Relation to the varieties of the components— it cannot exceed their
Sum (if we think in logarithms, as is more convenient here). Thus,
If a car may have any one of 10 ages, of 8 horse-powers, and of 12
colours, then the variety in the types of car cannot exceed 3.3 +
3.0 + 3.6 bits, i.e. 9.9 bits.
The components are independent when the variety in the
Whole of some given set of vectors equals the sum of the (logarith-
Mic) varieties in the individual components. If it were found, for
Instance, that all 960 types of car could be observed within some
128
Defined set of cars, then the three components would be said to be
“independent”, or to “vary independently”, within this defined set.
It should be noticed that such a statement refers essentially to
What is observed to occur within the set; it need contain no refer-
Ence to any supposed cause for the independence (or for the con-
Straint).
Ex. 1: When Pantagruel and his circle debated whether or not the time had come
For Panurge to marry, they took advisers, who were introduced thus: “...
Rondibilis, is married now, who before was not— Hippothadeus was not
Before, nor is yet— Bridlegoose was married once, but is not now— and
Trouillogan is married now, who wedded was to another wife before.” Does
This set of vectors show constraint ?
Ex. 2: If each component can be Head (H) or Tail (T), does the set of four vectors
H,H,H), (T,T,H), (H,T,T), (T,H,T) show constraint in relation to the set
Showing independence ?
Degrees of freedom. When a set of vectors does not show the
Full range of possibilities made available by the components (S.7/
The range that remains can sometimes usefully be measured
By saying how many components with independence would give
The same variety. This number of components is called the degrees
Of freedom of the set of vectors. Thus the traffic lights (S.7/8)
Show a variety of four. If the components continued to have two
States apiece, two components with independence could give the
Same variety (of four). So the constraint on the lights can be
Expressed by saying that the three components, not independent,
Give the same variety as two would if independent; i.e. the three
|
|
|
Lights have two degrees of freedom.
If all combinations are possible, then the number of degrees of
Freedom is equal to the number of components. If only one com-
Bination is possible, the degrees of freedom are zero.
It will be appreciated that this way of measuring what is left free
Of constraint is applicable only in certain favourable cases. Thus,
Were the traffic lights to show three, or five combinations, the
Equivalence would no longer be representable by a simple, whole,
Number. The concept is of importance chiefly when the compo-
Nents vary continuously, so that each has available an infinite
Number of values. A reckoning by degrees of freedom may then
Still be possible, though the states cannot be counted.
Ex. 1: If a dealer in second-hand cars boasts that his stock covers a range of 10
Ages, 8 horse powers, and 12 colours, in all combinations, how many degrees
Of freedom has his stock ?
Ex. 2: The angular positions of the two hands on a clock are the two components
Of a vector. Has the set of vectors (in ordinary working round the 12 hours)
A constraint if the angles are measured precisely?
129
A N I N T R O D UC T I O N T O C Y B E R NE T I C S
Q UA N TI TY O F V AR IE TY
Ex. 3: (Continued.) How many degrees of freedom has the vector? (Hint: Would
Removal of the minute-hand cause an essential loss ?)
Ex. 4: As the two eyes move, pointing the axes in various directions, they define
A vector with four components: the upward and lateral deviations of the right
And left eyes. Man has binocular vision; the chameleon moves his two eyes
Independently, each side searching for food on its own side of the body. How
Many degrees of freedom have the chameleonts eyes ? Man’s ?
Ex. 5: An arrow, of fixed length, lying in a plane, has three degrees of freedom
|
|
|
For position (for two co-ordinates will fix the position of its centre, say, and
Then one angle will determine its direction). How many degrees of freedom
Has it if we add the restriction that it must always point in the direction of a
Given point P?
Ex. 6: T is a given closed and single-valued transformation, and a any of its oper-
Дата добавления: 2019-11-16; просмотров: 232; Мы поможем в написании вашей работы! |
Мы поможем в написании ваших работ!