Smoothing Applications for Irregular Time Series with Measurement - - PowerPoint PPT Presentation

Smoothing Applications for Irregular Time Series with Measurement Errors

Jonathan Rathjens, Eva Becker, Arthur Kolbe, Katharina Olthoff, Michael Wilhelm, Katja Ickstadt, and Jürgen Hölzer

2 December 2016

1

Introduction

2

Data

3

Model Development Regression Kernel Smoothing Categorization

4

Results Regression Kernel Smoothing

5

Conclusions and Outlook

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Introduction

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Epidemiological Background

PFASs

per- and polyfluoroalkyl substances lead substances:

perfluorooctanoic acid (PFOA) perfluorooctane sulphonic acid (PFOS)

ubiquitous in industrial and household products persistent; accumulate in organisms ◮ internal exposure of general population

Importance of Drinking Water

food as most important source of human exposure to PFASs contaminated drinking water predominant ◮ surrogate marker for internal exposure

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Introduction

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Occasion

Contamination in North Rhine-Westphalia (NRW)

drinking water rivers Ruhr and Möhne affected by PFASs-polluted fertilizer prior to summer 2006 (very) high concentrations measured at water supply stations downstream motivated human biomonitoring studies

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Introduction

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Long-Term Objectives

modelling state-wide PFASs exposure both temporal and spatial find regions and time periods of increased exposure use as explanatory variable for spatio-temporal NRW birth data explore/infer possible dependencies

Current

PFASs measurements from water supply stations and areas: irregular sampled, non-stationary time series find realistic estimation of mean- and var-function

regression on time interpolation, smoothing extrapolation

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Introduction

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1

Introduction

2

Data

3

Model Development Regression Kernel Smoothing Categorization

4

Results Regression Kernel Smoothing

5

Conclusions and Outlook

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Data

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Drinking Water PFASs Data

provided by NRW state environmental agency LANUV drinking water samples from water supply stations and network

Spatial Structure

stations (data from ca. 250 of 650 stations) water supply areas (data from ca. 200 of 450 areas) complex assignment rivers

Temporal Structure

irregular time series (ca. one value per month or less) from summer 2006; partially ongoing

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Data

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Maximum PFOA 2006–2014 [ng/l]

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Data

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Characteristics

irregular sampling non-stationary different patterns need for extrapolation values < LoQ (varying) extremely high values (possible) change points

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Data

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Measurement Errors

additional variability depending on scale due to serial dilution in chemical analysis coefficient of variation (cv) ca. 20%

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Data

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1

Introduction

2

Data

3

Model Development Regression Kernel Smoothing Categorization

4

Results Regression Kernel Smoothing

5

Conclusions and Outlook

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Model Development

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Observations and Goal

measurements (xi)i=1,...,n, positive at times t1 ≤ . . . ≤ tn, unequally spaced estimate process X = f(t) (or distribution / posterior predictive) for arbitrary time t

Time-Dependent Regressions

simple regression, e.g. ln(x) = b0 + b1t + ǫ, too restrictive segmented according to change points (if known) P-splines

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Model Development: Regression

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Conjugate Γ-Γ-Model

Xt|βt ∼ Γ(α, βt) for arbitrary t fixed α representing measurement error:

cv = √

Var(X) E(X)

= √

α/β2 α/β

=

1 √α

e.g. cv = 0.2 ⇒ α = 25

prior βt ∼ Γ(θt, ηt) alternative to log N-Model (no transformation)

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Model Development: Kernel Smoothing

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„Weighted Posterior“

θt → θt + αn n

i=1 wi

ηt → ηt + n n

i=1 xiwi

Weights

wi ∈ [0, 1] with w := n

i=1 wi ∈ [0, 1]

from kernel, e.g. Gaussian: wi = fN(t,δ2)(ti) w ց 0: no informative data, retain prior w ր 1: data „almost everywhere“, usual Γ-Γ-update with ˜ xi := nwixi smoothing parameter δ depending on t-scale/resolution

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Model Development: Kernel Smoothing

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Prior Choices

empiral Bayes from whole sample vague, e.g. with prior predictive ∈ [0, 1000] informative:

high values prior to 2006 (for 2 to 4 years) afterwards decrease

sequential: adjacent time’s posterior

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Model Development: Kernel Smoothing

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Categorical Data

simplify xi’s to ordinal categories such as „not detected“, „low“, „increased“, „very high“ natural incorporation of values < LoQ magnitude less likely to be „false“ then value with measurement error useful as an epidemiological predictor ◮ model risk/rate of, e.g., increased or very high values for a period of time

Future Approaches

(cumulative) probit or logit P-splines simple case: binary regression use known change points

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Model Development: Categorization

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1

Introduction

2

Data

3

Model Development Regression Kernel Smoothing Categorization

4

Results Regression Kernel Smoothing

5

Conclusions and Outlook

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Results

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Simple Regression

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Results: Regression

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Spline Regression

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Results: Regression

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Empirical Bayes Prior

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Results: Kernel Smoothing

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Vague Prior with Large δ

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Results: Kernel Smoothing

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Informative Prior

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Results: Kernel Smoothing

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Sequential Prior with Uniform Kernel

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Results: Kernel Smoothing

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1

Introduction

2

Data

3

Model Development Regression Kernel Smoothing Categorization

4

Results Regression Kernel Smoothing

5

Conclusions and Outlook

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Conclusions and Outlook

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Findings

sufficient (local) fit to present data no useful extrapolation without specific prior knowledge difficult global modelling individual solution for each series preferable

Evaluation

non-parametric solutions to incorporate changes and nonlinear trends parametric regression useful if change points known kernel smoother able to include local prior information, if available

sequential prior weighting down important maxima too little variance for data ց 0

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Conclusions and Outlook

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Model Enhancement

estimate Γ-parameter α (globally, restricted) distinguish error- and process-related variability tune smoothing parameter δ

respect „data density“ in time locally adapted appropriate for chosen prediction interval (daily, monthly, . . . )

asymmetric weighting, e.g. f(t − a) < f(t + a) several series:

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Conclusions and Outlook

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Supply Stations and Areas

supply from the Ruhr

• ne station may supply several

areas

• ne area may be supplied by

several stations different periods observed measurements from stations and network:

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Conclusions and Outlook

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Network Samples

important for estimation of areas’ contamination verification of single station models unknown water source possibly mixture of stations’ waters, e.g.: X(t) = π1X1(t) + π2X2(t) + π3X3(t)

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Conclusions and Outlook

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Spatial Dependence

water from contaminated rivers, esp. the Ruhr

• ther spatial processes (e.g., for groundwater)

modelling X = f(t, s) two-dimensional Ruhr models (river × time) discrete space: supply areas (use, e.g., GMRF)

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Conclusions and Outlook

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Modelling Internal Exposure

extremely slow decrease after exposure very weak effects of random fluctuations What is important?

find times and values of exposure peaks (short-term) model the sum of subsequent exposures correctly (no loss by averaging, long-term) background exposure (equilibrium, long-term)

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Conclusions and Outlook

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Thanks to . . .

the NRW state environmental agency LANUV for providing water data Stiftung Mercator for funding our work all co-workers and participants

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Acknowledgements

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