Linear Algebra

Calculus I

Calculus II

Introduction to Differential Equations

Probability Statistics for Bioinformatics

Statistical Methods in Bioinformatics

Lecture 13  Occurrences of a Word in a Sequence o Consider a (long) single-stranded nucleotide sequence τ = τ 1 …τ N and a (short) word w = w 1 …w k :  τ = τ 1 …τ 19 = CTATAGATAGATAGACAGT  w = w 1 …w 9 = ATAGATAGA o Say w occurs in τ at position j when w is in τ ending at position j  So w occurs in τ at 11 and 15 o Let I j = { 1 if w occurs ∈ τ at j 0 otherwise  I j is an indicator variable (1 when condition is true, 0 when condition is false) o Y = I k + I k + 1 + … + I N is the number of times w occurs in τ  Here, Y = 2  Computing Mean Number of Occurrences μ = E ( Y ) o Suppose τ is generated by N independent rolls of a 4-sided die, whose sides have probabilities p A ,p C ,p G ,p T adding up to 1 o The probability of a word being generated by rolling such a die is the product of the probabilities of its nucleotides:  π ( w ) = p w 1 ∗ … ∗ p w k o The probability of w occurring at j = k,k + 1 ,…,N is π ( w ) o I j ’s are indicator variables, so:  E ( I j ) = 0 P ( I j = 0 ) + 1 P ( I j = 1 ) = P ( I j = 1 ) = π ( w )  For j = k,k + 1 ,…,N o Y = I k + I k + 1 + … + I N , so:  μ = E ( Y ) = E ( I k ) + … + E ( I N ) = ( N − k + 1 ) ∗ π ( w )  Dependencies Between Positions o Occurrences at different positions have dependencies  If AGAG occurs in τ at position 20, it CANNOT occur at 17, 19, 21, or 23, while the probability of overlapping occurrences at 18 or 22 is increased since 2 letters are already known to be correct in τ  Chapter 5.8 does the related problem of counting occurrences without overlaps  Chapters 4 and 11 do the much more general problem of Markov chains  Self-Overlaps of a Word o Define ϵ r = { 1 if the first r letters of w equal the last r letters of w ∈ the exact same order 0 otherwise o This lets us account for dependencies between I j and I j + k − r   Computing σ 2 = Var ( Y ) o Since the I j ’s have dependencies, the variance of their sum Y = I k + … + I N is NOT necessarily the sum of their variances  We must consider covariance terms as well:  Var ( Y ) = ∑ j = k N Var ( I j ) + 2 ∑ j,l : k≤j < l≤N Cov ( I j ,I l ) o First Sum : Note that I j 2 = I j , since I j = 0 ∨ 1 , so:  Var ( I j ) = E ( I j 2 ) − ( E ( I j ) ) 2 = π ( w ) − π ( w ) 2  And the first sum in Var ( Y ) is:  ∑ j = k N Var ( I j ) = ( N − k + 1 ) ( π ( w ) − π ( w ) 2 ) o The covariances sum is complicated:  If l − j≥k then I j ,I l are independent, and thus Cov ( I j ,I l ) = 0  If 0 < l − j < k , the words ending at l and j overlap by r = k − ( l − j ) = k − l + j letters  Rewrite l as l = j + k − r  Cov ( I j ,I l ) = Cov ( I j ,I j + k − r ) = E ( I j I j + k − r ) − E ( I j ) E ( I j + k − r )  I j I j + k − r = 1 iff w 1 …w k w r + 1 …w k occurs at position j + k − r in τ  E ( I j I j + k − r ) = ϵ r ∗ π ( w 1 …w k w r + 1 …w k )  Cov ( I j ,I j + k − r ) = E ( I j I j + k − r ) − E ( I j ) E ( I j + k − r )  Cov ( I j ,I j + k − r ) = ϵ r ∗ π ( w 1 …w k w r + 1 …w k ) − π ( w ) 2  Note that this depends on r but not j o The covariance sum becomes:  ∑ j,l : k≤j < l≤N Cov ( I j ,I l ) = ∑ r = 1 k − 1 ∑ j = k N − k + r ¿¿¿ a really ugly summation, see the lecture slides for this  Hypothesis Tests on Frequencies in SARS o We have not yet determined the complete distribution of Y  We will assume it is approximately normal with mean and standard deviation as computed above o That lets us compute Z and use it as a test statistic to see if the observed frequencies are consistent with a “random” sequence o Three Possible Hypothesis Tests

