Applications of stochastic analysis to sequential CUSUM procedures

**Authors:**Uys, Nadia**Date:**2010-02-23T10:20:22Z**Subjects:**Stochastic analysis , CUSUM technique**Type:**Thesis**Identifier:**uj:6631 , http://hdl.handle.net/10210/3032**Description:**Ph.D.**Full Text:**

**Authors:**Uys, Nadia**Date:**2010-02-23T10:20:22Z**Subjects:**Stochastic analysis , CUSUM technique**Type:**Thesis**Identifier:**uj:6631 , http://hdl.handle.net/10210/3032**Description:**Ph.D.**Full Text:**

Tests for constancy of the mean fall and rise times of a light curve

- Potgieter, Gert Diedericks Johannes

**Authors:**Potgieter, Gert Diedericks Johannes**Date:**2012-08-27**Subjects:**Stars , Light , Optics , Wave mechanics**Type:**Mini-Dissertation**Identifier:**uj:3162 , http://hdl.handle.net/10210/6579**Description:**M.Sc. , The brightness of many stars vary over time. A plot of the brightness against time is known as a light curve. The fall and rise times of the light curve can be modelled by two cross-correlated white noise processes. We propose four statistics to test the hypothesis of a constant mean fall and rise time of the light curve against the alternative that at least one of the fall or rise times exhibits a change in mean. The asymptotic null distributions of the test statistics are derived. The power of these four test statistics will be compared via Monte Carlo simulations for a few alternatives. The use of these test statistics is illustrated by application to data from the variable stars R Camelopardalis and R Cassiopeiae.**Full Text:**

**Authors:**Potgieter, Gert Diedericks Johannes**Date:**2012-08-27**Subjects:**Stars , Light , Optics , Wave mechanics**Type:**Mini-Dissertation**Identifier:**uj:3162 , http://hdl.handle.net/10210/6579**Description:**M.Sc. , The brightness of many stars vary over time. A plot of the brightness against time is known as a light curve. The fall and rise times of the light curve can be modelled by two cross-correlated white noise processes. We propose four statistics to test the hypothesis of a constant mean fall and rise time of the light curve against the alternative that at least one of the fall or rise times exhibits a change in mean. The asymptotic null distributions of the test statistics are derived. The power of these four test statistics will be compared via Monte Carlo simulations for a few alternatives. The use of these test statistics is illustrated by application to data from the variable stars R Camelopardalis and R Cassiopeiae.**Full Text:**

Statistical properties of sequential detonation systems

**Authors:**Winter, Theodor Daniël**Date:**2012-08-24**Subjects:**Blasting , Detonators , Mining engineering , Explosives , Statistical mechanics -- Mathematical models**Type:**Thesis**Identifier:**uj:3139 , http://hdl.handle.net/10210/6558**Description:**M.Sc. , At the very roots of this dissertation lies a commercial process with many as yet unexplored characteristics that will be thoroughly examined, using a rich variety of statistical methods and techniques. Broadly speaking, the main objective of this study involves the development of techniques to control the quality of advanced explosives detonators used in commercial mining operations. To accomplish this task, various statistical characteristics of this detonation process are described and examined in order to obtain a holistic understanding of the underlying process. The parameters of the process are introduced and estimates for unknowns are derived. Real-time quality control techniques based on these results are suggested. 1.2. The role of blasting in mining A major part of South Africa's economy is based on the mining of the rich mineral deposits that are to be found in the country. These mining operations are carried out both above ground (open-pit iron ore mines, for example) and below ground (gold, uranium and others). Open-pit mining, in particular, requires significant amounts of commercial blasting to dislodge the high volumes of material that have to be moved and processed. An average blasting block at Iscor's Sishen mine, for example, contains about 250 000 tons of material, although a world record was established in April 1981 when 7, 2 million tons of rock was broken during a single blast. The chemical quality of the final products is partly controlled by supplying the primary crusher at the mine with a suitable mixture of so-called run-of-mine ore. To determine which material from a specific blasting block may be sent to the plant, and to which waste dump the remaining material should be assigned, factors such as beneficiation properties of the raw material and the content of various by-products are considered. Samples are typically taken from alternate blast holes for every metre drilled. Each drill sample is divided into two parts by means of a riffler for a washed and unwashed sample. The washed samples are examined and the rock types noted. Subsequently, all the samples are grouped and analysed chemically and the densities of the different rock types are determined. The results are processed and those for the washed and unwashed samples correlated. The blasting blocks in the pit are demarcated by means of whitewash lines, according to the divisions on the blasting-block plans, and they are marked with signs to guide shovel operators. Primary drilling is performed by means of electrically-driven rotary drills. At the Sishen mine, 310 mm diameter blast holes are drilled in all rock types. The following table depicts typical drilling 2 patterns for various rock types: Rock type Pattern (m) Drill depth (m) Hard iron ore 2 x 8, 3 3, 0 Medium-hard iron ore 1 x 9, 3 2, 7 Quartzite 8,2 x 9,4 2, 5 Flagstone 8,2 x 9,4 5 Calcrete 8,1 x 9,3 0 Primary blasting is done at Sishen with Heavy Anfo, an explosive that is manufactured by mine personnel at the emulsion plant on site. The ingredients for the explosive blends are transported by pump trucks to the blasting blocks, where it is mixed and pumped down the blast holes. Good fragmentation of the blasted material is a prerequisite for high loading rates by the loading equipment. At Sishen and other similar mines, a blasting efficiency of 3, 2 tons of rock per kilogram of explosives used, is considered to be acceptable.**Full Text:**

