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3 Statistical Analysis
3.1 Chi-square Goodness-of-Fit Test
When an analyst attempts to fit a statistical model to ob-
served data, they may wonder how well the model actu-
ally reflects the data. How "close" are the observed values
to those which would be expected under the fitted model?
One statistical test that addresses this issue is the chi-square
goodness-of-fit test. The chi-square test is used to test if
a sample of data came from a population with a specific
distribution. The chi-square goodness-of-fit test is applied
to binned data. This is actually not a restriction since for
non-binned data you can simply calculate a histogram or
frequency table before generating the chi-square test. How-
ever, the value of the chi-square test statistic are dependent
on how the data is binned.
The chi-square test is defined for the hypothesis:
Ho: The data follow a specified distribution.
Ha: The data do not follow the specified distri-
bution.
Test Statistic: For the chi-square goodness-of-
fit computation, the data is divided into k bins and the test
statistic is defined as in
k (O - Eg)2
X2- E (4)
i=1
where Oi is the observed frequency for bin i and E is the
expected frequency for bin i. Put simply if the computed
test statistic X2 is large, then the observed and expected
values are not close and the model is a poor fit to the data.
Before understanding how to reject or accept a hy-
pothesis we need to understand two terms which help in
this process.
Degrees of Freedom: The number of independent pieces
of information that go into the estimate of a parameter is
called the degrees of freedom (df) [12]. In general, the
degrees of freedom of an estimate is equal to the number
of independent scores that go into the estimate minus the
number of parameters estimated as intermediate steps or
constraints in the estimation of the parameter itself. For
example, if the variance, o2, is to be estimated from a ran-
dom sample of N independent scores, then the degrees of
freedom is equal to the number of independent scores (N)
minus the number of parameters estimated as intermediate
steps (one, mean) and is therefore equal to N - 1.
In chi-square test statistic with k data bins, degrees
of freedom can be calculated as the number of data bins k
minus the number of constraints. For chi-square test statis-
tic with k data bins the only constraint is the number of
observed data points equal to the number of expected data
points. Therefore, df for a chi-square test statistic with k
data bins is equal to k - 1.
Level of Significance or p-value: The p-value or calcu-
lated probability is the estimated probability of rejecting
the null hypothesis (Ho) of a study question when that hy-
pothesis is true [12]. The term significance level is used
to refer to a pre-chosen probability and the term "p-value"
is used to indicate a probability that one calculates after a
given study.
The smaller the p-value, the more strongly the test
confirms the null hypothesis. A p-value of .05 or less con-
firms the null hypothesis "at the 5% level" that is, the sta-
tistical assumptions used imply that only 5% of the time
would the supposed statistical process produce a finding
this extreme if the null hypothesis were false. A p < 0.05
is considered statistically significant and p < 0.001 is con-
sidered statistically highly significant.
3.2 Curve Fitting
Field data is often accompanied by noise. Even though
all control parameters (independent variables) remain con-
stant, the resultant outcomes (dependent variables) vary. A
process of quantitatively estimating the trend of the out-
comes, also known as regression or curve fitting, therefore
becomes necessary.
The curve fitting process fits equations of approximat-
ing curves to the raw field data. Nevertheless, for a given
set of data, the fitting curves of a given type are generally
not unique. Thus, a curve with a minimal deviation from
all data points is desired. This best-fitting curve can be ob-
tained by the method of least squares.
For Log-distance Path Loss Model a linear least
squares curve fitting is used. The least-squares line uses
a straight line equation as shown below
Y = PX + P2,
(5)
to approximate the given set of data, (x1,yl), (z2,y2), *...
(xn,yn), where n >=2. The best fitting curve f(x) has the
least square error, i.e.,
n1
i=-1
n
f(x)]2 - ~
i=1
(Plxi + P2)]2. (6)
4 Experimental Setup
Four different scenarios are considered for measurements.
The scenarios used will help in developing signal loss equa-
tions, by which a generalization for propagation in an in-
door environment at 2.4 GHz can be obtained. The scenar-
ios are described as follows:
* Closed Corridor: A closed corridor is used for signal
measurements. This corridor is closed on both sides
with walls. This corridor is 9' high and 8'7" wide.
Signal measurements are taken at every five feet inter-
val in the middle of the corridor.
* Open corridor: An open corridor is used for signal
measurements. The corridor is open on one side and
closed with a wall on the other side. This corridor is
12'5" high and 14'7" wide. Signal measurements are
taken at every five feet interval in the middle of the
corridor.
