

In [2]:

Exploring the Data

First we answer basic questions about our data

Question: What is the data?



In [3]:

    
# Read in data
df = pd.read_csv('../output.csv')
print "Data types:"
print df.dtypes









    



Data types:
cx          int64
cy          int64
cz          int64
unmasked    int64
synapses    int64
dtype: object

Exploring the Data

Data set contents

The data set is a table consisting of the following columns: "cx", "cy", "cz", "unmasked", and "synapses". there are 61776 rows, each corresponding to a cortical volumn called "bins" henceforth.

"cx", "cy", and "cz" denote the unique location of the bin. "synapses" is an integer count of the number of synapses found within the bin. Each bin was comprised of many individual voxels of the EM image. A synapse could technically be found at any given voxel. However, a subset of these voxels were pre-determined by the experimenters to contain material that are not synapses (i.e. cell bodies). These voxels were considered "masked". Thus the "unmasked" voxels comprise the subset area of each bin in which a synapse may reside.

We found no values within our data set that were "bad": no NaNs, Infs or nonsensical values. All numbers were integers as expected. Synapses and unmasked values were nonnegative.

Statistics

There are roughly 7.7e6 total synapses across the entire cortical volume
The maximum number of synapses per bin is 507
The mean number of synapses per bin is 124.7
The median number of synapses per bin is 144
The standard deviation of synapses per bin is 92.0
The dimensions of each bin is 3.9 x 3.9 x 5.55 um^3
The dimensions of the entire scanned cortical volume is 421.2 x 202.8 x 61.1 um^3

Exploritory Analysis - Raw Synapse Count

Distribution of Synapses

Shown is a histogram of synapse count across all bins of the dataset.

There are a significant amount of zero or low-count bins. Later we realized that this was due to masking.
Bins with higher synapse counts seem to be roughly normally distributed

To get a sense of outliers, we boxplotted synapse count. Only two bins were determined to be outliers.

1-Dimensional Marginal Distributions

Shown are the marginal distributions of synapse count for along the cortical dimensions of x, y, and z A couple observations are apparent:

synapse count is relatively uniform across x and z dimensions
synapse count is not uniform across y dimension. The drop-off in synapse count at higher values of y is primarily due to edge effects of the data volume
However, even away from the edges, there seems to be a slight negative trend beween synapse density and y-coordinate

2-Dimensional Marginal Distributions

Shown are the heatmaps of synapse count.

The z-dimension is much courser than x and y. There are far less z-layers than x and y.
The edge effects affecting the y-dimension distribution in the previous section are apparent in the y-z heatmap. Each Z layers had a variable edge cutoff in the y-dimension
The x and z dimensions fairly "rectangular". There are significant edge effects here.
Certain z-layers appear to be more dense than others. We investigate this further later.

Exporatory Analysis - Weighted Synaptic Count

At this point, we realized the significance of the unmasked data. We incorporated unmasked considerations into our subsequent exploratory analysis.

Distribution of Unmasked Count

Shown is a box plot of the unmasked count:

There are no outliers
There is a lower tail

Synapse vs. Unmasked

Shown is a joing plot of Synapse vs. unmasked data.

note that the marginal distribution of synapse count is a repeat of what was shown before
now we can appreciate the degree to which synapse count is affected by unmasked count. Virtually all the bins with zero-to-low synapse counts were the unmasked count also being zero-to-low
There is an overall positive correlation between unmasked and synapse data, as one might expect.
Most of the data lies in a dense region of relatively high unmasked and synapse count. Within this region, there is less correlation between synapse and unmasked count.

Weighted Synapse

At this point we did two things with the data: 1) We defined "weighted" synapse count of a bin as the raw synapse count divided by the unmasked count 2) We discarded all bins with lower than 50% of voxels unmasked

Shown is the distribution of weighted synapses over all bins that met the 50% unmasked requirement: We have successfully pruned out the subset of bins with zero-to-low synapse/unmasked count. The remaining data looks fairly normally distributed. All subsequent analysis was carried out with weighted and thresholded synapse count, and is referred to as "weighted synapse count".

Hypothesis Testing

Statistical Test

We decided to test whether the weighted synapse count follows a poisson distribution.

Test Definition

$\lambda$ is average number of synapses per bin
$X_i$ is the number of synapses in a 3D bin $i \in \{1, 2, ..., N\}$
$\chi^2 = \sum{\frac{(Observed-Expected)^2}{Expected}}$
$H_0$: $X_i$ is Poisson (with rate $\lambda$)
$H_1$: $X_i$ is not Poisson

Algorithm

Combine the tails and/or coarsen bins of $X_i$ histogram so that all cells have at least 5 observations (http://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm)
New histogram $O_j$ is the number of observations (3D bins) with $k$ synapses, where $k \in j^{th}$ cell, $j \in \{1,2,...,m\}$ (m total cells)
Compute $E_j = \sum_{k \in cell(j)}{N\frac{\lambda^k exp(-\lambda)}{k!}}$, $j \in \{1,2,...,m\}$
$\chi^2 = \sum_{j=1}^{m}{\frac{(O_j-E_j)^2}{E_j}}$
Degrees of freedom = $m-2$
Significance level $\alpha = 0.05$
If $\chi^2 > \chi^2_{1-\alpha,m-2}$, we reject the null hypothesis.
Otherwise, there is not sufficient evidence to conclude that $X_i$ is not Poisson.