**Authors:**Winter, Theodor Daniël**Date:**2012-08-24**Subjects:**Blasting , Detonators , Mining engineering , Explosives , Statistical mechanics -- Mathematical models**Type:**Thesis**Identifier:**uj:3139 , http://hdl.handle.net/10210/6558**Description:**M.Sc. , At the very roots of this dissertation lies a commercial process with many as yet unexplored characteristics that will be thoroughly examined, using a rich variety of statistical methods and techniques. Broadly speaking, the main objective of this study involves the development of techniques to control the quality of advanced explosives detonators used in commercial mining operations. To accomplish this task, various statistical characteristics of this detonation process are described and examined in order to obtain a holistic understanding of the underlying process. The parameters of the process are introduced and estimates for unknowns are derived. Real-time quality control techniques based on these results are suggested. 1.2. The role of blasting in mining A major part of South Africa's economy is based on the mining of the rich mineral deposits that are to be found in the country. These mining operations are carried out both above ground (open-pit iron ore mines, for example) and below ground (gold, uranium and others). Open-pit mining, in particular, requires significant amounts of commercial blasting to dislodge the high volumes of material that have to be moved and processed. An average blasting block at Iscor's Sishen mine, for example, contains about 250 000 tons of material, although a world record was established in April 1981 when 7, 2 million tons of rock was broken during a single blast. The chemical quality of the final products is partly controlled by supplying the primary crusher at the mine with a suitable mixture of so-called run-of-mine ore. To determine which material from a specific blasting block may be sent to the plant, and to which waste dump the remaining material should be assigned, factors such as beneficiation properties of the raw material and the content of various by-products are considered. Samples are typically taken from alternate blast holes for every metre drilled. Each drill sample is divided into two parts by means of a riffler for a washed and unwashed sample. The washed samples are examined and the rock types noted. Subsequently, all the samples are grouped and analysed chemically and the densities of the different rock types are determined. The results are processed and those for the washed and unwashed samples correlated. The blasting blocks in the pit are demarcated by means of whitewash lines, according to the divisions on the blasting-block plans, and they are marked with signs to guide shovel operators. Primary drilling is performed by means of electrically-driven rotary drills. At the Sishen mine, 310 mm diameter blast holes are drilled in all rock types. The following table depicts typical drilling 2 patterns for various rock types: Rock type Pattern (m) Drill depth (m) Hard iron ore 2 x 8, 3 3, 0 Medium-hard iron ore 1 x 9, 3 2, 7 Quartzite 8,2 x 9,4 2, 5 Flagstone 8,2 x 9,4 5 Calcrete 8,1 x 9,3 0 Primary blasting is done at Sishen with Heavy Anfo, an explosive that is manufactured by mine personnel at the emulsion plant on site. The ingredients for the explosive blends are transported by pump trucks to the blasting blocks, where it is mixed and pumped down the blast holes. Good fragmentation of the blasted material is a prerequisite for high loading rates by the loading equipment. At Sishen and other similar mines, a blasting efficiency of 3, 2 tons of rock per kilogram of explosives used, is considered to be acceptable.**Full Text:**

The use of copulas in risk management.

