Please can any one explain this formula to me, because I don't have enough information about statistic.
where d is the length of the signal.
This equation represents Stein's Unbiased Risk Estimate. I don't have any information about symbol such as # or ^.
Although this question is posted long ago, I feel like I should answer it since I was struggling to make sense of it for a while last night.
SURE as you mentioned represent Stein's Unbiased Risk Estimate:
In the equation, there are 3 parts:
The first part is d which as mentioned in the paper is the dimension of data and if we are talking about time-series data, the dimension is effective the length of data(which OP already pointed out)
The second part is 2.#{i: |x(i)| <= t} is simply 2 times the number of elements that is smaller or equal to t. In python, np.sum((np.abs(y)<=t)).
The third part is simply saying do a min check for every element in y which is min(abs(y(i)),t). Then square this result and sum it up.
Technically, you should choose the value of t which produces the smallest SURE.
Stein's unbiased risk estimator is more general than the SURE estimator you have described. The SURE Shrink estimator is due to Donoho and Johnstone 1994 for wavelet-based denoising and represents a particular application of Stein's result. You can see how it is derived from Stein's result on Page 8 of their paper.
If you are just looking how to interpret the risk of the SURE Shrink estimator, note that d is the dimensionality of the vector, $\#\{i: |x_i| \leq t \}$ just counts the number of elements in the vector whose magnitude is less than the threshold $t$. As for $x \wedge t$, it shrinks x towards 0 by up to a magnitude of $t$. Thus $$ 5 \wedge 2=3\\ -5 \wedge 2=-3\\ -1 \wedge 2 = 0 $$ The actual estimator is the so-called soft thresholding estimator $x \wedge t^S$ in which $$t^S = \arg \min_t SURE(t; x)$$ to the value which minimizes the estimate of risk above.
