Chapter 1: Introduction

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1 1 Chapter 1: Introduction Stochastic Processes The concept of a random variable gives a mathematical formulation to the notion of a quantifiable event whose current value is not exactly known or whose future values are not predictable. The notion of a stochastic process generalizes that of a random variable to sequences of random phenomena, where any one event in a given sequence may possibly influence another and, furthermore, may display some form of time dependence. Consider the stylized classical experiments consisting of a set of throws of a dice or a sequence of draws of a card from a deck, where each card is drawn from a pack each minute but is immediately shuffled back into the deck. In such experiments the observed events form, respectively, the sequence of values shown by the top face of the dice and the sequence of drawn cards. Let us make the standard assumption that the outcomes in each of these idealized experiments do not influence each other and are independent of time; then the outcomes of any experiment is not exactly predictable and there is no information in a long sequence of observations which permits one to improve the quality of bets on the next throw or draw; in other words, repeated experiments simply generate sequence of unpredictable observations. Now let these experiments be modified, so that in the first case the observed event (as far as the experimenter is concerned) is the sum of the preceding and current dice face values, and in the second case any card which is drawn at the minute t is shuffled back into the pack just before the draw which takes place at the minute t + 2; then it is evident that earlier observed events of each type influence future events. Then exact predictability is still impossible but it is evident that past observations help to reduce the uncertainty in the predictions of the subsequent observation.

2 2 Probability Theory There are several general, non-mutually exclusive, situations which invite a probabilistic formulation and which motivate the notion of a stochastic process. They include the following: (i) Unpredictability of Complete Observations. There are many cases where the specified phenomena are directly (i.e. completely) observed but subsequent phenomena are not predictable. Some examples of this type are the following: the outcomes of the classical dice and card experiments above, the movement of stock, bond and financial markets, lotteries, local meteorological phenomena, internet traffic at a specific network node, the time instants at which radiation emissions take place in nuclear physics, and the outcome of experiments in a quantum mechanical context. It is in such settings that we often observe the crucial phenomenon that relative frequencies and other long term averages display a form stability in that they converge to constant values over the long term. In the case of nuclear and quantum mechanical events the phenomena may be viewed to be intrinsically probabilistic. However, in all the other examples above, the unpredictable nature of the phenomena may be partly ascribed to at least one of the three following causes: lack of information (about the actual entire state of the system), the complexity of a full dynamical description of the phenomena, or senstivity to initial conditions. (ii) Lack of Information. Although the values of measurements of certain phenomena may be exactly known to certain agents at a given time t, they may be viewed as random to other agents (at, possibly later times) for the simple reason that they are unknown to those agents. For instance, at any given moment, the status of the electrical generating sets in a power station on a continental network may be known to its operators, but not the status of all other sets on the network; a

3 3 similar situation occurs in the case of the meteorological state of the planet, and, on a rather smaller scale, for the hands in a bridge game. Further, there is a vast range of engineering problems where the values of a process are hidden in observation noise and the objective of filtering (or estimation) theory is to obtain the best estimates of such underlying processes; standard examples include: noise in radio, microwave and optical communication systems. Here we observe that improvements in the extent and quality of observations may improve the predictability of phenomena and hence, in this sense, reduce their degree of randomness. (iii) Complexity. The complexity of a set of phenomena may in itself may give rise to the non-inferability of current values or non-predictability of future values of the phenomena. For instance, the initial positions and velocities of the approximately molecules of a gas which occupy a volume of 1 cc cannot be stored in a computer system; neither could the dynamical equations of motion be integrated numerically to yield solution trajectories, since they involve approximately 10 9 collisions per second for a single molecule. A second example is given by the digital expansion of π. It is reasonable to regard all digits of π beyond the trillionth as random; computations at the time of writing (January, 2004) have not reached the trillionth digit and there are no known regularity patterns which make the expansion of π predictable (to any extent) without direct computation. The reader is referred to [Fine, 1973] for a presentation and analysis of various foundations of probability theory; in particular, one important such formulation uses the so-called Solomonoff-Chaitin-Kolmogorov measure of complexity (in which data compressibility corresponds to non-randomness) as a basis for the notion of probabilistic phenomena. An important result in this mathematical framework is that long term averages of sequences which have a high complexity actually converge (in an appropriate technical sense); we note that this holds without any of the mathematical structure which gives rise to the convergence results of form of the strong law of large numbers in standard probability theory.