**Authors:**Stander, Yolanda Sophia**Date:**2012-08-15**Subjects:**Copulas (Mathematical statistics) , Dependence (Statistics) , Risk management**Type:**Mini-Dissertation**Identifier:**uj:9418 , http://hdl.handle.net/10210/5852**Description:**M.Sc. , In this dissertation we take a closer look at how copulas can be used to improve the risk measurement at a financial institution. The focus is on market risk in a trading environment. In practice risk numbers are calculated with very basic measures that are easy to explain to senior management and to traders. It is important that traders understand the risk measure as that helps them to understand the risk inherent in any deal and may assist them in deciding on the optimal hedge. The purpose of a hedge is to reduce the risk in a portfolio. As senior management is responsible for deciding on the optimal risk limits and risk appetite of the financial institution, it is important for them to understand what the risks are and how to measure these. The simplicity of the risk measures leads to certain inadequacies that can have very negative consequences for a financial institution. If the risk measure does not adequately capture the risk of a deal, the financial institution may suffer big losses when there are stress events in the market. Alternatively, when the risk measure overestimates the risk of a deal, too much economic capital is tied up in the deal. This inhibits the trader from adding more deals to a portfolio that may potentially lead to big profits. Economic capital is the capital that has to be held against positions to protect the financial institution if and when extreme market moves occur. In this dissertation the focus is on how copulas can be used to improve current risk measures. We focus on bivariate copulas. Bivariate copulas are easier to depict graphically than multivariate copulas with more than two dimensions. It is also easier to prove that the fitted bivariate copulae do adequately describe the underlying dependence structure between risk factors. Even though the focus is on the bivariate case, all methodologies can easily be extended to higher dimensions. In Chapter 1 copulas are defined and some basic copula properties are shown. We consider the definition of elliptical copulas and discuss some drawbacks to using them in a financial application. Some useful Archimedean copula properties are discussed and it is shown how to generate the copula function for n 2 dimensions. The various ways in which to estimate the parameters of a copula are also discussed as well as goodness-of-fit tests that are used to test whether the copula fits the underlying data adequately. Finally the chapter ends with an example that illustrates the theory. A back-test is done to establish whether the copula adequately describes the dependence structure over time. It is also shown how the fitted copula can be used to generate stress scenarios that are used as an alternative to historical scenarios when calculating a value-at-risk (VaR) number. In chapter 2 the properties of a dependence measure are discussed and it is argued that linear correlation does not conform to these desired properties. Rank correlation measures have some additional properties that make them more efficient than linear correlation measures in certain instances. We also consider their relationship to copulas. Finally it is shown how copulas can be used in practice to get another view on the dependence structure between risk factors. In risk measurement we are mainly concerned with extreme moves that market variables may show. In chapter 3 some of the techniques used in risk management are discussed as well as some of their shortcomings. The shortcomings are addressed by applying extreme value theory to calculate stress factors and using copulas to model the dependence structure between risk factors. The theory underlying bivariate extreme copulas is discussed and illustrated with a practical example.**Full Text:**

