Professor Dimitris Kugiumtzis, Aristotle University of Thessaloniki, Greece, presented this workshop on nonlinear analysis of time series as part of the Summer School on Modern Statisitical Analysis and Computational Methods hosted by the Social Sciences Compuing Hub at the Whitaker Institute, NUI Galway on 17th-19th June 2013.
2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climat...NUI Galway
Professor Dimitris Kugiumtzis, Aristotle University of Thessaloniki, Greece, presented this workshop on time series analysis as part of the Summer School on Modern Statistical Analysis and Computational Methods hosted by the Social Sciences Computing Hub at the Whitaker Institute, NUI Galway on 17th-19th June 2013.
2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climat...NUI Galway
Professor Dimitris Kugiumtzis, Aristotle University of Thessaloniki, Greece, presented this workshop on linear stochastic processes as part of the Summer School on Modern Statistical Analysis and Computational Methods hosted by the Social Sciences Computing Hub at the Whitaker Institute, NUI Galway on 17th-19th June 2013.
state space representation,State Space Model Controllability and Observabilit...Waqas Afzal
State Variables of a Dynamical System
State Variable Equation
Why State space approach
Block Diagram Representation Of State Space Model
Controllability and Observability
Derive Transfer Function from State Space Equation
Time Response and State Transition Matrix
Eigen Value
2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climat...NUI Galway
Professor Dimitris Kugiumtzis, Aristotle University of Thessaloniki, Greece, presented this workshop on time series analysis as part of the Summer School on Modern Statistical Analysis and Computational Methods hosted by the Social Sciences Computing Hub at the Whitaker Institute, NUI Galway on 17th-19th June 2013.
2013.06.17 Time Series Analysis Workshop ..Applications in Physiology, Climat...NUI Galway
Professor Dimitris Kugiumtzis, Aristotle University of Thessaloniki, Greece, presented this workshop on linear stochastic processes as part of the Summer School on Modern Statistical Analysis and Computational Methods hosted by the Social Sciences Computing Hub at the Whitaker Institute, NUI Galway on 17th-19th June 2013.
state space representation,State Space Model Controllability and Observabilit...Waqas Afzal
State Variables of a Dynamical System
State Variable Equation
Why State space approach
Block Diagram Representation Of State Space Model
Controllability and Observability
Derive Transfer Function from State Space Equation
Time Response and State Transition Matrix
Eigen Value
Representation of signals & Operation on signals
(Time Reversal, Time Shifting , Time Scaling, Amplitude scaling, Signal addition, Signal Multiplication)
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
Computational Motor Control: Optimal Control for Deterministic Systems (JAIST...hirokazutanaka
This is lecure 2 note for JAIST summer school on computational motor control (Hirokazu Tanaka & Hiroyuki Kambara). Lecture video: https://www.youtube.com/watch?v=lNH1q4y1m-U
La introducción de la incertidumbre en modelos epidemiológicos es un área de incipiente actividad en la actualidad. En la mayor parte de los enfoques adoptados se asume un comportamiento gaussiano en la formulación de dichos modelos a través de la perturbación de los parámetros vía el proceso de Wiener o movimiento browiniano u otro proceso discretizado equivalente.
En esta conferencia se expone un método alternativo de introducir la incertidumbre en modelos de tipo epidemiológico que permite considerar patrones no necesariamente normales o gaussianos. Con el enfoque adoptado se determinará en contextos epidemiológicos que tienen un gran número de aplicaciones, la primera función de densidad de probabilidad del proceso estocástico solución. Esto permite la determinación exacta de la respuesta media y su variabilidad, así como la construcción de predicciones probabilísticas con intervalos de confianza sin necesidad de recurrir a aproximaciones asintóticas, a veces de difícil legitimación. El enfoque adoptado también permite determinar la distribución probabilística de parámetros que tienen gran importancia para los epidemiólogos, incluyendo la distribución del tiempo hasta que un cierto número de infectados permanecen en la población, lo cual, por ejemplo, permite tener información probabilística para declarar el estado de epidemia o pandemia de una determinada enfermedad. Finalmente, se expondrá algunos de los retos computacionales inmediatos a los que se enfrenta la técnica expuesta.