MATH | 283 | Statistical Methods in Bioinformatics | lecture notes | Lecture 13  Occurrences of a 

Calculus I © 2007 Paul Dawkins 11 http://tutorial.math.lamar.edu/terms.aspx 1. 2 14sin 5 3 t   - =   L l 2. ( ) ( ) 4cos 4 8 10 cos 4 x x + = - 3. ( ) 2tan 3 3 25 w + = 4. 3 7 1 2sin 5 5 5 x   - =   L l in [ ] 0,15 5. 1 3 8cos 2 w   = +   L l in [ ] 20,5 - 6. 45sin 9 7 sin 17 2 2 x x     - = +     L l L l in [ ] 10, 20 - 7. ( ) 2 4 3sec 11 3 x = - in [ ] 0,1 8. ( ) ( ) 3sin 4 18cos 4 0 v v + = in [ ] 2,5 9. 2 2 2 cos 3 7 cos 6 7 7 t t       + = +       L l L l L l in [ ] 10,30 - 10. 1 10 3 csc 2 3 7 14 y   - =   L l in [ ] 0,32 11. 31 1 40 cos 8 t   = +   L l in [ ] 50, 60 - 12. ( ) ( ) 15csc 15 14 20 12csc 15 x x + = - in [ ] 1, 2 13. ( ) ( ) 1 1 cos 6 3 1 cos 6 2 3 t t + = + in [ ] 0,5 14. 41 2sec 12 5 z     - =     L l L l in [ ] 0,15 Calculus I © 2007 Paul Dawkins 12 http://tutorial.math.lamar.edu/terms.aspx 15. 2 2 11 7 sin 23 19sin 13 13 x x     - = -     L l L l in [ ] 60, 60 - Review : Solving Trig Equations with Calculators, Part II Find all the solution(s) to the following equations. These will require the use of a calculator so use at least 4 decimal places in your work. 1. ( ) 22cos 8 10 0 x - + = 2. ( ) 10 tan 4 10 7 31 x + - = 3. ( ) 4 tan sin 2 tan 0 3 3 w w w     - =     L l L l 4. ( ) ( ) ( ) 3tan 4 sec 2 1 sec 2 1 0 z z z - + - = 5. ( ) ( ) 2 2 sin 2 3sin 2 y y - = 6. ( ) ( ) 2 4 cos 2 5 4cos 2 5 1 t t + - + =- 7. 2 6 5sin 7 sin 4 4 x x     - =     L l L l 8. ( ) ( ) 2 2 2 tan 8 3tan 8 t t = + 9. ( ) ( ) 3 35csc 4 csc 4 z z z = 10. ( ) 3 8 cos 5 t t t = + 11. ( ) 6 5 1 sin 25 5 0 2 y x x -   + + + =   L l 12. ( ) 2 2 4 5 20 8 2 sec 9 w w w   - = -   L l Calculus I © 2007 Paul Dawkins 13 http://tutorial.math.lamar.edu/terms.aspx Review : Exponential Functions Sketch the graphs of each of the following functions. 1. () 3 2 7 t gt - = 2. ( ) 4 1 3 5 x f x + = - 3. ( ) 2 1 6 3 x hx - = - e 4. () 3 2 5 7 9 t f t - = + e Review : Logarithm Functions Without using a calculator determine the exact value of each of the following. 1. 7 log 343 2. 4 log 1024 3. 3 8 27 log 512 4. 11 1 log 121 5. 0.1 log 0.0001 6. 16 log 4 7. log10000 8. 5 1 ln e Calculus I © 2007 Paul Dawkins 14 http://tutorial.math.lamar.edu/terms.aspx Write each of the following in terms of simpler logarithms 9. ( ) 7 3 8 7 log 10 abc - 10. ( ) 3 2 2 log 4 z x Ø ø + Œ œ º ß 11. 2 3 4 ln w t t w       + L l Combine each of the following into a single logarithm with a coefficient of one. 12. 7 ln 6 ln 5ln t s w - + 13. ( ) 1 log 1 2 log 4 log 3log 2 z x y z + - - - 14. ( ) 3 3 1 2 log 6 log 3 x y x + + - Use the change of base formula and a calculator to find the value of each of the following. 15. 7 log 100 16. 5 7 1 log 8 Review : Exponential and Logarithm Equations For problems 1 – 14 find all the solutions to the given equation. If there is no solution to the equation clearly explain why. 1. 10 7 15 12 5 w - = + e 2. 2 2 4 7 2 x x + - = e 3. 49 8 3 1 z - + = e

