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| [[Image:Total least squares.svg|right|thumb|200xp| Deming regression. The red lines show the error in both ''x'' and ''y''. This is different from the traditional least squares method which measures error parallel to the ''y'' axis. The case shown, with deviations measured perpendicularly, arises when ''x'' and ''y'' have equal variances.]]
| | The following post is about 5 important steps which usually guide we inside a pursuit of getting slim and may coach we on how to lose 70 pounds inside merely 2 months and all proper factors to do. If you apply these procedures daily, you can lose fat plus receive to the actual weight you desire.<br><br>Calculating your BMR with a [http://safedietplansforwomen.com/bmr-calculator bmr calculator] is an significant step. This tells you how numerous calories you burn a day by really existing. Breathing, hearts beating, kidneys working and everything the bodies do takes calories. Knowing how several calories these functions utilize is significant knowledge in we weight loss system.<br><br>There several factors which may influence ones TD. Factors like: basal metabolic rate (BR), activity level, Lean Body Mass (LM), weight, gender and age. To get the most exact measuring we have to take into account these factors. There are numerous methods which you could use, some more exact than others. So to provide we an example how you can calculate a TE, I may use the Katch-McArdle formula. It is a quite correct method compared to others.<br><br>Carbs, whenever converted to glucose, are utilized mainly for stamina. Foods which are categorized mainly because carbs include: grains plus their flours, potatoes, sugars (all forms), fruits, veggies, plus anything made of them.<br><br>The initially piece is the Basic stuff you do. This really is everything except the work outs we will add to a daily routine. But we don't like to count every time you sneeze, besides the fact that which does burn calories. If you are basically sedentary during the day, you are able to increase a bmr by .2, and that is a advantageous estimate of the Basic calories. Remember, for this step never count any additional exercise you're doing in order to lose weight. Count the small standard details we do each day. If you kind all day in an office, choose the sedentary level. If you chase small youngsters around the apartment all day, choose a higher level. Most of us that are struggling to lose fat will be starting at the sedentary level.<br><br>At any provided time, 25 percent of all guys and 33 % of all women are on several sort of formal diet in the United States. More than 55 % gain back all of their weight and over what they began with.1 Unfortunately, most diets are a one-size-fits-all approach. With any diet book you pick off the bookstore shelf, or any aged diets passed down by your excellent aunt, there are the same diet for everyone. Some of those are completely unsound nutritionally while others may be backed by advantageous nutrition principles. Yet, even those with wise nutrition principles don't personalize their approach to fit each person's body makeup. These are generally unluckily a one-size-fits-all dieting approach.<br><br>In summary, you could certainly do very a bit to better the metabolism. Lifestyle change and consistent frequent exercise, all may boost metabolic rate. |
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| In [[statistics]], '''Deming regression''', named after [[W. Edwards Deming]], is an [[errors-in-variables model]] which tries to find the [[line of best fit]] for a two-dimensional dataset. It differs from the [[simple linear regression]] in that it accounts for [[errors and residuals in statistics|errors]] in observations on both the ''x''- and the ''y''- axis. It is a special case of [[total least squares]], which allows for any number of predictors and a more complicated error structure.
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| Deming regression is equivalent to the [[maximum likelihood]] estimation of an [[errors-in-variables model]] in which the errors for the two variables are assumed to be independent and [[normal distribution|normally distributed]], and the ratio of their variances, denoted ''δ'', is known.<ref>{{harv|Linnet|1993}}</ref> In practice, this ratio might be estimated from related data-sources; however the regression procedure takes no account for possible errors in estimating this ratio.
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| The Deming regression is only slightly more difficult to compute compared to the [[simple linear regression]]. Many software packages used in clinical chemistry, such as [[Analyse-it]], EP Evaluator, [[MedCalc]] and [[S-PLUS]] offer Deming regression.
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| The model was originally introduced by {{harvtxt|Adcock|1878}} who considered the case ''δ'' = 1, and then more generally by {{harvtxt|Kummell|1879}} with arbitrary ''δ''. However their ideas remained largely unnoticed for more than 50 years, until they were revived by {{harvtxt|Koopmans|1937}} and later propagated even more by {{harvtxt|Deming|1943}}. The latter book became so popular in [[clinical chemistry]] and related fields that the method was even dubbed ''Deming regression'' in those fields.<ref>Cornbleet, Gochman (1979)</ref>
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| == Specification ==
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| Assume that the available data (''y<sub>i</sub>'', ''x<sub>i</sub>'') are measured observations of the "true" values (''y<sub>i</sub>*'', ''x<sub>i</sub>*''):
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| : <math>\begin{align}
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| y_i &= y^*_i + \varepsilon_i, \\
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| x_i &= x^*_i + \eta_i,
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| \end{align}</math>
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| where errors ''ε'' and ''η'' are independent and the ratio of their variances is assumed to be known:
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| : <math> \delta = \frac{\sigma_\varepsilon^2}{\sigma_\eta^2}. </math>
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| In practice the variance of the <math>x</math> and <math>y</math> parameters is often unknown which complicates the estimate of <math> \delta </math> but where the measurement method for <math>x</math> and <math>y</math> is the same they are likely to be equal so that <math> \delta = 1 </math> for this case.
