# ADF: Serial Dependence Diagrams In SDD: Serial Dependence Diagrams

## Description

The function computes (and by default plots) different types of serial dependence diagrams.

## Usage

 ```1 2 3 4 5 6 7 8``` ```ADF(x, dtype = c("ADF", "CADF", "RPADF", "DeltaADF", "ACF"), lag.max = floor(10 * log10(length(x))), alpha = 0.05, num.clas, B = 99, bandwidth, delta = "Delta_1", fres = ".Perm", fdenest = ".denest", fdiv, argacf, R = 1:lag.max, p.adjust.method = p.adjust.methods, plot = TRUE, ...) ## S3 method for class 'SDD' print(x, digits=3, ...) ```

## Arguments

 `x` an `"ADF"` object or a univariate numeric time series object or a numeric vector. `dtype` an optional character string. It specifies the type of autodependence function and must be: `"ADF"` (default; see Bagnato, Punzo, Nicolis, 2012) `"CADF"` (see Bagnato, Punzo, Nicolis, 2012) `"RPADF"` (see Bagnato, De Capitani, Punzo, 2014) `"DeltaADF"` (see Bagnato, De Capitani, Punzo, 2013) `"ACF"` `lag.max` maximum lag at which to calculate the ADF. Default is `10*log10(n)` where `n` is the length of the series . `alpha` significance level of the tests of lag-independence (related to each bar). Default value is 0.05. `num.clas` when `dtype="ADF"` or `"CADF"` or `"RPADF"`, it sets the number of equifrequency classes for each of the two marginal distributions of the contingency table. If not specified, it is determined internally using a rule of thumb described in Bagnato, Punzo, Nicolis (2012). `B` when `dtype="DeltaADF"`, it sets the number of permutations used. Default value is 99 (see Bagnato, De Capitani, Punzo, 2013a,b). `bandwidth` when `dtype="DeltaADF"`, it sets the bandwidth used for the Gaussian kernel density estimator. Default value is computed with likelihood cross-validation (see Bagnato, De Capitani, Punzo, 2013a,b). `delta` a character vector; when `dtype="DeltaADF"`, it specifies the type of divergence measure used (see Bagnato, De Capitani, Punzo, 2013b); for each element in `delta` a different plot is produced. Possible values are: `"Delta_1"` (default) `"Delta_0.5"` `"Delta_2"` `"Delta_3"` `"Delta_4"` `"Delta_SD"` `"Delta_L1"` `"Delta_ST"` `"Delta_fdiv"`; in this case, the external function named `fdiv` is used to compute divergence. `fres` an optional character string which specifies, when `dtype="DeltaADF"`, the name of the external `function(x,B)` specifying the resampling method from the raw series, where `x` is a time series and `B` the number of resamples; the function should return a matrix with `B` rows and `length(x)` columns. If not specified, permutations are randomly generated. `fdenest` an optional character string which specifies when `dtype="DeltaADF"`, the name of the external `function(x,m,ngrid,bandwidth)` to use for univariate and bivariate density estimation, where `x` is the time series, `m` is the lag considered, `ngrid` is the number of points in the grid, and `bandwidth` is the bandwidth; the function should return: `fi`, a matrix of dimension `ngrid` x `ngrid` containing conjoint density estimates for lag `m` `gi`, a matrix of dimension `ngrid` x `ngrid` containing conjoint density estimates in case of independence, for lag `m` `ngi`, is equal to `ngrid`. If `fdenest` is not specified, the Gaussian kernel density estimation is used (see Bagnato, De Capitani, Punzo, 2013a,b). `fdiv` an optional character string which specifies, when `dtype="DeltaADF"` and `delta="Delta_fdiv"` , the name of the external `function(fi,gi,ngi)` to use to compute divergence; its arguments are defined as in `fdenest`; the function should return a scalar. `plot` if `TRUE` (default), the specified ADF is displayed. `argacf` when `dtype="ACF"`, it is a list with optional arguments for function `acf()`. `R` a vector. It specifies the lags on which to test for simultaneous independence (see Bagnato, Punzo, 2010, 2012 and Bagnato, De Capitani, Punzo, 2013b). Default value is `1:lag.max` `p.adjust.method` a character string. It specifies the method to be used in the simultaneous independence test. It must be one of `p.adjust.methods`. `...` optional arguments to be passed to the `plot.SDD` method, such as graphical parameters. `digits` minimal number of significant digits.