4 4 (iv) Chaotic dynamics. Finally we mention chaotic dynamics. In this case the phenomena can be measured and stored with a high degree of accuracy. Consider, for example, the position and velocity (i.e. the state) of a billiard ball at a given instant on a billiard table with an irregular perimeter; this information can be measured with a high degree of accuracy and is concise enough to be stored. In contrast to (iii) above, the equations of motion are simple and can be computed over any time interval. However errors in the computation of the velocity vector increase geometrically with respect to the number of collisions with the boundary. If the collisions are assumed to be perfectly elastic the energy of the system is conserved and the system never enters a stationary (i.e. equilibrium) state. The trajectory in such cases is random in the sense that, over long periods, predictions via direct computation bear no relation to the true state of the system. Classic examples of chaotic systems are given by the Lorentz system and the Hénon map. (a) The Lorentz system. dx dt dy dt dz dt = σx + σy, (1) = xz + rx y, σ, r, b > 0, (2) = xy bz. (3) These equations are obtained by a truncation of an extreme simplification of the Navier-Stokes equation; they give an approximate description of a horizontal fluid layer heated from below. Warmer fluid rises to give convection-n currents which are very sensitive to initial conditions (see [ Ruelle, 1989]).

5 5 (b) The Hénon Map. where a, b > 0. We note that x(t + 1) = y(t) + 1 ax 2 (t), (4) y(t + 1) = bx(t), (5) J = (x(t + 1), y(t + 1)) (x(t), y(t)) = b, (6) and so the map is volume decreasing only if 0 b < 1. Again we refer the reader to [Ruelle, 1989] for a discussion of the chaotic properties of this set of equations. The trajectories of deterministic (continuous or discrete time) equations in R n which satisfy the standard conditions for the existence and uniqueness of solutions (see e.g. [XXX]), have the property that, until a possible time of escape to infinity, a given initial state x t0 will generate a unique trajectory φ(t, t 0, x t0 ), t t 0. For an exact calculation procedure the trajectories of such systems must evolve without diffusion, and in this sense prediction of the evolution of the trajectories is exact. However, if there is extreme sensitivity to changes in initial conditions, any initial interval, or more generally, any compact subset of the state space, will dissipate very rapidly, so that elements of this original volume may be found distributed throughout a large proportion of the state space after a relatively short interval of time. Within this framework, small errors in the specification of the initial condition (i.e initial state), or small errors in calculation, will lead to large divergences and hence large errors in prediction after the lapse of small periods of time. This will make individual trajectories impossible to extrapolate from earlier values. It is a fundamental feature of systems of this type that ensemble properties, such as the existence of invariant measures, will hold when the system satisfies certain general technical conditions. In other words, certain distributions

6 6 of mass over special subsets of the state space (the so-called strange attractors) will have the property that they are invariant under the chaotic flow. Hence, in the ensemble sense, there are stable properties associated with the extreme instability of the individual trajectories. Interpreting the masses in question as probability measures, or distributions, gives a basic connection between chaotic systems and the theory of stochastic processes. (See [Ruelle,1989] and the references therein). The Frequentest Foundation for Probability Theory Measure theoretic probability theory postulates the existence of an underlying abstract probability space (Ω, B, P ) with all random phenomena under consideration modelled by random variables, that is to say, by functions on Ω. Certain hypotheses on a given random variable x t then guarantee the existence of a density p(x) which yields the probability p(x)dx of the event x t dx, where dx is the volume element at x t. (Basic probability theory is reviewed in the Appendix.) Two of the great theoretical triumphs of measure theoretic probability theory were the strong law of large numbers and the rigorous definition of conditional probability. The importance of the first result is that the standard frequentist foundation for probability theory interprets probabilities as the limiting relative frequencies of idealized repeated independent identical experiments. Consider: (1) The observed or assumed convergence of relative frequencies of certain classes of observed phenomena. (2) The mathematical convergence of sample frequencies to expected values within the theory of probability.( The principal results of this type are the Law of Large Numbers, the Ergodic Theorems and the Central Limit Theorems, see e.g. [Chung, 1968] and the probability theory appendix in [Caines, 1988]).

