how to calculate variance on SVD parameter estimation? How do i Calculate the variance of a estimated parameter by SVD?
I know that there is an uncertainty on the dataset, but how can that be used to calculate variance of an parameter?
 A: Since you haven't introduced any notation, I'll simply assume that your regression model is of the form 
$X\beta = y + \epsilon$
where $X$ is of size $n$ by $m$, $\beta$ is a vector of length $m$ and the vectors $y$ and $\epsilon$ are of length $n$.  Here, the $\beta$ coefficients are to be obtained and $\epsilon$ has a multivariate normal distribution with mean 0 and covariance matrix $I$. If the measurement standard deviations are not all equal but they're known and the errors are independent, then you can scale the equations to get a covariance of $I$.  If you have correlated measurement errors in $y$ things get to be a bit more complicated, and you'll need to check out some more advanced reference.  
The least squares estimate of the parameters can be written as 
$\hat{\beta}=X^{\dagger}y$
where $X^{\dagger}$ is the pseudoinverse of $X$.  
The expected value of $\hat{\beta}$ is then 
$E[\hat{\beta}]=X^{\dagger} E[y]$
$E[\hat{\beta}]=X^{\dagger} X\beta$.   
also 
$Var(\hat{\beta})=Var(X^{\dagger}y)=X^{\dagger}IX^{\dagger^{T}}$.
The nicest case occurs when $X$ has full column rank.  In that case, since $X$ has full column rank, 
$X^{\dagger}=(X^{T}X)^{-1}X^{T}$.
The least squares estimate is unbiased in this case since 
$E[\hat{\beta}]=X^{\dagger}X\beta=(X^{T}X)^{-1}X^{T}X\beta=\beta$.
The variance is 
$Var(\hat{\beta})=(X^{T}X)^{-1}X^{T}X(X^{T}X)^{-1}=(X^{T}X)^{-1}$.
When we use the singular value decomposition, we can write $X$ in the "compact" form as 
$X=U_{p}S_{p}V_{p}^{T}$,
where $p$ is the number of nonzero singular values of $X$ (in practice, some tolerance must be set below which we discard small values), $S_{p}$ is a $p$ by $p$ diagonal matrix of singular values, $U_{p}$ is an $n$ by $p$ matrix with orthonormal columns, and $V_{p}$ is a $m$ by $p$ matrix with orthonormal columns.  
The pseudoinverse is 
$X^{\dagger}=V_{p}S_{p}^{-1}U_{p}^{T}$.
The expected value of $\hat{\beta}$ is now
$E[\hat{\beta}]=X^{\dagger} X\beta=V_{p}S_{p}^{-1}U_{p}^{T}U_{p}S_{p}V_{p}^{T}\beta$
and after a bit of simplification, 
$E[\hat{\beta}]=V_{p}V_{p}^{T}\beta$.
If it happens that $X$ is of full column rank (that is $p=m$), then $V_{p}V_{p}^{T}=I$, and $\hat{\beta}$ is still an unbiased estimate of $\beta$.  However, if $X$ is not of full column rank, then $\hat{\beta}$ will no longer be an unbiased estimate of $\beta$.  In fact, you can't even bound the magnitude of this bias without making further assumptions about $\beta$!
As before, we can also compute the variance of $\hat{\beta}$.  This is 
$Var(\hat{\beta})=X^{\dagger} IX^{\dagger^{T}}=V_{p}S_{p}^{-1}U_{p}^{T}U_{p}S_{p}^{-1}V_{p}^{T}=V_{p}S_{p}^{-2}V_{p}^{T}$.
You can easily check that when $X$ is of rank $m$ (that is, $p=m$), then $(X^{T}X)^{-1}=V_{p}S_{p}^{-2}V_{p}^{T}$.  Numerically, it's more accurate to use the SVD of $X$ then it is to compute $(X^{T}X)^{-1}$ because computing $X^{T}X$ squares the condition number of $X$. 
When $p$ is less than $m$, it is misleading to report this covariance matrix for $\hat{\beta}$, because of the fact that estimate is biased and the bias might be far larger than any uncertainty propagated from uncertainty in $y$.  
