The Closest Line to a Data Set in the Plane. David Gurney Southeastern Louisiana University Hammond, Louisiana

Transcription

1 The Closest Lie to a Data Set i the Plae David Gurey Southeaster Louisiaa Uiversity Hammod, Louisiaa ABSTRACT This paper looks at three differet measures of distace betwee a lie ad a data set i the plae: the sum of squares, the sum of the vertical deviatios, ad the sum of the orthogoal distaces. Each measure will be described ad the closest lies for various data sets will either be computed mathematically or approximated usig the computer. Fially, the actual values of these distace measures will be compared.

2 Throughout this paper we will suppose the data set we are workig with cotais ordered pairs of the form (x i, y i ). I each case, we will be lookig for a lie L of the form y = ax + b which is closest to the give data set. What we mea by closest will deped o the distace measure beig used. The first distace measure we cosider is the usual sum of squares. I this case, the closest lie L miimizes the sum of the squared differeces betwee the y-values of the data poits ad the y-values of L. I other words, L miimizes ( ) 2 (.) Square Sum = yi ( axi + b) i= Gauss proposed this measure, ad calculus ca be used to show the slope ad y-itercept of the closest lie i this case are give by (.2) ( xi x)( yi y) i= a = ad b= y ax 2 ( xi x) i= (Tamhae ad Dulap 2000) The other distace measures we will look at are actually simple modificatios of (.). For the ext measure, the squarig process is replaced by the absolute value. The goal is ow to miimize the sum of the deviatios from the actual y-values. I formula form, we have (.3) Deviatio Sum = yi ( axi + b) i= The fial measure defies the distace from a poit to the lie by the legth of the perpedicular segmet from that poit to the lie. If a = 0, the the lie is horizotal ad b must be chose to miimize the sum (.4) Orthog Sum* = y b. i= i

3 If a is ot zero, the the slope perpedicular to L is /a. The earest poit o the lie L to (x i, y i ) is thus the solutio ( ˆ, ˆ ) x y of the system i i (.5) y = ax+ b y yi = x xi a ( ). Hece, (.6) ( ) ( 2 ) ( 2 ) ( 2) xˆ i = xi + ayi ab /+ a yˆ i = axi + a yi + b /+ a. The sum we eed to miimize is thus the followig. (.7) 2 2 ( ( x ˆ ) ( ˆ i xi + yi yi) ) i= ( ) ( a ) /2 ( ) axi yi + b yi axi b = a i= ( a ) + + /2. This last sum ca be simplified to give the followig formula. (.8) Orthog Sum = yi ( axi + b) + 2 a i= Note that if a is zero, (.4) ad (.8) are equivalet. Also, ote that the oly differece from the deviatio sum (.3) is the factor 2 / + a i frot of (.8). Now we wat to test these distace formulas. To do this, we will look for the closest lies to various data sets usig these differet formulas. There is o ice meas of fidig the closest lie usig the last two distace formulas, but the followig fact meas we do ot have to cosider every pair of umbers a ad b. The sum. (.9) k j= w j m

4 is miimized whe m is the media of the set { w j} j give value of a, (.0) ( ) y ax + b = y ax b i i i i i= i= will be a miimum, whe b is the media of the set { } k =. (Tamhae & Dulop 2000) Thus for a i i i y ax =. This provides us with a method for approximatig the miimum deviatio sum. For each value of a, we let b be the i i i media of { } y ax = ad compute the sum of the deviatios for the give data set. Fially, we choose the value of a that miimizes these sums. A similar approach works with the miimum orthogoal distace. Usig these approaches, the lies that miimize the deviatio sums ad the orthogoal distace sums were estimated for three differet data sets. The first data set was based o a liear equatio with a small radom error added to the y-value. The secod data set was produced by makig the slope of a liear equatio somewhat radom. The third set was produced by geeratig pairs of radom umbers. The results are show o the pages that follow. Table below shows the close lies that result whe each of the three methods are applied to the radom error data. The last three colums give the distace betwee these close lies ad the data set usig the three differet distace measures. Table : Radom Error Data Distace Formula Close Lie Sum of Squares Distace Deviatio Sum Distace Orthogoal Distace Sum Sum of Squares y = 7.82x Deviatio Sum y = 7.86x Orthog. Distace y = 7.87x

5 Figure : Radom Error Data SqrSum DevSum OrthSum

6 Figure shows the radom error data with a plot of each close lie. I this case, the three lies plotted o the graph are almost idetical. Table 2 shows the close lies for the radom slope data together with the correspodig distace measures usig each method. Table 2: Radom Slope Data Distace Formula Close Lie Sum of Squares Distace Deviatio Sum Distace Orthogoal Distace Sum Sum of Squares y = 4.64x Deviatio Sum y = 4.7x Orthog. Distace y = 4.8x Figure 2 shows the graph of the radom slope data ad the three close lies. There is a greater differece betwee the three lies i this figure tha i Figure, but the lies are still very close together. Table 3 shows the close lies for the radom coordiate data ad the correspodig distace measures usig each method. I this case the orthogoal distace method resulted i two local miimums, so two lies are listed eve though oe lie defiitely has a smaller orthogoal distace sum tha the other. This situatio may idicate that for some data sets, the orthogoal distace measure could result i two distict lies with quite differet locatios with the same miimum distace. Of course, usig a lie to model radom data like this is ot a good idea.

