Logistic Growth, S-Curves, Bifurcations, & Lyapunov Exponents in R (Tidyversion)
This is an updated version of my 2015 (Base R) blog post using tidyverse. Original Rmd is available on request.
Category Archives: Education
How Linear Regression and Two-Way ANOVA are the Same
Why ANOVA and Linear Regression are the Same
A Decade of PhD: What I’ve Learned from Academia and Industry
Today is Cinco de Mayo! It’s also the 10th Anniversary of my PhD defense (in Quality Systems)…. something I carefully timed for late afternoon on this day in 2009. (I wanted to make sure I could celebrate the joyful occasion — or drown my sorrows — with 2-for-1 margaritas. Fortunately, the situation was liquid joy; unfortunately, I still got a hangover.)
I’m writing this post to share what I’ve learned about the value of getting a PhD (is there value?) and the applicability of PhD-level work to industry. If you’re considering more education, maybe this will help you decide whether it’s the right choice. If you’re in industry and trying to figure out whether to hire PhDs, some of what I write here might help. But first, some background!
I never even thought I’d get a PhD — it certainly didn’t happen out of intent or design. My family was poor (my dad was an East Prussian refugee whose family lost a couple hundred years’ worth of assets and had to start from scratch in the U.S. in the 1960s, and my mom’s grandparents were very poor Irish laborers who came to the U.S. in the early 1900s) so I studied ridiculously hard to “escape”. I didn’t think I was smart enough for a PhD, even though I started college at 16 taking half undergrad classes and half grad classes in meteorology. I aced my grad classes and very maturely ignored my required classes, so I got kicked out. (At the same time, I wasn’t really fitting in with people… my roommate called me “Nerdcole”.) When I was let back in the department head wouldn’t let me take any grad classes so I got bored and burned out… not surprising since I was supporting myself, and working three jobs to make that happen. I quit school to work at an e-commerce startup when I was 18. A few months later, thanks to (good) peer pressure, I took 3 credit by exams to see if it would get me over the finish line, and thanks to some side skills I had picked up in vector calculus and statistics, it worked and I got the BS. But I was still left with a pretty bad GPA, and even worse self esteem, and I was convinced no one would ever let me into grad school.
I figured I’d focus on industry and help companies grow. There was no other choice.
The Back Story
After spending a couple years building web sites and storefronts (a huge feat in 1995 and 1996!) I took a job at a national lab as a systems analyst, supporting established scientists and engineers and helping them get work done. The main lesson I learned during this time was: Alignment between strategy and objectives doesn’t come for free (teams of people have to spend dedicated time on it), and most people are really disorganized. There had to be a better way to get work done.
A few years later, I was a traveling Solutions Architect, parachuted once or twice a month into CRM software implementation fiascos around the globe. My job was to figure out what to do to turn these jobs around — was it a people problem? An architecture problem? A training problem? A systems thinking problem? A little of everything? I had a couple weeks to make a recommendation, and then I was on to the next project (results were usually pretty good). But since this required evaluating technology decisions in the context of business and financial constraints, my boss suggested that I use the tuition benefits offered by my job to get an MBA. I had taken 9 credits of science and industrial engineering classes since I’d graduated, so I contacted two of the local MBA schools to see if they’d accept me and my credits. Sure enough, one of them did! I took evening classes for a year and a half, and eventually ended up with an MBA. But I never thought I could (or would) go farther — I’m not that smart, I’d tell myself. Also, it’s expensive. Also, a PhD would probably make me less marketable. (All lies, spoken by a lack of confidence and a heavy dose of impostor syndrome.)
Shortly thereafter, the travel started to get to me (I was flying at least three days a week), so I looked for an opportunity to grow and cultivate a software development organization. (That’s how I ended up leading monitor and control systems and data management at NRAO.) A little management led to a lot of management. A few years later one of the organization’s leaders casually said it was “too bad I didn’t have a PhD” — because in a highly scientific and technical organization like NRAO, it would give me more credibility and make me a better leader.
“Will you pay for it?” I asked. “Sure,” they said. I just had to find a suitable program that wouldn’t require me to go full time. I’ve always loved learning, and I couldn’t resist the temptation of free education — even if it meant I’d have to balance the demands of a challenging full-time job and a first-time baby at the same time. That’s how much I love learning, just for learning’s sake! I still didn’t think a PhD had that much value, unless you were studying to be a lab scientist or you were dead set on becoming a historian and teaching for the rest of your life. None of these personas was me, but the free education thing sold me, and I didn’t really think about how relevant this step was to my career direction until much later.
