R Language: Drawing Function Plots (Part II - Basic Curves & Inflection Points)

For a previous post on inflection points I needed a few examples, so I thought to write the code in the R language, which I did. Here's the final output:

Examples of Inflection Points

And, here's the code used to generate the above graphic:

par(mfrow = c(2,2)) #2x2 matrix display

# Example A: Inflection point with bifurcation
curve(x^3+20, -3,3, col = "black", main="(A) Inflection Point with Bifurcation")
curve(-x^2+20, 0, 3, add=TRUE, col="blue")
text (2, 10, "f(x)=-x^2+20, [0,3]", pos=1, offset = 1) #label inflection point
points(0, 20, col = "red", pch = 19) #inflection point 
text (0, 20, "inflection point", pos=1, offset = 1) #label inflection point


# Example B: Inflection point with Up & Down Concavity
curve(x^3-3*x^2-9*x+1, -3,6, main="(B) Inflection point with Up & Down Concavity")
points(1, -10, col = "red", pch = 19) #inflection point 
text (1, -10, "inflection point", pos=4, offset = 1) #label inflection point
text (-1, -10, "concave down", pos=3, offset = 1) 
text (-1, -10, "f''(x)<0", pos=1, offset = 0) 
text (2, 5, "concave up", pos=3, offset = 1)
text (2, 5, "f''(x)>0", pos=1, offset = 0) 


# Example C: Inflection point for multiple curves
curve(x^3-3*x+2, -3,3, col ="black", ylab="x^n-3*x+2, n = 2..5", main="(C) Inflection Point for Multiple Curves")
text (-3, -10, "n=3", pos=1) #label curve
curve(x^2-3*x+2,-3,3, add=TRUE, col="blue")
text (-2, 10, "n=2", pos=1) #label curve
curve(x^4-3*x+2,-3,3, add=TRUE, col="brown")
text (-1, 10, "n=4", pos=1) #label curve
curve(x^5-3*x+2,-3,3, add=TRUE, col="green")
text (-2, -10, "n=5", pos=1) #label curve
points(0, 2, col = "red", pch = 19) #inflection point 
text (0, 2, "inflection point", pos=4, offset = 1) #label inflection point
title("", line = -3, outer = TRUE)


# Example D: Inflection Point with fast change
curve(x^5-3*x+2,-3,3, col="black", ylab="x^n-3*x+2, n = 5,7,9", main="(D) Inflection Point with Slow vs. Fast Change")
text (-3, -100, "n=5", pos=1) #label curve
curve(x^7-3*x+2, add=TRUE, col="green")
text (-2.25, -100, "n=7", pos=1) #label curve
curve(x^9-3*x+2, add=TRUE, col="brown")
text (-1.5, -100, "n=9", pos=1) #label curve
points(0, 2, col = "red", pch = 19) #inflection point 
text (0, 2, "inflection point", pos=3, offset = 1) #label inflection point

mtext("© sql-troubles@blogspot.com @sql_troubles, 2024", side = 1, line = 4, adj = 1, col = "dodgerblue4", cex = .7)
#title("Examples of Inflection Points", line = -1, outer = TRUE)

Mathematically, an inflection point is a point on a smooth (plane) curve at which the curvature changes sign and where the second derivative is 0 [1]. The curvature intuitively measures the amount by which a curve deviates from being a straight line.

In example A, the main function has an inflection point, while the second function defined only for the interval [0,3] is used to represent a descending curve (aka bifurcation) for which the same point is a maximum point.  

In example B, the function was chosen to represent an example with a concave down (for which the second derivative is negative) and a concave up (for which the second derivative is positive) section. So what comes after an inflection point is not necessarily a monotonic increasing function. 

In example C are depicted several functions based on a varying power of the first coefficient which have the same inflection point. One could have shown only the behavior of the functions after the inflection point, while before choosing only one of the functions (see example A).

In example D is the same function as in example C with varying powers of the first coefficient considered, though for higher powers than in example C. I kept the function for n=5 to offer a basis for comparison. Apparently, the strange thing is that around the inflection point the change seems to be small and linear, which is not the case. The two graphics are correct though, because as basis is considered the scale for n=5, while in C the basis is n=3 (one scales the graphic further away from the inflection point). If one adds n=3 as the first function in the example D, the new chart will resemble C. Unfortunately, this behavior can be misused to show something like being linear around the inflection point, which is not the case. 

