Automatic Fitting of Generalized Dynamic Principal Components

Description

Computes Generalized Dynamic Principal Components. The number of components can be supplied by the user or chosen automatically so that a given proportion of variance is explained. The number of lags is chosen automatically using one of the following criteria: Leave-one-out cross-validation, an AIC type criterion, a BIC type criterion or a criterion based on a proposal of Bai and Ng (2002). See Peña, Smucler and Yohai (2020) for more details.

Usage

auto.gdpc(Z, crit = 'LOO', normalize = 1, auto_comp = TRUE, expl_var = 0.9,
          num_comp = 5, tol = 1e-4, k_max = 10,
          niter_max = 500, ncores = 1, verbose = FALSE)

Arguments

Details

Suppose the data matrix consists of m series of length T. Let \bold{f} be the dynamic principal component defined using k lags, let R be the corresponding matrix of residuals and let \Sigma = (R^{\prime} R) / T.

If crit = 'LOO' the number of lags is chosen among 0,\dots, k_{max} as the value k that minimizes the leave-one-out (LOO) cross-validation mean squared error, given by

LOO = \frac{1}{T m}\sum\limits_{i=1}^{m}\sum\limits_{t=1}^{T}\frac{R_{t,i}^{2}}{(1-h_{t,t})^{2}},

where h_{t,t} are the diagonal elements of the hat matrix H = F(F^{\prime} F)^{-1} F^{\prime} , with F being the T \times (k+2) matrix with rows (f_{t-k}, f_{t-k+1}, \dots, f_{t}, 1).

If crit = 'AIC' the number of lags is chosen among 0,\dots, k_{max} as the value k that minimizes the following AIC type criterion

AIC = T \log(trace(\Sigma)) + 2 m (k+2) .

If crit = 'BIC' the number of lags is chosen among 0,\dots, k_{max} as the value k that minimizes the following BIC type criterion

BIC = T \log(trace(\Sigma)) + m (k+2) \log(T) .

If crit = 'BNG' the number of lags is chosen among 0,\dots, k_{max} as the value k that minimizes the following criterion

BNG = \min(T, m) \log(trace(\Sigma)) + (k+1) \log(\min(T, m)).

This is an adaptation of a criterion proposed by Bai and Ng (2002).

For problems of relatively small dimension, say T \geq m 10, 'AIC' can can give better results than the default 'LOO'.

If normalize = 1, the data is analyzed in the original units, without mean and variance standarization. If normalize = 2, the data is standardized to zero mean and unit variance before computing the principal components, but the intercepts and loadings are those needed to reconstruct the original series. If normalize = 3 the data are standardized as in normalize = 2, but the intercepts and the loadings are those needed to reconstruct the standardized series. Default is normalize = 1.

Value

An object of class gdpcs, that is, a list of length equal to the number of computed components. The i-th entry of this list is an object of class gdpc, that is, a list with entries

components, fitted, plot and print methods are available for this class.

Author(s)

Daniel Peña, Ezequiel Smucler, Victor Yohai

References

Bai J. and Ng S. (2002). “Determining the Number of Factors in Approximate Factor Models.” Econometrica, 70(1), 191–221.

Peña D., Smucler E. and Yohai V.J. (2020). “gdpc: An R Package for Generalized Dynamic Principal Components.” Journal of Statistical Software, 92(2), 1-23.

See Also

gdpc, plot.gdpc, plot.gdpcs, fitted.gdpcs, components.gdpcs

Examples

T <- 200 #length of series
m <- 200 #number of series
set.seed(1234)
f <- rnorm(T + 1)
x <- matrix(0, T, m)
u <- matrix(rnorm(T * m), T, m)
for (i in 1:m) {
    x[, i] <- 10 * sin(2 * pi * (i/m)) * f[1:T] + 10 * cos(2 * pi * (i/m)) * f[2:(T + 1)] + u[, i]
}
#Choose number of lags using the LOO criterion.
#k_max=3 to keep computation time low
autofit <- auto.gdpc(x, k_max = 3)
autofit
fit_val <- fitted(autofit, 1) #Get fitted values
resid <- x - fit_val #Residuals
plot(autofit, which_comp = 1) #Plot component

Generic Function for Getting Components From an Object

Description

Generic function for getting components from an object.

