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