Straight Lines & Math By Joanne Green
Co-ordinates & Points The image shows random points lying in a plane with co-ordinates on X and Y axis. Haeuser et al. (1985) describe in detail that in this Cartesian grid Model planes are dimensional having shape, depth and interact with substrate and systems. 04/05/2014 Joanne Green 2
Co-ordinates & Points Essentially the dimensions of the plane allow for quantitative measurement of anything you wish to measure. Haeuser et al. (1985) measured a combination of equations but this power point only discussed Cartesian. Ferrier (2002) notes the spatial distribution of species is able to pose dilemmas for conservation, both biotic and abiotic, noting systematic land planning has been requested for over two decades. 04/05/2014 Joanne Green 3
Plotting Points The plotting of points is mapping of something. Naidoo et al. (2008) define the importance of mapping articulating its relationship to the economic and social benefits of ecosystem services. They infer mapping is similar to a computer server system in that mapping has the ability to deliver information. The points in the image could be trees ready to harvest in a forest, allotments & lakes in a country, or countries in Europe. Each having products for sale and information for analysis. 04/05/2014 Joanne Green 4
Plotting Points Points are noted as being in quadrants. Noting Haeuser et al. (1985) dimensions with Naidoo et al. (2008) ecosystem services these coordinates can only be found on the image. When substituting the image with a map, the coordinates could be reservoirs within the tetrads of an OS Grid Reference. Alternatively the points could be the points of a fingerprint. 04/05/2014 Joanne Green 5
From Points to Lines This image shows a straight line from two coordinates. Coordinates have properties which are discussed in greater detail in the next slides. 04/05/2014 Joanne Green 6
From Points to Lines Facts/Properties of the line Point A lies in quadrant 3; Point B lies in quadrant 1; The line crosses both the horizontal and vertical axis through the origin; The line has a positive gradient. 04/05/2014 Joanne Green 7
More Points & Lines 04/05/2014 Joanne Green 8
More Points & Lines Properties of the line Point A lies in quadrant 2; Point B lies in quadrant 4; The line crosses both the horizontal and vertical axis through the origin; The line has a negative gradient. 04/05/2014 Joanne Green 9
Even more points to lines The straight line could be the direction of nuisances such as smells, the flight path of birds, ground water direction. 04/05/2014 Joanne Green 10
Even more points to lines Properties of the line Point A lies in quadrant 3; Point B lies in quadrant 1; The line crosses both the horizontal and vertical axis through the origin; The line has a positive gradient. 04/05/2014 Joanne Green 11
The equation of a straight line passing through the origin 04/05/2014 Joanne Green 12
The equation of a straight line passing through the origin X -3-2 -1 0 1 2 3 Y=-3X 9 6 3 0-2 -6-9 04/05/2014 Joanne Green 13
Several straight lines X -3-2 -1 0 1 2 3 Y=-3X Y=-1.5X 9 4.5 6 3 3 1.5 0 0-2 -1.5-6 -3-9 -4.5 04/05/2014 Joanne Green 14
Several straight lines 04/05/2014 Joanne Green 15
Insect example: Several straight lines X -3-2 -1 0 1 2 3 Y=-2X Y=-2X Y=1.5X -6 6-4.5-4 4-3 -2 2-1.5 0 0 0 2-2 1.5 4-4 3 6-6 4.5 04/05/2014 Joanne Green 16
Several straight lines If regression coefficient of Y on X is found to be +2 X -3-2 -1 0 1 2 3 Y=2X -6-4 -2 0 2 4 6 If regression coefficient of Y on X is found to be -2 X -3-2 -1 0 1 2 3 Y=-2X 6 4 2 0-2 -4-6 If regression coefficient of Y on X is found to be +1.5 X -3-2 -1 0 1 2 3 Y=1.5X -4.5-3 -1.5 0 1.5 3 4.5 04/05/2014 Joanne Green 17
