I am using 1.0.1 version of lyx and facing a bug problem in a lyx file.
I tried to resolve it by taking the help from help menu but could not.
Could you please help me in this regard.
The file is attached herewith.
Kind regards,
I. Arshad
#This file was created by <aairsh> Fri Jul 28 15:23:10 2000
#LyX 1.0 (C) 1995-1999 Matthias Ettrich and the LyX Team
\lyxformat 2.15
\textclass article
\language default
\inputencoding default
\fontscheme default
\graphics default
\paperfontsize default
\spacing single
\papersize Default
\paperpackage a4
\use_geometry 0
\use_amsmath 0
\paperorientation portrait
\secnumdepth 3
\tocdepth 3
\paragraph_separation skip
\defskip medskip
\quotes_language english
\quotes_times 2
\papercolumns 1
\papersides 1
\paperpagestyle default
\layout Title
MULTIPLE LINEAR REGRESSION LINE OF SATURATION/HUE/GREY VALUES ON COORDINATES
(X AND Y)
\layout Section
PURPOSE:
\layout Enumerate
To identify the blue-sky bits on days with mixture.
\layout Enumerate
To find the relationship between the Saturation/Hue/Grey values and the
coordinates(X and Y).
So that we can use this relation to estimate the parameters of a cloudy
day.
\layout Section
WHY THERE IS PROBLEM:
\layout Standard
We used Saturation, Hue and Grey values to find their relaionship with rows(X)
and columns(y) because all these values are useful to identify the blue-sky.
We observed in the frequency distribution of Saturation, Hue and Grey and
also by ploting these values according to their respective values that
these values have the multiple linear relation with their co-ordinates(rows
& columns).
But in fitting a multiple linear regression line a problem occurs, because
the appearance of blue-sky depends on:
\layout Enumerate
position of the Sun.
\layout Enumerate
Amount of haze.
\layout Standard
However for a fixed amount of haze and at a fixed time of day, the equation
\layout Standard
\begin_inset Formula \( s=\alpha \)
\end_inset
\begin_inset Formula \( +\beta (X-\overline{{X}})+\gamma (Y-\overline{{Y}} \)
\end_inset
\begin_inset Formula \( ) \)
\end_inset
\layout Standard
or
\layout Standard
\begin_inset Formula \( s=\alpha \)
\end_inset
\begin_inset Formula \( +\beta x+\gamma y \)
\end_inset
Where
\begin_inset Formula \( x=X-{}\bar{X} \)
\end_inset
and
\begin_inset Formula \( y=Y-{}\bar{Y} \)
\end_inset
\layout Standard
works well and gives the value of
\begin_inset Formula \( R^{2} \)
\end_inset
\begin_inset Formula \( \geq \)
\end_inset
\begin_inset Formula \( 0.90 \)
\end_inset
\layout Standard
For a given picture, with parameter estimated from that picture, we have
\begin_inset Formula \( {}\hat{s}\approx \)
\end_inset
\begin_inset Formula \( s \)
\end_inset
\layout Standard
While for a fixed amount of haze and at a fixed time of day, the equation
\layout Standard
\begin_inset Formula \( h=\alpha \)
\end_inset
\begin_inset Formula \( +\beta (X-\overline{{X}})+\gamma (Y-\overline{{Y}} \)
\end_inset
\begin_inset Formula \( ) \)
\end_inset
\layout Standard
or
\layout Standard
\begin_inset Formula \( h=\alpha \)
\end_inset
\begin_inset Formula \( +\beta x+\gamma y \)
\end_inset
\layout Standard
does not work well, for a given picture, with parameter estimated from that
picture, we have high variation in errors.
\layout Standard
However the equation
\layout Standard
\begin_inset Formula \( g=\alpha \)
\end_inset
\begin_inset Formula \( +\beta (X-\overline{{X}})+\gamma (Y-\overline{{Y}} \)
\end_inset
\begin_inset Formula \( ) \)
\end_inset
\layout Standard
or
\layout Standard
\begin_inset Formula \( g=\alpha \)
\end_inset
\begin_inset Formula \( +\beta x+\gamma y \)
\end_inset
\layout Standard
also gives the precise estimates of the parameters.
\layout Subsection
To identify the position of the Sun
\layout Standard
To identify the position of the Sun, we observed that saturation is much
low corresponding to the rays of the Sun.
so to find the column with minimum total of s-values will be useful to
identify the position of the Sun.
\layout Itemize
If that column is the first column of the plane then the sun is to the left
of the plane.
\layout Itemize
If that column is the last column of the plane then the Sun is to the right
of the plane.
