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Distractions and amusements, with a sandwich and coffee.

Here we are now at the middle of the fourth large part of this talk.
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I want to diverge from my traditional scope of topics and propose a new benchmark for weightlifting and CrossFit: the `k` index.

The `k` index is brainy and fun and gives you a new way to structure your goals. At high values, it's very hard to increase, making each increment a huge challenge.

Going around with a t-shirt that says "`k=40` at everything" is maximally geeky and athletic.

For a given movement (e.g. squat, deadlift, press), you achieve index rank `k` if you can perform `k` unbroken reps at `k`% of your current one rep max (1RM).

Unbroken means no more than 1–2 seconds of pause between each rep. Exact standards would need to be established for competition.

Suppose an athlete's one rep max (1RM) squat is 100 kg (220 lb).

The athlete will have a `k` index rating of `k=10` for squats if they can perform 10 unbroken reps at 10% 1RM, which is 10 kg (22 lb).

The athlete will have a `k` index rating of `k=20` for squats if they can perform 20 unbroken reps at 20% 1RM, which is 20 kg (44 lb).

The athlete will have an independent `k` index rating for other movements like deadlift and press. These ratings are likely to be different for each movement.

The minimum `k` index is `k=0` and it's a value we're born with.

The maximum theoretical `k` index is trivially `k=100` because, by definition, you cannot lift more than your current 1RM.

The maximum practical `k` index is much lower than `k=100` and probably in the range of `k = 50`.

If you perform `r` reps at weight `w` you can estimate your 1RM using the Epley formula `w(1+r/30)` assuming `r>1`. In other words, for each rep at a weight `w` your estimate of your 1RM increases by ~3.3% above `w`.

There are various 1RM estimate equations and generally they are designed for fewer than 20 reps.

In some estimates, like by Mayhew et al. and by Wathan, the rep counts appear as `e^{-xr}` for small `x` such as `x=0.055` which makes the formula converge to a value when `r` is large like `r>15`. For example, Mayhew et al. converges to to a maximum of 1.91× of the weight being tested and Wathan converges to 2.05× of the weight being tested.

Other estimates, like McGlothin, generate unreasonable values for high rep count (e.g. 185 lb at 25 reps suggests a 1RM of 535 lb, which is insanely high) and for `r>37` gives negative estimates.

Therefore, while I'm reluctant to apply any of these equations to get a rough idea of an expected `k` value, I'm going to go ahead any way and use the Epley formula.

But let's try it anyway. If you lift `k` reps at `k`% 1RM and we assume Epley applies, $$ k/100 * 1RM * (1+k/30) = 1RM $$

which gives `k` to the nearest rep $$ k = 41 $$

Again, if Epley applies (which it probably doesn't), this means that if your `k > 41` you have more endurance relative to your level of strength.

Similarly, if your `k < 41` then your strength is higher relative to your level of endurance.

In the table below I provide the 1RM estimate for each `k` index value, which should give you an idea of how difficult it is to increase the `k` index.

From personal experience—I haven't tried this yet—achieving `k=30` is going to be hard and `k=40` feels to me like it would be extremely challenging.

The `k` index requires more preparation and thought than testing your 1RM.

To accurately assess your `k` index you need to first have an accurate assessment of your 1RM. Thus, a proper 1RM test is a prerequisite.

Second, you need to pick a value of `k` to attempt to achieve and then attempt to perform the `k` reps at the required weight. If this is your first `k` index test, you need to pick `k` conservatively.

I suggest trying for `k=20` (20 reps at 20% 1RM). This should be doable for most people. Depending on how the 20th rep felt, you might want to take a 5 min rest and then try for `k=25` (25 reps at 25% 1RM). This should be exponentially harder than testing for `k=20`.

Everytime you retest your 1RM you need to retest your `k` index.

The `k` index is a measure of muscle endurance and movement efficiency.

It is not a measure of raw strength, since it is not strictly a function of your 1RM—an athelete with a lower 1RM may have a higher `k` index for that movement.

It attempts to simplify multi-rep benchmarks like 3RM, 5RM, 10RM and so on. Although these themselves are very useful and I would not advocate forgetting about them, the `k` index adds a measure of grit into the mix.

As a single number, the `k` index can be used to visualize performance, especially in combination with 1RM. A plot of `k` vs 1RM would very nicely distinguish different training regimes, such as powerlifting (low `k` high 1RM) and CrossFit (high `k` lower 1RM).