**Authors:**Stander, Yolanda Sophia**Date:**2012-08-15**Subjects:**Copulas (Mathematical statistics) , Dependence (Statistics) , Risk management**Type:**Mini-Dissertation**Identifier:**uj:9418 , http://hdl.handle.net/10210/5852**Description:**M.Sc. , In this dissertation we take a closer look at how copulas can be used to improve the risk measurement at a financial institution. The focus is on market risk in a trading environment. In practice risk numbers are calculated with very basic measures that are easy to explain to senior management and to traders. It is important that traders understand the risk measure as that helps them to understand the risk inherent in any deal and may assist them in deciding on the optimal hedge. The purpose of a hedge is to reduce the risk in a portfolio. As senior management is responsible for deciding on the optimal risk limits and risk appetite of the financial institution, it is important for them to understand what the risks are and how to measure these. The simplicity of the risk measures leads to certain inadequacies that can have very negative consequences for a financial institution. If the risk measure does not adequately capture the risk of a deal, the financial institution may suffer big losses when there are stress events in the market. Alternatively, when the risk measure overestimates the risk of a deal, too much economic capital is tied up in the deal. This inhibits the trader from adding more deals to a portfolio that may potentially lead to big profits. Economic capital is the capital that has to be held against positions to protect the financial institution if and when extreme market moves occur. In this dissertation the focus is on how copulas can be used to improve current risk measures. We focus on bivariate copulas. Bivariate copulas are easier to depict graphically than multivariate copulas with more than two dimensions. It is also easier to prove that the fitted bivariate copulae do adequately describe the underlying dependence structure between risk factors. Even though the focus is on the bivariate case, all methodologies can easily be extended to higher dimensions. In Chapter 1 copulas are defined and some basic copula properties are shown. We consider the definition of elliptical copulas and discuss some drawbacks to using them in a financial application. Some useful Archimedean copula properties are discussed and it is shown how to generate the copula function for n 2 dimensions. The various ways in which to estimate the parameters of a copula are also discussed as well as goodness-of-fit tests that are used to test whether the copula fits the underlying data adequately. Finally the chapter ends with an example that illustrates the theory. A back-test is done to establish whether the copula adequately describes the dependence structure over time. It is also shown how the fitted copula can be used to generate stress scenarios that are used as an alternative to historical scenarios when calculating a value-at-risk (VaR) number. In chapter 2 the properties of a dependence measure are discussed and it is argued that linear correlation does not conform to these desired properties. Rank correlation measures have some additional properties that make them more efficient than linear correlation measures in certain instances. We also consider their relationship to copulas. Finally it is shown how copulas can be used in practice to get another view on the dependence structure between risk factors. In risk measurement we are mainly concerned with extreme moves that market variables may show. In chapter 3 some of the techniques used in risk management are discussed as well as some of their shortcomings. The shortcomings are addressed by applying extreme value theory to calculate stress factors and using copulas to model the dependence structure between risk factors. The theory underlying bivariate extreme copulas is discussed and illustrated with a practical example.**Full Text:**

Frequency domain tests for the constancy of a mean

**Authors:**Shen, Yike**Date:**2012-08-28**Subjects:**Statistical hypothesis testing -- Asymptotic theory , Goodness-of-fit tests , Heteroscedasticity -- Testing**Type:**Thesis**Identifier:**uj:3361 , http://hdl.handle.net/10210/6761**Description:**D. Phil. , There have been two rather distinct approaches to the analysis of time series: the time domain approach and frequency domain approach. The former is exemplified by the work of Quenouille (1957), Durbin (1960), Box and Jenkins (1970) and Ljung and Box (1979). The principal names associated with the development of the latter approach are Slutsky (1929, 1934), Wiener (1930, 1949), Whittle (1953), Grenander (1951), Bartlett (1948, 1966) and Grenander and Rosenblatt (1957). The difference between these two methods is discussed in Wold (1963). In this thesis, we are concerned with a frequency domain approach. Consider a model of the "signal plus noise" form yt = g (2t — 1 2n ) + 77t t= 1,2,—. ,n (1.1) where g is a function on (0, 1) and Ti t is a white noise process. Our interest is primarily in testing the hypothesis that g is constant, that is, that it does not change over time. There is a vast literature related to this problem in the special case where g is a step function. In that case (1.1) specifies an abrupt change model. Such abrupt change models are treated extensively by Csorgo and Horvath (1997), where an exhaustive bibliography can also be found. The methods associated with the traditional abrupt change models are, almost without exception, time domain methods. The abrupt change model is in many respects too restrictive since it confines attention to signals g that are simple step functions. In practical applications the need has arisen for tests of constancy of the mean against a less precisely specified alternative. For instance, in the study of variables stars in astronomy (Lombard (1998a)) the appropriate alternative says something like: "g is non-constant but slowly varying and of unspecified functional form". To accommodate such alternatives within a time domain approach seems to very difficult, if at all possible. They can, however, be accommodated within a frequency domain approach quite easily, as shown by, for example, Lombard (1998a and 1998b). Tests of the constancy of g using the frequency domain characteristics of the observations have been investigated by a number of authors. Lombard (1988) proposed a test based on the maximum of squared Fourier cosine coefficients at the lowest frequency oscillations. Eubank and Hart (1992) proposed a test which is based on the maximum the averages of Fourier cosine coefficients. The essential idea underlying these tests is that regular variation in the time domain manifests itself entirely at low frequencies in the frequency domain. Consequently, when g is "high frequency" , that is consists entirely of oscillations at high frequencies, the tests of Lombard (1988) and of Eubank and Hart (1992) lose most of their power. The fundamental tool used in frequency domain analysis is the periodogram; see Chapter 2 below for the definition and basic properties of the latter. A new class of tests was suggested by Lombard (1998b) based on the weighted averages of periodogram ordinates. When 7i t in model (1.1) are i.i.d. random variables with zero mean and variance cr-2 , one form of the test statistic is T1r, = Etvk fiy (A0/0-2 - (1.2) k=1 where wk is a sequence of constants that decrease as k increases and m = [i]. The rationale for such tests is discussed in detail in Lombard (1998a and 1998b). The greater part of the present Thesis consists of an investigation of the asymptotic null distributions, and power, of such tests. It is also shown that such tests can be applied directly to other, seemingly unrelated problems. Three instances of the latter type of application that are investigated in detail are (i) frequency domain competitors of Bartlett's test for white noise, (ii) frequency domain-based tests of goodness-of-fit and (iii) frequency domain-based tests of heteroscedasticity in linear or non-linear regression. regression. The application of frequency domain methods to these problems are, to the best of our knowledge, new. Until now, most research has been restricted to the case where m in (1.1) are i.i.d. random variables. As far as the correlated data are concerned, the changepoint problem was investigated by, for instance, Picard (1985), Lombard and Hart (1994) and Bai (1994) using time domain methods. Kim and Hart (1998) proposed two test statistics derived from frequency domain considerations and that are modeled along the lines of the statistics considered by Eubank and Hart (1992) in the white noise case. An analogue of the type of test statistic given in (1.2) for use with correlated data was proposed, and used, by Lombard (1998a). The latter author does not, however, provide statements or proofs regarding the asymptotic properties of the proposed test.**Full Text:**