Usage

  components(object, which_comp)

Arguments

Value

A matrix whose columns are the desired components.

Author(s)

Daniel Peña, Ezequiel Smucler, Victor Yohai

Get Generalized Dynamic Principal Components From a gdpcs Object

Description

Get Generalized Dynamic Principal Components from a gdpcs object.

Usage

## S3 method for class 'gdpcs'
components(object, which_comp = 1)

Arguments

Value

A matrix whose columns are the desired dynamic principal components.

Author(s)

Daniel Peña, Ezequiel Smucler, Victor Yohai

See Also

gdpc, auto.gdpc, plot.gdpc

Examples

T <- 200 #length of series
m <- 200 #number of series
set.seed(1234)
f <- rnorm(T + 1)
x <- matrix(0, T, m)
u <- matrix(rnorm(T * m), T, m)
for (i in 1:m) {
    x[, i] <- 10 * sin(2 * pi * (i/m)) * f[1:T] + 10 * cos(2 * pi * (i/m)) * f[2:(T + 1)] + u[, i]
}
#Choose number of lags using the LOO criterion.
#k_max=2 to keep computation time low
autofit <- auto.gdpc(x, k_max = 2, auto_comp = FALSE, num_comp = 2) 
comps <- components(autofit, which_comp = c(1,2))

Get Reconstructed Time Series From a gdpcs Object

Description

Get reconstructed time series from a gdpcs object.

Usage

  ## S3 method for class 'gdpcs'
fitted(object, num_comp = 1, ...)

Arguments

Value

A matrix that is the reconstruction of the original series.

Author(s)

Daniel Peña, Ezequiel Smucler, Victor Yohai

See Also

gdpc, auto.gdpc, plot.gdpc

Examples

  T <- 200 #length of series
  m <- 200 #number of series
  set.seed(1234)
  f <- rnorm(T + 1)
  x <- matrix(0, T, m)
  u <- matrix(rnorm(T * m), T, m)
  for (i in 1:m) {
    x[, i] <- 10 * sin(2 * pi * (i/m)) * f[1:T] + 10 * cos(2 * pi * (i/m)) * f[2:(T + 1)] + u[, i]
  }
  #Choose number of lags using the LOO criterion.
  #k_max=2 to keep computation time low
  autofit <- auto.gdpc(x, k_max = 2, auto_comp = FALSE, num_comp = 2) 
  recons <- fitted(autofit, num_comp = 2)

Generalized Dynamic Principal Components

Description

Computes a single Generalized Dynamic Principal Component with a given number of lags.

Usage

gdpc(Z, k, f_ini = NULL, tol = 1e-4, niter_max = 500, crit = 'LOO')

Arguments

Details

See auto.gdpc for the definition of criterion that is part of the output of this function.

Value

An object of class gdpc, that is, a list with entries:

fitted, plot and print methods are available for this class.

Author(s)

Daniel Peña, Ezequiel Smucler, Victor Yohai

See Also

auto.gdpc, plot.gdpc

Examples

T <- 200 #length of series
m <- 500 #number of series
set.seed(1234)
f <- rnorm(T + 1)
x <- matrix(0, T, m)
u <- matrix(rnorm(T * m), T, m)
for (i in 1:m) {
    x[, i] <- 10 * sin(2 * pi * (i/m)) * f[1:T] + 10 * cos(2 * pi * (i/m)) * f[2:(T + 1)] + u[, i]
}
fit <- gdpc(x, k = 1) #find first DPC with one lag
fit
par(mfrow = c(1, 2)) #plot loadings
plot(fit, which = 'Loadings', which_load = 0, xlab = '', ylab = '') 
plot(fit, which = 'Loadings', which_load = 1, xlab = '', ylab = '') 

Industrial Production Index (IPI) of France, Germany, Italy, United Kingdom, USA and Japan

Description

Six series corresponding to the Industrial Production Index (IPI) of France, Germany, Italy, United Kingdom, USA and Japan. Monthly data from January 1991 to December 2012.

