Reconstruction of the phase space from data by time delay embedding
rather than with actual model variables, it is customary to call this state space a phase space. We will use a time-delay
embedding defined in Section 2 to reconstruct the phase- space from the observed streamflow signals. In principle,
the phase space contains the knowledge about the internal dynamics of the system and thus can be used as a predictive
tool. The basic idea here is that since the embedding map preserves the underlying dynamic structure, the future can
be predicted from the behavior of the past. As shown by Takens
2
, the phase space retains essential properties of the original state space including the dimensionality of the
underlying system. Now, if one can reconstruct the determi- nistic rules underlying the data in a phase space then one can
attempt to predict the future states from the history of the data embedded in the phase space. Several recent studies
have successfully used phase-space-based models for chaotic signal characterization
3–5
, prediction
6
, noise reduc- tion
7
and lake level prediction
8
. In addition, as we will see, a phase-space-based model can also be used to make short-
term prediction and provide a tentative distinction between low-dimensional determinism and noise
9,10
. Hydrologists have long maintained that large basins are
smoother in their streamflow response behavior than small basins. This assumption has not been really substantiated
from a quantitative, data based, point of view, although arguments based on the smoothing effect resulting from
larger storage property constitute a reasonable basis for its acceptance. Such a smoothing effect of large basins is fre-
quently translated in the assertion that because of their inherent larger degree of linearity, their response e.g. run-
off is easier compared to the smaller basins to predict. It is commonly argued that as the time and spatial averaging
increase, then the rainfall–streamflow relationships may become more linear and hence the streamflow becomes
more predictable. However, even if the above is true, it is not clear how much the predictability of streamflow will
increase in terms of accuracy and prediction lead time. Recent studies have shown possible presence of chaos in
streamflow
11,12
. If the underlying streamflow signal is chaotic, it is quite possible that its inherent predictability
will be quite limited irrespective of the basin area. In this study, we will describe an alternative model for
streamflow prediction. This model will be used to investi- gate the characteristic signatures of streamflow signals e.g.
low-order determinism vs stochastic noise at the daily scale. For example, does streamflow change dynamics non-
linear to linear with increasing basin area? What is the impli- cation of the nature of streamflow characteristics on its
predictability? We will use recent developments in nonlinear modeling, phase-space reconstruction from a time series and
related diagnostic tools to address the above issues.
2 STREAMFLOW MODELING: A DYNAMICAL SYSTEM PERSPECTIVE
Due to the dramatic expansion of digital data acquisition and processing, it is now possible to develop predictive
models for streamflow dynamics from a ‘theory-poor’ and ‘data-rich’ perspective. By theory-poor we mean that our
approach does not require explicit formulation of governing partial differential equations. The idea of data intensive
modeling is by no means new—an autoregressive model
13,14
is a good example. What is new is the emergence of a set of concepts and tools such as phase-space recon-
struction, neural network, etc. that combine broad approxi- mation abilities and few specific assumptions
15
. We will take this data-rich and theory-poor perspective to construct
a predictive model directly from streamflow time series. Building this type of dynamical model from a time series
involves two steps: i reconstruction of the phase space from data by time delay embedding; and ii development
of a methodology for phase-space prediction.