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Consider the following setting. Let

Y = f * ( X ) + W ,

where X is a random variable (r.v.) on X = [ 0 , 1 ] , W is a r.v. on Y = R , independent of X and satisfying

E [ W ] = 0 and E [ W 2 ] = σ 2 < .

Finally let f * : [ 0 , 1 ] R be a function satisfying

| f * ( t ) - f * ( s ) | L | t - s | , t , s [ 0 , 1 ] ,

where L > 0 is a constant. A function satisfying condition [link] is said to be Lipschitz on [ 0 , 1 ] . Notice that such a function must be continuous, but it is not necessarilydifferentiable. An example of such a function is depicted in [link] (a).

Example of a Lipschitz function, and our observations setting. (a) random sampling of f * , the points correspond to ( X i , Y i ) , i = 1 , ... , n ; (b) deterministic sampling of f * , the points correspond to ( i / n , Y i ) , i = 1 , ... , n .

Note that

E [ Y | X = x ] = E [ f * ( X ) + W | X = x ] = E [ f * ( x ) + W | X = x ] = f * ( x ) + E [ W ] = f * ( x ) .

Consider our usual setup: Estimate f * using n training examples

{ X i , Y i } i = 1 n i . i . d . P X Y , Y i = f * ( X i ) + W i , i = { 1 , ... , n } ,

where i . i . d . means independently and identically distributed . [link] (a) illustrates this setup.

In many applications we can sample X = [ 0 , 1 ] as we like, and not necessarily at random. For example we can take n samples uniformly on [0,1]

x i = i n , i = 1 , ... , n , Y i = f ( x i ) + W i = f i n + W i .

We will proceed with this setup (as in [link] (b)) in the rest of the lecture.

Our goal is to find f ^ n such that E [ f * - f ^ n 2 ] 0 , as n 0 (here · is the usual L 2 -norm; i.e., f * - f ^ n 2 = 0 1 | f * ( t ) - f ^ n ( t ) | 2 d t ).

Let

F = { f : f is Lipschitz with constant L } .

The Riskis defined as

R ( f ) = f * - f 2 = 0 1 | f * ( t ) - f ( t ) | 2 d t .

The Expected Risk (recall that ourestimator f ^ n is based on { x i , Y i } and hence is a r.v.) is defined as

E [ R ( f ^ n ) ] = E [ f * - f ^ n 2 ] .

Finally the Empirical Riskis defined as

R ^ n ( f ) = 1 n i = 1 n f i n - Y i 2 .

Let 0 < m 1 m 2 m 3 be a sequence of integers satisfying m n as n , and k n m n = n for some integer k n > 0 . That is, for each value of n there is an associated integer value m n . Define the Sieve F 1 , F 2 , F 3 , ... ,

F n = f : f ( t ) = j = 1 m n c j 1 { j - 1 m n t < j m n } , c j R .

F n is the space of functions that are constant on intervals

I j , m n j - 1 m n , j m n , j = 1 , ... , m n .

From here on we will use m and k instead of m n and k n (dropping the subscript n ) for notational ease. Define

f n ( t ) = j = 1 m c j * 1 { t I j , m } , where c j * = 1 k i : i n I j , m f * i n .

Note that f n F n .

Exercise 1

Upper bound f * - f n 2 .

f * - f 2 = 0 1 | f * ( t ) - f n ( t ) | 2 d t = j = 1 m I j , m | f * ( t ) - f n ( t ) | 2 d t = j = 1 m I j , m | f * ( t ) - c j * | 2 d t = j = 1 m I j , m f * ( t ) - 1 k i : i n I j , m f * i n 2 d t = j = 1 m I j , m 1 k i : i n I j , m f * ( t ) - f * i n 2 d t j = 1 m I j , m 1 k i : i n I j , m f * ( t ) - f * i n 2 d t j = 1 m I j , m 1 k i : i n I j , m L m 2 d t = j = 1 m I j , m L m 2 d t = j = 1 m 1 m L m 2 = L m 2 .

The above implies that f * - f n 2 0 as n , since m = m n as n . In words, with n sufficiently large we can approximate f * to arbitrary accuracy using models in F n (even if the functions we are using to approximate f * are not Lipschitz!).

