• Aims: Extract information from time series data and then give the description of complex behavior (irregular, chaotic, non-stationary and noise-corrupted signals).

• Two parts: modeling and identifying.

• Modeling: Applied mathematics identifies the key quantities in the problem, and connects them by differential equations or matrix equations. These equations are the starting point for scientific computing.

• Identifying: Discrete sequences of these (experimental) measurements become time series. Investigations of these sequences are known as “system identification”, namely reconstruction of the dynamical system.

• Both abstract (mathematical) and concrete (implementable) concetps related to these aims will be discussed and illustrated.

• Applications: Concerns of interdisciplinary topics rooted in microelectronics, neuroscience, biomedicine, statistics, demography, geography, econom(etr)ics, ecology, epidemiology, pharmacology, etc.

• Slides, codes, and supplements will be uploaded on Moodle https://moodleucl.uclouvain.be/ under the name LECON2601

• References: 1. Extracting Knowledge from Time Series 2. Stochastic Methods: A Handbook for the Natural and Social Sciences.

• Both books are from Springer series in Synergetics, downloadable from bibliotheque UCLouvain.

• Other references: 1 Probability, Markov Chains, Queues, and Simulation. 2 Computational Science and Engineering

• Exam: Take-home exam.

• Office hour: By appointment (email: zhengyuan.gao@uclouvain.be) Place: Voie du Roman Pays 34, Room C215

• The entire human knowledge is a model.

• Science is a sphere of human activity whose function evolves from reality.

• A scientific system of knowledge, namely a proper model, must allow experimental refutation. (Karl Popper)

• Meanwhile, in human society, human beings have all kinds of forces to “invent” a social experiment that can fit some sorts of “hypothesis” about the economic/social world.

• Such a “hypothesis” can morph into a “theory” if no refutation sucesses.

• My opinion of “time” and “time seires modeller”: Refute the pseudo-realistic “theories” and Illuminate verifiable possibilities.

• Principal of predictability: predictive ability determines which (mathematical) tools are appropriate.

• Deterministic model: each occurrence of an event \(A\) (a cause) inevitably leads to an occurrence of an event \(B\) (a consequence).

• Viewpoint of classic mechanistic determinism: a unique future corresponds to a given present.

• Stochastic model: a probability represents the chance of transmitting from a chosen current state \(A\) (a cause) to an occurrence state \(B\) in an event set \(\mathcal{B}\) (a set of consequences).

• Viewpoint of probabilistics: a set of prospects correspond to a given present.

1. Deterministic model: - Ordinary differential equations (ODEs).

2. Deterministic time series.

3. Computation.

4. Stochastic model: - Stochastic differential equations (SDEs).

5. Stochastic time series, stochastic process.

6. Estimation.

• Representative categories of realistic problems: Growth (continuous states), birth and death (discrete states), Suspectible-Infected-Recovered (SIR - a generalization of the previous two categories)

• The most important equation in mechanics is Newton’s law: force = mass times acceleration or \[F = m a.\]

• We have a mass \(m\) that obey this law. Its displacement \(u(t)\) is changing with time (but not near the speed of light…).

• Oscillation: The force \(F\) composes of external force \(f\) and spring force \(cu(t)\) (Hooke’s law, spring constant \(c\) times elongation \(u(t)\)).

• The acceleration of the mass is \[ a = \frac{d^2 u(t)}{dt^2}.\]

• Together, we have a second order differential equation \[ f - c u(t) = m \frac{d^2 u(t)}{dt^2}.\]

• The velocity of the mass is \(v(t) = \frac{d u(t)}{dt}.\) So we know the momentum: \[ p(t) = m v(t) = m\frac{d u(t)}{dt}. \]

• Newton’s second law: the rate of change of the momentum of a particle is equal to the instantaneous force acting on it: \[ F = \frac{dp(t)}{dt} = m\frac{d v(t)}{dt} = m a.\]

• Thus the previous second order ODE can be presented as a system of first order ODEs \[\begin{cases} \frac{du(t)}{dt} & =v(t),\\ m\frac{dv(t)}{dt} & =f-cu(t). \end{cases}\]

times = seq(0,10,by=0.01)
funct = function(t,integ,parms){
  x = integ[1]; y = integ[2]
  dx =  parms[1]*y; dy = -parms[2]*x
  list(c(dx,dy))}
require(deSolve)
cinit = c(1,1); parms = c(1,3)
xy = rk4(cinit,times,funct,parms) 
# Solve System of ODE by Euler's method/Runge-Kutta 4th order Integration
head(xy, 5)

##      time        1         2
## [1,] 0.00 1.000000 1.0000000
## [2,] 0.01 1.009850 0.9698515
## [3,] 0.02 1.019396 0.9394121
## [4,] 0.03 1.028637 0.9086908
## [5,] 0.04 1.037569 0.8776969

• In the 1920s, ecologists (Lotka and Volterra) began to study the populations of two Arctic species, lynx (a predator) and snowshoe hares (their prey).

