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Weight change and rules of thumb for a healthy lifestyle - a mathematical framework

Updated: May 7, 2023

1. Background

I have long been interested in fitness and health. I have been blessed with a high metabolism and a discipline related to eating. I also have parents that encouraged health and fitness when I was growing up. What follows is my approach to eating and health.

I’m not a professional mathematician but I do have a mathematical mind and regularly use math in practice. I have forgotten much of the more technical notation I used earlier in my life. What follows is a descriptive mathematical framework for the key drivers and their interactions related to human weight change. I use simple math notation. (I.e., the math I can remember!)

Note to the reader: If you find the math sections do not add value to you, you may safely skip to the interpretations and the rules of thumb summary in section 3.

The following is the article sections:

  1. Background

  2. The Math build

    1. Formula specification and relationships

    2. Variable definitions and interpretations

  3. Summary - Related rules of thumb for a healthy lifestyle

  4. Additional Resources

2 a. Formula specification and functional relationships

Weight =

f(QS(+), QUDs), {Calorie Supply}

f(QD(-), QUDd, MET(-)) , {Calorie Demand}

f(QOD(+), MET(-)). {Calorie Overflow}

2 b. Variable definitions and interpretation

The following is to help understand the variable relationships and the related interpretations.

A) Calorie Supply

QS = Quantity of calories consumed (supply). This is the total calories you regularly eat.

QUDs = quality of calories consumed, distributed across nutrition group vectors (e.g., carbs, proteins, fiber, fat, etc). For example, some carbs have little to no nutritional value, so would be an empty (or low-value) calories.

Interpretation: Calories supplied are both related to the quantity and quality of those calories. Quality is concerned with the mix of nutrients provided for a healthy life. Notice for the quality variables, I did not specify either a positive or a negative functional relationship. It is more of a parabolic relationship. Either you get the mix right and there is an effect on your weight appropriate to the healthy body, or you get it wrong and there could be a negative effect on your weight (weight loss or weight gain) related to overeating or undereating certain nutrition groups.

B) Calorie Demand

QD = Quantity of calories demanded via natural appetite. Natural appetite is the feeling of how hungry you are as a signal for eating.

QUDd = Quality of calories needed, distributed across nutrition group vectors (e.g., carbs, proteins, fiber, fats, etc)

MET = the body's natural metabolism rate. I.e., the body’s efficiency of caloric consumption. MET has a feedback loop to QD. I.e, QD = f(MET(+))

Interpretation: A higher quantity demanded means the body requires more calories and will reduce weight if supply is not increased to meet the quantity demanded. Calories demanded is about listening to your body and doing some research about a healthy mix of nutrition groups. In terms of healthy mix research, beware of 1) the USDA recommendations (see Freakonomics description in section V) and 2) any diet that encourages extreme changes in diet. (see Fit For Life reference in section V).

Metabolism is dynamic and should be both listened to and cared for.

C) Calorie Overflow

QOD - Quantity of calories overflow beyond quantity demanded. Empty calories, like carbs, will generally pass to overflow. This is specified by the net of the QUD vectors (QUDs - QUDd)

MET - the body's natural metabolism rate. I.e., the body’s processing efficiency for calories consumed.

Interpretation: It is the body’s calorie overflow that creates weight gain, it is the body’s calorie deficiency (negative overflow) that creates weight loss. The body uses or creates fat as a calorie energy store to manage overflow.

D) Additional interpretation

1) Exercise creates a positive feedback loop for MET. (Antifragile / Convex)

  • Support 1 - dMET / dExercise > 0). Also, weight is an indicator variable, likely causally related to overall health. Convex function suggests the gain in overall health (as indicated by weight) is greater than the pain of exercise.

  • Support 2 - Exercise leads to higher QD and MET, which reduces overflow (ceteris paribus) and increases MET to process overflow.

2) Proper management of one’s digestive system can lead to greater digestive efficiency that contributes to a higher metabolism. See Diamond, Fit For Life for digestive strategies including food type mixing, food type emphasis, time of day targeting, etc.

3) Think of overflow as an error or residual. (as in the least squares equation) The most disciplined bodies will have very low overflow (I.e., low error). The body’s ability to process overflow is enhanced by metabolism.

4) High MET is a great error correction for a high-calorie diet. That is, the high MET engine burns hot, thus more efficiently consuming overflow calories. High MET will also signal a QD increase via a bigger appetite. Given appetite is subject to signaling inertia, be aware of changes in metabolism that may have a delayed QD decrease.