Insect example: Several straight lines Site area: 15cm 2, dimensions 5cm (L 1 to L 2 ), 3cm (L 1 to L 3 ). Surrounding area is runnels and bogs containing marsh frogs (Pelophylax ridibundus) wanting to eat Southern Damselfly (Coenagrion mercuriale) when it rests. Gradient Y=2X shows it takes the insect flying left to right at 0.002ms - approximately 33.5 s to fly from site edge (0,0) to (3,-6) resting place. 04/05/2014 Joanne Green 18
The equation of a general straight line 04/05/2014 Joanne Green 19
The equation of a general straight line Mean deviation from a frequency distribution Site Time taken Number Xxf Deviation in hrs to of visits (D) from do an Frequency the value insect (f) (X=6.7)=Dx survey (value/x) SD (σ) Dxf (Dx) 2 1 3 2 6 3.7 7.4 13.69 2 5 4 20 1.7 6.8 2.89 3 7 5 35 0.3 1.5 0.09 4 1 1 1 5.7 5.7 32.49 5 9 7 63 2.3 16.1 5.29 Totals 25 19 125 13.7 37.5 54.45 X 125/19 6.5789474 6.7 Mean D 37.5/19 1.9736842 2 σ 3.3 (54.45/5) 04/05/2014 Joanne Green 20
The equation of a general straight line X -3-2 -1 0 1 2 3 3X -9-6 -3 0 3 6 9 +4 +4 +4 +4 +4 +4 +4 +4 Y=3X+4-5 -2 1 4 7 10 13 04/05/2014 Joanne Green 21
The equation of a general straight line Product moment correlation Years Value (X) Frequency (f) X Deviation (D) from the value (X=29.2)=D x Y Deviation (D) from the value (Y=14.8)=Dx X 2 Y 2 XY 1 25 19 4.2 4.2 17.64 17.64 17.64 2 40 12 10.8 2.8 116.64 7.84 30.24 3 23 4 6.2 10.8 38.44 116.64 66.96 4 46 23 16.8 8.2 282.24 67.24 137.76 5 12 16 17.2 1.2 295.84 1.44 20.64 Totals 146 74 55.2 27.2 750.8 210.8 273.24 X 146/5 29.2 Y 74/5 14.8 r 273.24/ 750.8X210. 8 273.24/397.8 0.7 This shows a strong positive correlation between time taken and frequency of visits 04/05/2014 Joanne Green 22
The equation of a general straight line- 2 nd example X -3-2 -1 0 1 2 3 4X -12-8 -4 0 4 8 12-3 -3-3 -3-3 -3-3 -3 Y=4X-3-15 -11-7 -3 1 5 9 04/05/2014 Joanne Green 23
The equation of a general straight line- 2 nd example 04/05/2014 Joanne Green 24
The equation of a general straight lineinsect example X -3-2 -1 0 1 2 3 2X -6-4 -2 0 2 4 6 +1 +1 +1 +1 +1 +1 +1 +1 Y=2X+1-5 -3-1 1 3 5 7 04/05/2014 Joanne Green 25
The equation of a general straight lineinsect example 04/05/2014 Joanne Green 26
Regression Coefficient If regression coefficient of Y on X is found to be +2X +1 X -3-2 -1 0 1 2 3 2-6 -4-2 0 2 4 6 Plus 1 Plus 1 Plus 1 Plus 1 Plus 1 Plus 1 Plus 1 Plus 1 Y=2X+1-5 -3-1 1 3 5 7 Co-ordinates of point are (2,5) If regression coefficient of Y on X is found to be -1.5X-1 X -3-2 -1 0 1 2 3-1.5 4.5 3 1.5 0-1.5-3 -4.5 Minus 1 Minus 1 Minus 1 Minus 1 Minus 1 Minus 1 Minus 1 Minus 1 Y=-1.5-1 3.5 2 0.5-1 -2.5-4 -5.5 Co-ordinates of point are (2,-4) If regression coefficient of Y on X is found to be +2.5-2 X -3-2 -1 0 1 2 3 2.5-7.5-5 -2.5 0 2.5 5 7.5 Minus 2 Minus 2 Minus 2 Minus 2 Minus 2 Minus 2 Minus 2 Minus 2 Y=2.5-1 -9.5-7 -4.5-2 0.5 3 5.5 Co-ordinates of point are (2,3) 04/05/2014 Joanne Green 27
References Haeuser, J., Paap, H.G., Eppel, D., & Mueller, A. (1985). Solution of shallow water equations for complex flow domains via boundary-fitted co-ordinates. International Journal for Numerical Methods in Fluids, Vol. 5 p.p. 527-744. Available at: http://www.hpccspace.com/publications/documents/haeuserpaapeppelmueller_shallowwat erequations.pdf Ferrier, S. (2002). Mapping Spatial Pattern in biodiversity for regional conservation planning: where to from here? Systematic Biology, Vol. 51 p.p. 331-363. Available at: http://labs.bio.unc.edu/peet/courses/bio255_2003f/papers/ferrier2002.pdf Naidoo, R., Balmford, A., Costanza, R. Fisher, B., Gree, R.E., Lehner, B., Malcolm, T.R., & Ricketts, T.H. (2008). Global mapping of ecosystem services and conservation priorities. PNAS. Vol. 105 no. 28 p.p. 9495-9500. Available at: http://www.pnas.org/content/105/28/9495.full Thank you very much 04/05/2014 Joanne Green 28