Modernism, Postmodernism, and Digitality
Primary Component Analysis (PCA)
We are now able to use Principal Component Analysis (PCA) to show where breakpoints are in our historical scenario. First, we are not able to find the key moment because there are many moments, and each PCA will be looking for a particular moment in a logical system. Second, from the above we can see why many Cultural Systems are in play at any given time therefore we must restrict any PCA to one particular Cultural System, and any that impact directly may include other Cultural Systems which are based on the Cultural System that we are looking for or preceded and therefore the PCA will be in response to it.
Because Primary Component Analysis is relatively old dating back to Pearson (1901) and Frisch (1929) with the primary focus on getting a line to a plane or a scattering of points or indeed any subspace of a higher dimension, it is a rich field for getting some number of vectors to another set of vectors and therefore has a rich set of features derived from a number of individuals. i
The first step is to explain how PCA works, and then how to fit PCA into the system of logical analysis which is being proposed here. Because this is often a model rather than an observable or theoretical system, it should be used with caution and with the knowledge that more data may make a PCA invalid for a particular time.
The fundamental secret of PCA is to select as few pieces of data as possible and then plot them on a scale and draw a simple line that captures as much of the data as possible. This then means that the variance of the two pieces of data can be calculated so as to see how much data the PCA captures. For example, let us take an election where there are numerous issues that can be voted on but the two largest are a lower taxation rate and where the candidate stands on abortion. Some people will vote for a party without questioning these areas, so many be regarded as safe votes, but as one gets into the middle many of the individuals who vote will decide on one of these two issues.
So how do we compute the PCA? We start with the PCA itself:
First, we take the raw data and place each issue for each individual and place them down the rows. So:
Where each row represents one individual with data on each issue that the individual cares about with each row representing one individual. This is a difference between SVD and PCA, with SVD representing individuals as columns and the PCA representing them as rows. This as far as we are concerned is a difference without distinction since the matrix can be inverted.
Because we are looking for the most data in the fewest number of data points, we can simply invert the rows by linear algebra. Therefore, we can see which pieces of data are the largest and convert them to the most relevant. The first step is to arrange it so that the data is around the origin. This is relatively simple.
The first step is to compute the mean row:
Where are all of the rows, in this case, the electorate? We can then make a mean X with 1 and the with n copies.
The next step is to subtract the mean from the data:
Which makes B centered around the [0,0].
The next step is slightly more complicated, in that we want to compute the covariance of B. Because this is PCA we use the notation C.
Which means that we square the matrix by multiplying it by its transpose.
The next step may seem yet more complicated because it involves calculating the eigenvalues and eigenvectors of C. In particular, we want to calculate the eigenvector of the matrix which is the largest, and so on. This means that we calculate the largest eigenvector or vn,
CV = VD
Where V is the vector and the is the largest eigenvalue.
We finally reach the principle component analysis by taking T, as the principal component analysis matrix:
T = BV
However, the PCA is different in history. Because time is experienced differently than other points of data. This means that the PCA for a marker needs to be calculated in two ways: one as time-dependent and one as time as part of the variable.
One of the primary differences is that in the beginning, one PCA was all that was required. But with increased computing power the variable of time becomes a matrix that is computed from the previous matrix. Foreword is to make PCA recursive: not just get one PCA, but once one has one PCA then call the next PCA in succession and produce a new T which is one larger than the old T.
So:
And that means that:
The trick is to find the function to a recurse on: not just the simple ones but any which can produce a recursive chain. The obvious one is of course time, so instead of diagramming out a whole sentence through the entire chain picking the value from the recursive chain. For example, a person who owns a product then can be a centerpiece for others who wish to purchase the product. That means instead of a uniform distribution it is in clumps around the centers where the first owners then convince other would-be owners to purchase. That is, the gunpowder is not uniformly distributed it gathers in clumps.
For example, in the 1960s in America, the number of color television sets went from almost to the dominant form of television. in some Cultural Systems, this would not be noticed at all, but in a few of them, it would be the marker that changed the way people viewed television in particular countries.
That it is important to some Cultural Systems is seen by how quickly color television was introduced: an experiment in the United States started in 1950 but did not catch on. Initially, the color television sets were expensive but when we look at the numbers, a different rationale is indicated. The “color revolution” took place, but not in terms of price:ii
One can see that color televisions skyrocketed from 6.51% in 1961 to a preponderant majority of 81.13%. but what is important from the Cultural Systems standpoint is that color television sets did not reduce in price during the time period, except by the amount of inflation. There is a slight uptick at the end that calls for further investigation.