Usage

data(ipi91)

Format

A matrix time series with 264 observations on the following 6 variables.

France

IPI of France.

Germany

IPI of Germany.

Italy

IPI of Italy.

United Kingdom

IPI of United Kingdom.

USA

IPI of USA.

Japan

IPI of Japan.

Examples

data(ipi91)
plot(ipi91, plot.type = 'multiple', main = 'Industrial Production Index')
## Not run: 
#Compute first GDPC with nine lags; this may take a bit.
gdpc_ipi <- gdpc(ipi91, 9, niter_max = 1500)
#Plot the component
plot(gdpc_ipi, which = 'Component', ylab = '')
#Get reconstruction of the time series and plot
recons <- fitted(gdpc_ipi)
colnames(recons) <- colnames(ipi91)
plot(recons, main = 'Fitted values')

## End(Not run)

Plot Generalized Dynamic Principal Components

Description

Plots a gdpc object.

Usage

## S3 method for class 'gdpc'
plot(x, which = 'Component', which_load = 0, ...)

Arguments

Author(s)

Daniel Peña, Ezequiel Smucler, Victor Yohai

See Also

gdpc, auto.gdpc, plot.gdpcs

Examples

T <- 200 #length of series
m <- 200 #number of series
set.seed(1234)
f <- rnorm(T + 1)
x <- matrix(0, T, m)
u <- matrix(rnorm(T * m), T, m)
for (i in 1:m) {
    x[, i] <- 10 * sin(2 * pi * (i/m)) * f[1:T] + 10 * cos(2 * pi * (i/m)) * f[2:(T + 1)] + u[, i]
}
#Choose number of lags using the LOO type criterion.
#k_max=3 to keep computation time low
autofit <- auto.gdpc(x, k_max = 3) 
plot(autofit[[1]], xlab = '', ylab = '')

Plot Generalized Dynamic Principal Components

Description

Plots a gdpcs object.

Usage

## S3 method for class 'gdpcs'
plot(x, which_comp = 1, plot.type = 'multiple', ...)

Arguments

Author(s)

Daniel Peña, Ezequiel Smucler, Victor Yohai

See Also

gdpc, auto.gdpc, plot.gdpc

Examples

T <- 200 #length of series
m <- 200 #number of series
set.seed(1234)
f <- rnorm(T + 1)
x <- matrix(0, T, m)
u <- matrix(rnorm(T * m), T, m)
for (i in 1:m) {
    x[, i] <- 10 * sin(2 * pi * (i/m)) * f[1:T] + 10 * cos(2 * pi * (i/m)) * f[2:(T + 1)] + u[, i]
}
#Choose number of lags using the LOO criterion.
#k_max=2 to keep computation time low
autofit <- auto.gdpc(x, k_max = 2, auto_comp = FALSE, num_comp = 2) 
autofit
plot(autofit, which_comp = c(1,2), xlab = '', ylab = '')

Stock Prices of the First 50 Components of S&P500

Description

Fifty series corresponding to the stock prices of the first 50 components of the Standard&Poor's 500 index. Five hundred daily observations starting 1/1/2010.

Usage

data(pricesSP50)

Format

A matrix time series with 500 observations on the stock prices of the first 50 components of the Standard&Poor's 500 index.

Examples

data(pricesSP50)
## Not run: 
#Plot the first four series
plot(pricesSP50[, 1:4], main = 'Four components of the S&P500 index')
#Compute GDPCs; this may take a bit.
fit_SP <- auto.gdpc(pricesSP50, normalize = 2, niter_max = 1000, ncores= 4)
fit_SP
#Get reconstruction and plot
recons <- fitted(fit_SP, num_comp = 2)
colnames(recons) <- colnames(pricesSP50)
plot(recons[, 1:4], main = 'Reconstruction of four components of the S&P500 index')

## End(Not run)