Kash022 asked . 2022-05-23

Mean and 3-sgima for Lognormal distributions

Hello,

I am trying to plot the lognormal distribution over 10 iterations and would like to see the mean and 3 sigma outliers. Somehow, doing this for lognormal plots does not look easy. Also, is it possible that I can skip the histogram in my plots and only look at the lognormal curves? This is the code snippet I am using. Thanks!

figure(113);
for i = 1:10
norm=histfit(Current_c(i,:),10,'lognormal'); %%Current_c is a 10*100 matrix %%
[muHat, sigmaHat] = lognfit(Current_c(i,:));
% Plot bounds at +- 3 * sigma.
lowBound = muHat - 3 * sigmaHat;
highBound = muHat + 3 * sigmaHat;
yl = ylim;
%line([lowBound, lowBound], yl, 'Color', [0, .6, 0], 'LineWidth', 3);
%line([highBound, highBound], yl, 'Color', [0, .6, 0], 'LineWidth', 3);
line([muHat, muHat], yl, 'Color', [0, .6, 0], 'LineWidth', 3);
grid on;
set(gcf, 'Toolbar', 'none', 'Menu', 'none');
% Give a name to the title bar.
set(gcf, 'Name', 'Line segmentation', 'NumberTitle', 'Off')
hold on;
end

#lognormal , #histogram , #plot , #normal , #gaussian , #3sigma

Expert Answer

Prashant Kumar answered . 2025-02-08 08:00:44

When plotting the lognormal distribution and marking the mean and $3σ3\sigma$ bounds, you need to handle the lognormal nature of the data carefully. For a lognormal distribution, the mean and standard deviation in the original (non-log) space differ from the parameters $μ\mu$ and $σ\sigma$ of the underlying normal distribution.

Key Points:

Lognormal Mean and $3σ3\sigma$ Bounds:
- For a lognormal distribution, if the parameters of the underlying normal distribution are $μ\mu$ $μ$ and $σ\sigma$ $σ$ :
  - Mean: $mean=eμ+σ22\text{mean} = e^{\mu + \frac{\sigma^2}{2}}$
  - $3σ3\sigma$ $3 σ$ Bounds:
    - Lower bound: $eμ−3σe^{\mu - 3\sigma}$
    - Upper bound: $eμ+3σe^{\mu + 3\sigma}$
Skipping Histogram:
- To avoid plotting the histogram while using histfit, remove the bar plot that represents the histogram. This can be done by hiding or deleting it after creating the plot.

Updated Code:

figure(113);
for i = 1:10
    % Fit the lognormal distribution
    [muHat, sigmaHat] = lognfit(Current_c(i,:));
    
    % Generate the x-values for the fitted lognormal curve
    x = linspace(min(Current_c(i,:)), max(Current_c(i,:)), 100);
    y = lognpdf(x, muHat, sigmaHat);
    
    % Plot the lognormal curve
    plot(x, y, 'LineWidth', 2); % No histogram here
    hold on;
    
    % Calculate lognormal mean and 3-sigma bounds
    lognormalMean = exp(muHat + (sigmaHat^2) / 2);
    lowBound = exp(muHat - 3 * sigmaHat);
    highBound = exp(muHat + 3 * sigmaHat);
    
    % Plot vertical lines for mean and 3-sigma bounds
    yl = ylim; % Get current y-axis limits
    line([lognormalMean, lognormalMean], yl, 'Color', [0, 0.6, 0], 'LineWidth', 3, 'LineStyle', '--'); % Mean
    line([lowBound, lowBound], yl, 'Color', [0.6, 0, 0], 'LineWidth', 2, 'LineStyle', ':'); % Lower bound
    line([highBound, highBound], yl, 'Color', [0.6, 0, 0], 'LineWidth', 2, 'LineStyle', ':'); % Upper bound
    
    grid on;
    set(gcf, 'Toolbar', 'none', 'Menu', 'none');
    set(gcf, 'Name', 'Lognormal Distribution', 'NumberTitle', 'Off');
end
hold off;

Key Changes:

Skipping Histogram:
- Removed histfit and replaced it with lognpdf to directly plot the lognormal curve.
Mean and $3σ3\sigma$ :
- Calculated the lognormal mean and bounds using their respective formulas.
Custom Plot:
- Plotted only the lognormal curve and marked the mean and bounds with vertical lines.

Output:

For each iteration:

The lognormal curve is plotted for the dataset in Current_c(i,:).
The mean ( $eμ+σ2/2e^{\mu + \sigma^2/2}$ ) is shown with a dashed green line.
The $3σ3\sigma$ bounds ( $eμ±3σe^{\mu \pm 3\sigma}$ ) are shown with dotted red lines.

This approach avoids the histogram entirely and focuses on the lognormal curve and its statistical features.

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