# Example E: Inflection Point with slow vs. fast change extended
curve(x^3-3*x+2,-3,3, col="black", ylab="x^n-3*x+2, n = 3,5,7,9", main="(E) Inflection Point with Slow vs. Fast Change")
text (-3, -10, "n=3", pos=1) #label curve
curve(x^5-3*x+2,-3,3, add=TRUE, col="brown")
text (-2, -10, "n=5", pos=1) #label curve
curve(x^7-3*x+2, add=TRUE, col="green")
text (-1.5, -10, "n=7", pos=1) #label curve
curve(x^9-3*x+2, add=TRUE, col="orange")
text (-1, -5, "n=9", pos=1) #label curve
points(0, 2, col = "red", pch = 19) #inflection point 
text (0, 2, "inflection point", pos=3, offset = 1) #label inflection point

Comments:
(1) I cheated a bit calculating the second derivative manually, which is an easy task for polynomials. There seems to be methods for calculating the inflection point, though the focus was on providing the examples. 
(2) The examples C and D could have been implemented as part of a loop, though I needed anyway to add the labels for each curve individually. Here's the modified code to support a loop:

# Example F: Inflection Point with slow vs. fast change with loop
n <- list(5,7,9)
color <- list("brown", "green", "orange")

curve(x^3-3*x+2,-3,3, col="black", ylab="x^n-3*x+2, n = 3,5,7,9", main="(F) Inflection Point with Slow vs. Fast Change")
for (i in seq_along(n))
{
ind <- as.numeric(n[i])
curve(x^ind-3*x+2,-3,3, add=TRUE, col=toString(color[i]))
}

text (-3, -10, "n=3", pos=1) #label curve
text (-2, -10, "n=5", pos=1) #label curve
text (-1, -5, "n=9", pos=1) #label curve
text (-1.5, -10, "n=7", pos=1) #label curve

Happy coding!

Previous Post <<||>> Next Post

References:
[1] Wikipedia (2023) Inflection point (link)

Book Review: Zhamak Dehghani's Data Mesh: Delivering Data-Driven Value at Scale (2021)

	Zhamak Dehghani's "Data Mesh: Delivering Data-Driven Value at Scale" (2021) is a must read book for the data professional. So, here I am, finally managing to read it and give it some thought, even if it will probably take more time and a few more reads for the ideas to grow. Working in the fields of Business Intelligence and Software Engineering for almost a quarter-century, I think I can understand the historical background and the direction of the ideas presented in the book. There are many good ideas but also formulations that make me circumspect about the applicability of some assumptions and requirements considered.  So, after data marts, warehouses, lakes and lakehouses, the data mesh paradigm seems to be the new shiny thing that will bring organizations beyond the inflection point with tipping potential from where organization's growth will have an exponential effect. At least this seems to be the first impression when reading the first chapters.
The book follows to some degree the advocative tone of promoting that "our shiny thing is much better than previous thing", or "how bad the previous architectures or paradigms were and how good the new ones are" (see [2]). Architectures and paradigms evolve with the available technologies and our perception of what is important for businesses. Old and new have their place in the order of things, and the old will continue to exist, at least until the new proves its feasibility.   The definition of the data mash as "a sociotechnical approach to share, access and manage analytical data in complex and large-scale environments - within or across organizations" [1] is too abstract even if it reflects at high level what the concept is about. Compared to other material I read on the topic, the book succeeds in explaining the related concepts as well the goals (called definitions) and benefits (called motivations) associated with the principles behind the data mesh, making the book approachable also by non-professionals.  Built around four principles "data as a product", "domain-oriented ownership", "self-serve data platform" and "federated governance", the data mesh is the paradigm on which data as products are developed; where the products are "the smallest unit of architecture that can be independently deployed and managed", providing by design the information necessary to be discovered, understood, debugged, and audited. It's possible to create Lego-like data products, data contracts and/or manifests that address product's usability characteristics, though unless the latter are generated automatically, put in the context of ERP and other complex systems, everything becomes quite an endeavor that requires time and adequate testing, increasing the overall timeframe until a data product becomes available.  The data mesh describes data products in terms of microservices that structure architectures in terms of a collection of services that are independently deployable and loosely coupled. Asking from data products to behave in this way is probably too hard a constraint, given the complexity and interdependency of the data models behind business processes and their needs. Does all the effort make sense? Is this the "agility" the data mesh solutions are looking for? Many pioneering organizations are still fighting with the concept of data mesh as it proves to be challenging to implement. At a high level everything makes sense, but the way data products are expected to function makes the concept challenging to implement to the full extent. Moreover, as occasionally implied, the data mesh is about scaling data analytics solutions with the size and complexity of organizations. The effort makes sense when the organizations have a certain size and the departments have a certain autonomy, therefore, it might not apply to small to medium businesses. Previous Post <<||>> Next Post References: [1] Zhamak Dehghani (2021) "Data Mesh: Delivering Data-Driven Value at Scale" (link) [2] SQL-troubles (2024) Zhamak Dehghani's Data Mesh - Monolithic Warehouses and Lakes (link)

