hadam | R Documentation |
Computes the Hadamard Variance
hadam(x, type = "mo")
x |
A |
type |
A |
The decomposition and the amount of time it takes to perform it depends on whether you are using the Tau Overlap or the Maximal Overlap.
Maximal Overlap Hadamard Variance
Given N
equally spaced samples with averaging time \tau = n\tau _0
,
where n
is an integer such that 1 \le n \le \frac{N}{2}
.
Therefore, n
is able to be selected from \left\{ {n|n < \left\lfloor {{{\log }_2}\left( N \right)} \right\rfloor } \right\}
Then, M = N - 3n
samples exist.
The Maximal-overlap estimator is given by:
\frac{1}{{6(M - 3m + 1)}}\sum\limits_{i = 1}^{M - 3m + 1} {{{[{y_{i + 2}} - 2{y_{i + 1}} + {y_i}]}^2}}
where {y_i} = {{\bar y}_t}\left( \tau \right) = \frac{1}{\tau }\sum\limits_{i = 0}^{\tau - 1} {{{\bar y}_{t - i}}} .
.
Tau-Overlap Hadamard Variance
Given N
equally spaced samples with averaging time \tau = n\tau _0
,
where n
is an integer such that 1 \le n \le \frac{N}{2}
.
Therefore, n
is able to be selected from \left\{ {n|n < \left\lfloor {{{\log }_2}\left( N \right)} \right\rfloor } \right\}
Then, a sampling of m = \left\lfloor {\frac{{N - 1}}{n}} \right\rfloor - 1
samples exist.
The tau-overlap estimator is given by:
\frac{1}{{6(M - 2)}}\sum\limits_{t = 1}^{M - 2} {{{\left( {{y_{t + 2}} - 2{y_{t + 1}} + {y_t}} \right)}^2}}
where {\bar y_t}\left( \tau \right) = \frac{1}{\tau }\sum\limits_{i = 0}^{\tau - 1} {{{\bar y}_{t - i}}}
.
Hadamard variance fixed
hadam A list
that contains:
"clusters"The size of the cluster
"hadamard"The Hadamard variance
"errors"The error associated with the variance estimation.
Avinash Balakrishnan, JJB
set.seed(999)
# Simulate white noise (P 1) with sigma^2 = 4
N = 100000
white.noise = rnorm(N, 0, 2)
#plot(white.noise,ylab="Simulated white noise process",xlab="Time",type="o")
#Simulate random walk (P 4)
random.walk = cumsum(0.1*rnorm(N, 0, 2))
combined.ts = white.noise+random.walk
hadam_mo = hadam(combined.ts)
hadam_to = hadam(combined.ts, type = "to")
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