Statnote 19: non-linear regression: fitting an exponential curve to data

Non-linear relationships are common in microbiological research and often necessitate the use of the statistical techniques of non-linear regression or curve fitting. In some circumstances, the investigator may wish to fit an exponential model to the data, i.e., to test the hypothesis that a quantity Y either increases or decays exponentially with increasing X. This type of model is straight forward to fit as taking logarithms of the Y variable linearises the relationship which can then be treated by the methods of linear regression.

Relationships between fundus image quantification and visual field status in age-related macular degeneration

Presentation Purpose:To relate structural change to functional change in age-related macular degeneration (AMD) in a cross-sectional population using fundus imaging and the visual field status. Methods:10 degree standard and SWAP visual fields and other standard functional clinical measures were acquired in 44 eyes of 27 patients at various stages of AMD, as well as fundus photographs. Retro-mode SLO images were captured in a subset of 29 eyes of 19 of the patients. Drusen area, measured by automated drusen segmentation software (Smith et al. 2005) was correlated with visual field data. Visual field defect position was compared to the position of the imaged drusen and deposits using custom software. Results:The effect of AMD stage on drusen area within the 6000µm was significant (One-way ANOVA: F = 17.231, p < 0.001), however the trend was not strong across all stages. There were significant linear relationships between visual field parameters and drusen area. The mean deviation (MD) declined by 3.00dB and 3.92dB for each log % drusen area for standard perimetry and SWAP, respectively. The visual field parameters of focal loss displayed the strongest correlations with drusen area. The number of pattern deviation (PD) defects increased by 9.30 and 9.68 defects per log % drusen area for standard perimetry and SWAP, respectively. Weaker correlations were found between drusen area and visual acuity, contrast sensitivity, colour vision and reading speed. 72.6% of standard PD defects and 65.2% of SWAP PD defects coincided with retinal signs of AMD on fundus photography. 67.5% of standard PD defects and 69.7% of SWAP PD defects coincided with deposits on retro-mode images. Conclusions:Perimetry exhibited a stronger relationship with drusen area than other measures of visual function. The structure-function relationship between visual field parameters and drusen area was linear. Overall the indices of focal loss had a stronger correlation with drusen area in SWAP than in standard perimetry. Visual field defects had a high coincidence proportion with retinal manifestations of AMD.Smith R.T. et al. (2005) Arch Ophthalmol 123:200-206.

Seasonal growth of the crustose lichen Rhizocarpon geographicum (L.) DC. in South Gwynedd, Wales

Seasonal growth was studied in the slow-growing crustose lichen Rhizocarpon geographicum (L.) DC. in an area of South Gwynedd, Wales. Radial growth rate (RGR) of a sample of 20 thalli was measured in situ at three-month intervals over 51 months on a southeast-facing rock surface. There were five periods of significant growth: July-September of 1993, 1994 and 1995, in January-March of 1996, and in April-June of 1997. In four of these periods, growth coincided with a mean temperature maximum (Tmax) over a three-month period exceeding 15°C and three of the maxima with greater than 450 sunshine hours. Two of the growth maxima coincided with periods of total rainfall exceeding 300 mm and one with greater than 50 rain days in a three-month period. There were no significant linear correlations between RGR and the climatic variables measured. However, there were significant non-linear relationships between RGR and Tmax, the mean temperature minimum (Tmin), the total number of air and ground frosts and the number of rain days in a growth period, the relationship with Tmax being the most significant. Hence, in south Gwynedd, maximum growth of R. geographicum occurs in any season although the period July-September appears to be the most favourable. Relationships between growth and climatic variables were non-linear, temperature having the most significant influence on seasonal growth. ©2006 Balaban.

Multi-frequency segmental bio-impedance device:design, development and applications

Bio-impedance analysis (BIA) provides a rapid, non-invasive technique for body composition estimation. BIA offers a convenient alternative to standard techniques such as MRI, CT scan or DEXA scan for selected types of body composition analysis. The accuracy of BIA is limited because it is an indirect method of composition analysis. It relies on linear relationships between measured impedance and morphological parameters such as height and weight to derive estimates. To overcome these underlying limitations of BIA, a multi-frequency segmental bio-impedance device was constructed through a series of iterative enhancements and improvements of existing BIA instrumentation. Key features of the design included an easy to construct current-source and compact PCB design. The final device was trialled with 22 human volunteers and measured impedance was compared against body composition estimates obtained by DEXA scan. This enabled the development of newer techniques to make BIA predictions. To add a ‘visual aspect’ to BIA, volunteers were scanned in 3D using an inexpensive scattered light gadget (Xbox Kinect controller) and 3D volumes of their limbs were compared with BIA measurements to further improve BIA predictions. A three-stage digital filtering scheme was also implemented to enable extraction of heart-rate data from recorded bio-electrical signals. Additionally modifications have been introduced to measure change in bio-impedance with motion, this could be adapted to further improve accuracy and veracity for limb composition analysis. The findings in this thesis aim to give new direction to the prediction of body composition using BIA. The design development and refinement applied to BIA in this research programme suggest new opportunities to enhance the accuracy and clinical utility of BIA for the prediction of body composition analysis. In particular, the use of bio-impedance to predict limb volumes which would provide an additional metric for body composition measurement and help distinguish between fat and muscle content.

