Smith–Waterman algorithm

In probability theory, a normalized fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion without independent increments. It is a continuous-time Gaussian process B_H(t) on [0, T], which starts at zero, has expectation zero for all t in [0, T], and has the following covariance function:

$E[B_{H}(t)B_{H}(s)]={\tfrac {1}{2}}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}),$

where H is a real number in (0, 1), called the Hurst index or Hurst parameter associated with the fractional Brownian motion. The Hurst exponent describes the raggedness of the resultant motion, with a higher value leading to a smoother motion. It was introduced by Template:Harvtxt.

The value of H determines what kind of process the fBm is:

• if H = 1/2 then the process is in fact a Brownian motion or Wiener process;
• if H > 1/2 then the increments of the process are positively correlated;
• if H < 1/2 then the increments of the process are negatively correlated.

The increment process, X(t) = B_H(t+1) − B_H(t), is known as fractional Gaussian noise.

Like the Brownian motion that it generalizes, fractional Brownian motion is named after 19th century biologist Robert Brown, and fractional Gaussian noise is named after mathematician Carl Friedrich Gauss.

Background and definition

Prior to the introduction of the fractional Brownian motion, Template:Harvtxt used the Riemann–Liouville fractional integral to define the process

$X^{H}(t)={\frac {1}{\Gamma (H+1/2)}}\int _{0}^{t}(t-s)^{H-1/2}dB(s)$

where integration is with respect to the white noise measure dB(s). This integral turns out to be ill-suited to applications of fractional Brownian motion because of its over-emphasis of the origin Template:Harv.

The idea instead is to use a different fractional integral of white noise to define the process: the Weyl integral

$B_{H}(t)=B_{H}(0)+{\frac {1}{\Gamma (H+1/2)}}\left\{\int _{-\infty }^{0}\left[(t-s)^{H-1/2}-(-s)^{H-1/2}\right]\,dB(s)+\int _{0}^{t}(t-s)^{H-1/2}\,dB(s)\right\}$

for t > 0 (and similarly for t < 0).

The main difference between fractional Brownian motion and regular Brownian motion is that while the increments in Brownian Motion are independent, the opposite is true for fractional Brownian motion. This dependence means that if there is an increasing pattern in the previous steps, then it is likely that the current step will be increasing as well. (If H > 1/2.)

Properties

Self-similarity

The process is self-similar, since in terms of probability distributions:

$B_{H}(at)\sim |a|^{H}B_{H}(t).$

This property is due to the fact that the covariance function is homogeneous of order 2H and can be considered as a fractal property. Fractional Brownian motion is the only self-similar Gaussian process.

Stationary increments

It has stationary increments:

$B_{H}(t)-B_{H}(s)\;\sim \;B_{H}(t-s).$

Long-range dependence

For H > ½ the process exhibits long-range dependence,

$\sum _{n=1}^{\infty }E[B_{H}(1)(B_{H}(n+1)-B_{H}(n))]=\infty .$

Regularity

Sample-paths are almost nowhere differentiable. However, almost-all trajectories are Hölder continuous of any order strictly less than H: for each such trajectory, there exists a constant c such that

$|B_{H}(t)-B_{H}(s)|\leq c|t-s|^{H-\varepsilon }$

for every ε > 0.

Dimension

With probability 1, the graph of B_H(t) has both Hausdorff dimension and box dimension of 2−H.Template:Cn

Integration

As for regular Brownian motion, one can define stochastic integrals with respect to fractional Brownian motion, usually called "fractional stochastic integrals". In general though, unlike integrals with respect to regular Brownian motion, fractional stochastic integrals are not Semimartingales.

Sample paths

Practical computer realisations of an fBm can be generated, although they are only a finite approximation. The sample paths chosen can be thought of as showing discrete sampled points on an fBm process. Three realisations are shown below, each with 1000 points of an fBm with Hurst parameter 0.75.

"H" = 0.75 realisation 1

"H" = 0.75 realisation 2

"H" = 0.75 realisation 3

Two realisations are shown below, each showing 1000 points of an fBm, the first with Hurst parameter 0.95 and the second with Hurst parameter 0.55.