For any f F n , f = j = 1 m c j 1 { t I j , m } , we have

R ^ n ( f ) = 1 n j = 1 m i : i n I j , m ( c j - Y i ) 2 .

Let f ^ n = arg min f F n R ^ n ( f ) . Then

f ^ n ( t ) = j = 1 m c ^ j 1 { t I j , m } , where c ^ j = 1 k i : i n I j , m Y i

Exercise 2

Show [link] .

Note that E [ c ^ j ] = c j * and therefore E [ f ^ n ( t ) ] = f n ( t ) . Lets analyze now the expected risk of f ^ n :

E [ f * - f ^ n 2 ] = E [ f * - f n + f n - f ^ n 2 ] = f * - f n 2 + E [ f n - f ^ n 2 ] + 2 E [ f * - f n , f n - f ^ n ] = f * - f n 2 + E [ f n - f ^ n 2 ] + 2 f * - f n , E [ f n - f ^ n ] = f * - f n 2 + E [ f n - f ^ n 2 ] ,

where the final step follows from the fact that E [ f ^ n ( t ) ] = f n ( t ) . A couple of important remarks pertaining the right-hand-side of equation [link] : The first term, f * - f n 2 , corresponds to the approximation error, and indicates how well can we approximate the function f * with a function from F n . Clearly, the larger the class F n is, the smallest we can make this term. This term is precisely the squared bias of the estimator f ^ n . The second term, E [ f n - f ^ n 2 ] , is the estimation error, the variance of our estimator. We will see that the estimation erroris small if the class of possible estimators F n is also small.

The behavior of the first term in [link] was already studied. Consider the other term:

E [ f n - f ^ n 2 ] = E 0 1 | f n ( t ) - f ^ n ( t ) | 2 d t = E j = 1 m I j , m | c j * - c ^ j | 2 d t = j = 1 m I j , m E [ | c j * - c ^ j | 2 ] d t = j = 1 m I j , m E [ W 2 ] k d t j = 1 m I j , m σ 2 k d t = j = 1 m 1 m σ 2 k = σ 2 k = m n σ 2 .

Combining all the facts derived we have

E [ f * - f ^ n 2 ] L 2 m 2 + m n σ 2 = O max 1 m 2 , m n .

This equation used Big-O notation.

What is the best choice of m ? If m is small then the approximation error ( i.e., O ( 1 / m 2 ) ) is going to be large, but the estimation error ( i.e., O ( m / n ) ) is going to be small, and vice-versa. This two conflicting goals provide a tradeoff that directs our choice of m (as a function of n ). In [link] we depict this tradeoff. In [link] (a) we considered a large m n value, and we see that the approximation of f * by a function in the class F n can be very accurate (that is, our estimate will have a small bias), but when we use the measured dataour estimate looks very bad (high variance). On the other hand, as illustrated in [link] (b), using a very small m n allows our estimator to get very close to the best approximating function in the class F n , so we have a low variance estimator, but the bias of our estimator ( i.e., the difference between f n and f * ) is quite considerable.

Approximation and estimation of f * (in blue) for n = 60 . The function f n is depicted in green and the function f ^ n is depicted in red. In (a)we have m = 60 and in (b) we have m = 6 .

We need to balance the two terms in the right-hand-side of [link] in order to maximize the rate of decay (with n ) of the expected risk. This implies that 1 m 2 = m n therefore m n = n 1 / 3 and the Mean Squared Error (MSE) is

E [ f n - f ^ n 2 ] = O ( n - 2 / 3 ) .

So the sieve F 1 , F 2 , with

F n = f : f ( t ) = j = 1 m n c j 1 { j - 1 m n t < j m n } , c j R ,

produces a F -consistent estimator for f * = E [ Y | X + x ] F .

It is interesting to note that the rate of decay of the MSE we obtainwith this strategy cannot be further improved by using more sophisticated estimation techniques (that is, n - 2 / 3 is the minimax MSE rate for this problem). Also, rather surprisingly, we are considering classes of models F n that are actually not Lipschitz, therefore our estimator of f * is not a Lipschitz function, unlike f * itself.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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Graphene has a hexagonal structure
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Source:  OpenStax, Statistical learning theory. OpenStax CNX. Apr 10, 2009 Download for free at http://cnx.org/content/col10532/1.3
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