Time Series Data Cheat Sheet

Time series data is a term often heard within media typically based on machine learning, big data, and stock trading; this article aspires to, at least clarify some key characteristics that define time series data within the context of stock charts (e.g. stock charts within the Dow Jones and the S&P 500).

The cause of spikes within stock charts is from the idea of?trends.Trends, as is described in math-speak, are known as "inflection points"; Inflection points are to be described as the transition from a positive slope, to a decreasing slope, and vice versa. The funny thing is, though, is that for each period of a traditional sinusoidal function, there is only one local maximum, and only one local minimum; and, as can be seen, the typical stock graph has many local maximums, and many local minimums. To traverse the entire graph, if we were to only take the first derivative, after, lets say for example, that a ball hits a local minimum, that ball has no reason to go from that point, go up the respective slope, and then correspondingly, go along the rest of the graph.

In order to give the ball this capability, we must take the second derivative of the ball; where, it could be said that the ball is a metaphor to a function of some kind of independent variable, outputting a dependent variable, where the process of converting the independent variable to the dependent variable is the function. This previous statement makes sense because the first derivative, as its main purpose, is to find the?single?absolute maximum or?single?absolute minimum of the graph in one fell-swoop, but when we want to traverse the entire graph, it is assumed that we will hit?many?local maximums and local minimums. Therefore, to not get stuck at the first local maximum or local minimum that we encounter, the second derivative must be taken.

Going back to the idea of trends, the smoothing of sinusoidal functions is the assumption that each data point on the graph is related to a 1:1 relationship (i.e. an?average?change in concavity, where averages are smooth, when represented in graphs). But, within the graph, the graph is not smooth; its sharp tips represent the?instantaneous?change of concavity. Meaning that, rather than having a 1:1 relationship, instantaneous rates of change (thus describing the sharp points) represent a 1:1 relationship?plus a phase shift.?This phase shift allows the instantaneous rate of change to occur, rather than defaulting to an averaged "smoothed out" rate of change (i.e. to say, instead of having an "instantaneous" rate of change, the "smoothness" of a rate is by taking the function's?averaged?rate of change); think of the phase shift allowing a data point to explicitly break-away from the previous-data point's "sphere of influence", while still retaining a relationship with that data point.

The "break away" of a data point's relationship with one another is represented as the shifting of trends (i.e. the phase shift); and as can be reasoned, trends are unique from any other respective trend, but all trends relate to one another through the variable of time; time being the independent variable, and a trend is the dependent variable for a given hypothesis function, which approximates the target function (i.e. the stock's true value). Because we are interested in an instantaneous slope, we choose the tangent function. In an effort to relate the slope of a sinusoidal function to a slope (i.e. a "tangential"), the sinusoidal function is divided by a cosine. This is done because the forces can be represented as the cause for the given hypothesis function (i.e. the stock's true value, including its trends) to bend, where these forces are represented as the sides of a right triangle, where "theta" is the phase shift.

The tangent function, which represents the slope, is the interpolation of the two forces, dictating a company's stock price (i.e. respectively, the trends, and the given "true worth" of a company). This is a right triangle because the tangent vector, which represents the instantaneous slope of a function, is conceived by the cross product of "trends" (i.e. vector_1), and company's "true value" (i.e. vector_2), where the theta is represented as the phase shift (i.e. the time series data), which simultaneously-affects the two forces at any given time. And in continuance, why trends are important in "bowing-out" a constant (i.e. linear) "true value" of a stock, is because trends cause the true value of a stock to either deflect up, or cause the stock to deflect down, which is considered to be a centripetal force, causing the function to create a series of peaks and valleys; because of this "rounding" behavior, it is obvious that we are dealing with the second derivative (i.e. "deflection"). Because the true value of a company and the trend of a given company is competing for the function's instantaneous direction, there comes into the system, relating to deflection, is the aspect of reflection; reflection is where as one force overpowers the other, there is a point where that overpowering force reaches a maximum limit of influence, where the max limits are indicative of a sinusoidal bounds of -1/+1, thus supporting our hypothesis that the figure is indeed a species of a sinusoidal function.

Because we are calculating the tangent, keep in mind that we are not assuming we are finding an absolute fact of a price of a stock within any given time, but calculating the probability that the tangential function will predict a future value; thus the hyperbolic tangent is introduced; we choose "hyperbolic", because it's a "transcendental" of the tangent function; we are assuming the output of a given function (i.e. the dependent variable), is the result of plugging a value, which represents the independent variable, into the function. But, as can be assumed, this would be a foolish thing to believe that we could achieve the true result within the context of the future because the future is not known to us yet; basically functions with absolute-certainty in their output, are functions evaluated from a past-perspective, where the present perspective is the inflection point of the past and the future. Therefore, instead of outputting an absolute value (thus suggesting a past-perspective), in other words, +/- 1, we assume a probability. Thus, this is where the hyperbolic tangent comes into play. From here, it can be seen that because the function within a stock's chart has a static-like trace, is because the function is tracing out a noisy "y" value output; the noisy-static is present because, out of all the inputs that goes into a function, which models the trace in the graph, is not a traditional, discrete-value of an input "x" that matches with a given "y", but it is the probability that a true value of "y" will be produced, in lieu of a noisy input of "x." Which is why, the issue of probability is involved; in order to counteract noisy inputs.

Note that we take "pi/2", and "(negative)pi/2" to represent the absolute-best and absolute-worst outcomes of a stock's price (not a stock's value); because we are speaking in a context of statistics, it is known that probabilities, when reaching integer values (i.e. +/- 1), the function is infinitely-close to a horizontal asymptote (i.e. a bound of the function, where in the case of a sinusoidal function, the upper bound is "y = +1", and the lower bound is "y = -1").

Offering a direct statement, the graph that is sketched within a stock's chart is a trace of a displaced function with respect to a stock's overall value (i.e. the true value of a stock, including the stock's trend-noise), describes an object in simple harmonic motion; notice that as previously stated, that the second derivative must be taken, within the context of simple harmonic motion, whereupon, the hypothesis function obeys the behavior of a second-ordered ordinary differential equation. And, because functions in simple harmonic motion reserve a restoring force, the this hypothesis function reserves the tendency of becoming convex to a stock's true value.