Instead, it is clear that color television was an important choice that people made freely to get the value of the television station. That is, we can see how a Cultural System made color television a priority even though the price had not fallen for color television sets and was reduced only in the black-and-white market. If the price of a color television set had dropped, one would be able to argue that it was simply cheaper. The number of black and white television sets did not decrease until the very end instead the number of color television sets increased enormously. The color television sets increase almost linearly: This is taking the Singular Value Decomposition and honing into the key point: color versus black and white television sets in the United States from 0 to 1. We can then reduce it to PCA by taking the level of color television sets as the horizontal and see how much the yearly output of color television sets varies from the norm by least squares. We can see that the marker in color television sets reached a breakpoint in 1967 and can therefore argue that the Postmodern era began around that time: there is a new way of looking at information that comes into the home over color rather than black and white.
This shows that the color TV led the programming because the color was in 1964. However, it was only 1 year and the data was known. We then compute the C matrix by least squares, and we get the covariance. There is a small adjustment to sync it to the origin. This shows that color television sets had a lift in production once the networks could broadcast in color. Since in order to advertise color programming requires a year, one can see that the inflection point occurred in 1964.
Remember, the variables are % of sets sold and time. This means that PCA is not merely an estimate or a statistical representation of a true value, but a reflexive point of a marker. That marker is a matrix of the logical system. This means that the marker is not time nor of percentage of color television sets sold but a marker of color %/time and how both time and color % vary from the marker. This means that when looking at time alone things speed by more quickly because color % at a certain point leads to time, at the same time color % is taking the population more quickly than it is willing to allow. So again, PCA is here a recursive measurement which is real because the individuals are conscious of it whereas electrons and individuals at a genetic level are not conscious. However, in a biological sense, there is some consciousness of the PCA, but it is not as sharp as with history. This means there is a “historical” analysis of PCA which is slightly different than merely the statistical representation. This is because the statistical representation does not move in terms of time alone, but the historical representation does because of the individual’s consciousness of events which includes the PCA itself.
Note that the trigger point on the X axis is one year before the introduction of full-color television programming by the major networks. We can hypothesize that there is a set of circumstances whereby the introduction of receiver proceeds be the introduction of full-color programming which puts the PCA on a new course. There needs to be a fit between the marker and the transmission of the marker. Because we do not have full information this must be discussed in a future section.
This is related to genes that have no fitness in Darwin’s Descent of Man and Darwin laid the foundations for the Comparison of Mental Powers. iii This means that the concept is old though the specific mathematics is new.iv There is nothing like discovering a feature that is old but unnoticed. There is also the delicious irony of something being hoped for and, the unfortunately, coming to pass: the manufacturers want more Color sets to be produced. The results were unforeseen. Now, in retrospect, we can see the effect on Cultural Systems even while the Modern was still in textbooks and would be for some time.
This is important because the Vietnam War, equality for African Americans, and everything else changed the way that the population viewed it. We should note that the variation can then be plotted against the color value as a PCA to see how this affected domestic politics. Again, the Tet Offensive while it was bloody and looked from the standpoint of casualties to be a massive defeat for the North Vietnamese was instead looked at as draining the blood of the American soldiers for a war which we did not have any immediate need to fight. In a logical system, the change is often what is measured and how. It also changes as a young demographic group goes to college. When Kent State happened, in 1970, it showed that there was a Cultural System change and that the president was no longer a Modern president in black and white but a Postmodern president in color. The rock band Crosby, Stills, Nash, and Young pulled the single that they were using and recorded “Ohio” written by Young, and released it.v
Though it did not change the next Presidential Election, it was perhaps one of the factors that made Nixon reach for criminal avenues to win not just a victory but a huge crushing defeat. This then became the “Watergate scandal” and showed again the difference between the Modern and the Postmodern. Nixon would have one under most circumstances because the Democratic party was divided, but because of Nixon’s obsession with having a crushing defeat individual people giving orders brought down the government. Thus, Woodward and Bernstein mined the criminal records and found odd disagreements. These disagreements were at the same time that the presidential campaign was going on and did not affect the outcome, but eventually other people with more clout, such as Walter Cronkite and his top-rated news program on CBS. Eventually, this caused the Senate to investigate, and when John Dean testified before the Senate subcommittee and brought together numerous threads into a complete narrative that was the moment when the Watergate became clear to many people. This again is distinct to its age and only by searching for the Cultural System and the underpinnings of that Cultural System can the entire edifice be understood.