Forecasting solid waste generation in Juba Town, South Sudan using Artificial...Premier Publishers
Prediction of solid waste generation is critical for any long term sustainable waste management, especially of a fast-growing municipality. Lack of, or inaccurate solid waste generation records poses unparalleled challenges in developing cohesive and workable waste management strategies for any concerned authorities, as this is influenced by several interlinked demo-graphic, economic, and socio-cultural factors. The objective of this study was to compare two models in forecasting of MSW generation and how this would be built into an effective MSW management program. Two models, the Autoregressive Moving Average (ARMA 1,1) and the Artificial Neural Networks (ANNs) were tested for their ability to predict weekly waste generation of 14 households in Juba Town, Central Equatoria State (CES), South Sudan. Results showed that both the artificial intelligence model ANNs and the traditional ARMA model had good prediction performances; where for ANNs the RMSE, MAPE and r² were 0.080, 10.64%, 0.238 respectively, whereas for ARMA the RMSE, MAPE and r² were 0.102, 6.98% and 0.274 respectively. Both models showed no significant differences and could be therefore be used for Solid Waste (SW) forecasting. Based on the results, the weekly SW generated 52 weeks later (end of year) had reached 0.365 kg/capita indicating a 18.2% rise from 0.3 kg/capita at the start of the study. Under the current consumption rate, the weekly SW per capita in Juba Town is expected to reach 0.596 kg by 2020.
Representation of signals & Operation on signals
(Time Reversal, Time Shifting , Time Scaling, Amplitude scaling, Signal addition, Signal Multiplication)
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
Computational Motor Control: Optimal Control for Deterministic Systems (JAIST...hirokazutanaka
This is lecure 2 note for JAIST summer school on computational motor control (Hirokazu Tanaka & Hiroyuki Kambara). Lecture video: https://www.youtube.com/watch?v=lNH1q4y1m-U
La introducción de la incertidumbre en modelos epidemiológicos es un área de incipiente actividad en la actualidad. En la mayor parte de los enfoques adoptados se asume un comportamiento gaussiano en la formulación de dichos modelos a través de la perturbación de los parámetros vía el proceso de Wiener o movimiento browiniano u otro proceso discretizado equivalente.
En esta conferencia se expone un método alternativo de introducir la incertidumbre en modelos de tipo epidemiológico que permite considerar patrones no necesariamente normales o gaussianos. Con el enfoque adoptado se determinará en contextos epidemiológicos que tienen un gran número de aplicaciones, la primera función de densidad de probabilidad del proceso estocástico solución. Esto permite la determinación exacta de la respuesta media y su variabilidad, así como la construcción de predicciones probabilísticas con intervalos de confianza sin necesidad de recurrir a aproximaciones asintóticas, a veces de difícil legitimación. El enfoque adoptado también permite determinar la distribución probabilística de parámetros que tienen gran importancia para los epidemiólogos, incluyendo la distribución del tiempo hasta que un cierto número de infectados permanecen en la población, lo cual, por ejemplo, permite tener información probabilística para declarar el estado de epidemia o pandemia de una determinada enfermedad. Finalmente, se expondrá algunos de los retos computacionales inmediatos a los que se enfrenta la técnica expuesta.
