Quantifying heteroskedasticity using slope of local variances index

In econometrics, heteroskedasticity refers to the case when the variances of the error terms of the data in hand are not equal. Heteroskedastic time series are challenging to different forecasting models. However, all available solutions adopt the strategy of accommodating heteroskedasticity in the time series and consider it as a type of noise. Some statistical tests were developed over the past three decades to determine whether a time series features heteroskedastic behaviour. This paper presents a novel strategy to handle this problem by deriving a quantifying measure for heteroskedasticity. The proposed measure relies on the definition of heteroskedasticity as a time-variant variance in the time series. In this work, heteroskedasticity is measured by calculating local variances using linear filters, estimating variance trends, calculating changes in variance slopes, and finally obtaining the average slope angle. The results confirm that the proposed index complies with the widely popular heteroskedasticity tests.

Quantifying heteroskedasticity metrics

This study proposes a quantification measure for heteroskedasticity in the time series. Two methods are introduced for quantifying heteroskedasticity: Slope of Local Variance Index (SoLVI) and a statistical divergence method using Bhattacharys coefficient. Both measures show reliability in measuring and quantifying heteroskedasticity in comparison to numerical and hypothesis heteroskedasticity tests.

Learning from large data : bias, variance, sampling, and learning curves

One of the fundamental machine learning tasks is that of predictive classification. Given that organisations collect an ever increasing amount of data, predictive classification methods must be able to effectively and efficiently handle large amounts of data. However, it is understood that present requirements push existing algorithms to, and sometimes beyond, their limits since many classification prediction algorithms were designed when currently common data set sizes were beyond imagination. This has led to a significant amount of research into ways of making classification learning algorithms more effective and efficient. Although substantial progress has been made, a number of key questions have not been answered. This dissertation investigates two of these key questions. The first is whether different types of algorithms to those currently employed are required when using large data sets. This is answered by analysis of the way in which the bias plus variance decomposition of predictive classification error changes as training set size is increased. Experiments find that larger training sets require different types of algorithms to those currently used. Some insight into the characteristics of suitable algorithms is provided, and this may provide some direction for the development of future classification prediction algorithms which are specifically designed for use with large data sets. The second question investigated is that of the role of sampling in machine learning with large data sets. Sampling has long been used as a means of avoiding the need to scale up algorithms to suit the size of the data set by scaling down the size of the data sets to suit the algorithm. However, the costs of performing sampling have not been widely explored. Two popular sampling methods are compared with learning from all available data in terms of predictive accuracy, model complexity, and execution time. The comparison shows that sub-sampling generally products models with accuracy close to, and sometimes greater than, that obtainable from learning with all available data. This result suggests that it may be possible to develop algorithms that take advantage of the sub-sampling methodology to reduce the time required to infer a model while sacrificing little if any accuracy. Methods of improving effective and efficient learning via sampling are also investigated, and now sampling methodologies proposed. These methodologies include using a varying-proportion of instances to determine the next inference step and using a statistical calculation at each inference step to determine sufficient sample size. Experiments show that using a statistical calculation of sample size can not only substantially reduce execution time but can do so with only a small loss, and occasional gain, in accuracy. One of the common uses of sampling is in the construction of learning curves. Learning curves are often used to attempt to determine the optimal training size which will maximally reduce execution time while nut being detrimental to accuracy. An analysis of the performance of methods for detection of convergence of learning curves is performed, with the focus of the analysis on methods that calculate the gradient, of the tangent to the curve. Given that such methods can be susceptible to local accuracy plateaus, an investigation into the frequency of local plateaus is also performed. It is shown that local accuracy plateaus are a common occurrence, and that ensuring a small loss of accuracy often results in greater computational cost than learning from all available data. These results cast doubt over the applicability of gradient of tangent methods for detecting convergence, and of the viability of learning curves for reducing execution time in general.

Working with 'housing variance' to model urban growth futures with inner metropolitan Melbourne

Continued population growth in Melbourne over the past decade has led to the development of a range of strategies and policies by State and Local levels of government to set an agenda for a more sustainable form of urban development. As the Victorian State government moves towards the development of 'Plan Melbourne', a new metropolitan planning strategy currently being prepared to take Melbourne forward to 2050, the following paper addresses the issue of how new residential built form will impact on and be accommodated in existing Inner Melbourne activity centres. Working with the prospect of establishing a more compact city in order to meet an inner city target of 90,000 new dwellings (Inner Metropolitan Action Plan - IMAP Strategy 5), the paper presents a 'Housing Variance Model' based on household structure and dwelling type. As capacity is progressively altered through a range of built form permutations, the research attempts to assess the impact on the urban morphology of a case study of four Major Activity Centres in the municipality of Port Phillip.