Your `k` index may change independently of your 1RM, or vice versa. For example, you can get stronger and increase your 1RM but now find that your `k` index is lower.

I think most people can get `k=20`, or close to it, on their first try. The range between `k=30` and `k=50` is where things get interesting and very painful.

The `k` index is similar to the *h* index, a metric commonly used in academic publishing.

The *h* index is defined as follows. "A scholar with an index of `h` has published `h` papers each of which has been cited in other papers at least `h` times.". For example, if I have an `h=10` then I have 10 papers that have at least 10 citations each.

To increase my `h` index to 11, it is not enough to publish a paper with 11 citations. Now, the other 10 papers also have to have an extra citation each.

In 1 hour, establish your `k` index for squats, deadlift and strict press, in that order. The total of your `k` index values is your `k` total.

Beginner athletes: attempt `k=20`, `k=24` and `k=27`.

Intermediate athletes: attempt `k=24`, `k=27` and `k=30`.

Elite athletes: attempt `k=30`, `k=32` and `k=34`.

The goal of this workout is to test all three `k` indexes in one session. You can test the individual movements on separate days and compare.

The `k` index is only applicable for movements in which an athelete moves weight and for which a 1RM can be measured.

It does not apply to body weight movements like unweighted pushups or pullups. It can, however, apply to weighted forms of those movements.

Achieving the same `k` index for different movements may not be equally easy. For example, it's probably easier to have a `k=30` for deadlift than for strict press, since the latter uses smaller muscles.

All of the following are equivalent.

squat `k30`

`k30` (squat)

squat `k=30`

A squat `k` index of 30.

The table below shows the requirement for achieving each index assuming a 1RM of 100 kg.

The performance column gives the 1RM estimate using the Epley formula based on your rep count relative to the actual 1RM used to calculate the weight. For example, for `k=30` your 1RM estimate is `30/100 * (1+30/30) = 0.6` of your 1RM, which is lower than you estimated and you can said to be underperforming. On the other hand, if you manage `k=50` then your 1RM estimate based on your rep scheme is 1.33× the value used to calculate the weight.

As I mentioned above, the use of the Epley formula here is almost definitely wrong. However, I haven't done enough research in this field to know what formula to use for accurate 1RM estimation from a very high rep count. It's possible that no such accurate assessment can be made and I need to try to at least try it for myself in the gym.

