**Introduction**

How do we lose (or gain) weight? Is it really as simple as “calories in, calories out” (i.e., eat less than you burn), or is *what* you eat more important than *how much*? Is “3500 calories equals one pound” a useful rule of thumb, or just a myth? I don’t think I would normally find these to be terribly interesting questions, except for the fact that there seems to be a lot of conflicting, confusing, and at times downright misleading information out there. That can be frustrating, but I suppose it’s not surprising, since there is money to be made in weight loss programs– whether they are effective or not– particularly here in the United States.

Following is a description of my attempt to answer some of these questions, using a relatively simple mathematical model, in an experiment involving daily measurement of weight, caloric intake, and exercise over 75 days. The results suggest that you can not only measure, but *predict* future weight loss– or gain– with surprising accuracy. But they also raise some interesting open questions about how all this relates to the effectiveness of some currently popular diet programs.

**The model and the experiment**

Here was my basic idea: given just my measured *starting* weight , and a sequence of measurements of subsequent daily caloric intake, how accurately could I estimate my resulting *final* weight, weeks or even months later?

More precisely, consider the sequence of *predicted* daily weights given by the following recurrence relation:

Intuitively, my weight tomorrow morning should be my weight this morning , plus the effect of my net intake of calories that day, assuming 3500 calories per pound. Net calorie intake is modeled with three components:

- is the number of calories consumed.
- is the number of calories burned due to normal daily activity. Note that this is a function of
*current* weight, with typical values for of 12 to 13 calories per pound for men, or 10 to 11 for women; I used 12.5 (more on this later).
- is the number of
*additional* calories burned while running (my favorite form of exercise), where is the number of miles run that day. Note that we don’t really have to account for exercise separately like this; especially if duration and intensity don’t change much over time, we could skip this term altogether and just roll up *all* daily activity into the (larger) value for .

(*Aside*: I am intentionally sticking with U. S. customary units of pounds, miles, etc., to be consistent with much of the related literature.)

So, at the start of my experiment, my initial weight was pounds (for reference, I am a little over 6’4″ tall, 40-ish years old). Over each of the next 75 days, I recorded:

- My actual weight , first thing in the morning after rolling out of bed, using a digital scale with a display resolution of 0.1 pound.
- My total calories consumed for the day .
- My running mileage for the day .

Plugging in and to the recurrence relation above, I computed the sequence of predicted weights , and compared with the sequence of my actual weights .

**Results**

The following figure shows the resulting comparison of predicted weight (in blue) with measured actual weight (in red). See the appendix at the end of this post for all of the raw data.

Predicted and actual weight over 75 days.

I was surprised at just how well this worked. Two and a half months and nearly 30 pounds later, the final predicted weight differed from the actual weight by less than a pound!

There are a couple of useful observations at this point. First, the “3500 calories per pound” rule of thumb is perfectly valid… as long as it is applied correctly. Zoë Harcombe, a “qualified nutritionist,” does a great job of demonstrating how to apply it incorrectly:

“Every person who didn’t have that [55-calorie] biscuit every day should have lost 141 pounds over the past 25 years.”

This seems to be a common argument– professor of exercise science Gregory Hand makes a similar but slightly more vivid claim using the same reasoning about a hypothetical dieter, that “if she will lose 1 lb for every 3,500 calorie *deficit* [my emphasis], our individual will completely disappear from the face of the earth in 300 days.”

The problem in both cases is the incorrect assumption that an *initial* calorie deficit, due to skipping a biscuit, for example, persists as the *same* deficit over time, causing a *linear* reduction in weight. But that’s not how it works: as weight decreases, calorie *expenditure* also decreases, so that an initial reduced diet, continued over time, causes an *asymptotic* reduction in weight. (In the context of the recurrence relation above, Harcombe and Hand’s calculation effectively replaces the varying in the numerator with the constant .)

**Estimating **

The second– and, I think, most important– observation is that I arguably “got lucky” with my initial choice of calories burned per pound of body weight. If I had instead chosen 12, or 13, the resulting predictions would not agree nearly as well. And *your* value of is likely not 12.5, but something different. This seems to be one of the stronger common arguments *against* calorie-counting: even if you go to the trouble of religiously measuring calories *in*, you can never know calories *out* exactly, so why bother?

The Harris-Benedict equation is often used in an attempt to remedy this, by incorporating not only weight, but also height, age, gender, and activity level into a more complex calculation to estimate total daily calorie expenditure. But I think the problem with this approach is that the more complex formula is merely a regression fit of a *population of varying individuals, none of whom are you*. That is, even two different people of exactly the same weight, height, age, gender, and activity level do not necessarily burn calories at the same rate.

But even if you don’t know your personal value of ahead of time, *you can estimate it*, by measuring calories in and actual weight for a few weeks, and then finding the corresponding that yields a sequence of *predicted* weights that best fits the actual weights over that same time period, in a least-squares sense.

The following figure shows how this works: as time progresses along the *x*-axis, and we collect more and more data points, the *y*-axis indicates the corresponding best estimate of so far.

Estimating burn rate (alpha) over time. Early estimates are overwhelmed by the noisy weight measurements.