\layout Itemize
If that column is any other, then we need to fit the separate planes, one
for those columns which are
\begin_inset Formula \( \leq \)
\end_inset
the column having the minimum total of s-values, and other for those columns
which are
\begin_inset Formula \( \geq \)
\end_inset
the column having the minimum total of s-values.
So on the left of the screen we fit the model
\begin_inset Formula \( s=\alpha \)
\end_inset
\begin_inset Formula \( +\beta (x-\overline{{x}})+\gamma (y-\overline{{y}} \)
\end_inset
\begin_inset Formula \( ) \)
\end_inset
and on the right of the screen we fit the model
\begin_inset Formula \( s=\alpha \)
\end_inset
\begin_inset Formula \( ^{/}+\beta ^{/}(x-\overline{{x}})+\gamma ^{/}(y-\overline{{y}}
\)
\end_inset
\begin_inset Formula \( ) \)
\end_inset
\layout Standard
where
\begin_inset Formula \( \alpha ^{/}=\alpha +2\gamma k \)
\end_inset
, where
\begin_inset Formula \( k \)
\end_inset
\begin_inset Formula \( = \)
\end_inset
Column having the minimum total of s-values.
\layout Standard
\begin_inset Formula \( \beta ^{/}=\beta \)
\end_inset
and
\begin_inset Formula \( \gamma ^{/}=-\gamma \)
\end_inset
\layout Subsection
Solution of the problems
\layout Standard
To overcome the problems mentioned above, we take the following measures:
\layout Enumerate
we gave the weight
\begin_inset Formula \( 1 \)
\end_inset
to saturation where it was greater than
\begin_inset Formula \( 0.12 \)
\end_inset
and
\begin_inset Formula \( 0 \)
\end_inset
elsewhere.
\layout Enumerate
We use the iteration to improve the goodness of fit.
\layout Section
PROBLEM:
\layout Standard
We want to use pre-estimated parameters because, on a cloudy day we can't
do otherwise.
(or we can do ?)
\layout Section
FOR A BALL-PARK FIGURE
\layout Standard
By using the xv program we observed that saturation vary from
\begin_inset Formula \( 30 \)
\end_inset
to
\begin_inset Formula \( 50 \)
\end_inset
against the bluesky and it vary from
\begin_inset Formula \( 0 \)
\end_inset
to
\begin_inset Formula \( 10 \)
\end_inset
against the cloudy.
\layout Itemize
For blue-sky saturation vary from
\begin_inset Formula \( 30 \)
\end_inset
to
\begin_inset Formula \( 50 \)
\end_inset
.
\layout Itemize
For cloudy saturation vary from
\begin_inset Formula \( 0 \)
\end_inset
to
\begin_inset Formula \( 10 \)
\end_inset
.
\layout Section*
RESULTS:
\layout Standard
We found that
\layout Itemize
hue values don't have the multiple linear relation with the Coordinates.
\layout Itemize
Grey and Saturation are most useful to identify the blue-sky bits.
\layout Standard
We fit the multiple linear regression line of hue on X and Y.
Which we found useless to identify the blue sky as this line was not a
good fit.
\layout Standard
The multiple regression line of grey values on X and Y is useful to identify
the blue-sky bits from the day with mixture.
\layout Standard
The multiple regression line of saturation on X and Y is more useful to
identify the blue-sky bits as it is best fit for the data.
\layout Standard
At the first step, we take only the blue-sky data to see how best the fit
is?
\layout Section
RELATIONSHIP BETWEEN SATURATION/HUE/GREY VALUES AND TIME
\layout Standard
To see the change in Saturation with the per unit change in time, we used
the different blue-sky data of the same day to fit the model
\begin_inset Formula \( s=\alpha \)
\end_inset
\begin_inset Formula \( +\beta (X-\overline{{X}})+\gamma (Y-\overline{{Y}} \)
\end_inset
\begin_inset Formula \( ) \)
\end_inset
\layout Standard
and then plot the estimates vs time and observed that:
\layout Itemize
The constant term
\begin_inset Formula \( \alpha \)
\end_inset
followed the pattern of increase in the morning upto some certain time
and then decrease.
\layout Itemize
The term
\begin_inset Formula \( \beta \)
\end_inset
Which measure the slope on row, followed the pattern of increase in the
morning upto some certain time and then decrease in the estimate of
\begin_inset Formula \( \beta \)
\end_inset
occurred.
\layout Itemize
The term
\begin_inset Formula \( \gamma \)
\end_inset
which measure the change in saturation with per unit change in column,
and followed
\layout Standard
As we observed in the frequency distribution of Saturation for a blue-sky
data and also by colouring each pixel of a blue-sky data according to its
saturation value that the saturation has the multiple linear relation with
its co-ordinates(X & Y).