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| We seek to find the line of "best fit" ''y*'' = ''β''<sub>0</sub> + ''β''<sub>1</sub>''x*'', such that the weighted sum of squared residuals of the model is minimized:<ref>Fuller, ch.1.3.3</ref>
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| : <math>SSR = \sum_{i=1}^n\bigg(\frac{\varepsilon_i^2}{\sigma_\varepsilon^2} + \frac{\eta_i^2}{\sigma_\eta^2}\bigg) = \frac{1}{\sigma_\varepsilon^2} \sum_{i=1}^n\Big((y_i-\beta_0-\beta_1x^*_i)^2 + \delta(x_i-x^*_i)^2\Big) \ \to\ \min_{\beta_0,\beta_1,x_1^*,\ldots,x_n^*} SSR</math>
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| == Solution ==
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| The solution can be expressed in terms of the second-degree sample moments. That is, we first calculate the following quantities (all sums go from ''i'' = 1 to ''n''):
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| : <math>\begin{align}
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| & \overline{x} = \frac{1}{n}\sum x_i, \quad \overline{y} = \frac{1}{n}\sum y_i, \\
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| & s_{xx} = \tfrac{1}{n-1}\sum (x_i-\overline{x})^2, \\
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| & s_{xy} = \tfrac{1}{n-1}\sum (x_i-\overline{x})(y_i-\overline{y}), \\
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| & s_{yy} = \tfrac{1}{n-1}\sum (y_i-\overline{y})^2.
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| \end{align}</math>
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| Finally, the least-squares estimates of model's parameters will be<ref>Glaister (2001)</ref>
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| : <math>\begin{align}
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| & \hat\beta_1 = \frac{s_{yy}-\delta s_{xx} + \sqrt{(s_{yy}-\delta s_{xx})^2 + 4\delta s_{xy}^2}}{2s_{xy}} \\
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| & \hat\nu_1=\frac{-1}{\hat\beta_1} = \frac {-2 \delta s_{xy}}{s_{yy}-\delta s_{xx} - \sqrt{(s_{yy}-\delta s_{xx})^2 + 4\delta s_{xy}^2}}, \\
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| & \hat\beta_0 = \overline{y} - \hat\beta_1\overline{x}, \\
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| & \hat{x}_i^* = x_i + \frac{\hat\beta_1}{\hat\beta_1^2+\delta}(y_i-\hat\beta_0-\hat\beta_1x_i).
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| \end{align}</math>
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| where <math>\hat\nu_1</math> is slope of the line normal (perpendicular) to <math>\hat\beta_1</math>.
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| ==The case of equal error variances==
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| When <math>\delta=1</math>, Deming regression becomes [[orthogonal regression]]: it minimizes the sum of squared perpendicular distances from the data points to the regression line. In this case, denote each observation as a point ''z''<sub>''j''</sub> in the complex plane (i.e., the point (''x''<sub>''j''</sub>, ''y''<sub>''j''</sub>) is written as ''z''<sub>''j''</sub> = ''x''<sub>''j''</sub> + ''iy''<sub>''j''</sub> where ''i'' is the [[imaginary unit]]). Denote as ''Z'' the sum of the squared differences of the data points from the [[centroid]] (also denoted in complex coordinates), which is the point whose horizontal and vertical locations are the averages of those of the data points. Then:<ref>Minda and Phelps (2008), Theorem 2.3.</ref>
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| *If ''Z'' = 0, then every line through the centroid is a line of best orthogonal fit.
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| *If ''Z'' ≠ 0, the orthogonal regression line goes through the centroid and is parallel to the vector from the origin to <math>\sqrt{Z}</math>.
