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, information, loss, sensitive, dependence, initial and/or conditions:
“Unless a group of workers know their work is under surveillance, that they are being rated as fairly as human beings, with the fallibility that goes with human judgment, can rate them, and that at least an attempt is made to measure their worth to an organization in relative terms, they are likely to sink back on length of service as the sole reason for retention and promotion.”
—Mary Barnett Gilson (1877–?)
“I have all my life been on my guard against the information conveyed by the sense of hearing—it being one of my earliest observations, the universal inclination of humankind is to be led by the ears, and I am sometimes apt to imagine that they are given to men as they are to pitchers, purposely that they may be carried about by them.”
—Mary Wortley, Lady Montagu (1689–1762)
“Quarrel not at all. No man resolved to make the most of himself, can spare time for personal contention. Still less can he afford to take all the consequences, including the vitiating of his temper, and the loss of self-control. Yield larger things to which you can show no more than equal right; and yield lesser ones, though clearly your own. Better give your path to a dog, than be bitten by him in contesting for the right. Even killing the dog would not cure the bite.”
—Abraham Lincoln (1809–1865)
“Violence upon the roads: violence of horses;
Some few have handsome riders, are garlanded
On delicate sensitive ear or tossing mane,
But wearied running round and round in their courses
All break and vanish, and evil gathers head:
Herodias’ daughters have returned again.”
—William Butler Yeats (1865–1939)
“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)
“No punishment has ever possessed enough power of deterrence to prevent the commission of crimes. On the contrary, whatever the punishment, once a specific crime has appeared for the first time, its reappearance is more likely than its initial emergence could ever have been.”
—Hannah Arendt (1906–1975)
“Ours is a culture based on excess, on overproduction; the result is a steady loss of sharpness in our sensory experience. All the conditions of modern life—its material plenitude, its sheer crowdedness—conjoin to dull our sensory faculties.”
—Susan Sontag (b. 1933)