**Authors:**Shen, Yike**Date:**2012-08-28**Subjects:**Statistical hypothesis testing -- Asymptotic theory , Goodness-of-fit tests , Heteroscedasticity -- Testing**Type:**Thesis**Identifier:**uj:3361 , http://hdl.handle.net/10210/6761**Description:**D. Phil. , There have been two rather distinct approaches to the analysis of time series: the time domain approach and frequency domain approach. The former is exemplified by the work of Quenouille (1957), Durbin (1960), Box and Jenkins (1970) and Ljung and Box (1979). The principal names associated with the development of the latter approach are Slutsky (1929, 1934), Wiener (1930, 1949), Whittle (1953), Grenander (1951), Bartlett (1948, 1966) and Grenander and Rosenblatt (1957). The difference between these two methods is discussed in Wold (1963). In this thesis, we are concerned with a frequency domain approach. Consider a model of the "signal plus noise" form yt = g (2t — 1 2n ) + 77t t= 1,2,—. ,n (1.1) where g is a function on (0, 1) and Ti t is a white noise process. Our interest is primarily in testing the hypothesis that g is constant, that is, that it does not change over time. There is a vast literature related to this problem in the special case where g is a step function. In that case (1.1) specifies an abrupt change model. Such abrupt change models are treated extensively by Csorgo and Horvath (1997), where an exhaustive bibliography can also be found. The methods associated with the traditional abrupt change models are, almost without exception, time domain methods. The abrupt change model is in many respects too restrictive since it confines attention to signals g that are simple step functions. In practical applications the need has arisen for tests of constancy of the mean against a less precisely specified alternative. For instance, in the study of variables stars in astronomy (Lombard (1998a)) the appropriate alternative says something like: "g is non-constant but slowly varying and of unspecified functional form". To accommodate such alternatives within a time domain approach seems to very difficult, if at all possible. They can, however, be accommodated within a frequency domain approach quite easily, as shown by, for example, Lombard (1998a and 1998b). Tests of the constancy of g using the frequency domain characteristics of the observations have been investigated by a number of authors. Lombard (1988) proposed a test based on the maximum of squared Fourier cosine coefficients at the lowest frequency oscillations. Eubank and Hart (1992) proposed a test which is based on the maximum the averages of Fourier cosine coefficients. The essential idea underlying these tests is that regular variation in the time domain manifests itself entirely at low frequencies in the frequency domain. Consequently, when g is "high frequency" , that is consists entirely of oscillations at high frequencies, the tests of Lombard (1988) and of Eubank and Hart (1992) lose most of their power. The fundamental tool used in frequency domain analysis is the periodogram; see Chapter 2 below for the definition and basic properties of the latter. A new class of tests was suggested by Lombard (1998b) based on the weighted averages of periodogram ordinates. When 7i t in model (1.1) are i.i.d. random variables with zero mean and variance cr-2 , one form of the test statistic is T1r, = Etvk fiy (A0/0-2 - (1.2) k=1 where wk is a sequence of constants that decrease as k increases and m = [i]. The rationale for such tests is discussed in detail in Lombard (1998a and 1998b). The greater part of the present Thesis consists of an investigation of the asymptotic null distributions, and power, of such tests. It is also shown that such tests can be applied directly to other, seemingly unrelated problems. Three instances of the latter type of application that are investigated in detail are (i) frequency domain competitors of Bartlett's test for white noise, (ii) frequency domain-based tests of goodness-of-fit and (iii) frequency domain-based tests of heteroscedasticity in linear or non-linear regression. regression. The application of frequency domain methods to these problems are, to the best of our knowledge, new. Until now, most research has been restricted to the case where m in (1.1) are i.i.d. random variables. As far as the correlated data are concerned, the changepoint problem was investigated by, for instance, Picard (1985), Lombard and Hart (1994) and Bai (1994) using time domain methods. Kim and Hart (1998) proposed two test statistics derived from frequency domain considerations and that are modeled along the lines of the statistics considered by Eubank and Hart (1992) in the white noise case. An analogue of the type of test statistic given in (1.2) for use with correlated data was proposed, and used, by Lombard (1998a). The latter author does not, however, provide statements or proofs regarding the asymptotic properties of the proposed test.**Full Text:**