3. Summary - Related rules of thumb for a healthy lifestyle

The following rules of thumb are just that. These are observations that generally work to help me live a healthy and happy lifestyle. Of course, sometimes I adapt and adjust to the situation.

  • Exercise has multiple benefits - 1) in the short run, it burns calories, so will help with calorie overflow. 2) in the longer run, it increases metabolism and is very helpful to increase “heat of the fire” to consume future calorie overflow. Exercise is a positive reinforcing loop in your personal health system.

  • Listen to your appetite - only eat if you are hungry. Don’t unnecessarily add to the calorie quantity supplied

  • Be aware of your stress levels (glucocorticoid production). Excess glucocorticoids can cause psychologically triggered “comfort eating.” Use the self-discipline from your executive function / prefrontal cortex to identify and remediate comfort eating.

  • Be aware of calorie quality - eat a mix of foods that are especially nutritious, and reduce intake of empty or low-value calories.

  • Be aware of food mixing and digestive efficiency, including all the nutrition groups but only as properly mixed in a single sitting. E.g., reduce the mixing of carbs and proteins.

  • Be careful with controlling negative overflow/deficiency, not to go too far, or too fast. Starving a body can cause negative health effects and lead to weight gain in the longer run. You are better off equalizing calorie supply and demand, exercising, building metabolism, and slowly letting your body adjust the weight.

  • Related to comfort eating and the impact on executive function, excessive dieting can create a cognitive tax that reduces your bandwidth for other activities. In effect, excessive dieting can negatively impact other parts of your life by overly increasing cognitive attention to dieting. (4) Practice “Boiling You Own Frog” by slowly changing your weight via healthy behaviors as described earlier.

  • Fat is a necessary part of your diet. Be wary of no or low-fat diets. You need fat intake.

  • Practice periodic fasting. Fasting is good to cleanse the body. Even the most disciplined diet can lead to a build-up of toxins over time. These toxins can reduce metabolism and reduce the body's ability to absorb nutrients.

4. Additional resources:

1) Freakonomics Podcast: How the Supermarket Helped America Win the Cold War (Ep. 386)

Abstract: Aisle upon aisle of fresh produce, cheap meat, and sugary cereal — a delicious embodiment of free-market capitalism, right? Not quite. The supermarket was in fact the endpoint of the U.S. government’s battle for agricultural abundance against the U.S.S.R. Our farm policies were built to dominate, not necessarily to nourish — and we are still living with the consequences.

Interpretation: U.S. Citizens were encouraged to consume excess that resulted from farm subsidies and price floors. The excess food consumption (which was wheat / grain-based) was encouraged by the USDA food pyramid diet recommendations but was not an appropriate healthy diet for the citizens. The result was significant weight gain and diseases like Diabetes and Celiac's disease in some citizens. Also, the economic policies reduced the resilience of our agriculture production by requiring scale to survive and reducing resilience by reducing the number of suppliers. Finally, the cultural preference for red meat/cattle is an extremely inefficient and environmentally destructive activity. This occurred as a result of the US Government’s policy encouraging the use of excess grain as cattle feed.

Winning the Cold War came at a tremendous cost. We (the US citizens) were unwitting foot soldiers.

2) TED Talk - The mathematics of weight loss | Ruben Meerman

Title of the TED talk is a bit of a misnomer. More like the chemistry, not the mathematics, of weight loss.


To lose weight

Turn it into CO2 and H2O

To do that

Eat less, move more, and keep breathing

3) Fit For Life by Harvey Diamond, Marilyn Diamond

Summary: Nutritional specialists Harvey and Marilyn Diamond explain how you can eat more kinds of food than you ever ate before without counting calories...and still lose weight! The natural body cycles, a permanent weight-loss plan that proves it's not only what you eat, but also when and how....

(4) In the book Scarcity: Why Having Too Little Means So Much, the authors observe several studies that demonstrate people perform cognitively worse when they are dieting. The following quote is from the Scarcity book. It is particularly on point as to why we should “Boil our own frog.”

“Thinking about that delicious dessert only makes things harder. Dieting creates a scarcity of calories, and that scarcity in turn places the dessert firmly top of mind. Studies have shown that food ends up top of mind of dieters and not just because they are hungry but because of the scarcity they face. In one study, the preoccupation with food grew only more intense among dieters who had just eaten a chocolate bar. Physiologically, they had more calories; psychologically, they had now exacerbated the trade-offs they needed to make. Diets prove dificult precisely because they focus us on that which we are trying to avoid.”

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