Forecasting solid waste generation in Juba Town, South Sudan using Artificial...Premier Publishers
Prediction of solid waste generation is critical for any long term sustainable waste management, especially of a fast-growing municipality. Lack of, or inaccurate solid waste generation records poses unparalleled challenges in developing cohesive and workable waste management strategies for any concerned authorities, as this is influenced by several interlinked demo-graphic, economic, and socio-cultural factors. The objective of this study was to compare two models in forecasting of MSW generation and how this would be built into an effective MSW management program. Two models, the Autoregressive Moving Average (ARMA 1,1) and the Artificial Neural Networks (ANNs) were tested for their ability to predict weekly waste generation of 14 households in Juba Town, Central Equatoria State (CES), South Sudan. Results showed that both the artificial intelligence model ANNs and the traditional ARMA model had good prediction performances; where for ANNs the RMSE, MAPE and r² were 0.080, 10.64%, 0.238 respectively, whereas for ARMA the RMSE, MAPE and r² were 0.102, 6.98% and 0.274 respectively. Both models showed no significant differences and could be therefore be used for Solid Waste (SW) forecasting. Based on the results, the weekly SW generated 52 weeks later (end of year) had reached 0.365 kg/capita indicating a 18.2% rise from 0.3 kg/capita at the start of the study. Under the current consumption rate, the weekly SW per capita in Juba Town is expected to reach 0.596 kg by 2020.
2012.09.27 Lessons Learned from Doing Qualitative ResearchNUI Galway
Dr. Professor Richard A. Krueger, University of Minnesota, USA and Dr. Mary Anne Casey, Consultant in Designing Research, USA presented this seminar "Lessons Learned from Doing Qualitative Research" at the Whitaker Institute on 27th September 2012.
2017.03.09 collaboration is key to thriving in the 21st centuryNUI Galway
Dr Bettina von Stamm, Innovation Leadership Forum, presented this masterclass entitled "Thriving in the 21st Century: Collaboration is Key" as part of the All-Island Innovation Programme at NUI Galway on the 9th of March 2017.
2017.02.08 The Darkside of Enterprise Social MediaNUI Galway
Dr Eoin Whelan, from the Agile & Open Innovation research cluster, presented this seminar entitled "The Darkside of Enterprise Social Media" as part of the Whitaker Institute's Ideas Forum seminar series on 8th February 2017.
2017.03.09 innovation and why it matters more in the 21st century than ever b...NUI Galway
Dr Bettina von Stamm, Innovation Leadership Forum, presented this masterclass entitled "Innovation: Why It Matters More In the 21st Century Than Ever Before" as part of the All-Island Innovation Programme at NUI Galway on the 9th of March 2017.
2012.09.18 exploring the glocalization of activism and empowermentNUI Galway
Dr. Peter Bloom, College of Business, Economics and Law, Swansea University, UK presented this seminar "Beyond the Laws of Gravity: Exploring the Glocalization of Activism and Empowerment" as part of the Visiting Fellows Seminar Series at the Whitaker Institute on 18th September 2012.
Dr Alma McCarthy, Discipline of Management, gave this workshop on how to manage the PhD journey at the 2017 Whitaker Institute PhD Forum on the 24th May 2017 at NUI Galway.
In these two lectures, we’re looking at basic discrete time representations of linear, time invariant plants and models and seeing how their parameters can be estimated using the normal equations.
The key example is the first order, linear, stable RC electrical circuit which we met last week, and which has an exponential response.
Vincenzo MacCarrone, Explaining the trajectory of collective bargaining in Ir...NUI Galway
Vincenzo MacCarrone, UCD, Explaining the trajectory of collective bargaining in Ireland: 2000-2017 presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Tom Turner, Tipping the scales for labour in Ireland? NUI Galway
Dr Tom Turner, University of Limerick, Tipping the scales for labour in Ireland? Collective bargaining and the industrial relations amendment) act 2015 presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Tom McDonnell, Medium-term trends in the Irish labour market and possibilitie...NUI Galway
Dr Tom McDonnell, NERI, Medium-term trends in the Irish labour market and possibilities for reform presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Stephen Byrne, A non-employment index for IrelandNUI Galway
Stephen Byrne, Central Bank of Ireland, A non-employment index for Ireland presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Sorcha Foster, The risk of automation of work in IrelandNUI Galway
Sorcha Foster, Oxford University, The risk of automation of work in Ireland – both sides of the border presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Sinead Pembroke, Living with uncertainty: The social implications of precario...NUI Galway
