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Functional clustering methods and marital fertility modelling
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two parts.The first part considers further development of a model used for marital fertility, the Coale-Trussell's fertility model, which is based on age-specific fertility rates. A new model is suggested using individual fertility data and a waiting time after pregnancies. The model is named the waiting model and can be understood as an alternating renewal process with age-specific intensities. Due to the complicated form of the waiting model and the way data is presented, as given in the United Nation Demographic Year Book 1965, a normal approximation is suggested together with a normal approximation of the mean and variance of the number of births per summarized interval. A further refinement of the model was then introduced to allow for left truncated and censored individual data, summarized as table data. The waiting model suggested gives better understanding of marital fertility and by a simulation study it is shown that the waiting model outperforms the Coale-Trussell model when it comes to estimating the fertility intensity and to predict the mean and variance of the number of births for a population.

The second part of the thesis focus on developing functional clustering methods.The methods are motivated by and applied to varved (annually laminated) sediment data from lake Kassj\"on in northern Sweden. The rich but complex information (with respect to climate) in the varves, including the shapes of the seasonal patterns, the varying varve thickness, and the non-linear sediment accumulation rates makes it non-trivial to cluster the varves. Functional representations, smoothing and alignment are functional data tools used to make the seasonal patterns comparable.Functional clustering is used to group the seasonal patterns into different types, which can be associated with different weather conditions.

A new non-parametric functional clustering method is suggested, the Bagging Voronoi K-mediod Alignment algorithm, (BVKMA), which simultaneously clusters and aligns spatially dependent curves. BVKMA is used on the varved lake sediment, to infer on climate, defined as frequencies of different weather types, over longer time periods.

Furthermore, a functional model-based clustering method is proposed that clusters subjects for which both functional data and covariates are observed, allowing different covariance structures in the different clusters. The model extends a model-based functional clustering method proposed by James and Suger (2003). An EM algorithm is derived to estimate the parameters of the model.

Place, publisher, year, edition, pages
Umeå: Umeå universitet , 2017. , 24 p.
Keyword [en]
censoring, Coale-Trussell model, EM-algorithm, functional data analysis, functional clustering, marital fertility, normal approximation, Poisson process, varved lake sediments, warping
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:umu:diva-130734ISBN: 978-91-7601-669-5 (print)OAI: oai:DiVA.org:umu-130734DiVA: diva2:1069809
Public defence
2017-02-24, Hörsal 1, Lindellhallen, Umeå Universitet, Umeå, 10:15 (English)
Opponent
Supervisors
Available from: 2017-02-03 Created: 2017-02-01 Last updated: 2017-02-09Bibliographically approved
List of papers
1. Aspects of the Coale-Trussell Model
Open this publication in new window or tab >>Aspects of the Coale-Trussell Model
1995 (English)Report (Other academic)
Abstract [en]

Coale-Trussell model for marital fertility is investigated. The assumption that the data follow a Poisson model is invalidated by empirical evidence. The data are less spread than assumed, which indicates a point process which is undersdispersed relative to the Poisson model. By generalizing the Poisson model, by using a more realistic assumption about spacing between births, eg. allowing for a constant delay after each birth, we produce a better and more natural descripton of human reproduction.

Place, publisher, year, edition, pages
Umeå: Institute of Mathematical Statistics, Umeå University, 1995. 35 p.
Series
Research Report, ISSN 1400-2701 ; 1
Keyword
Coale-Trussell model, Poisson process, marital fertility, parameter estimation
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-130730 (URN)
Available from: 2017-01-30 Created: 2017-01-30 Last updated: 2017-02-01Bibliographically approved
2. Approximation of the waiting model
Open this publication in new window or tab >>Approximation of the waiting model
1995 (English)Report (Other academic)
Abstract [en]

In an attempt to estimate the level of family planning in a population, Coale-Trussell suggested an intensity model based on five year summarized data as given in the reports of the United Nations. Their model is denoted Coale-Trussell model. To make inference in the model, it was assumed that the pregnancy data in the model followed a Poisson process. In Arnqvist (research report 1, 1995, Mathematical Statistics, Umeå University), a modification of the Poisson assumption in the intensity model was suggested, introducing waiting time after the pregnancies. The resulting model was named the waiting model. The aim of this paper is to compare the Coale-Trussell model with the waiting model when data of the form given in UN World Fertility Surveys are used (Table 1). By using a normal approximation of the first two moments of the number of pregnancies, the asymptotic variance of the estimators of the interesting parameters is given. Simulation studies show that the Coale-Trussell model and the normal approximation model both approximate the lower intensities in the waiting model well. However, the Coale-Trussell model gives essentially biased estimates of the intensities for high birth intensities.