The wave of “youth culture” was instrumental in changing television programming according to Aniko Bodroghko in her book Groove Tube: Sixties Television and the Youth Rebellion. In this, we can see that technical innovation in receiving based on the four-tube television, the rapid change in demographics, and the commercial drive to capture the new market all form the basis of a logical system. Instead of focusing only on one aspect, instead, we look at data and find the aspects of the data that fit within a logical system. For example, Bodroghko takes pains to emphasize that the demographic being captured is one of white college-educated individuals because this is the demographic that is at the intersection of both being young and having color television sets.vi
It also created a space where places were set up to listen to TV, stereo equipment, and other objects away from the home to gauge the sound as one would if one listened to it. This meant that in the 1970s the “audio file” movement happened in just such a place.vii This was all happening at the same time as a “marker” in certain logical systems. Whether it is important to your particular Cultural System will be shown by the PCA and its sigmoidal curve in relationship to other data points.
Of course, this is only one measurement but when multiple measurements happen at the same time then we can say that a Cultural System has changed in the United States. In 1964, Marshall McLuhan was prescient in his book Understanding Media: The Extensions of Man covering the transmission part of a logical system:
In a culture like ours, long accustomed to splitting and dividing all things as a means of control, it is sometimes a bit of a shock to be reminded that, in operational and practical fact, the medium is the message. This is merely to say that the personal and social consequences of any medium—that is, of any extension of ourselves—result from the new scale that is introduced into our affairs by each extension of ourselves, or by any new technology.viii
This then extends to the Cultural System in history: the view taken by key individuals will determine how events are going to be perceived. And the math is merely the way of hunting for the changes that the Cultural System employs. Because there will be an overturn in some aspect of a logical system, we are looking to mathematize those aspects of the chosen Logical system.
Then the rest of the world changed from black and white to color over the next 10 years, indicating that this change occurred overnight in the decision-making process in the rest of the world. This indicates that a Cultural System made color television important within only a few years even though there was a different standard, NTSC versus PAL, in operation. This is the force of a logical system: it changes the decision inside the brain not merely as a course of convenience.
By taking raw data and making a primary component analysis, we can see that there are decisions that are made on a few factors and these few factors drive the overall shape in certain contexts. When we find that a few factors make a large difference we can then ask if there is a motivation that lives inside the minds of the people making it. This is the historical power of why a Cultural System is important.
One of the signs of being a marker, as pointed out, in history it is often recursive. For example, the color television percentage aligns with the sigmoidal curve, but there is often a decline after the initial discovery, which tells historians that individuals perceive the next marker and try to bring it to market. The initial attempt fails, which means that the sigmoidal curve has a downside at its beginning.
This PCA is a time-dependent factorization of individual points. However, the individual points need to be praised in such a way as to make them numerical in some fashion. The example that we use in this book is the transatlantic cable which was finally made working in 1866 after three attempts since 1858. The final value is bytes per second, but this only occurred after an electric theory had been introduced. What measurement can we use for the paper as it affects the cable? It may seem that a paper does not have any good reference to the speed of a cable, but when we look at it, we can see the previous value of an electric field which was 18 hours for a short message. Thus, we can label the correct electric field theory as a potential against the previous theories’ potential. We can do this by using an imaginary number which indicates the modeling of the transmission rate. In mathematics take the i value as the theory against the real value. Since we can square the i and take the absolute value, we can then use a real value. This real value will be a calculation performed on paper because it was paper that these things were done in the 1850s, and that means that it can be a Fourier transform as opposed to a Laplace transform.
Since we must use ℝ for PCA this satisfies since we have shown that the imaginary value can be made real. Remember that in this particular context, a real value is distinguished by the transmission from one individual to another. But this may mean that the word “individual” must be taken in the broadest context including computers rather than merely human beings.
One way to see PCA is the way groups of people organize on one or two axes.
Key points on PCA
PCA is useful for determining the key vectors in a matrix.
The matrix shows where the logarithmic curve ends and the pressures.
The is a historical basis because individuals know a version of the PCA, and it is an extension of trait markers in biology.
i Rao, C. Radhakrishna. 1964, “The Use and Interpretation of Principal Component Analysis in Applied Research.” Sankhyā: The Indian Journal of Statistics, Series A (1961-2002) 26, no. 4: 329–58. http://www.jstor.org/stable/25049339. 333.
ii Frank F. Ward. “The Color Television Industry.” Financial Analysts Journal 22, no. 6 (1966): 39–47. http://www.jstor.org/stable/4470075.41. The data is from the US Department of Commerce.
iii Darwin, 829.
iv We should also note the PCA is old as well. This is a discovery that is long hidden in plain view.
vi Aniko Bodroghkozy. 2001. Groove Tube: Sixties Television and the Youth Rebellion. Durham, N.C.: Duke University Press, 9.
vii As noted in various places such as Kelvin, “Does vintage stereo sound better /and why.?”, Stereo review X,
viii McLuhan, Marshall. 1966. Understanding Media; the Extensions of Man. New York :Signet Books. 19