The next few years were pretty rough. By the time I got my PhD, I was in my 14th year of post-college professional employment. First lesson learned: it’s probably not the best move to start PhD coursework when you have a three-month old. I have no idea how I made it through.
Shortly after graduating, the impacts of the financial crisis hit our federally funded organization and I was able to segue into a second career as a college professor, teaching data science, manufacturing, and EHSQ classes. For the past year, I’ve been back in industry (maybe permanently; we’ll see) and have a better sense of the value of PhDs in industry.
Value of Getting a PhD
There are lots of reasons I’m happy with the time I spent getting a PhD, other than the fact that it helped me get an entirely new job when the economy was down:
- First and foremost, I’m a better critical thinker. It’s now my nature to look at all parts of a problem, examine the interactions between them, and make sure I have all the required background information needed to start working on a problem.
- I’m a better writer too. I look at reports and presentations I wrote years ago, and can see all the holes and places where I made assumptions that weren’t valid.
- I developed a new appreciation for clarity. Researchers want to make sure their messages, methodologies, and models are clear and unambiguous… through the contrast, I was able to recognize that in industry, there’s often pressure to skip due diligence and move fast to perform. This pressure leads to ambiguity, which tends towards what I call “intellectual waste” – people assuming that they see a problem or a project in the same way because they haven’t taken the time to guarantee clarity.
- It’s easier for me to quickly determine whether information might be true or false, or whether there are gaps that need to be closed before moving forward. (It’s possible that this skill is more from grading and evaluating student work… something that’s orders of magnitude harder than it seems.)
- I realized that words matter. Really thinking about how one person will respond to a word or phrase, and whether it conveys the meaning that you intend, is a craft — that’s enhanced by working with collaborators.
- And although I knew this one prior to the PhD, I found that data matters. Where did your data come from? Can you access the original sources? What kind of people (or instruments) gathered it? Can you trust them? The quality of your data — and the suitability of the methods you choose — will impact the quality and integrity of the conclusions you generate from it. Awareness of these factors is essential.
Value of Caution
One of the biggest lessons was the most surprising. Early on in the PhD program I was told that my opinion didn’t count — regardless of how many years of experience I had. Every statement I made had to be backed up and cited, preferably using material that had been peer-reviewed by other qualified people. At first I was kind of offended by this… didn’t these academics have ANY sense of the value of actual real-world employment? Apparently not.
But something funny happened as I developed the habit of looking for solid references, distilling their messages, and citing them accurately: I became more careful. And in the evolution of my caution and attention to detail, the quality of my work — ANY work — improved tremendously. I was able to learn from what other people had discovered, and anticipate (and resolve) problems in advance. I learned that “standing on the shoulders of giants” actually means figuring out when solved problems already exist so you don’t waste time reinventing wheels.
Something else funny happened as soon as I graduated: all of a sudden, people were asking me for my opinion. But the habit of due diligence was so ingrained that I couldn’t express my opinion… I was compelled to back up any opinion with facts!
(I think this was the point all along. Go figure.)
The beauty of going through the entire messy process of PhD coursework and comps and research and defense and editing — the entire end-to-end process, not cutting out in the middle anywhere — it gave me the discipline and process to root out accurate and complete answers to problems. Or at the very least, to be able to call out the gaps to close to get there.
There’s a lot of pressure in industry to move fast, but due diligence is still critical for accurate self-assessment and effective cross-functional communication. Slowing down and figuring out how you know what you know — and making sure everyone is literally on the same page — can help your organization achieve its goals more quickly.
Value of PhDs to Industry
So employers (especially in tech) — should you hire PhDs? Yes. Here’s why:
- PhDs are trained to find gaps in knowledge and understanding. Is your strategic plan grounded in reality, or is it just wishful thinking? Are your Project Charters well scoped, budgeted, and planned out? Is your workforce prepared to carry out your strategic initiatives?
- Many PhDs with experience teaching undergrads are great at making complex topics accessible to other audiences. This is fantastic for training, cross-training, and marketing.
- PhDs love research and writing, and can help you with gathering and interpreting data and content marketing.
- PhDs love learning. Want to be on the cutting edge? They’re great in R&D… they can help you distill new insights from research papers and interpret and apply them accurately.