We found on blue-sky day that the saturation increases as the number of
column increses and has almost the similar relation with rows except at
the bottom of the picture because of haze.
\layout Standard
The similer relation was observed for Hue and Grey values.
\layout Standard
Then we fit the multiple linear regression line of saturation on rows(
\begin_inset Formula \( X \)
\end_inset
) and columns(
\begin_inset Formula \( Y \)
\end_inset
) by using the following deviation model:
\layout Standard
\begin_inset Formula \( s=\alpha \)
\end_inset
\begin_inset Formula \( +\beta (X-\overline{{X}})+\gamma (Y-\overline{{Y}} \)
\end_inset
\begin_inset Formula \( ) \)
\end_inset
Where
\begin_inset Formula \( \overline{{X}} \)
\end_inset
\begin_inset Formula \( = \)
\end_inset
mean of number of rows and
\begin_inset Formula \( \overline{{Y}}= \)
\end_inset
mean of number of columns
\layout Standard
\begin_inset Formula \( \hat{{}\alpha =\bar{{}s \)
\end_inset
\layout Standard
\begin_inset Formula \( \hat{{}\beta
=\frac{S_{xs}S_{yy}-SysS_{xy}}{S_{xx}S_{yy}-S_{xy}S_{xy}} \)
\end_inset
\layout Standard
\begin_inset Formula \( \hat{{}\gamma
=\frac{S_{xs}S_{xy}-S_{ys}S_{xx}}{S_{xy}S_{xy}-S_{yy}S_{xx}} \)
\end_inset
\layout Standard
We also calculated the residuals (
\begin_inset Formula \( s-{}\hat{s} \)
\end_inset
) and we found that this model is good fit for a particular figure and at
a particular time but we cann't use the estimates of the parameters estimated
from any particulr picture to estimates the parameters of other pictures
because of the problem that how blue is a blue-sky depends on the following:
\layout Enumerate
position of the Sun.
\layout Enumerate
Amount of haze.
\layout Standard
To know the position of the Sun, we observed by using the xv programme on
a day when the Sun is very clear on the top of the picture that the saturation
is much low against the rays of Sun and high elsewhere.
So we found the column having the minimum total of saturaton.
It shows the position of the Sun, if that column is the first column of
the picture then the Sun is on the left and we fit the above model for
the whole plane.
If that column is the last column of the picture then the Sun is on the
right and again we fit the same model for the plane.
But if the column having minimum total of saturation is any other, then
we need to fit the separate planes on for those columns
\begin_inset Formula \( \leq \)
\end_inset
that particular column and the other for those column
\begin_inset Formula \( \geq \)
\end_inset
that particular plane.
\layout Standard
For the first plane we fit the model
\begin_inset Formula \( s=\alpha \)
\end_inset
\begin_inset Formula \( +\beta x+\gamma y \)
\end_inset
and for the second plane we fit the model
\begin_inset Formula \( s=\alpha \)
\end_inset
\begin_inset Formula \( ^{/}+\beta ^{/}x+\gamma ^{/}y \)
\end_inset
\layout Standard
We want some values on each plane when
\begin_inset Formula \( y=k \)
\end_inset
\layout Standard
So for first plane
\begin_inset Formula \( s=\alpha \)
\end_inset
\begin_inset Formula \( +\beta x_{1}+\gamma k \)
\end_inset
\layout Standard
\begin_inset Formula \( s=\alpha \)
\end_inset
\begin_inset Formula \( +\beta x_{2}+\gamma k \)
\end_inset
\layout Standard
and for the second plane
\begin_inset Formula \( s=\alpha ^{/} \)
\end_inset
\begin_inset Formula \( +\beta ^{/}x_{1}+\gamma ^{/}k \)
\end_inset
\layout Standard
\begin_inset Formula \( s=\alpha ^{/} \)
\end_inset
\begin_inset Formula \( +\beta ^{/}x_{2}+\gamma ^{/}k \)
\end_inset
\layout Standard
Bycomparing we got
\begin_inset Formula \( \beta (y_{1}-y_{2})=\beta ^{/}(y_{1}-y_{2}) \)
\end_inset
\layout Standard
therefore
\begin_inset Formula \( \beta =\beta ^{/} \)
\end_inset
\layout Standard
hence
\begin_inset Formula \( \alpha +\beta x_{1}+\gamma k= \)
\end_inset
\begin_inset Formula \( \alpha ^{/} \)
\end_inset
\begin_inset Formula \( +\beta ^{/}x_{1}+\gamma ^{/}k \)
\end_inset
\layout Standard
\begin_inset Formula \( \alpha +\gamma k= \)
\end_inset
\begin_inset Formula \( \alpha ^{/} \)
\end_inset
\begin_inset Formula \( +\gamma ^{/}k \)
\end_inset
\layout Standard
\begin_inset Formula \( \alpha ^{/} \)
\end_inset
\begin_inset Formula \( =\alpha +(\gamma -\gamma ^{/})k \)
\end_inset
\layout Standard
Consider two y values equi-distant from k.