Dr Sinéad Pembroke, TASC, Living with uncertainty: The social implications of precarious work presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Paul MacFlynn, A low skills equilibrium in Northern IrelandNUI Galway
Paul Mac Flynn, NERI, A low skills equilibrium in Northern Ireland presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Nuala Whelan, The role of labour market activation in building a healthy work...NUI Galway
Dr Nuala Whelan, Maynooth University & Ballymun Job Club, The role of labour market activation in building a healthy workforce: Enhancing well-being for the long-term unemployed through positive psychological interventions presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Michéal Collins, and Dr Michelle Maher, Auto enrolmentNUI Galway
Dr Michéal Collins, UCD and Dr Michelle Maher, Maynooth University, Auto enrolment: into what, for whom and how much? presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Michael Taft, SIPTU, A new enterprise model: The long march through the market economy presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Luke Rehill, Patterns of firm-level productivity in IrelandNUI Galway
Luke Rehill, Department of Finance, Patterns of firm-level productivity in Ireland presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Lucy Pyne, Evidence from the Social Inclusion and Community Activation ProgrammeNUI Galway
Ms Lucy Pyne, Pobal, Evidence from the Social Inclusion and Community Activation Programme (SICAP) presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Lisa Wilson, The gendered nature of job quality and job insecurityNUI Galway
Dr Lisa Wilson, NERI, The gendered nature of job quality and job insecurity presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Karina Doorley, axation, labour force participation and gender equality in Ir...NUI Galway
Dr Karina Doorley, ESRI, Taxation, labour force participation and gender equality in Ireland presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Jason Loughrey, Household income volatility in IrelandNUI Galway
Dr Jason Loughrey, Teagasc, Household income volatility in Ireland presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Ivan Privalko, What do Workers get from Mobility?NUI Galway
Ivan Privalko, Dublin City Council, What do Workers get from Mobility? presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Helen Johnston, Labour market transitions: barriers and enablersNUI Galway
Dr Helen Johnston, NESC, Labour market transitions: barriers and enablers presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Gail Irvine, Carnegie UK Trust, Fulfilling work in Ireland presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Frank Walsh, Assessing competing explanations for the decline in trade union ...NUI Galway
Dr Frank Walsh, UCD, Assessing competing explanations for the decline in trade union density in Ireland presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
Eamon Murphy, An overview of labour market participation in Ireland over the ...NUI Galway
Eamon Murphy, Social Justice Ireland, An overview of labour market participation in Ireland over the last two decades presented at the 6th Annual NERI Labour Market Conference in association with the Whitaker Institute, NUI Galway, 22nd May, 2018.
As Europe's leading economic powerhouse and the fourth-largest hashtag#economy globally, Germany stands at the forefront of innovation and industrial might. Renowned for its precision engineering and high-tech sectors, Germany's economic structure is heavily supported by a robust service industry, accounting for approximately 68% of its GDP. This economic clout and strategic geopolitical stance position Germany as a focal point in the global cyber threat landscape.
In the face of escalating global tensions, particularly those emanating from geopolitical disputes with nations like hashtag#Russia and hashtag#China, hashtag#Germany has witnessed a significant uptick in targeted cyber operations. Our analysis indicates a marked increase in hashtag#cyberattack sophistication aimed at critical infrastructure and key industrial sectors. These attacks range from ransomware campaigns to hashtag#AdvancedPersistentThreats (hashtag#APTs), threatening national security and business integrity.
🔑 Key findings include:
🔍 Increased frequency and complexity of cyber threats.
🔍 Escalation of state-sponsored and criminally motivated cyber operations.
🔍 Active dark web exchanges of malicious tools and tactics.
Our comprehensive report delves into these challenges, using a blend of open-source and proprietary data collection techniques. By monitoring activity on critical networks and analyzing attack patterns, our team provides a detailed overview of the threats facing German entities.
This report aims to equip stakeholders across public and private sectors with the knowledge to enhance their defensive strategies, reduce exposure to cyber risks, and reinforce Germany's resilience against cyber threats.
