Dyadic Transformation - Rate of Information Loss and Sensitive Dependence On Initial Conditions

Rate of Information Loss and Sensitive Dependence On Initial Conditions

One hallmark of chaotic dynamics is the loss of information as simulation occurs. If we start with information on the first s bits of the initial iterate, then after m simulated iterations (m<s) we only have (s-m) bits of information remaining. Thus we lose information at the exponential rate of one bit per iteration. After s iterations, our simulation has reached the fixed point zero, regardless of the true iterate values; thus we have suffered a complete loss of information. This illustrates sensitive dependence on initial conditions—the mapping from the truncated initial condition has deviated exponentially from the mapping from the true initial condition. And since our simulation has reached a fixed point, for almost all initial conditions it will not describe the dynamics in the qualitatively correct way as chaotic.

Equivalent to the concept of information loss is the concept of information gain. In practice some real-world process may generate a sequence of values {} over time, but we may only be able to observe these values in truncated form. Suppose for example that = .1001101, but we only observe the truncated value .1001 . Our prediction for is .001 . If we wait until the real-world process has generated the true value .001101, we will be able to observe the truncated value .0011, which is more accurate than our predicted value .001 . So we have received an information gain of one bit.

Read more about this topic:  Dyadic Transformation

Famous quotes containing the words rate of, rate, information, loss, sensitive, dependence, initial and/or conditions:

    We all run on two clocks. One is the outside clock, which ticks away our decades and brings us ceaselessly to the dry season. The other is the inside clock, where you are your own timekeeper and determine your own chronology, your own internal weather and your own rate of living. Sometimes the inner clock runs itself out long before the outer one, and you see a dead man going through the motions of living.
    Max Lerner (b. 1902)

    If you could choose your parents,... we would rather have a mother who felt a sense of guilt—at any rate who felt responsible, and felt that if things went wrong it was probably her fault—we’d rather have that than a mother who immediately turned to an outside thing to explain everything, and said it was due to the thunderstorm last night or some quite outside phenomenon and didn’t take responsibility for anything.
    D.W. Winnicott (20th century)

    If you have any information or evidence regarding the O.J. Simpson case, press 2 now. If you are an expert in fields relating to the O.J. Simpson case and would like to offer your services, press 3 now. If you would like the address where you can send a letter of support to O.J. Simpson, press 1 now. If you are seeking legal representation from the law offices of Robert L. Shapiro, press 4 now.
    Advertisement. Aired August 8, 1994 by Tom Snyder on TV station CNBC. Chicago Sun Times, p. 11 (July 24, 1994)

    No need to be sentimental to mourn the loss of Paradise.
    Mason Cooley (b. 1927)

    Peasants are a rude lot, and hard: life has hardened their hearts, but they are thick and awkward only in appearance; you have to know them. No one is more sensitive to what gives man the right to call himself a man: good-heartedness, bravery and virile brotherhood.
    Jacques Roumain (1907–1945)

    As, therefore, we can have no dependence upon morality without religion;Mso, on the other hand, there is nothing better to be expected from religion without morality;Mnevertheless, ‘tis no prodigy to see a man whose real moral character stands very low, who yet entertains the highest notion of himself, in the light of a religious man.
    Laurence Sterne (1713–1768)

    Capital is a result of labor, and is used by labor to assist it in further production. Labor is the active and initial force, and labor is therefore the employer of capital.
    Henry George (1839–1897)

    The depth and strength of a human character are defined by its moral reserves. People reveal themselves completely only when they are thrown out of the customary conditions of their life, for only then do they have to fall back on their reserves.
    Leon Trotsky (1879–1940)