Place, publisher, year, edition, pages
Umeå: Institute of Mathematical Statistics, Umeå University, 1995. 28 p.
Series
Research report, ISSN 1400-2701 ; 2
Keyword
Coale-Trussell model, Poisson model, normal approximation
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-130731 (URN)
Available from: 2017-01-30 Created: 2017-01-30 Last updated: 2017-02-01Bibliographically approved
3. Allowing Left Truncated and Censored Fertility Data in the Normal Waiting Model
Open this publication in new window or tab >>Allowing Left Truncated and Censored Fertility Data in the Normal Waiting Model
1995 (English)Report (Other academic)
Abstract [en]

Models describing marital fertility are under consideration. In Arnqvist (Research Report 2, Mathematical Statistics, Umeå University), a normal approximation of the waiting model was introduced. In this report a modification of the normal approximation is suggested. This specification allows the data to be left truncated and censored, which gives the possibility to apply the normally approximated waiting model in datasets as from the United Nations World Fertility Services. The model is appropriate except for extremely high fertility intensities, when it gives rise to bias in the parameter estimations. In this case, therefore, a bootstrap method is suggested to estimate and correct the bias. This means that the normal approximated waiting model is a good competitor to the well known Poisson or Coale-Trussell model. It also uses an understandable fertility specification.

Place, publisher, year, edition, pages
Umeå: Institute of Mathematical Statistics, Umeå University, 1995. 21 p.
Series
Research Report, ISSN 1400-2701 ; 5
Keyword
Coale-Trussell model, Poisson model, Waiting model, Normal approximated waiting model
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-130733 (URN)
Available from: 2017-01-30 Created: 2017-01-30 Last updated: 2017-02-01Bibliographically approved
4. Climatic influence on the inter-annual variability of late-Holocene minerogenic sediment supply in a boreal forest catchment
Open this publication in new window or tab >>Climatic influence on the inter-annual variability of late-Holocene minerogenic sediment supply in a boreal forest catchment
Show others...
2010 (English)In: Earth Surface Processes and Landforms, ISSN 0197-9337, E-ISSN 1096-9837, Vol. 35, no 4, 390-398 p.Article in journal (Refereed) Published
Abstract [en]

Processes controlling sediment yield vary over a range of timescales, although most process-based observations are extremely short. Lake sediments, however, can be used to extend the observational timescale and are particularly useful when annually laminated (varved) sediment is present. The sediment record at Kassjön (N. Sweden) consists of ∼6400 varves, each 0·5–1 mm thick. Image analysis was used to determine grey-scale variation and varve thickness from which annual minerogenic accumulation rate (MinAR) (mg cm−2 year−1) was inferred for the period 4486 BC – AD 1900. MinAR varies on annual to centennial scales and mainly reflects channel bank erosion by the inflow streams. The mineral input reflects the intensity of the spring run-off, which is dependent on the amount of snow accumulated during the winter, and hence MinAR is a long-term record of variability in past winter climate; other factors will be a variable response to catchment uplift, vegetation succession and pedogenesis. A major shift from low to high MinAR occurred ∼250 BC, and peaks occurred around AD 250, 600, 1000, 1350 and 1650. Wavelet power spectrum analysis (confirmed by Fourier analyses) indicated significantly different periodicities throughout the period 4000 BC – AD 1700, including 275 years for the period 4000 BC – 2900 BC, 567 years for the period 2901 BC – 1201 BC, and 350 and 725 years for the period 1200 BC – AD 1700. The long-term, centennial scale variability (∼350 years) may reflect solar forcing (cf the 385-year peak in tree-ring calibrated 14C activity) but interestingly, there is no obvious link to high frequency forcing, such as the North Atlantic Oscillation. The high resolution component of the record highlights the relevance of varved lake sediment records for understanding erosion dynamics in undisturbed forested catchments and their link to long-term climate dynamics and future climate change. 

