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| [[Image:Ley lines.svg|thumb|300px|right|80 4-point near-alignments of 137 random points]]
| | The tool is by Jay Stockwell, who functions with Allan Gardyne of Thi...<br><br>It is not each day that a computer software you have utilized for the 1st time blows you away. Nicely, SpeedPPC did that to me. In the event you require to identify extra information on [http://www.linkemperor.com/link-building-networks holoquinonic jotkexlzsophsogb cromwellian], we know about lots of libraries people can investigate. Wow is all I can say correct now. Initial off, I want to mention that this is not an low-cost computer software, BUT, it did pay itself off in 1 week! Now that left me in disbelief to say the least. SpeedPPC also does not demand any monthly costs, only that 1-time payment.<br><br>The tool is by Jay Stockwell, who works with Allan Gardyne of This is the very first product Allans released in the approx. ten years he"s been in affiliate advertising and marketing, and its a complete PPC tool and a high quality solution. Allan waited a extended time to release a solution, but boy did he hit gold with this a single.<br><br>What SpeedPPC basically does is it enables you to bid for competitive searches and makes it possible for you to bid for hundreds or thousands of low-competition phrases. Not only that, it helps you construct targeted AdWords Advertisements, helps you build hundreds of Adgroups, AND it makes use of a remedy to create highly targeted landing pages (all the above is completed AUTOMATICALLY by the computer software). With all that automation, you can go for thousands of search phrases that are low-search volumeand as a result low price per click.<br><br>You can also use the SpeedPPC Technique to automatically generate enormous "extended tail" keyword lists and then bind them together into hundreds of tight, modest ad groups that would generally take weeks to complete.<br><br>An instance...If you wanted an iPod Shuffle, wouldnt you be content material seeing an ad targeted to that exact keyword which, and when [https://www.Gov.uk/search?q=clicked clicked] on, took you to a page targeted to that precise iPod with a image, description and hyperlink to exactly where you could buy it? Thats what Speed PPC in fact does and it can do it quite quickly for thousands of ads and pages.<br><br>In the background, this will permit you to acquire good Google Top quality scores. As any PPC marketer is aware, if you get excellent Quality Scores, you spend a lot less per click.<br><br>To place all this into perspective...Everything that SpeedPPC can do for you in minutes, it would generally have taken you numerous weeks to achieve. It"s that strong!<br><br>If you have been struggling with PPC campaigns, by now, you happen to be possibly cramping on your sides right after reading what this software program can potentially do for your ROI.<br><br>SpeedPPC is extremely advisable if you happen to be significant about generating cash on the internet employing PPC advertising..<br><br>For those who have any issues regarding where in addition to how you can employ [http://storify.com/evasivedai810 buy health insurance], you can email us with our own webpage. |
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| '''Alignments of random points''', as shown by [[statistics]], can be found when a large number of [[randomness|random]] points are marked on a bounded flat surface. This might be used to show that [[ley line]]s exist due to chance alone (as opposed to supernatural or anthropological explanations).
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| One precise definition which expresses the generally accepted meaning of "alignment" as:
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| :''A set of points, chosen from a given set of landmark points, all of which lie within at least one straight path of a given width w''
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| "Straight path of width w" may be defined as the set of all points within a distance of ''w''/2 of a [[straight line]] on a plane, or a [[great circle]] on a sphere, or in general any [[geodesic]] on any other kind of [[manifold]]. Note that, in general, any given set of points that are aligned in this way will contain a large number of infinitesimally different straight paths. Therefore, only the existence of at least one straight path is necessary to determine whether a set of points is an alignment. For this reason, it is easier to count the sets of points, rather than the paths themselves.
