Title: EVALUATING THE STATISTICAL PROPERTIES OF TIME SERIES NON PARAMETRIC ESTIMATORS BY MEANS OF SMOOTHING
1EVALUATING THE STATISTICAL PROPERTIES OF TIME
SERIES NON PARAMETRIC ESTIMATORS BY MEANS OF
SMOOTHING MATRICES
- Estela Bee Dagum, Alessandra Luati
- Department of Statistics
- University of Bologna, Italy
- e-mail beedagum_at_stat.unibo.it,
luati_at_stat.unibo.it
2OUTLINE OF THE STUDY
- Contents theoretical and empirical study on the
algebraic and statistical properties of smoothing
matrices in time series. - Motivation the short-term trend-cycle estimation
of seasonally adjusted series. - Purpose to generalize the theory of symmetric
projection matrices to smoothing matrices (not
symmetric) and to define matrix-based measures of
goodness of fit and smoothness.
3THE PROBLEM
- Linear filters in time series smoothing methods
have always been studied by means of classical
spectral analysis techniques, i.e. gain and
phaseshift functions (Priestley, 1981). - Recently, weight-based measures of bias, variance
and mean square error have been proposed (Dagum
and Luati, 2002 SMA, 2004 SNDE).
4THE WEIGHT-BASED MEASURES
- are easy to apply
- are independent on the time series they are
applied on - provide information on the bias and variance
produced on the estimates by the smoothing
procedure - agree with the classical spectral ones
5MATRIX REPRESENTATION
- Linear estimators can be represented in matrix
form through the corresponding representative
matrix, called smoothing matrix. - Matrix-based measures of bias, variance and mean
square error can be defined - The theory of symmetric smoother matrices in
multivariate statistics (Hastie and Tibshirani,
1990) can be generalized to non symmetric
smoothing matrices in time series.
6SMOOTHING MATRICES IN TIME SERIES
- A linear operator acting on a time series
- to produce smooth estimates of the underlying
trend can be represented in matrix form as -
- where S is the SxS smoothing matrix of the form
(Dagum and Luati, 2000)
7SMOOTHING MATRICES IN TIME SERIES
- where
- Ws band matrix of 2m 1 symmetric weights to be
applied to central observations - Wa, Wa matrices whose row elements are the
asymmetric weights for first and last
observations, respectively.
8- With respect to symmetric filters,
- ASYMMETRIC FILTERS
- concern the most recent data
- produce phaseshifts in the output
- are not time invariant
- are often obtained based on different assumptions.
9- It is important to distinguish between symmetric
and asymmetric filters when studying the smoother
properties. -
10CENTROSYMMETRIC STRUCTURE OF SMOOTHING MATRICES
- The matrix S is centrosymmetric, i.e. wij
wN1-j,N1-j - Centrosymmetric matrices are symmetric with
respect to their geometric center. - The matrix Ws is rectangular centrosymmetric
- The matrices Wa and Ws are the t-transform of
each other, i.e. - Wa t (Wa)
11THE t-TRANSFORMATION
- t is the linear transformation (Dagum and Luati,
2004 LAA) - t Rmxn? Rmxn , A ? t (A), aij ? am1-,n1-j
- or equivalently
- t(A) EmAEn
- where En?Rn?n is the permutation matrix with
ones on the cross diagonal (bottom left to top
right) and zeros elsewhere.
12EXAMPLE smoothing matrix associated to a 13-term
Gaussian kernel estimator
0.347 0.302 0.199 0.100 0.038 0.011 0.002 0.000
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.232
0.267 0.232 0.153 0.077 0.029 0.008 0.002 0.000
0.000 0.000 0.000 0.000 0.000 0.000 0.133 0.201
0.231 0.201 0.133 0.066 0.025 0.007 0.002 0.000
0.000 0.000 0.000 0.000 0.000 0.062 0.125 0.189
0.217 0.189 0.125 0.062 0.024 0.007 0.001 0.000
0.000 0.000 0.000 0.000 0.023 0.061 0.122 0.184
0.212 0.184 0.122 0.061 0.023 0.007 0.001 0.000
0.000 0.000 0.000 0.007 0.023 0.060 0.121 0.183
0.210 0.183 0.121 0.060 0.023 0.007 0.001 0.000
0.000 0.000 0.001 0.007 0.023 0.060 0.121 0.183
0.210 0.183 0.121 0.060 0.023 0.007 0.001 0.000
0.000 0.000 0.001 0.007 0.023 0.060 0.121 0.183
0.210 0.183 0.121 0.060 0.023 0.007 0.001
0.000 0.000 0.000 0.001 0.007 0.023 0.060 0.121
0.183 0.210 0.183 0.121 0.060 0.023 0.007
0.001 0.000 0.000 0.000 0.001 0.007 0.023 0.060
0.121 0.183 0.210 0.183 0.121 0.060 0.023
0.007 0.000 0.000 0.000 0.000 0.001 0.007 0.023
0.061 0.122 0.184 0.212 0.184 0.122 0.061
0.023 0.000 0.000 0.000 0.000 0.000 0.001 0.007
0.024 0.062 0.125 0.189 0.217 0.189 0.125
0.062 0.000 0.000 0.000 0.000 0.000 0.000 0.002
0.007 0.025 0.066 0.133 0.201 0.231 0.201
0.133 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.002 0.008 0.029 0.077 0.153 0.232 0.267
0.232 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.002 0.011 0.038 0.100 0.199 0.302 0.347
13ALGEBRAIC PROPERTIES OFCENTROSYMMETRIC MATRICES
- Centrosymmetric matrices inherit desirable
properties from the properties of the permutation
matrices En which are - symmetric En ETn
- orthogonal E-1n ETn
- reflections E2n In
-
- Any set of square centrosymmetric matrices, Cn,
is an algebra (Dagum, Guidotti, Luati, 2004).