Place, publisher, year, edition, pages
John Wiley & Sons, 2010
Keyword
varves, image analysis, climate change, snow melt, temperature, minerogenic matter accumulation rate, sediment supply, stream erosion
National Category
Environmental Sciences
Identifiers
urn:nbn:se:umu:diva-31007 (URN)10.1002/esp.1933 (DOI)000276677100002 ()
Available from: 2010-01-26 Created: 2010-01-26 Last updated: 2017-01-30Bibliographically approved
5. Functional clustering of varved lake sediment to reconstruct past seasonal climate
Open this publication in new window or tab >>Functional clustering of varved lake sediment to reconstruct past seasonal climate
2016 (English)In: Environmental and Ecological Statistics, ISSN 1352-8505, E-ISSN 1573-3009, Vol. 23, no 4, 513-529 p.Article in journal (Refereed) Published
Abstract [en]

Annually laminated (varved) lake sediments constitutes excellent environmental archives, and have the potential to play an important role for understanding past seasonal climate with their inherent annual time resolution and within-year seasonal patterns. We propose to use functional data analysis methods to extract the relevant information with respect to climate reconstruction from the rich but complex information in the varves, including the shapes of the seasonal patterns, the varying varve thickness, and the non-linear sediment accumulation rates. In particular we analyze varved sediment from lake Kassjon in northern Sweden, covering the past 6400 years. The properties of each varve reflect to a large extent weather conditions and internal biological processes in the lake the year that the varve was deposited. Functional clustering is used to group the seasonal patterns into different types, that can be associated with different weather conditions. The seasonal patterns were described by penalized splines and clustered by the k-means algorithm, after alignment. The observed (within-year) variability in the data was used to determine the degree of smoothing for the penalized spline approximations. The resulting clusters and their time dynamics show great potential for seasonal climate interpretation, in particular for winter climate changes.

Place, publisher, year, edition, pages
Springer, 2016
Keyword
Climate, Clustering, Curve registration, Functional data analysis, Penalized least squares, Varved lake diment
National Category
Probability Theory and Statistics Environmental Sciences
Identifiers
urn:nbn:se:umu:diva-130095 (URN)10.1007/s10651-016-0351-1 (DOI)000387424200002 ()
Available from: 2017-01-13 Created: 2017-01-11 Last updated: 2017-01-30Bibliographically approved
6. Clustering misaligned dependent curves applied to varved lake sediment for climate reconstruction
Open this publication in new window or tab >>Clustering misaligned dependent curves applied to varved lake sediment for climate reconstruction
Show others...
2017 (English)In: Stochastic environmental research and risk assessment (Print), ISSN 1436-3240, E-ISSN 1436-3259, Vol. 31, no 1, 71-85 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we introduce a novel functional clustering method, the Bagging Voronoi K-Medoid Aligment (BVKMA) algorithm, which simultaneously clusters and aligns spatially dependent curves. It is a nonparametric statistical method that does not rely on distributional or dependency structure assumptions. The method is motivated by and applied to varved (annually laminated) sediment data from lake Kassjön in northern Sweden, aiming to infer on past environmental and climate changes. The resulting clusters and their time dynamics show great potential for seasonal climate interpretation, in particular for winter climate changes.

Place, publisher, year, edition, pages
Springer, 2017
Keyword
Functional data, Clustering, Dependence, Misalignment, Sediment data
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-130727 (URN)10.1007/s00477-016-1287-6 (DOI)000394278600006 ()
Available from: 2017-01-30 Created: 2017-01-30 Last updated: 2017-04-04Bibliographically approved
7. Model based functional clustering of varved lake sediments
Open this publication in new window or tab >>Model based functional clustering of varved lake sediments
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Climate and environmental changes are today widely discussed, and in particular the impact of human activity. To understand variations in past climate over longer time periods, historical documents, year rings from trees, ice cores from glaciers as well as lake and sea sediments are being used.In this paper we introduce a model based functional cluster analysis, giving us possibility to use both the functional form and covariates in our analysis. It also allow us to model the dependency of the chosen basis coefficients and the covariates. We also allow for different covariance structure within each cluster and give suggestions on how to determine how many clusters to use.In particular we analyze varved sediment from lake Kassjön (N Sweden) which cover more than 6400 years.

Keyword
Model based functional clustering, warping, annually lake sediment data
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-130767 (URN)
Available from: 2017-01-31 Created: 2017-01-31 Last updated: 2017-01-31

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