Data Science: Fractal (Definitions)

"A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales [...]" (Benoît Mandelbrot, "The Fractal Geometry of Nature", 1982)

"Objects (in particular, figures) that have the same appearance when they are seen on fine and coarse scales." (David Rincón & Sebastià Sallent, Scaling Properties of Network Traffic, 2008) "A collection of objects that have a power-law dependence of number on size." (Donald L Turcotte, "Fractals in Geology and Geophysics", 2009) 

"A fractal is a geometric object which is self-similar and characterized by an effective dimension which is not an integer." (Leonard M Sander, "Fractal Growth Processes", 2009) 

"A fractal is a structure which can be subdivided into parts, where the shape of each part is similar to that of the original structure." (Yakov M Strelniker, "Fractals and Percolation", 2009) 

"A fractal is an image that comprises two distinct attributes: infinite detail and self-similarity." (Daniel C. Doolan et al, "Unlocking the Hidden Power of the Mobile", 2009)

"A geometrical object that is invariant at any scale of magnification or reduction." (Sidney Redner, "Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks", 2009) 

[Fractal structure:] "A pattern or arrangement of system elements that are self-similar at different spatial scales." (Michael Batty, "Cities as Complex Systems: Scaling, Interaction, Networks, Dynamics and Urban Morphologies", 2009) 

"A set whose (suitably defined) geometrical dimensionis non-integral. Typically, the set appears selfsimilar on all scales. A number of geometrical objects associated with chaos (e. g. strange attractors) are fractals." (Oded Regev, "Chaos and Complexity in Astrophysics", 2009) 

[Fractal system:] "A system characterized by a scaling law with a fractal, i. e., non-integer exponent. Fractal systems are self-similar, i. e., a magnification of a small part is statistically equivalent to the whole." (Jan W Kantelhardt, "Fractal and Multifractal Time Series", 2009) 

"An adjective or a noun representing complex configurations having scale-free characteristics or self-similar properties. Mathematically, any fractal can be characterized by a power law distribution." (Misako Takayasu & Hideki Takayasu, "Fractals and Economics", 2009) 

"Fractals are complex mathematical objects that are invariant with respect to dilations (self-similarity) and therefore do not possess a characteristic length scale. Fractal objects display scale-invariance properties that can either fluctuate from point to point (multifractal) or be homogeneous (monofractal). Mathematically, these properties should hold over all scales. However, in the real world, there are necessarily lower and upper bounds over which self-similarity applies." (Alain Arneodo et al, "Fractals and Wavelets: What Can We Learn on Transcription and Replication from Wavelet-Based Multifractal Analysis of DNA Sequences?", 2009) 

"Mathematical object usually having a geometrical representation and whose spatial dimension is not an integer. The relation between the size of the object and its “mass” does not obey that of usual geometrical objects." (Bastien Chopard, "Cellular Automata: Modeling of Physical Systems", 2009) 

 "A fragmented geometric shape that can be split up into secondary pieces, each of which is approximately a smaller replica of the whole, the phenomenon commonly known as self similarity." (Khondekar et al, "Soft Computing Based Statistical Time Series Analysis, Characterization of Chaos Theory, and Theory of Fractals", 2013) 

 "A natural phenomenon or a mathematical set that exhibits a repeating pattern which can be replicated at every scale." (Rohnn B Sanderson, "Understanding Chaos as an Indicator of Economic Stability", 2016) 

 "Geometric pattern repeated at progressively smaller scales, where each iteration is about a reproduction of the image to produce completely irregular shapes and surfaces that can not be represented by classical geometry. Fractals are generally self-similar (each section looks at all) and are not subordinated to a specific scale. They are used especially in the digital modeling of irregular patterns and structures in nature." (Mauro Chiarella, Folds and Refolds: Space Generation, Shapes, and Complex Components, 2016)