- If you want to do AI or machine learning, or anything that uses Big Data, make sure you have at least one PhD statistician with practical analytical experience. They can prevent you from spending millions on dead ends and help you apply Occam’s Razor to avoid unnecessary complexity (the kind that can lead to technical debt later).
Will there be drawbacks? Sure. The habit of caution may need to be tempered somewhat — you don’t have to probe all the way to the bottom of an issue to generate useful information that a business can use to make progress. (This is where the perception comes from that “PhDs are slow.” If you’re a PhD, always ask yourself: How can I move this project forward as quickly as possible? Your job is to find a path forward, not find objections or cause the work to stagnate. Of course this is good guidance for anyone on a team or in a workgroup.)
Bottom line… don’t be afraid of PhDs! We are mere mortals who just happen to have spent several years trying to figure out how to get to the core — the fundamental truths — of a complex problem. As a result we know how to approach complex problems like this — problems that many businesses have lots of. (We are not overqualified at all… we just have an extra skill set in something you desperately need, but may not realize you need it.)
Getting a PhD was challenging, frustrating, and maddening at times (especially the final part of getting your camera-ready text ready for ProQuest). I never planned to do it, but I’d totally do it again. I think my only regret is that I got a PhD in a hybrid business/industrial engineering discipline… it allowed me the freedom to pursue my interests, but if I was at the same crossroads now, I’d get a PhD in statistics to complement my MBA. Overall, this is a pretty tiny regret.
Imperfect Action is Better Than Perfect Inaction: What Harry Truman Can Teach Us About Loss Functions (with an intro to ggplot)
One of the heuristics we use at Intelex to guide decision making is former US President Truman’s advice that “imperfect action is better than perfect inaction.” What it means is — don’t wait too long to take action, because you don’t want to miss opportunities. Good advice, right?
When I share this with colleagues, I often hear a response like: “that’s dangerous!” To which my answer is “well sure, sometimes, but it can be really valuable depending on how you apply it!” The trick is: knowing how and when.
Here’s how it can be dangerous. For example, statistical process control (SPC) exists to keep us from tampering with processes — from taking imperfect action based on random variation, which will not only get us nowhere, but can exacerbate the problem we were trying to solve. The secret is to apply Truman’s heuristic based on an understanding of exactly how imperfect is OK with your organization, based on your risk appetite. And this is where loss functions can help.
Along the way, we’ll demonstrate how to do a few important things related to plotting with the ggplot package in R, gradually adding in new elements to the plot so you can see how it’s layered, including:
- Plot a function based on its equation
- Add text annotations to specific locations on a ggplot
- Draw horizontal and vertical lines on a ggplot
- Draw arrows on a ggplot
- Add extra dots to a ggplot
- Eliminate axis text and axis tick marks
What is a Loss Function?
A loss function quantifies how unhappy you’ll be based on the accuracy or effectiveness of a prediction or decision. In the simplest case, you control one variable (x) which leads to some cost or loss (y). For the case we’ll examine in this post, the variables are:
- How much time and effort you put in to scoping and characterizing the problem (x); we assume that time+effort invested leads to real understanding
- How much it will cost you (y); can be expressed in terms of direct costs (e.g. capex + opex) as well as opportunity costs or intangible costs (e.g. damage to reputation)