k | reps | performance |
---|---|---|

1 | 1 @ 1% 1RM (1 kg) | 0.01 |

2 | 2 @ 2% 1RM (2 kg) | 0.02 |

3 | 3 @ 3% 1RM (3 kg) | 0.03 |

4 | 4 @ 4% 1RM (4 kg) | 0.05 |

5 | 5 @ 5% 1RM (5 kg) | 0.06 |

6 | 6 @ 6% 1RM (6 kg) | 0.07 |

7 | 7 @ 7% 1RM (7 kg) | 0.09 |

8 | 8 @ 8% 1RM (8 kg) | 0.10 |

9 | 9 @ 9% 1RM (9 kg) | 0.12 |

10 | 10 @ 10% 1RM (10 kg) | 0.13 |

11 | 11 @ 11% 1RM (11 kg) | 0.15 |

12 | 12 @ 12% 1RM (12 kg) | 0.17 |

13 | 13 @ 13% 1RM (13 kg) | 0.19 |

14 | 14 @ 14% 1RM (14 kg) | 0.21 |

15 | 15 @ 15% 1RM (15 kg) | 0.22 |

16 | 16 @ 16% 1RM (16 kg) | 0.25 |

17 | 17 @ 17% 1RM (17 kg) | 0.27 |

18 | 18 @ 18% 1RM (18 kg) | 0.29 |

19 | 19 @ 19% 1RM (19 kg) | 0.31 |

20 | 20 @ 20% 1RM (20 kg) | 0.33 |

21 | 21 @ 21% 1RM (21 kg) | 0.36 |

22 | 22 @ 22% 1RM (22 kg) | 0.38 |

23 | 23 @ 23% 1RM (23 kg) | 0.41 |

24 | 24 @ 24% 1RM (24 kg) | 0.43 |

25 | 25 @ 25% 1RM (25 kg) | 0.46 |

26 | 26 @ 26% 1RM (26 kg) | 0.49 |

27 | 27 @ 27% 1RM (27 kg) | 0.51 |

28 | 28 @ 28% 1RM (28 kg) | 0.54 |

29 | 29 @ 29% 1RM (29 kg) | 0.57 |

30 | 30 @ 30% 1RM (30 kg) | 0.60 |

31 | 31 @ 31% 1RM (31 kg) | 0.63 |

32 | 32 @ 32% 1RM (32 kg) | 0.66 |

33 | 33 @ 33% 1RM (33 kg) | 0.69 |

34 | 34 @ 34% 1RM (34 kg) | 0.73 |

35 | 35 @ 35% 1RM (35 kg) | 0.76 |

36 | 36 @ 36% 1RM (36 kg) | 0.79 |

37 | 37 @ 37% 1RM (37 kg) | 0.83 |

38 | 38 @ 38% 1RM (38 kg) | 0.86 |

39 | 39 @ 39% 1RM (39 kg) | 0.90 |

40 | 40 @ 40% 1RM (40 kg) | 0.93 |

41 | 41 @ 41% 1RM (41 kg) | 0.97 |

42 | 42 @ 42% 1RM (42 kg) | 1.01 |

43 | 43 @ 43% 1RM (43 kg) | 1.05 |

44 | 44 @ 44% 1RM (44 kg) | 1.09 |

45 | 45 @ 45% 1RM (45 kg) | 1.12 |

46 | 46 @ 46% 1RM (46 kg) | 1.17 |

47 | 47 @ 47% 1RM (47 kg) | 1.21 |

48 | 48 @ 48% 1RM (48 kg) | 1.25 |

49 | 49 @ 49% 1RM (49 kg) | 1.29 |

50 | 50 @ 50% 1RM (50 kg) | 1.33 |

k | reps | performance |
---|---|---|

51 | 51 @ 51% 1RM (51 kg) | 1.38 |

52 | 52 @ 52% 1RM (52 kg) | 1.42 |

53 | 53 @ 53% 1RM (53 kg) | 1.47 |

54 | 54 @ 54% 1RM (54 kg) | 1.51 |

55 | 55 @ 55% 1RM (55 kg) | 1.56 |

56 | 56 @ 56% 1RM (56 kg) | 1.61 |

57 | 57 @ 57% 1RM (57 kg) | 1.65 |

58 | 58 @ 58% 1RM (58 kg) | 1.70 |

59 | 59 @ 59% 1RM (59 kg) | 1.75 |

60 | 60 @ 60% 1RM (60 kg) | 1.80 |

61 | 61 @ 61% 1RM (61 kg) | 1.85 |

62 | 62 @ 62% 1RM (62 kg) | 1.90 |

63 | 63 @ 63% 1RM (63 kg) | 1.95 |

64 | 64 @ 64% 1RM (64 kg) | 2.01 |

65 | 65 @ 65% 1RM (65 kg) | 2.06 |

66 | 66 @ 66% 1RM (66 kg) | 2.11 |

67 | 67 @ 67% 1RM (67 kg) | 2.17 |

68 | 68 @ 68% 1RM (68 kg) | 2.22 |

69 | 69 @ 69% 1RM (69 kg) | 2.28 |

70 | 70 @ 70% 1RM (70 kg) | 2.33 |

71 | 71 @ 71% 1RM (71 kg) | 2.39 |

72 | 72 @ 72% 1RM (72 kg) | 2.45 |

73 | 73 @ 73% 1RM (73 kg) | 2.51 |

74 | 74 @ 74% 1RM (74 kg) | 2.57 |

75 | 75 @ 75% 1RM (75 kg) | 2.62 |

76 | 76 @ 76% 1RM (76 kg) | 2.69 |

77 | 77 @ 77% 1RM (77 kg) | 2.75 |

78 | 78 @ 78% 1RM (78 kg) | 2.81 |

79 | 79 @ 79% 1RM (79 kg) | 2.87 |

80 | 80 @ 80% 1RM (80 kg) | 2.93 |

81 | 81 @ 81% 1RM (81 kg) | 3.00 |

82 | 82 @ 82% 1RM (82 kg) | 3.06 |

83 | 83 @ 83% 1RM (83 kg) | 3.13 |

84 | 84 @ 84% 1RM (84 kg) | 3.19 |

85 | 85 @ 85% 1RM (85 kg) | 3.26 |

86 | 86 @ 86% 1RM (86 kg) | 3.33 |

87 | 87 @ 87% 1RM (87 kg) | 3.39 |

88 | 88 @ 88% 1RM (88 kg) | 3.46 |

89 | 89 @ 89% 1RM (89 kg) | 3.53 |

90 | 90 @ 90% 1RM (90 kg) | 3.60 |

91 | 91 @ 91% 1RM (91 kg) | 3.67 |

92 | 92 @ 92% 1RM (92 kg) | 3.74 |

93 | 93 @ 93% 1RM (93 kg) | 3.81 |

94 | 94 @ 94% 1RM (94 kg) | 3.89 |

95 | 95 @ 95% 1RM (95 kg) | 3.96 |

96 | 96 @ 96% 1RM (96 kg) | 4.03 |

97 | 97 @ 97% 1RM (97 kg) | 4.11 |

98 | 98 @ 98% 1RM (98 kg) | 4.18 |

99 | 99 @ 99% 1RM (99 kg) | 4.26 |

100 | 100 @ 100% 1RM (100 kg) | 4.33 |

Inference creates a mathematical model of the datageneration process to formalize understanding or test a hypothesis about how the system behaves. Prediction aims at forecasting unobserved outcomes or future behavior. Typically we want to do both and know how biological processes work and what will happen next. Inference and ML are complementary in pointing us to biologically meaningful conclusions.

