How do I do a regress like this?

I have a series of numbers that decrease by about the same percetentage rate with time. How do I do the regression to find the exact percentage rate it is changing?

For example, it can be something like the below:

1 100

2 95

3 91

4 86

5 80

Many thanks.

[Walter Roberson]

think about logarithm

Answer 1

[Image Analyst]

Not sure why you need a regression. How about this to just find out the percentage decrease at each time point. Then compute the average percentage decrease over all time points if you want.

y = [100, 95, 91, 86, 80];
deltaY = diff(y)

deltaY = 1×4

-5 -4 -5 -6

percentageDecreases = 100 * deltaY ./ y(1:end-1)

percentageDecreases = 1×4

-5.0000 -4.2105 -5.4945 -6.9767

averagePercentageDecrease = mean(percentageDecreases)

averagePercentageDecrease = -5.4204

[Leon]

Thanks for the reply.

In my real world example, I would have missing values in lots of the time periods, so it is not as straightforward as this. That's why I decide to do regression for the calculation.

[Image Analyst]

You can use fitnlm. I'm attaching some examples for an exponential decay and a power law. Adapt as needed.

[Leon]

Wow, this is amazing! Thank you so much.

Would you please help me with a few questions? Is the final modeled equation strongly influenced by the initial guessed values? I notice that the coefficnets are very close to the guessed values. Why would we need guessed values? What if we use some really weird numbers for the guessed values?

Thanks!

[Image Analyst]

fitnlm just needs a starting point for some reason. So what I usually do is to just put anything there. It will do it's best and then I use the returned, estimated coordinates for the guess next time and do it again. After a few times of this, the values should converge. You could put it in a loop and do it like 5 or 10 times and see how the values settle on the final, good values.

[Leon]

Thank you so much!

Answer 2

[Sam Chak]

Hi @Leon

I'm not exactly sure what you want. If you're looking to find the exact percentage drop, as @Image Analyst has shown you, that's one approach. If you want to use MATLAB to create a linear regression model fit, you can do so using the 'fitlm()' function.

y   = [100, 95, 91, 86, 80];    % data
x   = 1:numel(y);
mdl = fitlm(x, y)               % linear regression model

mdl = 
Linear regression model:
    y ~ 1 + x1

Estimated Coefficients:
                   Estimate      SE        tStat       pValue  
                   ________    _______    _______    __________

    (Intercept)     105.1      0.63509     165.49    4.8652e-07
    x1               -4.9      0.19149    -25.589    0.00013089


Number of observations: 5, Error degrees of freedom: 3
Root Mean Squared Error: 0.606
R-squared: 0.995,  Adjusted R-Squared: 0.994
F-statistic vs. constant model: 655, p-value = 0.000131

plot(mdl)

[Leon]

Thanks for the reply.

The change should be exponential (e.g., 5% decrease during each time period), instead of linear (a fixed decrease rate of 5). In this case, how should I do the regression?

[Sam Chak]

Hi @Leon, you can refer to the example of the exponential regression shared by @Image Analyst.