We expect the saturation to be the same (for the same y)
\layout Standard
i.e.
we want
\layout Standard
\begin_inset Formula \( \alpha +\gamma (k-l)=\alpha ^{/}+\gamma ^{/}(k+l) \)
\end_inset
\layout Standard
i.e.
\begin_inset Formula \( \alpha +\gamma (k-l)=\alpha ^{/}+(\gamma -\gamma ^{/})+\gamma
^{/}(k+l) \)
\end_inset
\layout Standard
\begin_inset Formula \( \gamma (k-l-k)=\gamma ^{/}(k+l-k) \)
\end_inset
\layout Standard
\begin_inset Formula \( -\gamma l=\gamma ^{/}l \)
\end_inset
\layout Standard
therefore
\begin_inset Formula \( \gamma ^{/}=-\gamma \)
\end_inset
\layout Standard
and
\begin_inset Formula \( \alpha ^{/}=\alpha +2\gamma \)
\end_inset
\layout Comment
Rsquare,Partial Total Regression sum of squares
\layout Standard
Fitted model:
\layout Standard
\begin_inset Formula \( \hat{{}s=\hat{{}\alpha +\hat{{}\beta
(X-\bar{{}X)+\hat{{}\gamma (Y-\bar{{}Y) \)
\end_inset
\layout Standard
\begin_inset Formula \( R^{2}=\frac{Regr.\: S.\: S.}{Total\: S.\: S.} \)
\end_inset
\layout Standard
\begin_inset Formula \( =1-\frac{Error\: S.\: S.}{Total\: S.\: S.} \)
\end_inset
\layout Standard
\begin_inset Formula \( =1-\frac{\Sigma (s-\hat{{}s)^{2}}{\Sigma (s-\bar{{}s)^{2}} \)
\end_inset
\layout Standard
\begin_inset Formula \( \sigma ^{\bigwedge 2}=Error\: M.\: S.=\frac{\Sigma
(s-\hat{{}s)^{2}}{n-3} \)
\end_inset
\layout Standard
\begin_inset Formula \( Regr.\: S.\: S.\: for\: x=\frac{(\Sigma sx)^{2}}{(\Sigma
x)^{2}} \)
\end_inset
\layout Standard
\begin_inset Formula \( Regr.\: S.\: S.\: for\: y=\frac{(\Sigma sy)^{2}}{(\Sigma
y)^{2}} \)
\end_inset
\layout Standard
\begin_inset Formula \( Regr.\: S.\: S.\: for\: xy(both)=total\: S.\: S.-Error\: S.\:
S. \)
\end_inset
\layout Standard
\begin_inset Formula \( F-ratio\: for\: X=\frac{Regr.\: S.\: S.\: for\: x}{Error.\:
M.\: S.} \)
\end_inset
\latex latex
\backslash
\backslash
\layout Standard
\begin_inset Formula \( F-ratio\: for\: y=\frac{Regr.\: S.\: S.\: for\: y}{Error.\:
M.\: S.} \)
\end_inset
\latex latex
\backslash
\backslash
\layout Standard
\begin_inset Formula \( partial\: F-ratio\: for\: x=\frac{Regr.\: S.\: S.\: for\:
xy-Regr.\: S.\: S.\: for\: Y}{Error.\: M.\: S.} \)
\end_inset
\latex latex
\backslash
\backslash
\layout Standard
\begin_inset Formula \( partial\: F-ratio\: for\: y=\frac{Regr.\: S.\: S.\: for\:
xy-Regr.\: S.\: S.\: for\: x}{Error.\: M.\: S.} \)
\end_inset
\layout Comment
residual
\layout Standard
\begin_inset Formula \( Residual=error=s-\hat{{}s \)
\end_inset
\layout Standard
76
\layout Standard
42.
30
\layout Standard
79
\layout Standard
41
\layout Standard
40.
\layout Standard
99
\layout Standard
85
\layout Standard
39.
\layout Standard
83
\layout Standard
88
\layout Standard
38.
\layout Standard
83
\layout Standard
91
\layout Standard
37.
\layout Standard
34
\layout Standard
94
\layout Standard
35 .42
\layout Standard
163
\layout Standard
30.
69
\layout Standard
166
\layout Standard
32.
12
\layout Standard
169
\layout Standard
32.91
\the_end