Here we can see the effect of the noisiness of the measured actual weights; it can take several weeks just to get a reasonably settled estimate. But keep in mind that we don’t necessarily need to be trying to *lose* weight during this time. This estimation approach should still work just as well whether we are losing, maintaining, or even gaining weight. But once we have a reasonably accurate “personal” value for , then we can predict *future* weight changes assuming any particular planned diet and exercise schedule.

(One final note: recall the constant 0.63 multiplier in the calculation of calories burned per mile run. I had hoped that I could *estimate* this value as well using the same approach… but the measured weights turned out to be simply too noisy. That is, the variability in the weights outweighs the relatively small contribution of running to the weight loss on any given day.)

*Edit*: In response to several requests for a more detailed description of a procedure for estimating , I put together a simple Excel spreadsheet demonstrating how it works. It is already populated with the time series of my recorded weight, calories, and miles from this experiment (see the Appendix below) as an example data set.

Given a particular calories/pound value for , you can see the resulting sequence of predicted weights, as well as the sum of squared differences (SSE) between these predictions and the corresponding actual measured weights.

Or you can *estimate* by minimizing SSE. This can either be done “manually” by simply experimenting with different values of (12.0 is a good starting point) and observing the resulting SSE, trying to make it as small as possible; or automatically using the Excel Solver Add-In. The following figure shows the Solver dialog in Excel 2010 with the appropriate settings.

Excel Solver dialog showing the desired settings to estimate alpha minimizing SSE.

**Conclusions and open questions**

I learned several interesting things from this experiment. I learned that it is really hard to accurately measure calories consumed, even if you are trying. (Look at the box and think about this the next time you pour a bowl of cereal, for example.) I learned that a chicken thigh loses over 40% of its weight from grilling. And I learned that, somewhat sadly, mathematical curiosity can be an even greater motivation than self-interest in personal health.

A couple of questions occur to me. First, how robust is this sort of prediction to abrupt changes in diet and/or exercise? That is, if you suddenly start eating 2500 calories a day when you usually eat 2000, what happens? What about larger, more radical changes? I am continuing to collect data in an attempt to answer this, so far with positive results.

Also, how much does the burn rate vary over the population… and even more interesting, how much control does an individual have over *changing* his or her own value of ? For example, I intentionally paid zero attention to the *composition* of fat, carbohydrates, and protein in the calories that I consumed during this experiment. I ate cereal, eggs, sausage, toast, tuna, steak (tenderloins and ribeyes), cheeseburgers, peanut butter, bananas, pizza, ice cream, chicken, turkey, crab cakes, etc. There is even one Chipotle burrito in there.

But what if I ate a strict low-carbohydrate, high-fat “keto” diet, for example? Would this have the effect of increasing , so that even for the same amount of calories consumed, I would lose more weight than if my diet were more balanced? Or is it simply hard to choke down that much meat and butter, so that I would tend to decrease , without any effect on , but with the same end result? These are interesting questions, and it would be useful to see experiments similar to this one to answer them.

**Appendix: Data collection**

The following table shows my measured actual weight in pounds over the course of the experiment:

Mon Tue Wed Thu Fri Sat Sun
251.8 251.6 250.6 249.8 248.4 249.8 249.0
250.4 249.0 247.8 246.6 246.6 247.8 246.2
246.6 244.0 244.6 243.6 243.6 244.0 244.8
242.0 240.6 240.4 240.2 240.2 239.4 238.6
238.0 238.0 237.6 238.0 238.0 238.6 238.6
237.4 239.0 237.6 235.8 236.0 235.0 236.0
233.8 232.4 232.6 233.4 233.4 232.0 233.2
232.6 231.6 232.2 232.2 231.2 231.2 229.6
229.6 229.6 230.6 230.4 229.8 228.0 227.4
227.6 226.2 226.4 225.6 225.8 225.8 226.0
228.0 225.8 225.4 224.6 223.8

The following table shows my daily calorie intake:

Mon Tue Wed Thu Fri Sat Sun
1630 1730 1670 1640 2110 2240 1980
1630 1560 1690 1700 2010 1990 2030
1620 1710 1590 1710 2180 2620 2100
1580 1610 1610 1620 1690 2080 1930
1620 1680 1610 1610 1810 2550 2430
1710 1660 1630 1710 1930 2470 1970
1660 1750 1710 1740 2020 2680 2100
1740 1750 1750 1610 1990 2290 1940
1950 1700 1730 1640 1820 2230 2280
1740 1760 1780 1650 1900 2470 1910
1570 1740 1740 1750

And finally, the following table shows the number of miles run on each day:

Mon Tue Wed Thu Fri Sat Sun
2.5 0.0 2.5 0.0 0.0 2.5 0.0
2.5 0.0 2.5 0.0 0.0 2.5 0.0
2.5 0.0 2.5 0.0 0.0 3.0 0.0
2.5 0.0 2.5 0.0 0.0 3.0 0.0
2.5 0.0 3.0 0.0 0.0 3.0 0.0
2.5 0.0 3.0 0.0 0.0 3.0 0.0
3.0 0.0 3.0 0.0 0.0 3.0 0.0
3.0 0.0 3.0 0.0 0.0 3.0 0.0
3.0 0.0 3.0 0.0 0.0 3.5 0.0
3.0 0.0 3.0 0.0 0.0 3.5 0.0
3.0 0.0 3.5 0.0