Chatty Kathy - UNC Bootcamp Final Project Presentation - Final Version - 5.23...John Andrews
SlideShare Description for "Chatty Kathy - UNC Bootcamp Final Project Presentation"
Title: Chatty Kathy: Enhancing Physical Activity Among Older Adults
Description:
Discover how Chatty Kathy, an innovative project developed at the UNC Bootcamp, aims to tackle the challenge of low physical activity among older adults. Our AI-driven solution uses peer interaction to boost and sustain exercise levels, significantly improving health outcomes. This presentation covers our problem statement, the rationale behind Chatty Kathy, synthetic data and persona creation, model performance metrics, a visual demonstration of the project, and potential future developments. Join us for an insightful Q&A session to explore the potential of this groundbreaking project.
Project Team: Jay Requarth, Jana Avery, John Andrews, Dr. Dick Davis II, Nee Buntoum, Nam Yeongjin & Mat Nicholas
2013.06.18 Time Series Analysis Workshop ..Applications in Physiology, Climate Change and Finance, part 3
1. Nonlinear analysis of time series
ARMA(p,q) model qtqttptptt zzzxxx 1111
Linear analysis / linear models
Advantages:
1. Simple
2. Gaussian process, established
theory for stochastic processes
and statistical inference
3. Useful in applications
Shortcomings:
1. Cannot explain irregular patterns
in the time series
- data (distribution) asymmetry
- time irreversibility
- «bursts»
2. Deterministic part:
- stable fixed point system
- unstable system
- periodic system
autocorrelation AR model
description of irregular
patterns
explanation / detection of complex
deterministic patterns
Time series, Part 3
Nonlinear analysis of time series
2.
3. ),,,,( 21 tptttt XXXfX A general
nonlinear model
tptttt XXXfX ),,,( 21
additive
noise
p
ptttt
'XXX ,,, 211 X p
f :
f ?
4. tptpttt XXXX 2211
Linear AR
model
Generalizations / extensions of the ΑR model
p ,,, 21
constant (linear ΑR)
random coefficients
- RCA
- BL
constant (linear ΑR, ARMA)
function of Xt
- ARCH
- GARCH
piecewise models
- SETAR
- Markovian
)1()1(
2
)1(
1 ,,, p
)2()2(
2
)2(
1 ,,, p
)()(
2
)(
1 ,,, l
p
ll
5. Self-excited threshold autoregressive models (SETAR)
ll rrrr ,,,, 110
lrrr 10
lRRR 21
lirrR iii ,,1],,( 1
p
Partition of
selection of a lag d,
partition of for dtX
t
j
pt
j
pt
j
t
j
t XXXX )()(
2
)(
21
)(
1
jdt RX
SETAR
when
7. AR models with probabilistic selection of threshold
Exponential autoregressive models (EAR)
tt
j
t
j
t XXX 2
)(
21
)(
1
1με2
με1
j
tt
j
t
j
t XXX 2
)(
21
)(
1
AR models with periodic coefficients
12όταν2
2όταν1
kt
kt
j
1
)1(
1 0)1(
2 0)2(
1 2
)2(
2
Example
8. Markov chain driven AR models
ljJt ,,2,1
The selection of the threshold
is determined by a Markov chain
)|( 1 iJjJP tt
Transition matrix
Example
tt
J
t XX t
1
)( 9.0)1(
9.0)2(
8.02.0
9.01.0
)|( 1 iJjJP tt =
9. Piecewise polynomial models
tptttt XXXfX ),,,( 21
1 2( , , , )t m t t t p tX p X X X
polynomial of
order p and
degree m
Example
2
1 1 1 1(1 )t t t t tX aX X aX aX logistic map1a
aa /)1( Two fixed points: 0 and
Fractional autoregressive models
tq
j
j
tj
p
j
j
tj
t
Xbb
Xaa
X
1
10
1
10
10 qp
0pa
0qb
Example
Fraction of two polynomials
10. random coefficients autoregressive models (RCA)
1 ttt XX AR(1) with multiplicative errors
p
i
titiit XtBbX
1
)( RCA
ib constant
)(tBb iii
)(,),(),( 21 tBtBtB p
independent of
t
tXrandom with mean 0
Example
titit XtBX )(1.0 )9.0,0(~ 2
tB
11. Bilinear models (BL)
BL of order 1: ttttt XbaXX 11
p
i
titiit XtAaX
1
)(
s
k
ktjki btA
1
)(
)(tAa iii coefficients
ts XXts const, tss ,- If linear w.r.t.
