| Title: | Detecting New Signals under Background Mismodelling |
|---|---|
| Description: | Given a postulated model and a set of data, the comparison density is estimated and the deviance test is implemented in order to assess if the data distribution deviates significantly from the postulated model. Finally, the results are summarized in a CD-plot as described in Algeri S. (2019) <arXiv:1906.06615>. |
| Authors: | Sara Algeri <[email protected]>, Haoran Liu<[email protected]> |
| Maintainer: | Sara Algeri <[email protected]> |
| License: | GPL-3 |
| Version: | 1.2 |
| Built: | 2025-02-13 03:57:28 UTC |
| Source: | https://github.com/cran/LPBkg |
Computes the deviance p-values considering different sizes of the polynomial basis and selects the one for which the deviance p-value is the smallest.
BestM(data, g, Mmax = 20, range = c(min(data),max(data)))BestM(data, g, Mmax = 20, range = c(min(data),max(data)))
data |
A vector of data. See details. |
g |
The postulated model from which we want to assess if deviations occur. |
Mmax |
The maximum size of the polynomial basis from which a suitable value |
range |
Range of the data/ search region considered. |
The argument data collects the data for which we want to test if deviations occur from the postulated model specified in the argument g. As in Algeri, 2019, the sample specified under data corresponds to the source-free sample in the background calibration phase and to the physics sample in the signal search phase.
The value M selected determines the smoothness of the estimated comparison density, with smaller values of M leading to smoother estimates. The deviance test is used to select the value M which leads to the most significant deviation from the postulated model. The default value for Mmax is set to 20. Notice that numerical issues may
arise for larger values of Mmax.
pvals |
The deviance test p-value obtained for each values of |
minp |
The minimum value of the deviance p-values observed. |
Msel |
The value of |
Sara Algeri
S. Algeri, 2019. Detecting new signals under background mismodelling <arXiv:1906.06615>.
#Generating data x<-rnorm(1000,10,7) data<-x[x>=10 & x<=20] #Create suitable postulated quantile function of data G<-pnorm(20,5,15)-pnorm(10,5,15) g<-function(x){dnorm(x,5,15)/G} Mmax=10 range=c(10,20) BestM(data,g,Mmax,range)#Generating data x<-rnorm(1000,10,7) data<-x[x>=10 & x<=20] #Create suitable postulated quantile function of data G<-pnorm(20,5,15)-pnorm(10,5,15) g<-function(x){dnorm(x,5,15)/G} Mmax=10 range=c(10,20) BestM(data,g,Mmax,range)
Approximates the quantiles of the supremum of the comparison density estimator using tube formulae and assuming that $H_0$ is true.
c_alpha2(M, IDs, alpha = 0.05, c_interval = c(1, 10))c_alpha2(M, IDs, alpha = 0.05, c_interval = c(1, 10))
M |
The size of the polynomial basis used to estimate the comparison density. |
IDs |
The IDs of the polynomial terms to be used out of the |
alpha |
Desired significance level. |
c_interval |
Lower and upper bounds for the quantile being computed. |
Approximated quantile of order 1-alpha of the supremum of the comparison density estimator.
Sara Algeri
S. Algeri, 2019. Detecting new signals under background mismodelling <arXiv:1906.06615>.
L.A. Wasserman, 2005. All of Nonparametric Statistics. Springer Texts in Statistics.
c_alpha2(5, c(2,4), alpha = 0.05, c_interval = c(1, 10))c_alpha2(5, c(2,4), alpha = 0.05, c_interval = c(1, 10))
Selects the largest coefficients according to the AIC or BIC criterion.
denoise(LP, n, method)denoise(LP, n, method)
LP |
Original vector of coefficients estimates. See details. |
n |
The dimension of the sample on which the estimates in |
method |
Either “AIC” or “BIC”. See details. |
Give a vector of M coefficient estimates, the largest is selected according to the AIC or BIC criterion as described in Algeri, 2019 and Mukhopadhyay, 2017.
Selected coefficient estimates.
Sara Algeri
S. Algeri, 2019. Detecting new signals under background mismodelling. <arXiv:1906.06615>.
S. Mukhopadhyay, 2017. Large-scale mode identification and data-driven sciences. Electronic Journal of Statistics 11 (2017), no. 1, 215–240.
Legj.
#generating data x<-rnorm(1000,10,7) xx<-x[x>=10 & x<=20] #create suitable postulated quantile function G<-pnorm(20,5,15)-pnorm(10,5,15) g<-function(x){dnorm(x,5,15)/G} #Vectorize quantile function g<-Vectorize(g) u<-g(xx) Mmax=20 S<- as.matrix(Legj(u=u,m=Mmax)) n<-length(u) LP <- apply(S,FUN="mean",2) denoise(LP,n=n,method="AIC")#generating data x<-rnorm(1000,10,7) xx<-x[x>=10 & x<=20] #create suitable postulated quantile function G<-pnorm(20,5,15)-pnorm(10,5,15) g<-function(x){dnorm(x,5,15)/G} #Vectorize quantile function g<-Vectorize(g) u<-g(xx) Mmax=20 S<- as.matrix(Legj(u=u,m=Mmax)) n<-length(u) LP <- apply(S,FUN="mean",2) denoise(LP,n=n,method="AIC")
Construction of CD-plot and adjusted deviance test. The confidence bands are also adjusted for post-selection inference.