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| The width ''w'' is important: it allows the fact that real-world features are not mathematical points, and that their positions need not line up exactly for them to be considered in alignment. [[Alfred Watkins]], in his classic work on ley lines ''[[The Old Straight Track]]'', used width of a pencil line on a map as the threshold for the tolerance of what might be regarded as an alignment. For example, using a 1 mm pencil line to draw alignments on an 1:50,000 [[Ordnance Survey]] map, a suitable value of ''w'' would be 50 m.<ref>{{cite isbn|0349137072}}</ref>
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| == An estimate of the probability of alignments existing by chance ==
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| [[Contrary to intuition]], finding alignments between randomly placed points on a landscape gets progressively easier as the geographic area to be considered increases. One way of understanding this phenomenon is to see that the increase in the number of possible [[combination]]s of sets of points in that area overwhelms the decrease in the probability that any given set of points in that area line up.
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| The number of alignments found is very sensitive to the allowed width ''w'', increasing approximately proportionately to ''w''<sup>''k''-2</sup>, where ''k'' is the number of points in an alignment.
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| The following is a very approximate order-of-magnitude estimate of the likelihood of alignments, assuming a plane covered with uniformly distributed "significant" points.
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| Consider a set of ''n'' points in a compact area with approximate diameter ''d'' and area approximately ''d''². Consider a valid line to be one where every point is within distance ''w''/2 of the line (that is, lies on a track of width ''w'', where ''w'' << ''d'').
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| Consider all the unordered sets of ''k'' points from the ''n'' points, of which there are:
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| :<math> \frac {n!} {(n-k)!k!}. </math>
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| What is the probability that any given set of points is [[collinearity|collinear]] in this way? Let us very roughly consider the line between the "leftmost" and "rightmost" two points of the ''k'' selected points (for some arbitrary left/right axis: we can choose top and bottom for the exceptional vertical case). These two points are by definition on this line. For each of the remaining ''k''-2 points, the probability that the point is "near enough" to the line is roughly ''w''/''d'', which can be seen by considering the ratio of the area of the line tolerance zone (roughly ''wd'') and the overall area (roughly ''d''²). | |
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| So, the expected number of k-point alignments, by this definition, is very roughly:
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| :<math> \frac {n!} {(n-k)!k!} \left({\frac{w}{d}}\right)^{k-2}.</math>
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| For ''n'' >> ''k'' this is approximately:
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| :<math> \frac {n^k} {k!} \left({\frac{w}{d}}\right)^{k-2}.</math>
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| Now assume that area is equal to <math>d^2</math>, and say there is a density α of points such that <math>n = \alpha d^2</math>.
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| Then we have the expected number of lines equal to:
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| :<math> \frac {\alpha^k d^{2k}} {k!} \left( {\frac{w}{d}} \right)^{k-2}</math>
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| and an area density of k-point lines of:
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| :<math> \frac 1 {d^2} \frac {\alpha^k d^{2k}} {k!} \left( {\frac{w}{d}} \right)^{k-2}.</math>
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| Gathering the terms in ''k'' we have an areal density of k-point lines of:
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| :<math> d^k \frac {\alpha^k} {k!} w^{k-2}.</math>
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| Thus, contrary to intuition, the number of k-point lines expected from random chance increases much more than linearly with the size of the area considered.