Math 20F A00, Linear Algebra, Winter 2012 Midterm Exam 2 Review Midterm2 will cover section 2.1-2.3, 3.1-3.3, 4.1-4.6, close-book, close-notes. The cheat sheet is not allowed. • Matrix operations : addition, scalar multiple, matrix multiplication (row-column rule for computing AB), transpose and inverse • The inverse of a matrix A . 1. If A is a 2 × 2 matrix • a b c d ‚ and ad - bc 6 = 0, use the formula in Thm 4 in section 2.2 (pg 103) to compute the inverse, i.e., A - 1 = 1 ad - bc • d - b - c a ‚ 2. If A is a 3 × 3 matrix, use the formula derived by Cramer’s rule, i.e., Thm 8 in section 3.3 (pg 179) 3. For larger matrix A , do row reduction to make [ A | I ] ∼ [ I | B ], then matrix B = A - 1 . • Determinant 1. For a 2 × 2 or 3 × 3 matrix, there is a trick to compute the determinant. 2. The determinant of a (upper or lower) triangle matrix is the product of the entries on the main diagonal. 3. Remember how elementary row operations change determinant, which can be used to compute the determinant of any matrix by transforming it to triangle matrix. 4. For large matrices, if it contains many “0”, then use co-factor expansion across one row or one column that contains most 0. Pay attention to the plus or minus sign as in a checkerboard pattern. 5. For a structured matrix, ﬁrst guess the determinants of a few smaller matrices and prove it’s true for any size by principle of induction. • Vector space : 1. A subspace is a subset of vector space, which contains zero vector and is closed under vector addition and multiplication by scalars. 2. Important subspaces and how to ﬁnd bases: for a m × n matrix A and B is row equivalent to A in echelon form. Null space is a subspace of R n . Computing the solution of A x = 0 in the parametric vector form will produce a spanning set for Nul A ; Column space is a subspace of R m . The pivot columns of matrix A form a basis for Col A ; Row space is a subspace of R n . The non-zero rows of B form a basis for Row A . 3. A basis for subspace H is a linearly independent set that spans H . Suppose dim H = n . If a set of vectors satisfy two out of (1) linearly independent, (2) spans H and (3) contains n vectors, then it is a basis for H 1 4. Coordinate mapping: is a one-to-one linear transformation from V onto R n . Its applica- tion: the vector space P n (polynomial of degree at most n) is isomorphic to R n +1 . The linear dependency in P n can be veriﬁed by matrix algebra. Show some polynomials form a basis and ﬁnd the coordinate vector relative to this basis. 5. Rank: for a m × n matrix A Rank A = dimCol A = dimRow A = Rank A T 6 min( m, n ) Rank A + dimNul A = n 2

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