dhatL2(data, g, M = 6, Mmax = NULL, smooth = TRUE, criterion = "AIC", hist.u = TRUE, breaks = 20, ylim = c(0, 2.5), range = c(min(data),max(data)), sigma = 2)dhatL2(data, g, M = 6, Mmax = NULL, smooth = TRUE, criterion = "AIC", hist.u = TRUE, breaks = 20, ylim = c(0, 2.5), range = c(min(data),max(data)), sigma = 2)
data |
A vector of data. See details. |
g |
The postulated model from which we want to assess if deviations occur. |
M |
The desired size of the polynomial basis to be used. |
Mmax |
The maximum size of the polynomial basis from which |
smooth |
A logical argument indicating if a denoised solution should be implemented. The default is |
criterion |
If |
hist.u |
A logical argument indicating if the CD-plot should be displayed or not. The default is |
breaks |
If |
ylim |
If |
range |
Range of the data/search region considered. |
sigma |
The significance level (in sigmas) with respect to which the confidence bands should be constructed. See details. |
The argument data collects the data for which we want to test if its distribution deviates from the one of the postulated model specified in the argument g. In Algeri, 2019, the sample specified under data corresponds to the source-free sample in the background calibration phase and to the physics sample in the signal search phase.
The value M selected determines the smoothness of the estimated comparison density, with smaller values of M leading to smoother estimates. The deviance test is used to select the value M which leads to the most significant deviation from the postulated model. The default value for Mmax is set to 20. Notice that numerical issues may
arise for larger values of Mmax.
If smooth=TRUE the largest coefficient estimates are selected according to either the AIC or BIC criterion as described in Algeri, 2019 and Mukhopadhyay, 2017.
If Mmax>1 and/or smooth=TRUE, post-selection Bonferroni's correction is automatically implemented to both the deviance test p-value and the confidence bands. The desired level of significance can be expressed as one minus the cdf of a standard normal evaluated at sigma (see Algeri, 2019).
Deviance |
Value of the deviance test statistic. |
Dev_pvalue |
Unadjusted p-value of the deviance test. |
Dev_adj_pvalue |
Post-selection Bonferroni adjusted p-value of the deviance test. |
kstar |
Number of coefficients selected by the denoising process. If |
dhat |
Function corresponding to the estimated comparison density in the u domain. |
dhat.x |
Function corresponding to the estimated comparison density in the x domain. |
SE |
Function corresponding to the estimated standard errors of the comparison density in the u domain. |
LBf1 |
Function corresponding to the lower bound of the confidence bands under in u domain. |
UBf1 |
Function corresponding to the upper bound of the confidence bands in u domain. |
f |
Function corresponding to the estimated density of the data. |
u |
Vector of values corresponding to the cdf of the model specified in |
LP |
Estimates of the coefficients. |
G |
Cumulative density function of the postulated model specified in the argument |
Sara Algeri
S. Algeri, 2019. Detecting new signals under background mismodelling. <arXiv:1906.06615>.
S. Mukhopadhyay, 2017. Large-scale mode identification and data-driven sciences. Electronic Journal of Statistics 11 (2017), no. 1, 215–240.
#generaing data x<-rnorm(1000,10,7) xx<-x[x>=10 & x<=20] #create suitable postulated quantile function of data G<-pnorm(20,5,15)-pnorm(10,5,15) g<-function(x){dnorm(x,5,15)/G} #Choose best M Mmax=20 range=c(10,20) m<-BestM(data=xx,g, Mmax,range) # vectorize postulated quantile function g<-Vectorize(g) u<-g(xx) #M has to be sufficient big, otherwise dhatL2 function will crush. #So,here we set m eqaul 6 as an example m<-6 comp.density<-dhatL2(data=xx,g, M=m, Mmax=Mmax,smooth=FALSE,criterion="AIC",hist.u=TRUE,breaks=20, ylim=c(0,2.5),range=range,sigma=2)#generaing data x<-rnorm(1000,10,7) xx<-x[x>=10 & x<=20] #create suitable postulated quantile function of data G<-pnorm(20,5,15)-pnorm(10,5,15) g<-function(x){dnorm(x,5,15)/G} #Choose best M Mmax=20 range=c(10,20) m<-BestM(data=xx,g, Mmax,range) # vectorize postulated quantile function g<-Vectorize(g) u<-g(xx) #M has to be sufficient big, otherwise dhatL2 function will crush. #So,here we set m eqaul 6 as an example m<-6 comp.density<-dhatL2(data=xx,g, M=m, Mmax=Mmax,smooth=FALSE,criterion="AIC",hist.u=TRUE,breaks=20, ylim=c(0,2.5),range=range,sigma=2)
Evaluates the a basis of normalized shifted Legendre polynomials over a specified data vector.
Legj(u, m)Legj(u, m)
u |
Data vector on which the polynomials are to be evaluated. |
m |
The size of the basis to be considered. |
Numerical values of the first m normalized shifted Legendre polynomials.
x<-rnorm(1000,10,7) xx<-x[x>=10 & x<=20] G<-pnorm(20,5,15)-pnorm(10,5,15) g<-function(x){dnorm(x,5,15)/G} g<-Vectorize(g) u<-g(xx) Mmax=20 s<-as.matrix(Legj(u,Mmax))x<-rnorm(1000,10,7) xx<-x[x>=10 & x<=20] G<-pnorm(20,5,15)-pnorm(10,5,15) g<-function(x){dnorm(x,5,15)/G} g<-Vectorize(g) u<-g(xx) Mmax=20 s<-as.matrix(Legj(u,Mmax))