2008年11月30日 星期日

Master equation

Consider the homogenious continuous Markov process Y(t) with transition probability:
p_{\tau}(y_2|y_1)=\Pr(Y(\tau)=y_2|Y(0)=y_1)
The Chapman-Kolmogorov equation
p_{t+\tau}(y_3|y_1)=\int p_{\tau}(y_3|y_2)p_{t}(y_2|y_1) dy_2
tells us how to find the transition probability at time t+tau.

Consider the change of state in a very short time interval h, either Y(t) stays at its present state or it changes to another state. If the state changes from y1 to y2, we can approximate the transition probability as a linear function of h:
p_{h}(y_2|y_1)=W(y_2|y_1)h
The coefficient W is the probability of transition per unit time. It's obvious that the probability that the state didn't change is the remaining probability:
p_{h}(y_1|y_1)=1-h\int_{y_2\neq y_1} W(y_2|y_1) dy_2=1-ha(y_1)
Therefore we can write the transition probability as
p_{t}(y_2|y_1)=[1-a(y_1)h]\delta(y_2-y_1)+hW(y_2|y_1)

Denotes the probability distribution at time t by \pi_t(y). By total probability:
\pi_t(y)=\int p_t(y|x)\pi_0(x)dx
Now we can rewrite the Chapman-Kolomogorov equation
\begin{align*} p_{t+h}(y_3|y_1)&=\int p_{t}(y_2|y_1) p_{h}(y_3|y_2)dy_2\\ &=\int p_{t}(y_2|y_1)\left[ (1-a(y_2)h)\delta(y_3-y_2)+hW(y_3|y_2) \right]dy_2\\ &=hp_t(y_3|y_1)(1-a(y_3))+h\int p_t(y_2|y_1)W(y_3|y_2)dy_2 \end{align*}
differentiate it with respect to h, we get
\partial_t p_t(y_3|y_1) = \int [W(y_3|y_2)p_t(y_2|y_1)-W(y_2|y_3)p_t(y_3|y_1)] dy_2
This is the master equation.

If we assume the initial distribution to be a delta function \pi_0(y)=\delta(y-y_0), then the solution of the distribution at time t can be found firstly by total probability
\pi_t(y)=\int p_t(y|x)\pi_0(x)dx
and then apply master equation to get
\partial_t\pi_t(y)=\int [W(y|y')\pi_t(y')-W(y'|y)\pi_t(y)]dy'
This equation has prominent physical interpretation, it's a gain-loss equation in the sense that the change of probability Y(t)=y is sum of the probability that y' transfer to y substract the probability that Y(t) leaves y to y'.

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2008年11月9日 星期日

Multivariate Normal Distribution

We want to characterize multivariate normal distribution.

Consider the random vector
\mathbf{X}=\begin{pmatrix} X_1,&\dots{}&,X_N \end{pmatrix}^\text{T}

There are several equivalent definitions of multivariate normal distribution. One simple definition is the explicite formula of its distribution function:
\frac{1}{\sqrt{(2\pi)^N|\Sigma|}}\exp{\left\{ -\frac{1}{2}(\mathbf{X}-\mathbf{\mu})^\text{T}\Sigma^{-1}(\mathbf{X}-\mathbf{\mu}) \right\}}

However, this definition relies on the invertability of the covariance matrix. If the covariance didn't have full rank, then the formula is not well-defined.

Other definitions get rid of the problem:
  1. All the linear combination of its entries is normal distributed
  2. X can be writen as the affined linear transformation of an independent standard normal vectors.
  3. Defined via the characteristic
Nevertheless, the distribution formula definition is the clearest one. Therefore, we want to extend the formula definition so that it is well-defined even if covariance matrix is rank-defected.
The idea is to approximate the distribution by the sequence of well-defined distributions. The theory behind it is weak convergence. For the one-dimensional case, when the variance of the sequence of normal random variables decreases to 0, the distribution tends to the Dirac delta function. That is
\lim_{n\rightarrow \infty}\mathcal{N}(0,\frac{1}{n})\xrightarrow{\mathcal{D}}\delta(x)

Similarly, for the multivariate case, we can approximate the rank-defected covariance matrx by full rank matrices. For the moment, assume the covariance matrix has full-rank. Since the covariance matrix is symmetirc, its eigen-decomposition always exist and it's unitary:
\Sigma = U\Lambda U^{\textrm{T}}
Suppose there are r positive eigenvalues in the rank-defected matrix, reorder the diagonal matrix such that the eigenvalues are in decreasing order. The decompostion can be partitioned as:
\Sigma= \begin{bmatrix} U_1 & U_2 \end{bmatrix} \begin{bmatrix} \Lambda_1 & 0 \\ 0 & \Lambda_2 \end{bmatrix} \begin{bmatrix} U_1^{\textrm{T}}\\ U_2^{\textrm{T}} \end{bmatrix}
Plug the above expression into the distribution formula:
\frac{1}{\sqrt{(2\pi)^N|\Lambda_1||\Lambda_2|}}\exp{\left\{ -\frac{1}{2}u^TU_1\Lambda_1^{-1}U_1^Tu-\frac{1}{2}u^TU_2\Lambda_2^{-1}U_2^Tu \right\}}
where u is x-\mu.
Now we can divide the distribution into two parts: the first part corresponds to the r positive eigenvalues:
\frac{1}{\sqrt{(2\pi)^r|\Lambda_1|}}\exp{\left\{ -\frac{1}{2}u^T\Sigma^+u\right\}}
And the second part:
\frac{1}{\sqrt{(2\pi)^{N-r}|\Lambda_2|}}\exp{\left\{ -\frac{1}{2}v^T\Lambda_2^{-1}v\right\}}
where v=U_2^Tu
As Lumbda2 tends to 0, the second part converges to multi-dimentional delta function:
\frac{1}{\sqrt{(2\pi)^{N-r}|\Lambda_2|}}\exp{\left\{ -\frac{1}{2}v^T\Lambda_2^{-1}v\right\}} \xrightarrow{\mathcal{D}}\delta(v_1)\delta(v_2)\dots\delta(v_{N-r})=\delta(\mathbf{v})
The final result is
f_{\mathbf{X}}(\mathbf{x})=f_{\mathbf{v}_1}(U_1^T\mathbf{(x-\mathbf{\mu})})\delta_{\mathbf{v}_2}(U_2^T\mathbf{(x-\mathbf{\mu})})
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2008年11月7日 星期五

Compact space

There are several equivalent statements of a compact space X:
  1. Every open cover of X has a finite subcover
  2. Every family of closed subsets of X with finite intersection properity has a nonempty intersection.
  3. Every net in X has a limit point
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