14THE ALGEBRA OF SQUARECENTROSYMMETRIC MATRICES
- Being Cn an algebra is crucial on an (1)
algebraic and a (2) statistical point of view. In
fact - (1) centrosymmetric matrices are the sole
structured matrices, respect to some symmetry, to
constitute an algebra -
- (2) the convolution of smoothing matrices gives
a smoothing matrix as well.
15LIMITATIONS OF CENTROSYMMETRICMATRICES (respect
to symmetric, algebraically)
- The degrees of freedom of a smoother (df, Hastie
and Tibshirani, HT, 1990) - trS
- do not make sense for non symmetric matrices for
- complex eigenvalues and
- asymmetric filters.
- Using trSTS trSST overcomes (a) but not (b).
16THE DECOMPOSITION trSST trWsWsT2trWaWaT
- trWsWsT df of the symmetric filter
-
- trWaWaT df of the asymmetric filters
17THE STANDARDIZED DECOMPOSITION
- (N-2m2m) trSST
- (N-2m) trWsWsT/(N-2m)(2m)2trWaWaT/(2m)
-
- (N-2m) (trSST/N- trWsWsT/(N-2m))
- (2m) (trSST/N-2trWaWaT/2m) 0
- trWsWsT/(N-2m) df of the symmetric filter
-
- trWaWaT/2m df of the asymmetric filters
18LOCAL BIAS OF A SYMMETRIC SMOOTHER
- HT a smoother introduces zero bias if
- We define the local bias of a symmetric smoother
as - D? R (N 2m)N 2k 1-band matrix such that
dij1 if j - i m k,m k, dij0 otherwise
19- The local bias measures provides a measure of
bias for symmetric smoothers that cannot be
inferred by classical spectral analysis.
20LOCAL VARIANCE OF A SYMMETRIC SMOOTHER
- We define the local variance of a symmetric
smoother as - Global variance
- PN NxN symmetric (centrosymmetric) Toeplitz
matrix of the process.
21LOCAL VARIANCE OF A SYMMETRIC SMOOTHER
- To evaluate the goodness of the local
approximation - Best approximation for I1?1 and I0 ?0
22- Analogous measures of local bias and local
variance for asymmetric smoothers have been
defined, as well as a local mean square error
given by square local bias plus local variance.
We restrict here to measures for symmetric
filters.
23EMPRICAL ANALYSIS
- Local bias we compare standardized Hastie and
Tibshirani degrees of freedom with our local bias
measures - Local variance we apply I0 and I1 to different
smoothers (L1, L2, CSS, GK, H13) and different
ARMA(p,q) models (p,q1,0 2,0 0,1 0,21,1).
24LOCAL BIAS
25LOCAL VARIANCE
- Best approximation for
- Smoothers ? concentrated (unbiased)
- Models ? white noise, AR
- In particular,
- AR(1) with ??0
- MA (1) with ??0
- ARMA(1,1) with ?gt0
- MA(2) with ?2gt?1
- AR(2) with ?1gt?2
26CONCLUSIONS
- A theory for centrosymmetric smoothing matrix is
required due to asymmetry. - A decomposition of the smoothing matrix allows to
separately consider the effects of symmetric and
asym-metric weights. - Matrix-based measures of bias and variance can be
defined such that - local bias can be defined also for symmetric
filters - local variance is a good approximation of the
global variance left by the smoothing procedure
27FURTHER RESEARCH
- To relate the spectral properties of the
smoothing matrix with the fitting and smoothing
properties of the associated filters.