Transcript Investigating Estonian foreign trade balance with

Investigating Estonian foreign
trade balance with “Mathematica”
Tõnu Tõnso, Tallinn University,
[email protected]
Problem:
We have some statistical data (Estonian foreign trade balance,
exports and imports). Main questions are - how must we interpolate or approximate these data with functions and witch
representation is “ the best”?
About data:
These data comes from
database of Statistical Office
of Estonia. In the database,
the data is represented as
accurate to one kroon (is
represented with one kroon
accuracy, but unfortunately
sometimes the data is
changing behindhand
(apparently the source data
is corrected and
recalculations are made).
Typical changes in initial data of a single year vary from few kroons
to some twenty million kroons, in some particular cases the later
adjustment has been even about hundred million kroons.
Consequently, since the source data is approximate, there is
no reason to consider the digits less than million.
About refinement of the data:
The question arises perhaps it is worth to refine the table and use the
data of months and quarters instead of using the data of the year.
Unfortunately, when
using months or quarter’s data, new problems
arise — the data
becomes influenced by
seasonal fluctuations
and noise.
Estonian economy is so small that for example, when Tallink buys or
sells a ship, this single deal is visible on the graphics. On this reason we
gave up the idea of refinement of the data.
About trends in economy I:
Considering the export and import data graphics of 14 years, it is obvious that
you cannot describe it with the single simple rule. Export and import are
influenced by changes in foreign trade laws, by economical crises and bank
crashes. Those influences are visible on the graphic.
Right in the beginning,
Estonia unilaterally gave
up custom duties and
opened owns market to
the foreign trades. Until
1992, the foreign trade
balance was about
equilibrium, but as a
result of the one-party
open-door policy, the
import started increase
faster than the export.
About trends in economy II:
The first bigger change came in 1996 and 1997. Then the course to the joining
to European Union was taken and step-by-step the European requirements
were applied to the agriculture, industry and trade. At the same time, European
market was still closed to our products.
As a result, a lot of our
manufacturing enterprises
were extinguished and
the import grew sharply.
The export (mainly to the
non-EU countries) grew
also, but much slowly.
The stock market crash in
1997 and the following
bank crashes influenced
the foreign trade about
two years.
About trends in economy III:
The next bigger change came in 1999. The Russian economical crises
reached to us, the export of the food products (milk and fish products)
decreased almost to nonexistent.
Though in 2000-2003
Estonia assiduously applied all kind of eurodirectives to the manufacturing
and trading companies,
the EU market still remained closed to our
products. As a result, the
import from EU countries
increased, but we were
able to export only wood,
peat and other natural
resources.
About trends in economy IV:
There was a kind of shock while joining with EU – it appeared it was not so
simple to get to EU market as it had been hoped. To the products imported form
non-EU countries (Far Eastern cars and electronics, for example) the custom
taxes were added. As a result of it, both the export and import decreased.
To sum up, we cannot
claim that the period between 1993 – 2006 was a
uniform period with stable
conditions. Now question
arises – how to describe
the data mathematically?
How to interpolate and
approximate the data so
that the method chosen by
us would describe the
situation adequately and
without
remarkable
distortion?
Lagrange interpolating polynomials:
With interpolation we represent a set of points with a curve that passes exactly
through all of the points. On the next figure we can see Estonian export and
import data with Lagrange interpolating polynomials.
As can be seen, the polynomial goes through all of the points and is quite a good
representation of the data in an interval of, for example, [1995; 2004]. Outside of
this interval, that is, near the endpoints, the polynomial behaves badly. Indeed,
high-order interpolating polynomials should be used with caution.
Least squares fitting:
Least squares fitting is a mathematical procedure for finding the best-fitting
curve to a given set of points by minimizing the sum of the squares of the offsets
of the points from the curve. If we use 7-degree polynomials with least squares
fitting, we can see results on next figure:
The only conclusion we can make from the least-square method is that between
1993 and 2006 Estonian export grow slower than the import and with every year
Estonian foreign trade deficit increased. This plot confirms that the polynomial fit
to the data is not adequate; the residuals contain more information, in some
places the difference between initial data and least square fitting curve is more
than ten milliards.
Cubic spline interpolation :
If we use cubic spline interpolation, we get “better” results (Figure below). Cubic
spline graphics go through all points and they are smooth.
Unfortunately, the smoothness of the graphics is not important (economical
processes are not smooth). The question arises – is there any functions, which
represent the data better than the cubic splines?
Piecewise interpolation with cubic polynomials :
If we have many points and thus an interpolating polynomial of a high order, the
result may be bad, that is, the polynomial behaves badly, particularly near the
endpoints. Often it is better to proceed piecewise: calculate low-order polynomials between successive points. If we calculate the piecewise-cubic interpolating function for the data, the results are a little bit better then cubic splines.
Linear splines :
And of course, if we use linear spline interpolation, then the resultant spline is
just a polygon – it is very good and simple method.
Comparing I: Cubic splines vrs linear splines:
We can compare the graphs of cubic splines with linear splines.
Comparing II: Piecewise cubic interpolation vrs linear splines:
Using “MATHEMATICA ver 4.2 :
Lagrange interpolating polynomials of the 13th order, least square fitting
polynomials of the 7th order, linear and cubic splines and piecewise
interpolated cubic polynomials are found from the data and their
graphics representation is prepared with computer algebra package
Mathematica 4.2.
The author tried to compute the numerical differences between different
interpolations by calculating definite integral of the absolute value of
their difference. Unfortunately, built-in Mathematica functions were
surprisingly weak in numerical integration. It shows computer algebra
packages work well with classical school examples, but often fail with
real-life examples.
Thank you for your attention!
Tõnu Tõnso, Tallinn University
[email protected]