“Bilinear” because:
ts Xts const, tsXs ,- If linear w.r.t.
12. AR models with conditional heteroscedasticity
tX ~ ARCH ~ BL 2
tX
ARCH ttt VX
22
11 ptptt XXV 0
0i
Model of multiplicative noise
),0(~ 2
t
GARCH
q
i
iti
p
i
itit VXV
11
2
0i
ttt VX
0
0i
13. Analysis with nonlinear models
1. Model selection
2. Parameter estimation
- maximum likelihood method
- method of ordinary least squares
3. Diagnostic checking
uncorrelated
following normal distribution
rgm m 2)(ˆ|ln2)(AIC xθx
Μ candidate models, m = 1,...,M
errors (rediduals):
14. Real world time series
mechanics
physiology
geophysics economics
15. Nonlinear time series analysis and dynamical systems
Time series 1 2, , , nx x x
Assumption:
: trajectory of the dynamical systemd
ts
0s : state vector at time 0
dd
: t
f system function
t : continuous or discrete time
For time series we assume underlying systems to be dissipative
Trajectory in
d
attractor
d
:h observation function
( )t tx h sobservation :
0( )t
t s f sNonlinear dynamical system
16. Attractor:
● stable fixed (equilibrium) point
● finite set of equilibrium points
● limit cycle
● torus
● strange attractor
self similarity - fractals
chaossensitivity to initial conditions
can be derived by
a linear system
cannot be derived by
a linear system
17. Nonlinear dynamical systems, maps (discrete time)
si = 1 – 1.4 si-1
2 + 0.3si-2
chaotic map Hénon
2
1
1
1
6
4.0exp9.01
k
kk
s
i
iss
chaotic map Ikeda
si = a si-1(1 - si-1)
periodic a=3.52
chaotic a=4
Logistic map
19. Noise in the time series
( )t tx h s
0( )t
t s f s
noise
( )t t tx h w s
observational noise
noise
Observation
Dynamical system
0( )t
t tf s s
dynamic (system) noise
tw : white noise, uncorrelated to andtx ts
t : white noise, uncorrelated to us tu
20. Noise: dynamic (system) ε observational (measurement) w
si = a si-1(1 - si-1)
xi = si + wi, wi ~ N(0,s)
logistic map
si = a si-1(1 - si-1) + εi , εi ~ N(0,s2)
xi = si
chaotic
periodic
23. - Other topics:
- Hypothesis testing for linearity / nonlinearity
- Control system evolution
- Synchronization
- …
- State space reconstruction
in order to observe the complexity / stochasticity / structure
of the system
- Estimation of characteristics of the system / attractor
measuring the complexity / dimension of the system
- Modeling / Prediction
Use nonlinear models to improve predictions
Topics in
the analysis of time series and dynamical systems
24. xi = [xi , xi-t ,…, xi-(m-1)t ]
Method of delays
Parameters
embedding dimension m
delay time t
time window length tw
tw = (m-1)t
We assume that
the studied system
is deterministic
State space reconstruction
initial state
space
M
is
1is
)(1 ii sfs
x
R
observed
quantity
xi = h(si )
h
Embedding
?
1ix
ix
)(1 ii xFx
Rm
reconstructed
state space
xi = F(si )Φ
condition: 12 Dm
27. • From the autocorrelation r(τ)
(measures linear correlation)
τ r(τ) =1/e ή τ r(τ) =0
Estimation of τ
)()(
),(
log),(),(
, ypxp
yxp
yxpYXI
YX
XY
yx
XY
)(),( t
t
IYXI
xYxX ii
• From the mutual information I(τ)
(measures linear and
nonlinear correlation)
τ first local minimum I(τ)
28. • Close points on the attractor are:
- either real neighboring points due to system dynamics
- or false neighboring points due to self-intersections and insufficiently low m
Method of false nearest neighbors (FNN)
Estimation of m
Optimal m ?
