Como avaliar a semelhança de dois histogramas?

Dados dois histogramas, como avaliamos se são semelhantes ou não?

É suficiente simplesmente olhar para os dois histogramas? O mapeamento simples de um para um tem o problema de que, se um histograma for ligeiramente diferente e ligeiramente alterado, não obteremos o resultado desejado.

Alguma sugestão?

Um artigo recente que pode valer a pena ser lido é:

Cao, Y. Petzold, L. Limitações de precisão e medição de erros na simulação estocástica de sistemas de reação química, 2006.

Embora o foco deste artigo esteja na comparação de algoritmos de simulação estocástica, essencialmente a idéia principal é como comparar dois histogramas.

Você pode acessar o pdf na página do autor.

Existem várias medidas de distância entre dois histogramas. Você pode ler uma boa categorização dessas medidas em:

K. Meshgi e S. Ishii, "Expandindo o histograma de cores com grade para melhorar a precisão do rastreamento", em Proc. de MVA'15, Tóquio, Japão, maio de 2015.

As funções de distância mais populares estão listadas aqui para sua conveniência:

• $L_0$　Distância L 0 ou Hellinger

$D_{L0} = \sum\limits_{i} h_1(i) \neq h_2(i)$

• $L_1$Distância L 1 , Manhattan ou City Block

$D_{L1} = \sum_{i}\lvert h_1(i) - h_2(i) \rvert$

• $L=2$ ou distância euclidiana

$D_{L2} = \sqrt{\sum_{i}\left( h_1(i) - h_2(i) \right) ^2 }$

• Distância L ∞ ou Chybyshev$_{\infty}$

$D_{L\infty} = max_{i}\lvert h_1(i) - h_2(i) \rvert$

• L p ou distância fracionária (parte da família de distância Minkowski)$_p$

$D_{Lp} = \left(\sum\limits_{i}\lvert h_1(i) - h_2(i) \rvert ^p \right)^{1/p}$ e$0<p<1$

• Interseção do histograma

$D_{\cap} = 1 - \frac{\sum_{i} \left(min(h_1(i),h_2(i) \right)}{min\left(\vert h_1(i)\vert,\vert h_2(i) \vert \right)}$

• Distância Cosine

$D_{CO} = 1 - \sum_i h_1(i)h2_(i)$

• Distância Canberra

$D_{CB} = \sum_i \frac{\lvert h_1(i)-h_2(i) \rvert}{min\left( \lvert h_1(i)\rvert,\lvert h_2(i)\rvert \right)}$

• Pearson's Correlation Coefficient

$D_{CR} = \frac{\sum_i \left(h_1(i)- \frac{1}{n} \right)\left(h_2(i)- \frac{1}{n} \right)}{\sqrt{\sum_i \left(h_1(i)- \frac{1}{n} \right)^2\sum_i \left(h_2(i)- \frac{1}{n} \right)^2}}$

• Kolmogorov-Smirnov Divergance

$D_{KS} = max_{i}\lvert h_1(i) - h_2(i) \rvert$

• Match Distance

$D_{MA} = \sum\limits_{i}\lvert h_1(i) - h_2(i) \rvert$

• Cramer-von Mises Distance

$D_{CM} = \sum\limits_{i}\left( h_1(i) - h_2(i) \right)^2$

• $\chi^2$ Statistics

$D_{\chi^2} = \sum_i \frac{\left(h_1(i) - h_2(i)\right)^2}{h_1(i) + h_2(i)}$

• Bhattacharyya Distance

$D_{BH} = \sqrt{1-\sum_i \sqrt{h_1(i)h_2(i)}}$ & hellinger

• Squared Chord

$D_{SC} = \sum_i\left(\sqrt{h_1(i)}-\sqrt{h_2(i)}\right)^2$

• Kullback-Liebler Divergance

$D_{KL} = \sum_i h_1(i)log\frac{h_1(i)}{m(i)}$

• Jefferey Divergence

$D_{JD} = \sum_i \left(h_1(i)log\frac{h_1(i)}{m(i)}+h_2(i)log\frac{h_2(i)}{m(i)}\right)$

• Earth Mover's Distance (this is the first member of Transportation distances that embed binning information $A$ into the distance, for more information please refer to the abovementioned paper or Wikipedia entry.

