Description Usage Arguments Details Value Author(s) References See Also Examples

Calculate the p-value for a hypothesis test regarding a linear relationship between wavelet coefficients from two data series. See Details for a precise description.

1 2 3 4 |

`X` |
vector of equally-spaced data. This must be of length |

`Y` |
vector of equally-spaced data. This must be of length |

`J0` |
wavelet coefficient threshold level. Allowed values are integers between 0 and |

`R` |
number of WiSE bootstrap samples. Allowed value is a positive integer. Default is 100. |

`popParam` |
hypothesized parameter values. Allowed input is a vector of length 2 which is non-missing and contains numeric entries. The first entry of the vector is the hypothesized value of the population intercept. The second entry of the vector is the hypothesized value of the population slope. |

`XParam` |
estimated linear parameter values (in time) from |

`YParam` |
estimated linear parameter values (in time) from |

`TauSq` |
scale parameter for the bootstrap. Allowed values are |

`bootDistn` |
the distribution for the bootstrap. Allowed values are |

`wavFam` |
wavelet family. Allowed values are |

`wavFil` |
wavelet filter number. Allowed values are integers between 4 and 10 when |

`wavBC` |
wavelet boundary condition. Allowed values are |

`plot` |
logical. If |

`...` |
additional graphical arguments. See |

Given 2 vectors of equally-spaced data of length *T=2^J* for a positive integer, *J*, we assume the following models:

*X = γ_{x0} 1 + γ_{x1} t + Wγ_x + e_x*

*Y = γ_{y0} 1 + γ_{y1} t + Wγ_y + e_y*

where *Y* and *X* are the data vectors, linear parameters in time (*t*) are *γ_{x0}, γ_{x1}, γ_{y0}* and *γ_{y1}*.

The *γ_x* and *γ_y* are the wavelet coefficients (scaling and filter) and W is the DWT for a fixed wavelet basis. Note, in many cases of the DWT, the scaling coefficient is equivalent to *γ_{x0}, γ_{y0}*, and thus, estimated there.

In this function, we consider a linear relationship between the wavelet coefficients. Specifically, we hypothesize a relationship

*γ_y = α 1 + β γ_x*

The null hypothesis is

*H_0: α = m, β=n*

for real numbers *m, n*. The user specifies `popParam=c(m, n)`

.

The WiSE bootstrap sample is created under the null hypothesis for a set threshold, `J0=j`

. The sampling scheme is described in detail in Braverman et al. The distributon of the bootstrap sample of the parameters allows for calculation of a p-value associated with the null hypothesis.

Some notation to aid in understanding outputs:

1) *a, b*: estimates of *α, β* from the data wavelet coefficients

2) *a*, b**: estimates of *α, β* from the bootstrap wavelet coefficients

3) *g_{xj}, g_{yj}*: estimates of *γ_x, γ_y* from the data at the threshold `J0=j`

4) *g*_{xj}, g*_{yj}*: estimates of *γ_x, γ_y* from the bootstrap sample at the threshold `J0=j`

`AsymptoticPValue ` |
the asymptotic p-value based upon Hotelling's T^2. |

`BootstrapPValue ` |
the bootstrap p-value. |

`dataSlope` |
the estimated slope of the wavelet coefficients from the data. In the notation from Details, |

`dataIntercept` |
the estimated intercept of the wavelet coefficients from the data. In the notation from Details, |

`bootSlope` |
the estimated slopes of the wavelet coefficients from the bootstrap samples. In the notation from Details, |

`bootIntercept` |
the estimated intercepts of the wavelet coefficients from the bootstrap samples. In the notation from Details, |

`YWavelet` |
the estimated wavelet coefficients from the |

`XWavelet` |
the estimated wavelet coefficients from the |

`bootYWavelet` |
the estimated wavelet coefficients from the |

`bootXWavelet` |
the estimated wavelet coefficients from the |

Megan Heyman

The WiSE bootstrap hypothesis test is implemented as an analysis tool in Braverman, A. et al. "Probabilistic Climate Model Evaluation" (in progress).

`padMatrix`

, `padVector`

, `wavethresh-package`

1 2 3 4 5 6 7 | ```
##Test whether \alpha=0 and \beta=1 for AIRS and IPSL Run 1 at 60E
## R=10 bootstrap samples is not recommended. For demonstration only.
data(CM20N20S60E)
padData <- padMatrix(CM20N20S60E)
hypTest <- WiSEHypothesisTest(padData$xPad[,1], padData$xPad[,2], J0=5, R=10,
XParam=padData$linearParam[,1], YParam=padData$linearParam[,2],
plot=TRUE)
``` |

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