7 Figure 2: Radom Slope Data SqrSum OrthSum DevSum

8 Table 3: Radom Coordiate Data Distace Formula Close Lie Sum of Squares Distace Deviatio Sum Distace Orthogoal Distace Sum Sum of Squares y = 0.074x Deviatio Sum y = 0.3x Orthog. Dist. y =.2x Orthog. Dist. 2 y =.9x The radom coordiate data ad the resultig lies are plotted i Figure 3. Notice that these lies are very differet i slope. This holds true eve for the two orthogoal distace lies. To check these distace measures for stability, poits with x-values betwee 20 ad 30 were removed from the radom error data to leave a ifluetial observatio o the right side. Close lies were foud usig the three distace measures, ad the results are show i Table 4 below. Notice that the deviatio sum lie ad the orthogoal distace lie are the same to oe decimal place. Table 4: Data with Ifluetial Observatio Method Close Lie Sum of Squares Deviatio Sum Orthogoal Distace Sum Sum of Squares y =7.850x Deviatio Sum y = 7.89x Orthogoal Distace y = 7.89x This modified data ad the three close lies are plotted i Figure 4. Notice that as i the origial data set, the three lies are almost the same.

9 Figure 3: Radom Coordiate Data Data SqrSum OrthSum OrthSum2 DevSum

10 Figure 4: Data with Ifluetial Observatio Data Square Dev./Orth

11 We will ow test the resistace of the three measures to raisig ad lowerig the ifluetial observatio the isolated poit o the right side of the graph (Rossma 995). Table 5 shows the results of raisig the ifluetial observatio. Just lookig at the slope, the sum of squares lie shows the most chage, followed by the orthogoal distace method. I the graph show i Figure 5, these two lies are oticeably closer to the ifluetial observatio. Table 5: Effect of Raisig Ifluetial Observatio Method Close Lie Sum of Squares Deviatio Sum Orthogoal Distace Sum Sum of Squares y = 8.498x Deviatio Sum y = 8.25x Orthogoal Distace y = 8.5x Table 6 shows the results of fidig the earest lie for the lowered ifluetial observatio. I this case, the oly lie that chaged was the sum of squares lie. The Figure 6 shows the lies geerated usig the lowered ifluetial observatio. Table 6: Effect of Lowerig Ifluetial Observatio Method Close Lie Sum of Squares Deviatio Sum Orthogoal Distace Sum Sum of Squares y = 7.72x Deviatio Sum y = 7.89x Orthogoal Distace y = 7.89x

12 Figure 5: Effect of Raisig Ifluetial Obsservatio Data Square Dev. Orth

13 Figure 6: Effect of Lowerig If. Obs Data Square Dev./Orth

14 For those of you who like correlatio coefficiets, the table below lists them alog with the correspodig P-values for the five data sets used i this paper. Table 7: Correlatio Coefficets ad P-values for the Data i this Paper Radom Error Data R = P-value < Radom Slope Data R = P-value < Radom Coordiate Data R = P-value = Data with Ifluetial Observatio R = P-value < Data with Raised Ifluetial Observatio R = 0.99 P-value < Data with Lowered Ifluetial Observatio R = P-value < I coclusio, we have looked at three distace measures betwee a lie ad a data set i the plae: the usual sum of squares distace, the vertical deviatio sum, ad the orthogoal distace sum. Fidig the closest lie to a data set usig the sum of squares is a stadard procedure i statistics. Fidig the closest lie usig the other two methods is ot a stadard procedure ad the results are ot ecessarily uique. The other two methods, however, do seem to be more resistat to chages i ifluetial observatios. The mai poit remais that there is more tha oe way to measure distace betwee a lie ad a data set. REFERENCES: Alla J. Rossma, Workshop Statistics: Discovery with Data, Spriger-Verlag New York, Ic, New York, 995. Ajit C. Tamhae & Dorothy D. Dulop, Statistics ad Data Aalysis for Elemetary to Itermediate, Pretice-Hall, Ic., Upper Saddle River, NJ 2000.