"H" = 0.95

"H" = 0.55

Method 1 of simulation

One can simulate sample-paths of an fBm using methods for generating stationary Gaussian processes with known covariance function. The simplest method relies on the Cholesky decomposition method of the covariance matrix (explained below), which on a grid of size $n$ has complexity of order $O(n^{3})$. A more complex, but computationally faster method is the circulant embedding method of Dietrich and Newsam Template:Harvtxt, which is implemented in Matlab.

Suppose we want to simulate the values of the fBM at times $t_{1},\ldots ,t_{n}$ using the Cholesky decomposition method.

• Form the matrix $\Gamma ={\bigl (}R(t_{i},\,t_{j}),i,j=1,\ldots ,\,n{\bigr )}$ where $\,R(t,s)=(s^{2H}+t^{2H}-|t-s|^{2H})/2$.

• Compute $\,\Sigma$ the square root matrix of $\,\Gamma$, i.e. $\,\Sigma ^{2}=\Gamma$. Loosely speaking, $\,\Sigma$ is the "standard deviation" matrix associated to the variance-covariance matrix $\,\Gamma$.

• Construct a vector $\,v$ of n numbers drawn according a standard Gaussian distribution,

• If we define $\,u=\Sigma v$ then $\,u$ yields a sample path of an fBm.

In order to compute $\,\Sigma$, we can use for instance the Cholesky decomposition method. An alternative method uses the eigenvalues of $\,\Gamma$:

• Since $\,\Gamma$ is symmetric, positive-definite matrix, it follows that all eigenvalues $\,\lambda _{i}$ of $\,\Gamma$ satisfy $\,\lambda _{i}\geq 0$, ($i=1,\dots ,n$).

• Let $\,\Lambda$ be the diagonal matrix of the eigenvalues, i.e. $\Lambda _{ij}=\lambda _{i}\,\delta _{ij}$ where $\delta _{ij}$ is the Kronecker delta. We define $\Lambda ^{1/2}$ as the diagonal matrix with entries $\lambda _{i}^{1/2}$, i.e. $\Lambda _{ij}^{1/2}=\lambda _{i}^{1/2}\,\delta _{ij}$.

Note that the result is real-valued because $\lambda _{i}\geq 0$.

• Let $\,v_{i}$ an eigenvector associated to the eigenvalue $\,\lambda _{i}$. Define $\,P$ as the matrix whose $i$-th column is the eigenvector $\,v_{i}$.

Note that since the eigenvectors are linearly independent, the matrix $\,P$ is inversible.

• It follows then that $\Sigma =P\,\Lambda ^{1/2}\,P^{-1}$ because $\Gamma =P\,\Lambda \,P^{-1}$.

Method 2 of simulation

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$B_{H}(t)=\int _{0}^{t}K_{H}(t,s)\,dB(s)$

where B is a standard Brownian motion and

$K_{H}(t,s)={\frac {(t-s)^{H-{\frac {1}{2}}}}{\Gamma (H+{\frac {1}{2}})}}\;_{2}F_{1}\left(H-{\frac {1}{2}};\,{\frac {1}{2}}-H;\;H+{\frac {1}{2}};\,1-{\frac {t}{s}}\right).$

Where $_{2}F_{1}$ is the Euler hypergeometric integral.

Say we want simulate an fBm at points $0=t_{0}<t_{1}<\cdots <t_{n}=T$.

• Construct a vector of n numbers drawn according a standard Gaussian distribution.

• Multiply it component-wise by √(T/n) to obtain the increments of a Brownian motion on [0, T]. Denote this vector by $(\delta B_{1},\ldots ,\delta B_{n})$.

• For each $t_{j}$, compute

$B_{H}(t_{j})={\frac {n}{T}}\sum _{i=0}^{j-1}\int _{t_{i}}^{t_{i+1}}K_{H}(t_{j},\,s)\,ds\ \delta B_{i}.$

The integral may be efficiently computed by Gaussian quadrature. Hypergeometric functions are part of the GNU scientific library.

See also

• Brownian surface
• Autoregressive fractionally integrated moving average
• Multifractal: The generalized framework of fractional Brownian motions.
• Pink noise
• Tweedie distributions

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Template:Cite thesis
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Samorodnitsky G., Taqqu M.S. (1994), Stable Non-Gaussian Random Processes, Chapter 7: "Self-similar processes" (Chapman & Hall).

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