R
R2
• Takens theorem:
… but D is unknown
12 Dm
• At a larger m where there are no self-intersections all false neighboring points
will be resolved as they will no longer be close
• The optimal m’ is the one for which there are no longer any false nearest
neighbors as the dimension increases by one from m’ to m’+1.
• Too small m
self-intersection in the attractor
• Too large m
“curse of dimensionality”
29. An example of estimating m by the method FNN
The estimation of m with the method FNN depends on:
- the delay τ
- noise
x-Lorenz without noise
2 4 6 8 10
0
5
10
15
20
25
30
35
40
m
%FNN
FNN, x-lorenz, no-noise
t=2
t=5
t=10
t=20
x-Lorenz + 10% noise
2 4 6 8 10
0
5
10
15
20
25
30
35
40
m
%FNN
FNN, x-lorenz 10% noise
t=2
t=5
t=10
t=20
31. The correlation dimension ν characterizes the fractal structure of the
attractor (self-similarity at different scales) using the density of the points
of the attractor in the reconstructed state space
The basic idea is that the probability of two points being
closer than a distance r
Correlation dimension ν
rji xx
changes w.r.t. r as a power of r
i : number of points lying in a sphere with
radius r and center ix
i i jx
r x x
scaling law
rxi ~
ν integer the attractor is a regular geometric object
ν non-integer attractor is a fractal
holds for
0r N
xi
xi
32. xi
xi
rrC )(Scaling law for small r
Convergence of ν(m) for m sufficiently large
Estimation
dlog ( )
dlog
C r
r
for a range of r
If ν small and non-integer and the system is deterministic
small dimension and fractal (chaotic) structure
Estimation of the correlation dimension ν
Correlation sum
N
i
N
ij
jr
NN
rC
1 1)1(
2
)( xxi
Nii ,,1, xreconstructiontime series , 1, , ( 1)ix i N m t
Estimation of
xi
0 when 0
( )
1 when 0
x
x
x
Heaviside function
33. x-Lorenz + 10% observational noise, τ=2
x-Lorenz + 10% observational noise, τ=10
log C(r) vs log r local slope vs log r ν vs m
x-Lorenz without noise, τ=2
34. The estimation of ν is affected by the following factors:
- correlation time wji
- selection of τ and m
- noise
- time series length
36. The Lyapunov exponents measure the average rate of divergence and convergence
of the trajectories on the attractor at the directions of the local state space
Lyapunov spectrum: m ...21
λi > 0 divergence
λi < 0 convergence
λi = 0 direction of flow
If λ1 > 0 and the system is deterministic
chaos
Lyapunov exponents
Dissipative system :
m
i
i
1
0
37. xi
xi’
xi+t
xi’+t
d0
dt
Largest Lyapunov exponent λ1
Initial distance d0= xi - xi’ of two nearby trajectories is
expected to increase exponentially with time
If
t
t e 1
0
λ1 is the largest
Lyapunov exponent
N
j j
jt
Nt 1 ,0
,
1 ln
1
Computation:
After time t: dt= xi+t - xi’+t
39. The true system generating the time series: )(1 ii sfs
Prediction models
2
1, 1 1, 2,
2, 1 1,
1 1.4
0.3
i i i
i i
s s s
s s
Hénon map
1
1, 2, 1, 1( , ) f
i i is s s
2
1, 2, 2, 1( , ) f
i i is s s
1i if
s s
40. The true system generating the time series: unknown)(1 ii sfs
The problem of modeling and prediction of time series:
given x1, x2, … xi , to estimate / predict xi+1
State space reconstruction
with the method of delays:
xi = [xi, xi-t …, xi-(m-1)t]
Prediction models
The reconstructed system from the time series: estimation?)(1 ii xFx
The function that is relevant to