Statistics asks us to choose a model that incorporates our knowledge of the system, and ML requires us to choose a predictive algorithm by relying on its empirical capabilities. Justification for an inference model typically rests on whether we feel it adequately captures the essence of the system. The choice of pattern-learning algorithms often depends on measures of past performance in similar scenarios.

Bzdok, D., Krzywinski, M. & Altman, N. (2018) Points of Significance: Statistics vs machine learning. Nature Methods 15:233–234.

Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: a primer. Nature Methods 14:1119–1120.

Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: supervised methods. Nature Methods 15:5–6.

Celebrate `\pi` Day (March 14th) and go to brand new places. Together with Jake Lever, this year we shrink the world and play with road maps.

Streets are seamlessly streets from across the world. Finally, a halva shop on the same block!

Intriguing and personal patterns of urban development for each city appear in the Boonies, Burbs and Boutiques series.

No color—just lines. Lines from Marrakesh, Prague, Istanbul, Nice and other destinations for the mind and the heart.

The art is featured in the Pi City on the Scientific American SA Visual blog.

Check out art from previous years: 2013 `\pi` Day and 2014 `\pi` Day, 2015 `\pi` Day, 2016 `\pi` Day and 2017 `\pi` Day.

We examine two very common supervised machine learning methods: linear support vector machines (SVM) and k-nearest neighbors (kNN).

SVM is often less computationally demanding than kNN and is easier to interpret, but it can identify only a limited set of patterns. On the other hand, kNN can find very complex patterns, but its output is more challenging to interpret.

We illustrate SVM using a data set in which points fall into two categories, which are separated in SVM by a straight line "margin". SVM can be tuned using a parameter that influences the width and location of the margin, permitting points to fall within the margin or on the wrong side of the margin. We then show how kNN relaxes explicit boundary definitions, such as the straight line in SVM, and how kNN too can be tuned to create more robust classification.

Bzdok, D., Krzywinski, M. & Altman, N. (2018) Points of Significance: Machine learning: a primer. Nature Methods 15:5–6.

Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: a primer. Nature Methods 14:1119–1120.

In a Nature graphics blog article, I present my process behind designing the stark black-and-white Nature 10 cover.

Nature 10, 18 December 2017

In this primer, we focus on essential ML principles— a modeling strategy to let the data speak for themselves, to the extent possible.

The benefits of ML arise from its use of a large number of tuning parameters or weights, which control the algorithm’s complexity and are estimated from the data using numerical optimization. Often ML algorithms are motivated by heuristics such as models of interacting neurons or natural evolution—even if the underlying mechanism of the biological system being studied is substantially different. The utility of ML algorithms is typically assessed empirically by how well extracted patterns generalize to new observations.

We present a data scenario in which we fit to a model with 5 predictors using polynomials and show what to expect from ML when noise and sample size vary. We also demonstrate the consequences of excluding an important predictor or including a spurious one.

Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: a primer. Nature Methods 14:1119–1120.

Just in time for the season, I've simulated a snow-pile of snowflakes based on the Gravner-Griffeath model.

The work is described as a wintertime tale in In Silico Flurries: Computing a world of snow and co-authored with Jake Lever in the Scientific American SA Blog.

Gravner, J. & Griffeath, D. (2007) Modeling Snow Crystal Growth II: A mesoscopic lattice map with plausible dynamics.