# Minimum amount of explained variance after Principal Component Analysis [duplicate]

I am running a PCA against two datasets of biological data (gene expression data).

The first has 20K genes (dimensions) x 450 samples, while the second has 11K genes (dimensions) x 185 samples.

Running PCA on these two datasets I get a very low amount of explained variance by the first principal components. Specifically, in the case of the first dataset I get

                             PC1       PC2       PC3       PC4       PC5
Standard deviation     1.175e+05 1.132e+05 9.022e+04 8.412e+04 7.002e+04
Proportion of Variance 1.396e-01 1.296e-01 8.236e-02 7.161e-02 4.961e-02
Cumulative Proportion  1.396e-01 2.692e-01 3.516e-01 4.232e-01 4.728e-01
PC6       PC7       PC8       PC9      PC10
Standard deviation     6.857e+04 6.816e+04 6.147e+04 5.796e+04 4.940e+04
Proportion of Variance 4.758e-02 4.701e-02 3.823e-02 3.399e-02 2.470e-02
Cumulative Proportion  5.204e-01 5.674e-01 6.056e-01 6.396e-01 6.643e-01
PC11      PC12      PC13      PC14      PC15
Standard deviation     4.434e+04 4.196e+04 3.953e+04 3.667e+04 3.374e+04
Proportion of Variance 1.989e-02 1.782e-02 1.581e-02 1.361e-02 1.152e-02
Cumulative Proportion  6.842e-01 7.020e-01 7.178e-01 7.315e-01 7.430e-01
PC16      PC17      PC18      PC19      PC20
Standard deviation     3.324e+04 3.237e+04 3.155e+04 3.080e+04 2.955e+04
Proportion of Variance 1.118e-02 1.060e-02 1.007e-02 9.600e-03 8.830e-03
Cumulative Proportion  7.541e-01 7.648e-01 7.748e-01 7.844e-01 7.933e-01


In this case, the first principal component amounts for the $13.96$% of total variance.

For the second dataset, which is smaller in size, the first principal component amount for the $41.71$% of total variance, the second for the $10.56$%.

In particular:

• in the first dataset, cumulative explained variance for the first $10$ principal component is $66.43$%
• in the second dataset, cumulative explained variance for the first $10$ principal component is $78.43$%

I have two questions:

• Could it be that the amount of explained variance is this low because of the high number of dimensions?
• Could the result of this PCAs be used for subsequent analysis, even if the amount of variance is very low?
• Re your first Q: in general if a data matrix is pure noise, then the larger the dimensionality and the larger the number of data points, the smaller fraction of variance will be explained by the first PC. But I think this effect is much weaker than what you observe. You would expect to have around 0.7% explained for pure noise 180x11000 matrix and around 0.3% for pure noise 450x20000 matrix. – amoeba Apr 1 '16 at 13:00
• Yes @amoeba, my second question may be a duplicate of the question you cited. Why do you say that only the 0.7% or the 0.3% of the variance could be explained by pure noise? – user21 Apr 1 '16 at 13:19
• Just generate a random matrix of this size, do PCA on it, and check how much variance is explained by the PC1. This of course assumes that the noise is uncorrelated in the population which does not have to be the case for any real data. I gave these numbers to show that there is indeed some dependency on the matrix size. – amoeba Apr 1 '16 at 16:53
• Thanks, I tried using a random matrix with the same shape and I get ~1% explained by the first PC, to get ~2% I need 7 PCs. – user21 Apr 6 '16 at 10:52