Estimation and testing in location-scale families of distributions

**Authors:**Potgieter, Cornelis Jacobus**Date:**2011-10-11T08:08:30Z**Subjects:**Parameter estimation , Statistical hypothesis testing , Distribution (Probability theory)**Type:**Thesis**Identifier:**uj:7244 , http://hdl.handle.net/10210/3898**Description:**D.Phil. , We consider two problems relating to location-scale families of distributions. Firstly, we consider methods of parameter estimation when two samples come from the same type of distribution, but possibly differ in terms of location and spread. Although there are methods of estimation that are asymptotically efficient, our interest is in fi nding methods which also have good small-sample properties. Secondly, we consider tests for the hypothesis that two samples come from the same location-scale family. Both these problems are addressed using methods based on empirical distribution functions and empirical characteristic functions.**Full Text:**

**Authors:**Potgieter, Cornelis Jacobus**Date:**2011-10-11T08:08:30Z**Subjects:**Parameter estimation , Statistical hypothesis testing , Distribution (Probability theory)**Type:**Thesis**Identifier:**uj:7244 , http://hdl.handle.net/10210/3898**Description:**D.Phil. , We consider two problems relating to location-scale families of distributions. Firstly, we consider methods of parameter estimation when two samples come from the same type of distribution, but possibly differ in terms of location and spread. Although there are methods of estimation that are asymptotically efficient, our interest is in fi nding methods which also have good small-sample properties. Secondly, we consider tests for the hypothesis that two samples come from the same location-scale family. Both these problems are addressed using methods based on empirical distribution functions and empirical characteristic functions.**Full Text:**