Here is an example of what this might look like, if you have a situation where overestimating (putting in too much x) OR underestimating (putting in too little x) are both equally bad. In this case, x=10 is the best (least costly) decision or prediction:
# describe the equation we want to plot
parabola <- function(x) ((x-10)^2)+10
# initialize ggplot with a dummy dataset
library(ggplot)
p <- ggplot(data = data.frame(x=0), mapping = aes(x=x))
p + stat_function(fun=parabola) + xlim(-2,23) + ylim(-2,100) +
xlab("x = the variable you can control") +
ylab("y = cost of loss ($$)")In regression (and other techniques where you’re trying to build a model to predict a quantitative dependent variable), mean square error is a squared loss function that helps you quantify error. It captures two facts: the farther away you are from the correct answer the worse the error is — and both overestimating and underestimating is bad (which is why you square the values). Across this and related techniques, the loss function captures these characteristics:
Not all loss functions have that general shape. For classification, for example, the 0-1 loss function tells the story that if you get a classification wrong (x < 0) you incur all the penalty or loss (y=1), whereas if you get it right (x > 0) there is no penalty or loss (y=0):
# set up data frame of red points
d.step <- data.frame(x=c(-3,0,0,3), y=c(1,1,0,0))
# note that the loss function really extends to x=-Inf and x=+Inf
ggplot(d.step) + geom_step(mapping=aes(x=x, y=y), direction="hv") +
geom_point(mapping=aes(x=x, y=y), color="red") +
xlab("y* f(x)") + ylab("Loss (Cost)") +
ggtitle("0-1 Loss Function for Classification")Use the Loss Function to Make Strategic Decisions
So let’s get back to Truman’s advice. Ideally, we want to choose the x (the amount of time and effort to invest into project planning) that results in the lowest possible cost or loss. That’s the green point at the nadir of the parabola:
p + stat_function(fun=parabola) + xlim(-2,23) + ylim(-2,100) +
xlab("Time Spent and Information Gained (e.g. person-weeks)") + ylab("$$ COST $$") +
annotate(geom="text", x=10, y=5, label="Some Effort, Lowest Cost!!", color="darkgreen") +
geom_point(aes(x=10, y=10), colour="darkgreen")Costs get higher as we move up the x-axis:
p + stat_function(fun=parabola) + xlim(-2,23) + ylim(-2,100) +
xlab("Time Spent and Information Gained (e.g. person-weeks)") + ylab("$$ COST $$") +
annotate(geom="text", x=10, y=5, label="Some Effort, Lowest Cost!!", color="darkgreen") +
geom_point(aes(x=10, y=10), colour="darkgreen") +
annotate(geom="text", x=0, y=100, label="$$$$$", color="green") +
annotate(geom="text", x=0, y=75, label="$$$$", color="green") +
annotate(geom="text", x=0, y=50, label="$$$", color="green") +
annotate(geom="text", x=0, y=25, label="$$", color="green") +
annotate(geom="text", x=0, y=0, label="$ 0", color="green")And time+effort grows as we move along the x-axis (we might spend minutes on a problem at the left of the plot, or weeks to years by the time we get to the right hand side):
p + stat_function(fun=parabola) + xlim(-2,23) + ylim(-2,100) +
xlab("Time Spent and Information Gained (e.g. person-weeks)") + ylab("$$ COST $$") +
annotate(geom="text", x=10, y=5, label="Some Effort, Lowest Cost!!", color="darkgreen") +
geom_point(aes(x=10, y=10), colour="darkgreen") +
annotate(geom="text", x=0, y=100, label="$$$$$", color="green") +
annotate(geom="text", x=0, y=75, label="$$$$", color="green") +
annotate(geom="text", x=0, y=50, label="$$$", color="green") +
annotate(geom="text", x=0, y=25, label="$$", color="green") +
annotate(geom="text", x=0, y=0, label="$ 0", color="green") +
annotate(geom="text", x=2, y=0, label="minutes\nof effort", size=3) +
annotate(geom="text", x=20, y=0, label="months\nof effort", size=3)Planning too Little = Planning too Much = Costly
What this means is — if we don’t plan, or we plan just a little bit, we incur high costs. We might make the wrong decision! Or miss critical opportunities! But if we plan too much — we’re going to spend too much time, money, and/or effort compared to the benefit of the solution we provide.