• What is the cause of oscillations? Intuitive answer: Interactions.

• Predators eat some of the prey \(\rightarrow\) the predator numbers will go up and the prey numbers will go down

• High numbers of predators and low numbers of prey \(\rightarrow\) the system cannot continue at the same pace (consquence: predator numbers will decline).

• Feedback in this system: The prey population positively affects the number of predators, while the predator population negatively affects the number of prey.

• What is the cause of oscillations? From this system is: time delays or say negative feedback.

• Positive feedback (leverage effect): a positive value of a variable leads to an increase in that variable, and a negative value of a variable leads to a decrease (more negative) in that variable.

• Population growth: A young society has an increasing population. The larger the population is, the more babies are born, which makes the population even larger.

• Market bubbles: Investors buy into a stock, which causes the price to rise, which encourages more investors to buy, on the grounds that the stock is “going up.”

• Market crashes: Investors sell the stock, which lowers the price, which convinces others to sell because the stock is going down.

• Lowering the number of sharks takes the pressure off the tuna population, which grows to a higher level than before. The higher tuna population then gives rise to an even higher shark population. Thus, removing sharks dramatically actually results in a higher peak shark population.

• Population variable at time \(t\): \(x(t)\). What changes the population \(x\)? Births and deaths.

• If a species has birth rate \(b\) and they don’t die: \(\frac{dx(t)}{dt} = b x(t).\)

• If a species has die rate \(d\) and there is no birth: \(\frac{dx(t)}{dt} = - dx(t).\)

• Combine birth and death: \[\frac{dx(t)}{dt} = bx(t)- dx(t)=(b-d)x(t)=rx(t)\] where \(r\) is the net per capita growth rate. This relates to conservation law. When \(dx(t)/dt=0\), flow in equals flow out, the equation is called balance equation.

• Lanchester model during World War I: two armies, Red and Blue, in combat. \[\frac{dR(t)}{dt} = - c_1 B(t), \, \, \frac{dB(t)}{dt} = - c_2 R(t).\]

• \(dR(t)/dt\): the loss rate of Red soldiers depends on the existing Blue soldiers.

• A similar Richardson model is used for analyzing arm races: \[\begin{align} \frac{dR}{dt} & = c_1 B - c_2 R + c_3 ,\\ \frac{dB}{dt} & = c_4 R - c_5 B + c_6. \end{align}\] \(R=R(t)\) represents the total munitions stockpiles for country Red at time \(t\). \(dR(t)/dt\) corresponds to the rate of change of munition levels.

• Note that any effects of crowding, such as competition for scarce resources, would limit growth.

• A competition factor (some number less than \(1\)): \[1-\frac{x(t)}{N}\] where \(N\) is the largest number of population which the enviornment can afford. So \((1-x(t)/N)\) is the fraction of resources that are available.

• Multiply the net growth rate by the factor \[\frac{dx(t)}{dt} = bx(t)\left(1-\frac{x(t)}{N}\right)=bx(t) - cx^2(t)\] this is called a logistic growth model (Verhulst, 1838).

• For each speices, the population growth rate is affected by an interaction term (negative for tuna and positive for shark).

• Shark has a death rate but death of tuna is mainly caused by sharks. (The death rate of tuna is neligible.) \[\begin{align} \frac{d\mbox{Tuna}}{dt} & =b_{tuna}\mbox{Tuna}-c_{1}\mbox{Tuna}\times\mbox{Shark}\\ \frac{d\mbox{Shark}}{dt} & =c_{2}\mbox{Tuna}\times\mbox{Shark}-d_{shark}\mbox{Shark} \end{align}\]

• In fact, this is the classic Lotka-Volterra model (Predator-Prey model), a system of coupled differential equations.

• Infectious diseases in populations: Contact between susceptible and infected people increases the transmission of the disease. \[\mbox{Suspectible} \longrightarrow \mbox{infected} \longrightarrow \mbox{Recovered} \] \[\begin{align} \frac{dS}{dt} =&& c_{1}(N-S)-c_{2}\frac{I\times S}{N},\\ \frac{dI}{dt} =&& c_{2}\frac{I\times S}{N}-(c_{1}+c_{3})I,\\ \frac{dR}{dt} =&& c_{4}I-c_{1}R. \end{align}\]

• Memetic modelling: Propagation of memes. (Capitalism is often thought as a disease…)

• Efficiency and feasibility of mathematics deserve a special discussion.

• (Applied) Mathematics is a science studying quantitative relationships and space forms of the real world.

• A set of useful rules and formulas for the solution of practical tasks encountered by people in their everyday life.

• Admittedly there are unknowns that are impossible for mathematical modelling (at least nowadays).

• However, the pseudo-realistic theories/laws set by chiefs and peoples in the historical scene were inevitably vanishing, but mathematics developed together with the mankind and the nature evolves.