Integrating over a contour might sound intimidating, so let’s start with something a bit simpler. Suppose we want to integrate the function f(x) over the curve Γ, and suppose M ∈ ℂ1[I] defines a curve such that Γ = M(I).
That’s all well and good. But what if we want to integrate over a contour which is defined by M1,…Ml ∈ C1[I]? We could describe our contour this way:
Convergence of random variables (sometimes called stochastic convergence) is where a set of numbers settle on a particular number. It works the same way as convergence in everyday life; For example, cars on a 5-line highway might converge to one specific lane if there’s an accident closing down four of the other lanes. In the same way, a sequence of numbers (which could represent cars or anything else) can converge (mathematically, this time) on a single, specific number. Certain processes, distributions and events can result in convergence— which basically mean the values will get closer and closer together.
When Random variables converge on a single number, they may not settle exactly that number, but they come very, very close. In notation, x (xn → x) tells us that a sequence of random variables (xn) converges to the value x. This is only true if the https://www.statisticshowto.com/absolute-value-function/#absolute of the differences approaches zero as n becomes infinitely larger. In notation, that’s:
Each of these definitions is quite different from the others. However, for an infinite series of independent random variables: convergence in probability, convergence in distribution, and almost sure convergence are equivalent (Fristedt & Gray, 2013, p.272).
If you toss a coin n times, you would expect heads around 50% of the time. However, let’s say you toss the coin 10 times. You might get 7 tails and 3 heads (70%), 2 tails and 8 heads (20%), or a wide variety of other possible combinations. Eventually though, if you toss the coin enough times (say, 1,000), you’ll probably end up with about 50% tails. In other words, the percentage of heads will converge to the expected probability.
The concept of a limit is important here; in the limiting process, elements of a sequence become closer to each other as n increases. In simple terms, you can say that they converge to a single number.
Convergence in distribution (sometimes called convergence in law) is based on the distribution of random variables, rather than the individual variables themselves. It is the convergence of a sequence of cumulative distribution functions (CDF). As it’s the CDFs, and not the individual variables that converge, the variables can have different probability spaces.
In more formal terms, a sequence of random variables converges in distribution if the CDFs for that sequence converge into a single CDF. Let’s say you had a series of random variables, Xn. Each of these variables X1, X2,…Xn has a CDF FXn(x), which gives us a series of CDFs {FXn(x)}. Convergence in distribution implies that the CDFs converge to a single CDF, Fx(x) (Kapadia et. al, 2017).
Several methods are available for proving convergence in distribution. For example, Slutsky’s Theorem and the Delta Method can both help to establish convergence. Convergence of moment generating functions can prove convergence in distribution, but the converse isn’t true: lack of converging MGFs does not indicate lack of convergence in distribution. Scheffe’s Theorem is another alternative, which is stated as follows (Knight, 1999, p.126):
Let’s say that a sequence of random variables Xn has probability mass function (PMF) fn and each random variable X has a PMF f. If it’s true that fn(x) → f(x) (for all x), then this implies convergence in distribution. Similarly, suppose that Xn has cumulative distribution function (CDF) fn (n ≥ 1) and X has CDF f. If it’s true that fn(x) → f(x) (for all but a countable number of X), that also implies convergence in distribution.
Almost sure convergence (also called convergence in probability one) answers the question: given a random variable X, do the outcomes of the sequence Xn converge to the outcomes of X with a probability of 1? (Mittelhammer, 2013).
As an example of this type of convergence of random variables, let’s say an entomologist is studying feeding habits for wild house mice and records the amount of food consumed per day. The amount of food consumed will vary wildly, but we can be almost sure (quite certain) that amount will eventually become zero when the animal dies. It will almost certainly stay zero after that point. We’re “almost certain” because the animal could be revived, or appear dead for a while, or a scientist could discover the secret for eternal mouse life. In life — as in probability and statistics — nothing is certain.
The difference between almost sure convergence (called strong consistency for b) and convergence in probability (called weak consistency for b) is subtle. It’s what Cameron and Trivedi (2005 p. 947) call “…conceptually more difficult” to grasp. The main difference is that convergence in probability allows for more erratic behavior of random variables. You can think of it as a stronger type of convergence, almost like a stronger magnet, pulling the random variables in together. If a sequence shows almost sure convergence (which is strong), that implies convergence in probability (which is weaker). The converse is not true — convergence in probability does not imply almost sure convergence, as the latter requires a stronger sense of convergence.
Convergence in mean is stronger than convergence in probability (this can be proved by using Markov’s Inequality). Although convergence in mean implies convergence in probability, the reverse is not true.
