hadam: Calculate the Hadamard Variance

In SMAC-Group/gmwm: Generalized Method of Wavelet Moments

Calculate the Hadamard Variance

Description

Computes the Hadamard Variance

Usage

hadam(x, type = "mo")

Arguments

x	A vec containing the time series under observation.
type	A string containing either "mo" for Maximal Overlap or "to" for Tau Overlap

Details

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}}} .

Value

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.

Author(s)

Avinash Balakrishnan, JJB

Examples

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")

SMAC-Group/gmwm documentation built on June 10, 2025, 6:10 a.m.