Change detection in monitoring time series - Changes...

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1Challenge the future

Change detection in monitoring time series

Thom BogaardDelft University of Technology

Formose - Changes Workshop

Change detection in monitoring series

Content

What is a time series? What are we monitoring? What is a change? How can we detect the time series changes?

Example for streams/rivers Example for landslides

time series is a sequence of data points, measured typically at successive points in time spaced at uniform time intervals

Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data

What is a time series?

Observation interval = 5 daysObservation frequency = 6 times / month

Plot of hydrograph

0 72 144 216 288 360 432 504 576 648 720 792

Time interval (5 days)

Characteristics of time series

Mean: central tendency

h n1 = h i

Characteristics of time series

Variance: variation around mean

)h - h( 1-n

1 = s2

0 12 24 36 48 60

Time scale effect on time series

Time and spatial scale effect on discharge time series

0 10 20 30 40 50

Distance (km)

r(d) Daily Rainfall Monthly rainfall

Scale effect on rainfall time seriesExample of 9 rain gauges Luxembourg

Decompose a time series

Time series

Periodicity

Catastrophic event

Noise (random)

How to test periodicity?Serial correlation

Random

Autoregressive, Markov process

Periodicity

How to test periodicity?Serial correlation and confidence limits

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42

Time lag (15 days)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42

Time lag (15 days)

Lower limit Correlation coefficient Upper limit

Correlogram of seasonal time series

0 72 144 216 288 360 432 504 576 648 720 792

Time (5 days)

0 36 72 108 144 180

Time lag (5 dyas)

Lower limit Correlogram Upper limit

Floods: discharge, water level height, bed topography, …

What are we monitoring in natural hazards?

Landslide: displacement, groundwater level, precipitation

What is the difference between a cause and a trigger?

How to detect a change?

Methods

Visual inspection Double mass (residual mass) Statistics Physical modelling

… data quality and extremesHow to detect a change?

How to detect a change?… data quality and extremes

Patterns of movement in reactivated landslidesMassey ey al, 2013 Engineering Geology

Visual inspectionHow to detect a change?

Patterns of movement in reactivated landslidesMassey ey al, 2013 Engineering Geology

Visual inspectionHow to detect a change?

Double mass plot

Double mass analysis

100000

0 20000 40000 60000 80000

Cumulative Average Discharge (n-1 stations)C

Plot cumulative observation time series against another (averaged) cumulative time series

Double Mass curve

0 2000 4000 6000 8000Station 1

Double mass plot

Characteristics of Time SeriesStationarity

• Stationary: probability distribution doesn’t change with time

• First-order stationary: mean is a constant• Second-order stationary: mean is a constant and

covariance is only a function of time lag, not actual time• Non-stationary in the mean: presence of a trend or

periodicity

A stationary time series

0 12 24 36 48 60 72 84 96 108 120

Characteristics of Time Series

Non-stationary time series with a trend

0 12 24 36 48 60 72 84 96 108 120

Groundwater level Linear trend

Non-stationary time series with periodic changes

0 12 24 36 48 60 72 84 96 108 120

Groundwater level Seasonal trend

0 12 24 36 48 60 72 84 96 108 120

3738394041424344

Groundwater level Step trend

µ1 µ2

• Step trend

tn1>t121t + ) - ( + = h

Detection of a trend

• Hypothesis testH0 µ1 = µ2Ha µ1 µ2

t statistic

• t test resultGiven α(5%), find tα/2(n-2) from Student tableIf t > tα/2(n-2) accept Ha, step trend is significantIf t tα/2(n-2) accept H0, step trend is not significant

2)-t(n n/s2|x-x|=t

Detection of a step trend

t0.025(118) = 1.96

0 12 24 36 48 60 72 84 96 108 120

36373839404142434445

Groundwater level Step trend

9.89= 1201.1/*2

|40.0-42.0|=t

0.42h 601 = h i

1=i1 0.40h

601 = h i

1.1] )0.40 - h( + )0.42 - h( [ 2-120

1 = s 2i

Detection of a step trend

0 12 24 36 48 60 72 84 96 108 120

Detection of a linear trend

t10t + t + = htb + b = h 10t

• Hypothesis testH0 β1 = 0Ha β1 0

• t statistic

• t test resultGiven α(5%), find tα/2(n-2) from Student tableIf t > tα/2(n-2) accept Ha, linear trend is significantIf t tα/2(n-2) accept H0, linear trend is not significant

2)-t(n 1)-1)(n+n(n/s12

|b|=tl

Example of detecting a linear trend

0 12 24 36 48 60 72 84 96 108 120

05.0)t - (t

)t - )(th - h( = b

0.32tb - h = b 10

t0.025(118) = 1.96

0 12 24 36 48 60 72 84 96 108 120

18.97= 119*121*1201.0/*12

|-0.05|=t

0.1)tb - b - h( 2-n

1 = s2

Example of detecting a linear trend

Characteristics of a harmonic function

0 2 4 6 8 10 12 14 16 18 20 22 24

T ime (month)

Hramonic seriesA=2 amplitude

A0=0 base level

0= 0 initial phase T=1/f =12 period

)tfin(2s A + A = h 00t

Detection of a periodic trend

0 12 24 36 48 60 72 84 96 108 120Time interval (15 days)

) Measurements Harmonic se ries

Detection of a periodic trend

t2(sin0.53+)12

t2(cos0.86+)24

t2(sin1.68+)24

t2(cos2.23+49.97=ht

Fit of harmonic series with 2 harmonics

Shifts in the Mean

1. Student’s t-test2. Bayesian analysis 3. Mann–Whitney U-test4. Wilcoxon rank sum5. Pettitt test 6. Mann-Kendall test7. Lepage test8. Standard normal homogeneity test

9. Regression-based approach10. CUSUM test11. Oerlemans method12. Signal-to-noise ratio13. Intervention analysis14. Markov chain Monte Carlo15. Lanzante method

Shifts in the Variance1. Downton-Katz test

Shifts in the Spectrum1. Nikiforov method

Shifts in the System1. Principal component analysis2. Average standard deviates3. Fisher information4. Vector autoregressive method

Example: Min Tu – Assessment of the effects of climate variability and land use change on the hydrology of the Meuse river basin (2006)

Change detection in monitoring time series

Summary

What is a time series? What are we monitoring? What is a change?

• Step, linear, periodicity, etc

How can we detect the time series changes? • Visual, double mass, statistical, ….

Change detection in monitoring time series - Changes...

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