$D_{EM} = \frac{min_{f_{ij}}\sum_{i,j}f_{ij}A_{ij}}{sum_{i,j}f_{ij}}$ $\sum_j f_{ij} \leq h_1(i) , \sum_j f_{ij} \leq h_2(j) , \sum_{i,j} f_{ij} = min\left( \sum_i h_1(i) \sum_j h_2(j) \right)$ and $f_{ij}$ represents the flow from $i$ to $j$

• Quadratic Distance

$D_{QU} = \sqrt{\sum_{i,j} A_{ij}\left(h_1(i) - h_2(j)\right)^2}$

• Quadratic-Chi Distance

$D_{QC} = \sqrt{\sum_{i,j} A_{ij}\left(\frac{h_1(i) - h_2(i)}{\left(\sum_c A_{ci}\left(h_1(c)+h_2(c)\right)\right)^m}\right)\left(\frac{h_1(j) - h_2(j)}{\left(\sum_c A_{cj}\left(h_1(c)+h_2(c)\right)\right)^m}\right)}$ and $\frac{0}{0} \equiv 0$

A Matlab implementation of some of these distances is available from my GitHub repository: https://github.com/meshgi/Histogram_of_Color_Advancements/tree/master/distance Also you can search guys like Yossi Rubner, Ofir Pele, Marco Cuturi and Haibin Ling for more state-of-the-art distances.

Update: Alternative explaination for the distances appears here and there in the literature, so I list them here for sake of completeness.

• Canberra distance (another version)

$D_{CB}=\sum_i \frac{|h_1(i)-h_2(i)|}{|h_1(i)|+|h_2(i)|}$

• Bray-Curtis Dissimilarity, Sorensen Distance (since the sum of histograms are equal to one, it equals to $D_{L0}$)

$D_{BC} = 1 - \frac{2 \sum_i h_1(i) = h_2(i)}{\sum_i h_1(i) + \sum_i h_2(i)}$

• Jaccard Distance (i.e. intersection over union, another version)

$D_{IOU} = 1 - \frac{\sum_i min(h_1(i),h_2(i))}{\sum_i max(h_1(i),h_2(i))}$

The standard answer to this question is the chi-squared test. The KS test is for unbinned data, not binned data. (If you have the unbinned data, then by all means use a KS-style test, but if you only have the histogram, the KS test is not appropriate.)

You're looking for the Kolmogorov-Smirnov test. Don't forget to divide the bar heights by the sum of all observations of each histogram.

Note that the KS-test is also reporting a difference if e.g. the means of the distributions are shifted relative to one another. If translation of the histogram along the x-axis is not meaningful in your application, you may want to subtract the mean from each histogram first.

As David's answer points out, the chi-squared test is necessary for binned data as the KS test assumes continuous distributions. Regarding why the KS test is inappropriate (naught101's comment), there has been some discussion of the issue in the applied statistics literature that is worth raising here.

An amusing exchange began with the claim (García-Berthou and Alcaraz, 2004) that one third of Nature papers contain statistical errors. However, a subsequent paper (Jeng, 2006, "Error in statistical tests of error in statistical tests" -- perhaps my all-time favorite paper title) showed that Garcia-Berthou and Alcaraz (2005) used KS tests on discrete data, leading to their reporting inaccurate p-values in their meta-study. The Jeng (2006) paper provides a nice discussion of the issue, even showing that one can modify the KS test to work for discrete data. In this specific case, the distinction boils down to the difference between a uniform distribution of the trailing digit on [0,9],

You can compute the cross-correlation (convolution) between both histograms. That will take into account slight traslations.