p + stat_function(fun=parabola) + xlim(-2,23) + ylim(-2,100) +
xlab("Time Spent and Information Gained (e.g. person-weeks)") + ylab("$$ COST $$") +
annotate(geom="text", x=10, y=5, label="Some Effort, Lowest Cost!!", color="darkgreen") +
geom_point(aes(x=10, y=10), colour="darkgreen") +
annotate(geom="text", x=0, y=100, label="$$$$$", color="green") +
annotate(geom="text", x=0, y=75, label="$$$$", color="green") +
annotate(geom="text", x=0, y=50, label="$$$", color="green") +
annotate(geom="text", x=0, y=25, label="$$", color="green") +
annotate(geom="text", x=0, y=0, label="$ 0", color="green") +
annotate(geom="text", x=2, y=0, label="minutes\nof effort", size=3) +
annotate(geom="text", x=20, y=0, label="months\nof effort", size=3) +
annotate(geom="text",x=3, y=85, label="Little (or no) Planning\nHIGH COST", color="red") +
annotate(geom="text", x=18, y=85, label="Paralysis by Planning\nHIGH COST", color="red") +
geom_vline(xintercept=0, linetype="dotted") + geom_hline(yintercept=0, linetype="dotted")The trick is to FIND THAT CRITICAL LEVEL OF TIME and EFFORT invested to gain information and understanding about your problem… and then if you’re going to err, make sure you err towards the left — if you’re going to make a mistake, make the mistake that costs less and takes less time to make:
arrow.x <- c(10, 10, 10, 10)
arrow.y <- c(35, 50, 65, 80)
arrow.x.end <- c(6, 6, 6, 6)
arrow.y.end <- arrow.y
d <- data.frame(arrow.x, arrow.y, arrow.x.end, arrow.y.end)
p + stat_function(fun=parabola) + xlim(-2,23) + ylim(-2,100) +
xlab("Time Spent and Information Gained (e.g. person-weeks)") + ylab("$$ COST $$") +
annotate(geom="text", x=10, y=5, label="Some Effort, Lowest Cost!!", color="darkgreen") +
geom_point(aes(x=10, y=10), colour="darkgreen") +
annotate(geom="text", x=0, y=100, label="$$$$$", color="green") +
annotate(geom="text", x=0, y=75, label="$$$$", color="green") +
annotate(geom="text", x=0, y=50, label="$$$", color="green") +
annotate(geom="text", x=0, y=25, label="$$", color="green") +
annotate(geom="text", x=0, y=0, label="$ 0", color="green") +
annotate(geom="text", x=2, y=0, label="minutes\nof effort", size=3) +
annotate(geom="text", x=20, y=0, label="months\nof effort", size=3) +
annotate(geom="text",x=3, y=85, label="Little (or no) Planning\nHIGH COST", color="red") +
annotate(geom="text", x=18, y=85, label="Paralysis by Planning\nHIGH COST", color="red") +
geom_vline(xintercept=0, linetype="dotted") +
geom_hline(yintercept=0, linetype="dotted") +
geom_vline(xintercept=10) +
geom_segment(data=d, mapping=aes(x=arrow.x, y=arrow.y, xend=arrow.x.end, yend=arrow.y.end),
arrow=arrow(), color="blue", size=2) +
annotate(geom="text", x=8, y=95, size=2.3, color="blue",
label="we prefer to be\non this side of the\nloss function")Moral of the Story
The moral of the story is… imperfect action can be expensive, but perfect action is ALWAYS expensive. Spend less to make mistakes and learn from them, if you can! This is one of the value drivers for agile methodologies… agile practices can help improve communication and coordination so that the loss function is minimized.
## FULL CODE FOR THE COMPLETELY ANNOTATED CHART ##
# If you change the equation for the parabola, annotations may shift and be in the wrong place.
parabola <- function(x) ((x-10)^2)+10
my.title <- expression(paste("Imperfect Action Can Be Expensive. But Perfect Action is ", italic("Always"), " Expensive."))
arrow.x <- c(10, 10, 10, 10)
arrow.y <- c(35, 50, 65, 80)
arrow.x.end <- c(6, 6, 6, 6)
arrow.y.end <- arrow.y
d <- data.frame(arrow.x, arrow.y, arrow.x.end, arrow.y.end)
p + stat_function(fun=parabola) + xlim(-2,23) + ylim(-2,100) +
xlab("Time Spent and Information Gained (e.g. person-weeks)") + ylab("$$ COST $$") +
annotate(geom="text", x=10, y=5, label="Some Effort, Lowest Cost!!", color="darkgreen") +
geom_point(aes(x=10, y=10), colour="darkgreen") +
annotate(geom="text", x=0, y=100, label="$$$$$", color="green") +
annotate(geom="text", x=0, y=75, label="$$$$", color="green") +
annotate(geom="text", x=0, y=50, label="$$$", color="green") +
annotate(geom="text", x=0, y=25, label="$$", color="green") +
annotate(geom="text", x=0, y=0, label="$ 0", color="green") +
annotate(geom="text", x=2, y=0, label="minutes\nof effort", size=3) +
annotate(geom="text", x=20, y=0, label="months\nof effort", size=3) +
annotate(geom="text",x=3, y=85, label="Little (or no) Planning\nHIGH COST", color="red") +
annotate(geom="text", x=18, y=85, label="Paralysis by Planning\nHIGH COST", color="red") +
geom_vline(xintercept=0, linetype="dotted") +
geom_hline(yintercept=0, linetype="dotted") +
geom_vline(xintercept=10) +
geom_segment(data=d, mapping=aes(x=arrow.x, y=arrow.y, xend=arrow.x.end, yend=arrow.y.end),
arrow=arrow(), color="blue", size=2) +
annotate(geom="text", x=8, y=95, size=2.3, color="blue",
label="we prefer to be\non this side of the\nloss function") +
ggtitle(my.title) +
theme(axis.text.x=element_blank(), axis.ticks.x=element_blank(),
axis.text.y=element_blank(), axis.ticks.y=element_blank()) Now sometimes you need to make this investment! (Think nuclear power plants, or constructing aircraft carriers or submarines.) Don’t get caught up in getting your planning investment perfectly optimized — but do be aware of the trade-offs, and go into the decision deliberately, based on the risk level (and regulatory nature) of your industry, and your company’s risk appetite.
