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T H E B O U L EV A RD O F B RO KE N G E N E S : H I DD E N M A R KOV M O D E L S in .NET Drawer QR in .NET T H E B O U L EV A RD O F B RO KE N G E N E S : H I DD E N M A R KOV M O D E L S

T H E B O U L EV A RD O F B RO KE N G E N E S : H I DD E N M A R KOV M O D E L S using barcode generating for .net framework control to generate, create qrcode image in .net framework applications. Recommended GS1 barcodes for mobile apps 0.10 0.90 0.

10 1 (0.16 visual .net QR-Code 67) 2 (0.

1667) 3 (0.1667) 4 (0.1667) 5 (0.

1667) 6 (0.1667) 1 (0.1000) 2 (0.

1000) 3 (0.1000) 4 (0.1000) 5 (0.

1000) 6 (0.5000) 0.90.

symbols can range betw een two and many; the most common number in sequence analysis is either four (for DNA or RNA) or 20 (for amino acids). We now give a simple (and classic) example of a 2-state HMM to explore some of the basic ideas..

Fig. 4.1 The HMM assoc Quick Response Code for .

NET iated with Example 4.1 : transitions between fair and loaded dice are modeled as a Markov chain, and outcomes of rolls as independent emissions of a multinomial model. Example 4.1 Switching between fair and loaded dice. Imagine that we have a pair of dice, one that is fair, but the other is loaded (i.

e. it does not roll each side with equal probability). A Markov model decides which one of the two dice to roll, and depending on the state of the model either the emission probabilities for the fair or the loaded die is used.

In this way we generate a sequence of symbols that is the result of being in two different states. We will specify our transition parameter so that in either of the two states there is a 90% chance of remaining in that state, and a 10% chance of changing states, as shown in Figure 4.1.

For our fair die, the probability of rolling the numbers between 1 and 6 is equal and is given by:. 0.1667 0.1667 0.

1667 0 QR Code for .NET .1667 0.

1667 0.1667,. where each column desc ribes the probability for each of the six numbers; this is our multinomial distribution. For our loaded die, the emission probabilities are:. 0.1000 0.1000 0.

1000 0 QR Code 2d barcode for .NET .1000 0.

1000 0.5000..

Here the probabilities of rolling 1 5 are still equal, but there is a much higher probability of rolling a 6 (50%). The visible sequence produced by such an HMM might look like this: s = 4553653163363555133362665132141636651666. If we know the properties of the two dice and of the underlying Markov chain, can we nd the most likely sequence of hidden states behind it In other words, can we guess which die was used at each time point in the sequence This is a task referred to as segmentation (and what we earlier referred to as change point analysis in 1).

Later in this chapter we will describe how to infer these hidden states; here we will simply show you the hidden sequence that generated our visible sequence:. Hidden: Visible: h = 1 .net framework Denso QR Bar Code 111111111111111111122221111111222222222 s = 4553653163363555133362665132141636651666..

Notice that the symbol 6 occurs with high frequency when the sequence is in hidden state 2, corresponding to the loaded die, but this dependency is just probabilistic. Often in biological applications we will train our HMM on one set of data where the hidden states are known, even when we do not know the exact transition and emission probabilities. This allows our model to calculate the most likely transition and emission matrices based on data, and so to better infer the hidden states on novel data.

This is especially useful in gene nding where models are often trained on a well-annotated genome and then run on related genomes.. 4 . 2 H I DD E N M A R KOV M O D E L S Computing with HMMs. I t is an important feature of HMMs that the probability of a sequence (the likelihood of a model) is easily computed, as are all the fundamental quantities necessary to use them. The ef cient algorithms that made HMMs popular are based on the same dynamic programming principles behind global and local alignment methods discussed in 3 and will be discussed in depth separately in Section 4.

6. Here we simply review the basics of HMM computations and the parameters used in them. As a strict generalization of multinomial and Markov models, we have already seen that HMMs need to maintain parameters for both a Markov transition matrix and a number of multinomial models (one for each state of the system).

These parameters are generally presented as matrices, one for transitions, T , and one for emissions, E. The transition matrix has dimension N N where N is the size of the hidden alphabet, H (i.e.

the number of hidden states). The emission matrix has dimension N M, where M is the size of the observable alphabet (typically the nucleotide, N , or amino acid, A, alphabet). They are de ned as.

T (k, l) = P(h i = l h i 1 = k) E(k, b) = P(si = b h i = k).. In words: the probabil ity of being in hidden state l, given that the previous position was in state k, is given in the transition matrix by entry T (k, l). And the probability of emitting symbol b is determined by the multinomial model associated with state k, which is given in the emission matrix by entry E(k, b). We denote the sequence of hidden states created by the Markov process by h, and the sequence of symbols generated by the HMM by s.

We will assume the sequences have length n. In order for the model to be fully speci ed, we also need to declare the initial probabilities for the state of the Markov process, denoted T (0, k) = P(h 1 = k)..

Basic quantities compu qr codes for .NET table with HMMs. We would now like to make inferences about genome sequences with HMMs.

The most common task will be to infer the hidden states of the genome in order to better annotate it, or to better understand its dynamics. Our end goal, therefore, will be to nd hidden sequence with the highest likelihood; in order to do this, we must rst understand how to calculate probabilities associated with HMMs. The stochastic process assumed to be generating the sequence is the following: a hidden sequence is generated by the Markov process.

Then in each different state, a different multinomial model is used according to the emission parameters associated with that state to produce the observable sequence. This means that, for each position of the hidden sequence, the system emits a visible symbol independently from the appropriate multinomial distribution. The probabilities of these sequences can hence be written as.

P(h) = P(h 1 ).
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