Seasonal growth and growth rate-colony size relationships in six species of saxicolous lichens

The pattern of seasonal growth and the relation of growth rate to colony size were studied in four foliose and two crustose species of saxicolous lichens. A new method of measuring growth was used whereby the advance of a sample of lobes along millimetres marked on the substrate was measured under a magnification of x10. Three peaks of growth were found(in March, June and November) for the foliose species and a single peak (in May to August) for the crustose species. THe peaks of growth corresponded approximately to peaks of rainfall. Growth rate in relation to increasing colony size fell in a smooth exponential curve when expressed on a cm squared/ cm squared/ unit time basis. The result is consistent with a linear radial rate for most of the thallus sizes for the six species. There is also evidence for an exponential incresae in growth rate initially until about 1.5 cm thallus diameter in two of the sepcies when the linear radial rate is achieved.

Approximating differentiable relationships between delay embedded dynamical systems with radical basis functions

This thesis is about the study of relationships between experimental dynamical systems. The basic approach is to fit radial basis function maps between time delay embeddings of manifolds. We have shown that under certain conditions these maps are generically diffeomorphisms, and can be analysed to determine whether or not the manifolds in question are diffeomorphically related to each other. If not, a study of the distribution of errors may provide information about the lack of equivalence between the two. The method has applications wherever two or more sensors are used to measure a single system, or where a single sensor can respond on more than one time scale: their respective time series can be tested to determine whether or not they are coupled, and to what degree. One application which we have explored is the determination of a minimum embedding dimension for dynamical system reconstruction. In this special case the diffeomorphism in question is closely related to the predictor for the time series itself. Linear transformations of delay embedded manifolds can also be shown to have nonlinear inverses under the right conditions, and we have used radial basis functions to approximate these inverse maps in a variety of contexts. This method is particularly useful when the linear transformation corresponds to the delay embedding of a finite impulse response filtered time series. One application of fitting an inverse to this linear map is the detection of periodic orbits in chaotic attractors, using suitably tuned filters. This method has also been used to separate signals with known bandwidths from deterministic noise, by tuning a filter to stop the signal and then recovering the chaos with the nonlinear inverse. The method may have applications to the cancellation of noise generated by mechanical or electrical systems. In the course of this research a sophisticated piece of software has been developed. The program allows the construction of a hierarchy of delay embeddings from scalar and multi-valued time series. The embedded objects can be analysed graphically, and radial basis function maps can be fitted between them asynchronously, in parallel, on a multi-processor machine. In addition to a graphical user interface, the program can be driven by a batch mode command language, incorporating the concept of parallel and sequential instruction groups and enabling complex sequences of experiments to be performed in parallel in a resource-efficient manner.

Non-linear hierarchical visualisation

This thesis applies a hierarchical latent trait model system to a large quantity of data. The motivation for it was lack of viable approaches to analyse High Throughput Screening datasets which maybe include thousands of data points with high dimensions. High Throughput Screening (HTS) is an important tool in the pharmaceutical industry for discovering leads which can be optimised and further developed into candidate drugs. Since the development of new robotic technologies, the ability to test the activities of compounds has considerably increased in recent years. Traditional methods, looking at tables and graphical plots for analysing relationships between measured activities and the structure of compounds, have not been feasible when facing a large HTS dataset. Instead, data visualisation provides a method for analysing such large datasets, especially with high dimensions. So far, a few visualisation techniques for drug design have been developed, but most of them just cope with several properties of compounds at one time. We believe that a latent variable model (LTM) with a non-linear mapping from the latent space to the data space is a preferred choice for visualising a complex high-dimensional data set. As a type of latent variable model, the latent trait model can deal with either continuous data or discrete data, which makes it particularly useful in this domain. In addition, with the aid of differential geometry, we can imagine the distribution of data from magnification factor and curvature plots. Rather than obtaining the useful information just from a single plot, a hierarchical LTM arranges a set of LTMs and their corresponding plots in a tree structure. We model the whole data set with a LTM at the top level, which is broken down into clusters at deeper levels of t.he hierarchy. In this manner, the refined visualisation plots can be displayed in deeper levels and sub-clusters may be found. Hierarchy of LTMs is trained using expectation-maximisation (EM) algorithm to maximise its likelihood with respect to the data sample. Training proceeds interactively in a recursive fashion (top-down). The user subjectively identifies interesting regions on the visualisation plot that they would like to model in a greater detail. At each stage of hierarchical LTM construction, the EM algorithm alternates between the E- and M-step. Another problem that can occur when visualising a large data set is that there may be significant overlaps of data clusters. It is very difficult for the user to judge where centres of regions of interest should be put. We address this problem by employing the minimum message length technique, which can help the user to decide the optimal structure of the model. In this thesis we also demonstrate the applicability of the hierarchy of latent trait models in the field of document data mining.