Designing Experiences for Authentic Engagement: The Design for STEAM Canvas
As Industry 4.0 and Digital Transformation efforts bear their first fruits, capabilities, business models, and the organizations that embody them are transforming. A century ago, we thought of organizations as machines to be rigidly designed and controlled. In the latter part of the 20th century, organizations were thought of as knowledge to be cultivated, shared, and expanded. But “as intelligent systems gain traction, we are once again at a crossroads – where organizations must create complete and meaningful experiences” for their customers, stakeholders, and employees.
How do you design those complete, meaningful, and radically engaging experiences? To provide a starting point, check out “Design for Steam: Creating Participatory Art with Purpose” by my former student Nick Kamienski and me. It was just published today by the STEAM Journal.
“Participatory Art” doesn’t just mean creating things that are pretty to look at in your office lobby or tradeshow booth. It means finding ways to connect with your audience in ways that help them find meaning, purpose, and self-awareness – the ultimate ingredient for authentic engagement.
Designing experiences to make this happen is challenging, but totally within reach. Learn more in today’s new article!
Selecting Pages from a PDF to Make a New (Smaller) PDF
Sometimes small, simple tasks perplex me. Today’s challenge: I’m on Windows 10, and have a 53 page PDF of a journal. I need to make a NEW PDF that only contains pages 26 to 43 (my article) so I can send my article to a researcher who is requesting it. I know you can do this with Acrobat, but I don’t have Acrobat, and still would like to figure out how to make the smaller PDF. Here’s what I learned how to do today:
The Easy Way
- Go to https://www.pdflabs.com/tools/pdftk-the-pdf-toolkit/
- Find the GREEN button that says “Download PDFtk FREE” and click it
- After the download finishes, right click on the .exe file and Run it
- When the installer starts, click on all the default options all the way through “Finish”
- When installation is finished, go to the search box in the bottom left of your screen
- Type “cmd” and hit Enter to open the terminal window
- Navigate to the directory that contains your original PDF. I first typed D: to get to my auxiliary hard drive, and then cd Scratch to get to D:\Scratch where my full journal PDF was stored.
- Use this code:
pdftk yourlargefilename.pdf cat 26-43 output youroutputfilename.pdf
(replacing YOUR filenames and YOUR starting and ending page numbers instead of 26 and 43)
- Launch a File Explorer window and navigate to the directory you used in Step 7 above. Open the PDF file and check to make sure it contains only the pages you expect.
- There are LOTS more things you can do with PDFtk from the Windows command line, like you did in Step 8. Lots of other options are described at https://www.pdflabs.com/docs/pdftk-cli-examples/
After I finished, I stumbled upon another way that didn’t require downloading a free program, and only uses Google Chrome. (Sometimes those free programs bother me. What a great way to infiltrate computers… offer a totally useful utility completely for free. Consequently, my advice to you is to download it at your own risk. Including these instructions is in no way a guarantee from me that PDFtk is safe.)
The Even Easier Way
- Open Google Chrome
- Type Ctrl-O (that’s the letter O, not the number zero)
- Select the large PDF file that you want to snip
- Your PDF will open in the browser… click on the beginning and ending pages, and capture the page numbers
- Click the print icon in the far upper right corner of your browser
- Click the “Change” button to change destination to “Microsoft Print to PDF”
- Click the second radio button under Pages, and specify the start and end pages separated by a dash (for me, 26-43)
- Click Print, and select a filename for your new, snipped file
- After the PDF is generated, navigate to the directory you saved it in during Step 8. Open the file and check it to make sure the pages are as you expect.
The Sad News
I tried to use the staplr package in R to snip my PDFs, but I couldn’t get it to work. Will try again some other time 😦