The overlap between the two functions can be evaluated by a convolution integral, which is a “generalized product” of two functions when one of the functions is reversed and shifted.
Other names for the convolution integral include faltung (German for folding), composition product, and superposition integral (Arkshay et al., 2014). These integrals have many applications anywhere solutions for differential equations arise, like engineering, physics, and statistics.
If the Laplace transforms (F(s) = L{f(t)} and G(s) = L{g(t)}) both exist for s > a ≥ 0, then H(s) = F(s)G(s) = L{h(t)} for s > a, where:
Note that the integral is commutative, i.e. f * g = g * f: it doesn’t matter which function is evaluated first. This useful fact means that you can try placing either of the functions first or second, to see which resulting integral is easier to evaluate.
Statistics MagicIn general, integrating convolution integrals can be challenging except in the simplest of cases. It can be easier to work first in the Laplace Domain, then transform back into the whatever domain you’re working with, like the time domain (Cheever, 2020).
For someone recovering from complex mental health issues, this amount of detail is about the right amount for initial first introduction to this topic of study . Thank you..
A correlogram (also called Auto Correlation Function ACF Plot or Autocorrelation plot) is a visual way to show serial correlation in data that changes over time (i.e. time series data). Serial correlation (also called autocorrelation) is where an error at one point in time travels to a subsequent point in time. For example, you might overestimate the value of your stock market investments for the first quarter, leading to an overestimate of values for following quarters.
Correlograms can give you a good idea of whether or not pairs of data show autocorrelation. They cannot be used for measuring how large that autocorrelation is (for a mathematical way to test for serial correlation, try the Durbin Watson test).
A correlogram gives a summary of correlation at different periods of time. The plot shows the correlation coefficient for the series lagged (in distance) by one delay at a time. For example, at x=1 you might be comparing January to February or February to March. The horizontal scale is the time lag and the vertical axis is the autocorrelation coefficient (ACF). The plot is often combined with a measure of autocorrelation like Moran’s I; Moran’s values close to +1 indicate clustering while values close to -1 indicate dispersion.
The above image shows relatively small Moran’s I (between about -0.2 and 0.35). In addition, there is no pattern in the autocorrelations (i.e. no consistent upward or downward pattern as you travel across the x-axis). This set of data likely has no significant autocorrelation.
On the other hand, this next image shows fairly high Moran’s I values and an upward trend. This indicates that autocorrelation is highly likely for your set of data.
I created this graph with Desmo.com’s “Unraveling Trig Functions” calculator, which allows you to change values like the arc lengths. The calculator gives a host of great visuals to show how the covercosine compares to more familiar functions, like the sine function and cosine function. It’s well worth a look if you’re a visual learner.
Another way to look at the covercosine is as a shifted function. As an example, the more familiar cosine function can be thought of as a shifted sine function, as the following image illustrates:
Of course, you’d have to be familiar with the vercosine to fully understand what it means to shift it, but hopefully you get the idea: dozens of trigonometric functions were once in use that could measure any angle and any distance for segments of a trip. They were invaluable to seafarers, who could calculate how far and in what direction, based on the simple idea of breaking apart a circle into smaller components.
Covercosine is one of a group of archaic functions (along with hacoversine, vercosine and others) used historically for navigation. With the advent of modern navigation techniques, these functions have fallen out of practical use along with the typewriter, abacus, and slide rule. Thanks to computing, most trigonometric functions still in use are derived from sine functions and cosine functions or from complex exponential functions.
This archaic trig function was primarily used for global navigation, back in the days when bulging books of navigational tables were a must-have on ships. They can also can be found in older text books. If your calculations involved finding half of the square of cosine, you could look that up in a table and not have to deal with finding halves of squares and square roots. Nowadays, this function is of little practical use, thanks to modern computing. However, it does pop up now and again in the scholarly literature, like in 2013’s On the Trigonometric Loophole or 2019’s Computational Aspects of q-Method.
Criterion validity (or criterion-related validity) measures how well one measure predicts an outcome for another measure. A test has this type of validity if it is useful for predicting performance or behavior in another situation (past, present, or future). For example:
A job applicant takes a performance test during the interview process. If this test accurately predicts how well the employee will perform on the job, the test is said to have criterion validity.
A graduate student takes the GRE. The GRE has been shown as an effective tool (i.e. it has criterion validity) for predicting how well a student will perform in graduate studies.
The first measure (in the above examples, the job performance test and the GRE) is sometimes called the predictor variable or the estimator. The second measure is called the criterion variable as long as the measure is known to be a valid tool for predicting outcomes.