Transition probabilities:
    Y     N
Y  0.4   0.6
N  0.2   0.8

1. (3 points) Calculate the probability that in 4 consecutive days, it rains on days 2 and 4 and it doesn't rain on days 1 and 3. That is, calculate p(NYNY).
2. (2 points) Calculate the probability that in 4 consecutive days, it doesn't rain on day 4, given that it did't rain on day 1 but it rained on days 2 and 3. That is, calculate p(N|NYY).

Transition probabilities:
    C     D
C  0.9   0.1
D  0.1   0.9

Emission probabilities:

    H     T
C  0.5   0.5
D  0.75  0.25

1. (20 points) Follow the Forward and the Backward algorithms by hand for the following observed sequence x = TTHH. Show your work and record intermediate results of the dynamic programming algorithms in tables F (Forward) and B (Backward), as the algorithms would. Note that:
   • the F table would have an additional column 0, and additional row 0, corresponding to the fake Start state.
   • as discussed in class, multiplying small probabilities can create underflow errors. If you do run into underflow errors, redo your calculations in log₂ space (see Prof. Kellis' Algorithms for Computational Biology course (MIT) lecture notes and Prof. Mneimneh's Computational Biology course (Hunter College) lecture notes), or scaling (see Rabiner's tutorial Section V).

2. (5 points) Once that those tables have been completed, calculate the probability that the 3rd hidden state visited (i.e., the state that produced the leftmost H) was C (the fair coin). That is, calculate:

   p(s₃= C | TTHH) = ?

   Remember that p(s₃= C | TTHH) = p(TTHH, s₃=C)/p(TTHH). Don't forget to divide by p(TTHH), whose value you can easily calculate from the Forward table.

• What you need to do:
  Follow the instructions in Prof. Subramanian's homework assignment. Include in your written report answers to all the questions in that homework assignment. Credit points are as shown in that assignment.
  • See Prof. Subramanian's slides on Markov models
  • See Prof. Subramanian's slides on hidden Markov models pages 63-87.

• Data Files:
  Note that the data files have been updated since the date of Prof. Subramanian's assignment (2009).

  • Human chromosome 22 contigs:
    Below are the start, stop, and Symbol information for the 4 contigs in the human chromosome 22 (as of Sept. 2013), taken from the NCBI's human chromosome 22 webpage:

    Region Displayed: 0-51M bp Download/View Sequence/Evidence Download Data
    Total Contigs On Chromosome: 4
    Contigs in Region: 0
    	start 		stop		Symbol		O
    	16050001 	16697850	NT_028395.3	+
    	16847851 	20509431	NT_011519.10	+
    	20609432 	50364777	NT_011520.12	+
    	50414778 	51244566	NT_011526.7	+
    

    • You can upload any of these contigs onto Matlab using the "getgenbank" command as described in the homework assignment. Matlab will fetch the correct NCBI file by itself.
    • If you'd like to look at the original files online, you can find them at by looking for "Contigs" on the NCBI's human chromosome 22 webpage.

  • CpG island locations on the human chromosome 22:
    These locations are contained in the file "ncbimapAnnotation Release 104_22_cpg_0K.txt", which specifies the start and stop location of each CpG island.

    • You can download this file by clicking on the "Download Data" link next to "CpG Island" on the NCBI's CpG island for human chromosome 22 webpage.
    • You can upload these start-stop location pairs onto Matlab using the "[cpgStart,cpgStop]=textread('ncbimapAnnotation Release 104_22_cpg_0K.txt','%d %d','headerlines',6)" command as described in the homework assignment. (Note that you can use the file name as is, without using backslashes '\'.)