinnitus.

EPFL2017