time series prediction:
)(1 ii xFx
)(1 ii Fx x
mm
:F
m
F :
1 1( , )i i ix F x x m = 2, τ = 1
41. • Semi-local models, e.g. neural networks
the form of function F is derived as a weighted sum of
local basic functions
Nonlinear prediction models
• Global models, e.g. polynomials
function F bears the same analytic expression
for the whole domain
• Local models, e.g. the local linear model
function F is defined differently at each point of the
reconstructed state space
42. Prediction using similar segments of the time series
Prediction at time i+T from the mappings Τ step ahead of
“similar” segments from the past of the time series
43. Local prediction models
Implementation of the idea of “similar” segments:
time series segments reconstructed points
},...,,{ )()2()1( Kiii xxxThe nearest neighboring points to xi:
Prediction of xi+T from the mappings of the neighbors: },...,,{ )()2()1( TKiTiTi xxx
Zeroth order prediction: TiiTi xTxx )1()(ˆ
Average prediction:
K
j
Tjii x
K
Tx
1
)(
1
)(
44. Local linear prediction
We assume that for the neighbor of xi the local linear model is valid :
i
mimii
miiiii
'a
xaxaxaa
xxxFFx
xa
x
0
)1(210
)1(1 ),,,()(
tt
tt
xi(1)+T = a0 + a’ xi(1)
xi(2)+T = a0 + a’ xi(2)
xi(K)+T = a0 + a’ xi(K)
The model holds for
)()2()1( ,...,, Kiii xxx
K
j
mjimjiji
aaa
xaxaax
m
1
2
)1()()(101)(
,,,
)(min
10
t
Estimation of parameters
(method of ordinary least squares)
maaa ,,, 10
45. Estimation of prediction error
We split the time series in two parts:
1 11 2, 1, , , , ,N N Nx x x x x
learning set test set
1 1
ˆ ˆ, ,N Nx xpredictions
ˆi T i T i Te x x
prediction error
N
i
i
TN
Nt
TtTt
xx
N
xx
NTN
T
1
2
1
2
1
1
ˆ
1
)(NRMSE 1
statistic for
prediction error
( )ix T
46. Example: x-Lorenz
• local linear prediction model (LLP)
Prediction with:
• local average prediction model (LAP)
11,5,1 Kmt
without noise
with 10%-noise
47. 0 2 4 6 8 10
0.7
0.8
0.9
1
1.1
m
nrmse(m)
()
AR
LAM(K=15)
LLM(K=15)
Prediction error (nrmse) for the
last 30 quarters
annual- quarter growth rate of GNP of USE in the period 1947 – 1991
164 166 168 170 172 174 176
-0.01
-0.005
0
0.005
0.01
0.015
0.02
()
real
AR(3)
LAM(m=5,K=15)
LLM(m=5,K=15)
Predictions starting at the first
quarter of 1989 with prediction
horizon being the last 6 years
Prediction with
- linear model, AR
- local average model, LAM
- local linear model, LLM
48. Prediction starting at 20/9/2005
and prediction horizon is up to 16 days ahead
ASE index in the period 1/1/2002 – 20/9/2005
Predict index with
- linear model, AR
- local average model, LAM
returns 1
1
t t
t
t
x x
y
x
18 25 02 09 16
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
day
returnsofindex
()
general index returns
y
n
(T), AR(7)
y
n
(T), LAM(m=7,K=20)
index
18 25 02 09 16
3200
3250
3300
3350
3400
3450
day
closeindex
()
general index
xn
(T), AR(7)
xn
(T), LAM(m=7,K=20)
49. One step ahead prediction
in the period 21/9/2005 – 12/10/2005
ASE index in the period 1/1/2002 – 20/9/2005
Predict index with
- linear model, AR
- local average model, LAM
returns 1
1
t t
t
t
x x
y
x
18 25 02 09 16
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
day
indexreturn
()
general index
y
n
(1) AR(7)
y
n
(1) LAM(m=7,K=20)
index
18 25 02 09 16
3200
3250
3300
3350
3400
3450
day
closeindex
()
general index
xn
(1) AR(7)
xn
(1) LAM(m=7,K=20)