7 Then the frequentist philosophical position is that the notion that a class of phenomena 7 are probabilistic is expressed by the assertion that (1) corresponds to (2). Here the term corresponds to is to be interpreted to mean that (1) is accounted for (or is modeled by ) the theoretical framework of measure theoretic probability theory. Once this position is adopted, it is meaningful to perform statistical inference which consists in estimating the underlying probability distributions by processing the available finite length observation records. Within the probabilistic framework one is then justified in using the estimated probability distributions to calculate the conditional distributions of future events. An example of the convergence of relative frequencies is given by the simulated repeated selection of a ball numbered 0, 1 or 2 from an urn. Let the computer simulations be designed so that the assumptions of independent and identically distributed experimental trials are closely approximated. One sample path (i.e. trajectory) {x j ; 1 j 50} resulting from the simulations is shown in Figure 1. In Figure 2 the sample paths for 1 N 50 of the relative frequencies f k (N) 1 N N I k (x j ), k = 0, 1, 2, j=1 are displayed, where I k (x) is the characteristic function of the set {k; 0 k 2}, which takes the value 1 when x = k and is 0 otherwise. Although the convergence of the sample paths towards 1 3 in Figure 2 is suggestive, Figure 3, displaying the trajectories for 1 N 500, is more convincing since the stochastic process f (f 1, f 2, f 3 ) appears to be stably contained in the neighborhood of 1 3 after 225 observations. However, there can be no guarantee that the sample path of the process f will not oscillate significantly at some later stage. If such an observed divergence from 1 3 is calculated (within the probability framework) to have a very small probability, then strictly speaking one can only conclude one has observed a rare event. Evidently, the philosophical choice to use a probabilistic theory for a set of phenomena cannot be made on the grounds of any computed probable or improbable behaviour it displays

8 8 with respect to any given model. It can only be based on considerations external to the mathematical theory of probability, such as the complexity issues discussed earlier. We observe that this issue is distinct from the more familiar one of which probabilistic model to use once a probabilistic framework has been adopted; this question often falls within the area of model identification or, in an even more specific context, that of model order estimation. We note that by their definition, the relative frequencies here have the positivity, unit sum and additive properties which form the axioms for the probability measures (on finite sets) which model them in probability theory (see the Appendix). Probabilistic versus Deterministic Models We have emphasized that by the nature of probabilistic models the decision to use them to account for observed phenomena is necessarily a philosophical choice to be made outside of a probablistic framework. The following two simple examples illustrate this fact. First, consider a sequence σ1 N, N Z 1, of zeros and ones which has been generated by an ideal random coin toss with probability of, say, 1/2 for each outcome. Then for any given value of N it is possible to specify a finite state machine with at most N states which will yield the output sequence σ1 N as the initial segment of all output sequences σ1 R, R N. Hence it gives a zero error fit to the observed record and a meaningful set of predictions of future values. From a decision theory perspective, however, such predictions would be useless since they would lead one to bet on certain infinite sequences with finite periods. Second, consider a process {y t ; t Z 1 } generated by the deterministic system y t 1 2 y t 1 = j= 10 {a 1 (j) cos (2πjt + θ 1 (j)) + a 2 (j) sin (2πjt + θ 2 (j))}, (7) with the initial condition y 0 and the sets of amplitudes a 1, a 2, and phase shifts θ 1, θ 2, given. Let us denote the right hand side of (0.7) by {v t ; t Z }. Now suppose it is assumed that

