

Subject:
Weighted Least Linear Square Method simple example
Category: Science > Math Asked by: justin_championga List Price: $2.00 
Posted:
03 Sep 2006 03:22 PDT
Expires: 03 Oct 2006 03:22 PDT Question ID: 761806 
Hello I am looking for a simple example which I can follow showing the Weighted Least linear square method. I am hoping to find an example I can follow as I do not have a strong math background and I want to use this for a 2 dimensional graph with about 5 results on it. Thanks, Justin. 

There is no answer at this time. 

Subject:
Re: Weighted Least Linear Square Method simple example
From: mathtalkga on 06 Sep 2006 13:07 PDT 
Hi, Justin: Is it fair to assume you want to apply the weighted leastsquares method to fit a straight line to some points on a graph? regards, mathtalkga 
Subject:
Re: Weighted Least Linear Square Method simple example
From: justin_championga on 07 Sep 2006 00:17 PDT 
Yes it is that is exactly what I want :) Thanks, Justin. 
Subject:
Re: Weighted Least Linear Square Method simple example
From: berkeleychocolatega on 07 Sep 2006 19:45 PDT 
Suppose the weights are w1, w2, ... the independent values are x1, x2, ... and the dependent values are y1, y2, .... Form the following sums: I = the sum of the weights, II = the sum of the weights times the x's (w1*x1+w2*x2+...), III = the sum of the weights times the squares of the x's (w1*x1^2+w2*x2^2+...), IV = the sum of the weights times the y's (w1*y1+w2*y2+...), V = the sum of the weights times the x's times the y's (w1*x1*y1+w2*x2*y2+...). If m = the slope of the line and b = its intercept, then they are the solutions to the following two equations: I*b + II*m = IV and II*b + III*m = V. The solution to these two equations is the following: Let D = I*IIIII^2. Then b = [III*IV  II*V ]/D and m = [I*V  II*IV]/D. I'm surprized none of the paid answerers have responded to this question. Hope this helps. 
Subject:
Re: Weighted Least Linear Square Method simple example
From: mathtalkga on 08 Sep 2006 05:17 PDT 
As berkeleychocolatega points out, the use of weights in a (linear) weighted least squares fit amounts to introducing the weights as factors on each corresponding term of a summation in the usual formulas for an "ordinary" least squares fit. Let's do an example to make sure the concepts are clear. These calculations are easily organized in a spreadsheet. For example we might put the weights w_i, the independent observed variables x_i, and the dependent observed variables y_i in parallel rows: 1.0 2.0 1.0 2.0 4.0 (weights) 1.0 2.0 3.0 4.0 5.0 (x values) 2.35 2.29 2.24 2.27 2.25 (y values) This is a madeup example, motivated by a week in which a stock price falls. The sums defined by berkeleychocolatega are now: I 10 II 36 III 150 IV 22.71 V 81.39 which I used a spreadsheet to compute after adding rows for the products of weights times x values, xsquared values, y values, and x*y values (since "I" is just the sum of weights, it didn't need an auxiliary row). The denominator D = I*IIIII^2 = 204, and: yintercept: b = 2.34 slope: m = 0.02 (slight downward trend as expected) Perhaps most illustrative is to compute the "model fit" values of y predicted: 2.32 2.30 2.28 2.26 2.25 (y predicted) and to compare these with our starting observations: 2.35 2.29 2.24 2.27 2.25 (y observed) Although the input data bounces up and down a bit after the first "day", the larger weight assigned to the final observation (w_5 = 4) resulted in a close fit to that point. regards, mathtalkga 
Subject:
Re: Weighted Least Linear Square Method simple example
From: mathtalkga on 09 Sep 2006 05:42 PDT 
The coefficients of the line as I gave them above are rounded. More precise is: m = 0.01794 b = 2.33559 Adding another digit to the predicted y values may also clarify the results: 2.318 2.300 2.282 2.264 2.246 (y predicted) 2.35 2.29 2.24 2.27 2.25 (y observed)      0.032 0.010 0.042 0.006 0.004 (y errors) So the observations with the larger weights in this case wound up with the smaller errors. Because the weights were all (positive) integers in this problem, another way to formulate the example would be to repeat the observations as many times as their weight and use the "ordinary" least squares method. Of course repeating terms in the summations would lead in this fashion to the same calculations as above. For background note that justin_championga has recently posted a Question (Answered by elmartoga) about the ordinary least squares method: [Q: Least Linear Square Method simple example] http://answers.google.com/answers/threadview?id=760922 regards, mathtalkga 
Subject:
Re: Weighted Least Linear Square Method simple example
From: justin_championga on 09 Sep 2006 14:52 PDT 
trhank you very much for the replies I will try these out tomorrow morning. I again aplogise for not replying quickly as I had sort of given up on getting a reply and have not checked often enough :) Thanks, Justin. 
Subject:
Re: Weighted Least Linear Square Method simple example
From: justin_championga on 11 Sep 2006 00:41 PDT 
Thank you every much the answer you gave was perfect and very clear. I have followed it along now and put it into Excel and it works perfectly. :) Thank you very much for your time and effort, it is really appreciated. Justin Champion 
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