Bivariate scatter plots and densities

Transcription

1 Multivariate descriptive statistics Bivariate scatter plots and densities Plotting two (related) variables is often called a scatter plot. It is a bivariate version of the rug plot. It can show something about the dependence between the variables. In R, simply plot a data frame and you will get all pairwise scatter plots. The point density in a small area of the plane is an estimate of the density in that area. Various bivariate kernel methods can be used to provide smooth estimates of the density. Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

2 Expectation of positive variables Theorem (Fact) There exists an expectation operator E that assigns to any positive random variable X a number EX [0, ] called the expectation or mean of X, such that: 1 If X is a Bernoulli random variable then EX = P(X = 1). (1) 2 If c 0 is a positive constant and X is a positive random variable then E(cX )=cex. (2) 3 If X and Y are two positive random variables then E(X + Y )=EX + EY. (3) Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

3 Word counts Word counts Let X 1,..., X n be iid random variables from E = {A, C, G, T}. Let w = w 1 w 2... w m denote a word, and define N = n m+1 1(X i X i+1... X i+m 1 = w). N is the number of times word w occurs in the sequence. It follows from the facts that the expectation of N is EN = = n m+1 n m+1 E1(X i X i+1... X i+m 1 = w) P(X i X i+1... X i+m 1 = w) Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

4 Word counts Word counts Since the random variables are independent we have with p(a), p(c), p(g) and p(t) the point probabilities for the distribution of the X -variables P(X i X i+1... X i+m 1 = w) = P(X i = w 1 )P(X i+1 = w 2 )... P(X i+m 1 = w m ) = p(w 1 )p(w 2 )... p(w m ) = p(a) nw (A) p(c) nw (C) p(g) nw (G) nw (T) p(t) where n w (A), n w (C), n w (G) and n w (T) are the number of A s, C s, G s and T s in w Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

5 Word counts Word counts It follows that EN =(n m + 1)p(A) nw (A) p(c) nw (C) p(g) nw (G) p(t) nw (T). Taking the logarithm yields that log EN = log(n m+1)+n w (A) log p(a)+n w (C) log p(c)+n w (G) log p(g)+n w (T) log p(t). This suggests a strategy for how the relative frequencies of nucleotides may affect general patterns of words that the log-expectation should be a linear combination of log-probabilities. Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

6 Expectation of real valued random variables Define X + = max{x, 0} and X = max{ X, 0}, as the positive and negative part of X. Then and we will then define X = X + X EX = E(X + X )=EX + EX. (4) However, both terms EX + and EX need to be finite to subtract them. Since X = X + + X. and X is a positive random variable E X exists and E X = EX + + EX. Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

7 Expectation of real valued random variables Since E X = EX + + EX the sum is finite if and only if both terms are finite. This leads us to the definition: Definition If X is a real valued random variable we say that it has finite expectation if E X <. In this case the expectation of X, EX, is well defined by EX = EX + EX. Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

8 Expectation of real valued variables A messy and uninteresting derivation shows the following: Theorem If X and Y are two real valued random variables with finite expectation then X + Y has finite expectation and E(X + Y )=EX + EY. Furthermore, if c R is a real valued constant then cx has finite expectation and E(cX )=cex. Moreover, if X and Y are independent real valued random variables with finite expectation then E(XY )=EX EY. Although the derivation is not interesting the result is extremely useful. Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

9 Expectations and transformations But how do we compute the expectation? And what is the relation to the previously defined mean? The computational tools at our disposal are Theorem If X is a real valued random variable with density f and finite expectation then EX = xf (x)dx. If X is a discrete random variable taking values in E R with point probabilities (p(x)) x E and finite expectation then EX = x E xp(x). Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

10 Expectations and transformations If X is a random variable with values in a discrete sample space E and h : E R is a function Example gives that Eh(X )= x E h(x)p(x) where p(x) for x E are the point probabilities for the distribution of X. If X is a random variable with values in R n and h : R n R is a function Eh(X ) = h(x)f (x)dx = h(x 1,..., x n )f (x 1,..., x n )dx 1 dx n. }{{} n where f : R n [0, ) is the density for the distribution of X. Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

11 Expectations and transformations But can we always just mindlessly use these formulas? What about finiteness of the expectation? In all cases the expectation of h(x ) needs to be finite for these formulas to be meaningful. You can check finiteness by checking that either h(x) p(x) < or x E h(x) f (x)dx = h(x 1,..., x n ) f (x 1,..., x n )dx 1 dx n < }{{} n Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

12 Expectations and transformations How do we compute the expectation of h(x ) in practice? First try to use the fundamental rules for the expectation operator. This may not in all cases give the final result but can reduce technical computations to a minimum. Try computing the result using the computational formulas for the expectation of a transformed random variable. The last thing to try is to find the distribution of h(x ) and compute the expectation from first principle. Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

13 Binomial distribution Let X 1,..., X n denote n independent Bernoulli random variables with P(X i = 1) = p. The Binomial distribution is by definition the distribution of the transformed random variable S = n X i. The distribution of S has point probabilities (which have to be shown) ( ) n p(k) = p k (1 p) n k, k =0,..., n k which from first principle leaves us with the formula ES = n ( ) n k p k (1 p) n k. k k=0 Albeit one can compute the sum, it is not the easiest path. Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

14 Binomial distribution To compute ES = Eh(X ) where X =(X 1,..., X n ) and h : {0, 1} n R is we can also use the formula Eh(X ) = = = = h(x) =h(x 1,..., x n )= n h(x 1,..., x n )p x 1 (1 p) 1 x1... p xn (1 p) 1 xn x 1,...,x n n x i p x 1 (1 p) 1 x1... p xn (1 p) 1 xn x 1,...,x n n x i p x i (1 p) 1 x i x i n p = np x i Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

15 Binomial distribution The easiest computation is found as ( n ) E X i = = = n EX i n P(X i = 1) n p = np Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16

16 Higher order moments Definition We say that a real valued random variable X has finite k th moment if X k has finite expectation and we call EX k the k th moment. If X has finite k th moment we call the k th central moment. E(X EX ) k For k = 2 the second central moment is called the variance and usually written VX = E(X EX ) 2. Niels Richard Hansen (Univ. Copenhagen) Statistics BI/E lecture December 3, / 16