Simulated Data

We simulated data from poisson as well as geometric distributions under a range of sample sizes. We tested each set of simulated data under th Null Hypothesis that the data set was a poisson distribution. Thus, data generated by a poisson process was considered the "Null Model", while data generated by a geometric distribution was considered the "Alt Model"

Simulated Data Result

Shown is a the power of our statistical test over a range of sample sizes

As expected, data under the null model was rejected at alpha level over all sample sizes
The power of our statistical test increased with sample size, approaching 1.0

Statistical Test on Weighted Data

We applied our test to our weighted synapse data by setting $\lambda$ to the observed mean of weighted synapse count.

Shown is a comparison of the observed vs. expected weighted synapse counts:

chi-squared statistic: inf
p-value: 0.0
Clearly the data was not poisson distributed
We should have used a model with more than one degree of freedom (i.e. normal distribution). That way we could individually vary/fit a parameter for mean as well as a parameter for spread.

Classification

Problem Definition

We divided the bins according to Z layer, and attempted to classify the Z-layers based on weighted synapse count.

Assumptions

The set of mean synapse density of an entire Z-layer can be classified into 2 groups: $W_i$
$W_i = \{\text{High density},\text{ Low density}\}$
The distribution of synapse density of each group is normally distributed

Classification Probelm

We then randomly choose a small grid, and use the $X$ and $Y$ position to predict this grid belongs to high or low density.
$N$ is the number of synapses (observations)
$X_i = X\text{ position} $
$Y_i = Y\text{ position} $
$H_0 = N \perp\!\!\!\perp X, Y$ positions
$H_1 = N \text{ is not} \perp\!\!\!\perp X, Y$ positions
The objective is to minimize the expected error:
$E[l] = \sum_{n=1}^{\infty}I(\hat{W_i} \neq W_i)$ where $I$ is the indicator function.

Classification methods:

lda (Linear Discriminant Analysis): No parameter.
qda (Quadratic Discriminant Analysis): No parameter.
svm (Support Vector Machine): Linear kernel, penalty parameter set to 0.001, to improve computation time.
knn (K-Nearest Neighbours): Number of neighbors set to 3.
rf (Random Forest): Default values except maximum depth set to 5.

Implementation / Setup

Example x-y grid

Shown is an example of a 5-by-5 grid of bins.

We average the weighted synapse density across the 25 bins to produce a single "grid mean".

Grid Means by Z-layer,

We randomly sample 1000 grids from each Z-layer and visualize the distribution of grid means

Separating the Z-layers into 2 Groups

We then took the means of the grid-means of each Z-layer (to have 11 means total), and performed K-means clustering to isolate two groups: one "high-response" group and one "low-response" group.

The low-response group comprised of 8 Z-layers, while the high-response group comprised of 3 Z-layers

Grid Means Across the 2 Groups

Shown is the distribution of grid-means of each group

It is immediately obvious that the two groups have a significant overlap in grid means. Classification is not expected to be very successful.

Classification Results

Accuracy of each classifier

Accuracy of Nearest Neighbors: 0.75 (+/- 0.01) Accuracy of Linear SVM: 0.80 (+/- 0.01) Accuracy of Random Forest: 0.80 (+/- 0.01) Accuracy of Linear Discriminant Analysis: 0.80 (+/- 0.01) Accuracy of Quadratic Discriminant Analysis: 0.80 (+/- 0.01)

Interpretation

The five classifiers tested had accuracies between 75-80%, which is better than chance. However, these numbers are only slightly at or above the maximum prior probability of 73%. This means that our classifiers are only slightly better than just choosing the class with the maximum prior 100% of the time, assuming we trust the priors. Taking into account the observed synapse density provides little added information, which is not surprising given the large overlap in observed densities from the two classes. If we wanted to distinguish between similar populations of Z layers but with different priors from another dataset, our accuracy would decrease accordingly.

Question: How many Nan, Inf, or other "bad" data values are there?



In [5]:

    
isNan = df.isnull()
isInf = np.isinf(df)
isNeg = df < 0

print "Number of nan values by column:"
print isNan.sum(), "\n"
print "Number of rows with nan values:", isNan.sum(1).sum(), "\n"

print "Number of inf values by column:"
print isInf.sum(), "\n"
print "Number of rows with inf values:", isInf.sum(1).sum(), "\n"

print "Number of negative values by column:"
print isNeg.sum(), "\n"
print "Number of rows with negative values:", isNeg.sum(1).sum(), "\n"









    



Number of nan values by column:
cx          0
cy          0
cz          0
unmasked    0
synapses    0
dtype: int64 

Number of rows with nan values: 0 

Number of inf values by column:
cx          0
cy          0
cz          0
unmasked    0
synapses    0
dtype: int64 

Number of rows with inf values: 0 

Number of negative values by column:
cx          0
cy          0
cz          0
unmasked    0
synapses    0
dtype: int64 

Number of rows with negative values: 0

Conclusion:

There is no "bad" data in our data set

Question: What is the "size" of our data set?

how many synapses are there?
how many bins are there?



In [6]:

    
nSyn = df['synapses'].sum()
print "There are", nSyn, "total synapses in the data."









    



There are 7704178 total synapses in the data.



In [7]:

    
nBins = df['synapses'].count()
print "There are", nBins, "total 3D bins."









    



There are 61776 total 3D bins.



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