blocks: Computes number of observations for each block
Description
In Rufibach and Walther (2010) a new multiscale mode hunting procedure is presented
that compares the local test statistics with critical values given by blocks. Blocks are collection
of intervals on a given grid that contain roughly the same number of original observations.
Usage
blocks(n, m0 = 10, fm = 2)
Arguments
n
Number of observations.
m0
Initial parameter that determines the number of observations in one block.
fm
Factor by which $m$ is increased from block to block.
Value
$b \times 2$--matrix, where $b$ is the number of blocks and the columns contain the lower
and the upper number of observations that form each block.
Details
In our block procedure, we only consider a subset $\mathcal{I}_{app}$ of all possible intervals
$\mathcal{I}_{all}$ where
$$\mathcal{I}_{all} = \Bigl{(j, \ k ) \ : \ 0 \le j < k \le n+1, \ k - j > 1\Bigr}.$$
This subset $\mathcal{I}_{app}$ is computed as follows:
Set $d_1, m_1, f_m > 1$. Then:
$for \ \ r = 1,\ldots,\#blocks$
$d_r := round(d_1 f_m^{(r-1)/2}), \ m_r := m_1 f_m^{r-1}.$
Include $(j,k)$ in $\mathcal{I}_{app}$ if
(a) $j, k \in {1+i d_r, \ i = 0, 1, \dots }$ (we only consider every $d$--th observation) and
(b) $m_r \le k-j-1 \le 2m_r-1$ ($\mathcal{I}_{jk}$ contains between $m_r$ and $2m_r - 1$ observations)
$end \ \ for$
References
Rufibach, K. and Walther, G. (2010).
A general criterion for multiscale inference.
J. Comput. Graph. Statist., 19, 175--190.