Applications of time series modelling to variable star astronomy

**Authors:**Koen, Marthinus Christoffel**Date:**2012-09-11**Subjects:**Time series analysis , Variable stars - Observations , Statistical astronomy**Type:**Thesis**Identifier:**uj:10017 , http://hdl.handle.net/10210/7408**Description:**D.Phil. , During the last few years the number of known variable stars which show periodic light level changes has grown by several tens of thousands. The aim of the research reported here was to extend the suite of statistical methods available for the analysis of periodic variable star time series. Solution techniques for five problems are discussed. The first is an automated method for detecting periodic variable stars from a database containing of the order of 100 000 time series of observations. Typically only 100-200 brightness measurements of each star were obtained, spread irregularly over an interval of about 3 years. The proposed method is based on a signal to noise ratio. Percentiles for the statistic are found by studying randomisations of a large number of the observed time series. It is shown that the percentiles depend strongly on the number of observations in a given dataset, and the dependence is calibrated empirically. The estimation of the frequency, amplitude and phase of a sinusoid from observations contaminated by correlated noise is the second problem considered. The study of the observational noise properties of nearly 200 real datasets of the relevant type is reported: noise can almost always be characterised as a random walk with superposed white noise. A scheme for obtaining weighted nonlinear least squares estimates of the parameters of interest, as well as standard errors of these estimates, is described. Simulation results are presented for both complete and incomplete data, and an application to real observations is also shown. In the third topic discussed it is assumed that contemporaneous measurements of the light in-tensity of a pulsating star is obtained in several colours. There is strong theoretical interest in a comparison of the amplitudes and phases of the variations in the different colours. A general scheme for calculating the covariance matrix of the estimated amplitude ratios and phase differences is described. The first step is to fit a time series model to the residuals after subtracting the best-fitting sinusoid from the observations. The residuals are then crosscorrelated to study the interdependence between the errors in the different colours. Once the multivariate time series structure can be modelled, the covariance matrix can be found by bootstrapping. An illustrative application is described in detail. The times between successive instances of maximum brightness, or the times between successive brightness minima, serve as estimates for the periods of the so-called "long period variables" (stars with pulsation periods of the order of months). The times between successive maxima (or minima) vary stochastically, and are also subject to measurement errors, which poses a problem for tests for systematic period changes — the topic of the fourth problem studied. A simple statistical model for the times between successive maxima, or minima, of such stars is used to calculate the auto-correlation properties of a new time series, which is non-stationary in its variance. The new series consists of an alternation of cycle lengths based on respectively the times between maxima, and those between minima of the light curve. Two different approaches to calculating the theoretical spectrum of the non-stationary time series, as required in the proposed statistical hypothesis test, are given. Illustrative applications complete the relevant chapter.**Full Text:**

**Authors:**Koen, Marthinus Christoffel**Date:**2012-09-11**Subjects:**Time series analysis , Variable stars - Observations , Statistical astronomy**Type:**Thesis**Identifier:**uj:10017 , http://hdl.handle.net/10210/7408**Description:**D.Phil. , During the last few years the number of known variable stars which show periodic light level changes has grown by several tens of thousands. The aim of the research reported here was to extend the suite of statistical methods available for the analysis of periodic variable star time series. Solution techniques for five problems are discussed. The first is an automated method for detecting periodic variable stars from a database containing of the order of 100 000 time series of observations. Typically only 100-200 brightness measurements of each star were obtained, spread irregularly over an interval of about 3 years. The proposed method is based on a signal to noise ratio. Percentiles for the statistic are found by studying randomisations of a large number of the observed time series. It is shown that the percentiles depend strongly on the number of observations in a given dataset, and the dependence is calibrated empirically. The estimation of the frequency, amplitude and phase of a sinusoid from observations contaminated by correlated noise is the second problem considered. The study of the observational noise properties of nearly 200 real datasets of the relevant type is reported: noise can almost always be characterised as a random walk with superposed white noise. A scheme for obtaining weighted nonlinear least squares estimates of the parameters of interest, as well as standard errors of these estimates, is described. Simulation results are presented for both complete and incomplete data, and an application to real observations is also shown. In the third topic discussed it is assumed that contemporaneous measurements of the light in-tensity of a pulsating star is obtained in several colours. There is strong theoretical interest in a comparison of the amplitudes and phases of the variations in the different colours. A general scheme for calculating the covariance matrix of the estimated amplitude ratios and phase differences is described. The first step is to fit a time series model to the residuals after subtracting the best-fitting sinusoid from the observations. The residuals are then crosscorrelated to study the interdependence between the errors in the different colours. Once the multivariate time series structure can be modelled, the covariance matrix can be found by bootstrapping. An illustrative application is described in detail. The times between successive instances of maximum brightness, or the times between successive brightness minima, serve as estimates for the periods of the so-called "long period variables" (stars with pulsation periods of the order of months). The times between successive maxima (or minima) vary stochastically, and are also subject to measurement errors, which poses a problem for tests for systematic period changes — the topic of the fourth problem studied. A simple statistical model for the times between successive maxima, or minima, of such stars is used to calculate the auto-correlation properties of a new time series, which is non-stationary in its variance. The new series consists of an alternation of cycle lengths based on respectively the times between maxima, and those between minima of the light curve. Two different approaches to calculating the theoretical spectrum of the non-stationary time series, as required in the proposed statistical hypothesis test, are given. Illustrative applications complete the relevant chapter.**Full Text:**

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