## Details

There are print and data.frame methods for objects of class "ADF".

## Value

Returned from this function is a `SDD` object which is a list with the following components:

 `res` a data frame. According to `dtype`, it may contain: `lag`, a numeric vector containing the lags at which the bars of the diagrams are computed `vbar`, height of the bars of the diagram `pvalue`, p-values associated to the bars of the diagram `pstar`, transformed p-values associated to the bars of the diagram. If `dtype="DeltaADF"` transformed p-values are `vbar` `n`, vector of length `lag.max`, containing the effective number of pairs considered for each lag `crit.val`, vector, of length lag.max, with the critical values `xmin` vector of length `lag.max`, containing the non-centrality parameters for each bar of the RP-ADF `dtype` a character string. It specifies the type of serial dependence diagram generated. `delta` a character string. It specifies, when `type="DeltaADF"`, the type divergence measure used. `num.clas` a scalar. It is the number of classes in each contingency table. `alpha` a scalar. It specifies the significance level of the tests of lag independence (related to each bar). `df` a scalar. It contains the degrees of freedom of the reference chi-square distribution used when `dtype` is one of: `"ADF", "RPADF",` or `"CADF"`. `bandwidth` a scalar. It is the bandwidth used in kernel density estimation. `series` the name of the series `x`. `R` a vector. It specifies the lags to test in the simultaneous independence tests. `p.adjust.method` a character string. It specifies the method to be used in the simultaneous independence tests. It must be one of `p.adjust.methods`. `p.adjust` a vector. It contains the adjusted probabilities for the simultaneous independence tests.

## Author(s)

Luca Bagnato, Lucio De Capitani, Angelo Mazza and Antonio Punzo

## References

Bagnato L, De Capitani L, Mazza A, Punzo A (2015). SDD: An R Package for Serial Dependence Diagrams. Journal of Statistical Software, 64(2), 1-19. URL: http://www.jstatsoft.org/v64/c02/

Bagnato L, De Capitani L, Punzo A (2013a). Improving the autodependogram using the Kulback-Leibler divergence. arXiv:1306.5006 [stat.ME], URL: http://arxiv.org/pdf/1306.5006v1.pdf

Bagnato L, De Capitani L, Punzo A (2013b). Testing Serial Independence via Density-Based Measures of Divergence. Methodology and Computing in Applied Probability, 16(3), 627-641.

Bagnato L, De Capitani L, Punzo A (2014). Detecting Serial Dependencies with the Reproducibility Probability Autodependogram. Advances in Statistical Analysis, 98(1), 35-61.

Bagnato L, Punzo A (2010). On the Use of χ^2 Test to Check Serial Independence. Statistica & Applicazioni, VIII(1), 57-74.

Bagnato L, Punzo A (2012). Checking Serial Independence of Residuals from a Nonlinear Model. In W Gaul, A Geyer-Shulz, L Schmidt-Thieme, J Kunze (eds.), Challenges at the Interface of Data Analysis, Computer Science, and Optimization, volume XIV of Studies in Classification, Data Analysis and Knowledge Organization, pp. 203-211. Springer-Verlag, Berlin Heidelberg.

Bagnato L, Punzo A, Nicolis O (2012). The autodependogram: a graphical device to investigate serial dependencies. Journal of Time Series Analysis, 33(2), 233-254.

Bagnato L, Punzo A (2013). Using the Autodependogram in Model Diagnostic Checking. In N Torelli, F Pesarin, A Bar-Hen (eds.), Advances in Theoretical and Applied Statistics, volume XIX of Studies in Theoretical and Applied Statistics, pp. 129-139. Springer-Verlag, Berlin Heidelberg.

`SDD-package`, `plot.SDD`, `SMI`, `acf`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```# Dependence Diagrams on raw data data("SMI") ADF(SMI^2, dtype="ACF", main="") ADF(SMI, main="") ADF(SMI, dtype="RPADF", main="") # Dependence Diagrams on residuals from a fitted model library("tseries") residuals <- garch(SMI, order=c(1,1))\$residuals[-1] ADF(residuals^2, dtype="ACF", main="") ADF(residuals, dtype="RPADF", main="") ```