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| A [[trigonometry|trigonometric]] representation of the orthogonal regression line was given by Coolidge in 1913.<ref>Coolidge, J. L. (1913).</ref>
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| ===Application===
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| In the case of three [[Line (geometry)|non-collinear]] points in the plane, the [[triangle]] with these points as its [[vertex (geometry)|vertices]] has a unique [[Steiner inellipse]] that is tangent to the triangle's sides at their midpoints. The [[Ellipse#Elements of an ellipse|major axis of this ellipse]] falls on the orthogonal regression line for the three vertices.<ref>Minda and Phelps (2008), Corollary 2.4.</ref>
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{cite journal|last=Adcock|first=R. J.|year=1878|title=A problem in least squares|journal=The Analyst|volume=5|issue=2|pages=53–54|publisher=Annals of Mathematics|doi=10.2307/2635758|jstor=2635758}}
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| * {{cite journal|author=Coolidge, J. L.|year=1913|title=Two geometrical applications of the mathematics of least squares|journal=The [[American Mathematical Monthly]]|volume=20|issue= 6|pages=187–190}}
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| * {{cite journal|author=Cornbleet, P.J.|coauthor=Gochman, N.|year=1979|title=Incorrect Least–Squares Regression Coefficients|journal=Clin. Chem.|volume=25|issue=3|pages=432–438|pmid=262186}}
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| * {{cite book|last=Deming|first=W. E.|authorlink=W. Edwards Deming|year=1943|title=Statistical adjustment of data|publisher=Wiley, NY (Dover Publications edition, 1985)|isbn=0-486-64685-8}}
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| * {{cite book|last=Fuller|first=Wayne A.|year=1987|title=Measurement error models|publisher=John Wiley & Sons, Inc|isbn=0-471-86187-1}}
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| * Glaister, P. (March 2001). "Least squares revisited". ''[[The Mathematical Gazette]]'' 85: 104-107.
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| * {{cite book|last=Koopmans|first=T. C.|year=1937|title=Linear regression analysis of economic time series|publisher=DeErven F. Bohn, Haarlem, Netherlands}}
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| * {{cite journal
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| | last = Kummell
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| | first = C. H.
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| | year = 1879
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| | title = Reduction of observation equations which contain more than one observed quantity
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| | journal = The Analyst
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| | volume = 6
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| | issue = 4
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| | pages = 97–105
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| | publisher = Annals of Mathematics
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| | doi = 10.2307/2635646
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| | jstor = 2635646
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| }}
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| * {{cite journal
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| | last = Linnet
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| | first = K.
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| | year = 1993
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| | title = Evaluation of regression procedures for method comparison studies
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| | journal = Clinical Chemistry
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| | volume = 39
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| | issue = 3
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| | pages = 424–432
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| | url = http://www.clinchem.org/cgi/reprint/39/3/424
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| | pmid = 8448852
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| }}
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| *{{cite journal
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| | last1 = Minda | first1 = D. | author1-link = David Minda | |
| | last2 = Phelps | first2 = S.
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| | issue = 8
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| | journal = [[American Mathematical Monthly]]
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| | mr = 2456092
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| | pages = 679–689
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| | title = Triangles, ellipses, and cubic polynomials
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| | url = http://www.geogebra.org/en/upload/files/english/steve_phelps/minda%20phelps.pdf
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| | volume = 115
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| | year = 2008}}
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| {{DEFAULTSORT:Deming Regression}}
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| [[Category:Regression analysis]]
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The following post is about 5 important steps which usually guide we inside a pursuit of getting slim and may coach we on how to lose 70 pounds inside merely 2 months and all proper factors to do. If you apply these procedures daily, you can lose fat plus receive to the actual weight you desire.
Calculating your BMR with a bmr calculator is an significant step. This tells you how numerous calories you burn a day by really existing. Breathing, hearts beating, kidneys working and everything the bodies do takes calories. Knowing how several calories these functions utilize is significant knowledge in we weight loss system.
There several factors which may influence ones TD. Factors like: basal metabolic rate (BR), activity level, Lean Body Mass (LM), weight, gender and age. To get the most exact measuring we have to take into account these factors. There are numerous methods which you could use, some more exact than others. So to provide we an example how you can calculate a TE, I may use the Katch-McArdle formula. It is a quite correct method compared to others.
Carbs, whenever converted to glucose, are utilized mainly for stamina. Foods which are categorized mainly because carbs include: grains plus their flours, potatoes, sugars (all forms), fruits, veggies, plus anything made of them.
The initially piece is the Basic stuff you do. This really is everything except the work outs we will add to a daily routine. But we don't like to count every time you sneeze, besides the fact that which does burn calories. If you are basically sedentary during the day, you are able to increase a bmr by .2, and that is a advantageous estimate of the Basic calories. Remember, for this step never count any additional exercise you're doing in order to lose weight. Count the small standard details we do each day. If you kind all day in an office, choose the sedentary level. If you chase small youngsters around the apartment all day, choose a higher level. Most of us that are struggling to lose fat will be starting at the sedentary level.
At any provided time, 25 percent of all guys and 33 % of all women are on several sort of formal diet in the United States. More than 55 % gain back all of their weight and over what they began with.1 Unfortunately, most diets are a one-size-fits-all approach. With any diet book you pick off the bookstore shelf, or any aged diets passed down by your excellent aunt, there are the same diet for everyone. Some of those are completely unsound nutritionally while others may be backed by advantageous nutrition principles. Yet, even those with wise nutrition principles don't personalize their approach to fit each person's body makeup. These are generally unluckily a one-size-fits-all dieting approach.
In summary, you could certainly do very a bit to better the metabolism. Lifestyle change and consistent frequent exercise, all may boost metabolic rate.