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| == A more precise estimate of the expected number of alignments ==
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| A more precise expression for the number of 3-point alignments of maximum width ''w'' and maximum length ''d'' expected by chance among ''n'' points placed randomly on a square of side ''L'' is <ref>Edmunds, M.G. & George, G.H., ''Random Alignment of Quasars'', Nature, vol. 290, pages 481-483, 1981 April 9</ref>
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| :<math> \mu = \frac {\pi } {3} \frac {w}{L}
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| \left( {\frac {d}{L}} \right)^{3} n \left( n-1 \right)
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| \left( n-2 \right) </math>
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| If edge effects (alignments lost over the boundaries of the square) are included, then the expression becomes
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| :<math> \mu = \frac {\pi } {3} \frac {w}{L}
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| \left( {\frac {d}{L}} \right)^{3} n \left( n-1 \right)
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| \left( n-2 \right)
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| \left( 1 - \frac {3}{\pi } \left( \frac {d}{L} \right)
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| + \frac {3}{5} \left( \frac {4}{\pi } - 1 \right)
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| \left( \frac {d}{L} \right)^{2} \right) </math>
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| A generalisation to ''k''-point alignments (ignoring edge effects) is<ref>G.H. George, [http://www.engr.mun.ca/~ggeorge/astron/thesis.html] "Alignment of Quasars", ''Summary of Ph.D. Thesis'', 1983</ref>
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| :<math> \mu = \frac {\pi n \left( n-1 \right) \left( n-2 \right)
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| \cdots \left( n - \left( k-1 \right) \right) }
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| {k \left( k-2 \right) !} \left( \frac {w}{L} \right)^{k-2}
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| \left( {\frac {d}{L}} \right)^{k} </math>
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| == Computer simulation of alignments ==
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| [[image:leylines.png|Image of ley line simulation|thumb|300px|right|''607 4-point alignments of 269 random points'']]
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| [[Computer simulation]]s show that points on a plane tend to form alignments similar to those found by ley hunters in numbers consistent with the order-of-magnitude estimates above, suggesting that ley lines may also be generated by chance. This phenomenon occurs regardless of whether the points are generated pseudo-randomly by computer, or from data sets of mundane features such as pizza restaurants.
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| It is easy to find alignments of 4 to 8 points in reasonably small data sets with ''w'' = 50 m.
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| Choosing large areas or larger values of ''w'' makes it easy to find alignments of 20 or more points.
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| == References ==
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| {{Reflist}}
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| == See also ==
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| * [[Ley line]]s
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| * ''[[The Old Straight Track]]''
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| * [[Coincidence]]
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| * [[Procrustes analysis]]
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| [[Category:Euclidean geometry]]
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| [[Category:Statistical randomness]]
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The tool is by Jay Stockwell, who functions with Allan Gardyne of Thi...
It is not each day that a computer software you have utilized for the 1st time blows you away. Nicely, SpeedPPC did that to me. In the event you require to identify extra information on holoquinonic jotkexlzsophsogb cromwellian, we know about lots of libraries people can investigate. Wow is all I can say correct now. Initial off, I want to mention that this is not an low-cost computer software, BUT, it did pay itself off in 1 week! Now that left me in disbelief to say the least. SpeedPPC also does not demand any monthly costs, only that 1-time payment.
The tool is by Jay Stockwell, who works with Allan Gardyne of This is the very first product Allans released in the approx. ten years he"s been in affiliate advertising and marketing, and its a complete PPC tool and a high quality solution. Allan waited a extended time to release a solution, but boy did he hit gold with this a single.
What SpeedPPC basically does is it enables you to bid for competitive searches and makes it possible for you to bid for hundreds or thousands of low-competition phrases. Not only that, it helps you construct targeted AdWords Advertisements, helps you build hundreds of Adgroups, AND it makes use of a remedy to create highly targeted landing pages (all the above is completed AUTOMATICALLY by the computer software). With all that automation, you can go for thousands of search phrases that are low-search volumeand as a result low price per click.
You can also use the SpeedPPC Technique to automatically generate enormous "extended tail" keyword lists and then bind them together into hundreds of tight, modest ad groups that would generally take weeks to complete.
An instance...If you wanted an iPod Shuffle, wouldnt you be content material seeing an ad targeted to that exact keyword which, and when clicked on, took you to a page targeted to that precise iPod with a image, description and hyperlink to exactly where you could buy it? Thats what Speed PPC in fact does and it can do it quite quickly for thousands of ads and pages.
In the background, this will permit you to acquire good Google Top quality scores. As any PPC marketer is aware, if you get excellent Quality Scores, you spend a lot less per click.
To place all this into perspective...Everything that SpeedPPC can do for you in minutes, it would generally have taken you numerous weeks to achieve. It"s that strong!
If you have been struggling with PPC campaigns, by now, you happen to be possibly cramping on your sides right after reading what this software program can potentially do for your ROI.
SpeedPPC is extremely advisable if you happen to be significant about generating cash on the internet employing PPC advertising..
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