9 9 y is generated by a system of the form y t αy t 1 = w t, t Z 1, (8) where w is a white noise process (see Chapter 2, Section 1), and suppose 101 observations, y 100 0, of y are generated by (0.7). Now estimate α by the least squares procedure by which ˆα 100 argmin α 100 t=1 (y t αy t 1 ) 2. Then it will be found that the mean square error of the probability based model predictions ŷ t+1 = ˆα 100 y t, over the interval t = 100,, 120, given by t=101 (ŷ t y t ) 2 = 1 20 is of the same order of magnitude as t= t=101 v2 t. (ˆα 100 y t 1 ( 1 2 y t 1 + v t )) 2, (9) Here the complexity of the exact deterministic model (0.7) is exchanged for the simplicity of a prediction error model y t ˆαy t 1 = ɛ t, t Z, [Caines,1988], based upon (0.8), but at the cost of (i) the probabilistic hypothesis, and (ii) mean square prediction errors of the order of magnitude of the average energy of the driving process v in (0.7). Furthermore, a histogram of ɛ t ˆαy t 1 y t = (ˆα )y t 1 v t, t = 101,, 120, will be approximately symmetrically distributed about 0. This corresponds to the probabilistic formula Eαy t 1 = Ey t, t = 101,, 120, which states that the average prediction error is zero when α is known. Finally, we observe that if the simple stochastic model (0.8) is rejected as unsatisfactory because of the magnitude of the prediction errors {ɛ t ; t Z}, one may increase the order of the autoregressive model and use AR(M) models of the form y t = M j=1 α jy t j, t Z, with M 1. In general, however, no AR(M) model will give an exact fit or perfect predictions of long runs of data generated by (0.7). This example illustrates the system identification problem which occurs in an enormous range of applications: there may be solid grounds to believe that the functional dependence of the y process on a driving or disturbance process v has a specific linear, or some other, functional form, but ignorance of the mechanism generating v motivates the adoption of a probabilistic modelling framework. (See [Caines, 1988].)

10 10 An Econometric Example Along with stock, bond and monetary time series, the macro-econometric time series of gross national and domestic product, unemployment, investment, inflation rates, etc, form a natural area for the application of the theory of stochastic processes. Macro-econometric time series are particularly difficult to study since (i) the experiment generating them cannot be repeated (there is only one run of history), (ii) the desired variables (GDP in this case) must be extracted from a mass of noisy, error filled, institutional data, and (iii) there is reason to believe that over periods of the order of a decade the structure of the generating mechanism (the economy) undergoes significant changes which are themselves unpredictable processes (e.g. changes in trade relations, deregulation, patterns of consumer behaviour). Figures 4 and 5 display daily changes in the value of the Canadian dollar (with respect to the US dollar) over the period 4 September, November, 1996, and 4 September, December, 1996, respectively, while Figure 6 shows the daily changes in the Canadian dollar over the period 4 September, December, 1996, respectively. Figures 7 and 8 show the movement of the Dow Jones index and the time series of its daily changes over the time interval 4 September, December, Figure 9 gives the latter series over the period 4 September, December, Finally, Figure 10 shows a sequence of 75 values of a deterministic process generated by combining 150 sinusoids with distinct frequencies and large phase shifts between the various sinusoids. Concerning these graphs we observer the following: 1. The random, that is to say unpredictable, nature of the series: the downturn after the 46th day in Figure 4 is not consistent with the increasing trend over the days 1-45, and the extent of the fall shown in Figure 5 is not to be expected from the movements in Figure 4. Similar observations apply to the Dow-Jones series in Figure The dates of the so-called turning points show no obvious predictable behaviour.

11 11 3. The time series of changes, or differences, given in Figures 6, 8 and 9 are very rough and appear to be highly unpredictable. They correspond to the derivatives of the original time series. We note that the derivative time series are on average positive (respectively, negative) over periods when the original, smoother, time series is increasing (respectively, decreasing). Finally, despite the fact that Figure 10 corresponds to a computed series which is predictable (when its complex generating equation is known), and Figures 6, 8 and 9 are taken from natural time series, they may be seen to display significant similarities. This reinforces the view that there is something in common between natural unpredictable series and deterministic processes with complex generating mechanisms.

THREE DIMENSIONAL SYSTEMS. Lecture 6: The Lorenz Equations

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