4. SLICED INVERSE REGRESSION
4.1 A LGORITHM OF S LICED INVERSE REGRESSION
1. Arrange our data in the form as shown below, where Y in our data are neuron categories, and X are features extracted from fly calyx images. In our data, n is 113 and p is number of features (histogram 256 + RST-invariant 7 + volume 1 + skeleton neuron (RST- invariant 7 + volume 1) = 272).
Y 1 X1 =
(
X11,X12, ,… X ′1p)
Y 2 X2 =
(
X21,X22, ,… X2p)
′ .. .
. . .
Y n Xn =
(
Xn1,Xn2, ,… Xnp)
′Table 4.1 Data frame 2. Standardize x.
( ) ( )
ˆ 1/2 1, , (4 1)
i xx i
z = Σ− x − x i = … n −
3. Divide Y into k non-overlapping slices according to categories of neurons and compute the proportion of Yi in slice s; denote pˆs, that
ˆs #of Y in slice s, 1, , (4 2)
p = s = … k −
Let Is be the indicate function for slice s. Then ˆpscan be rewritten as
( )
1
ˆs 1 n S i , 1, , (4 3)
i
p I Y s k
n =
=
∑
= … −4. Compute the sample mean z within each slice. i
1
( )
1 n 1, , (4 4)
s i s i
s i
z z I Y s k
n =
=
∑
= … −5. Form the weighted covariance matrix Vˆ.
1
ˆ k ˆs s s (4 5)
s
V p z z
=
=
∑
′ −6. Find the eigenvalue λˆ i and eigenvector ηˆi of Vˆ. ηˆi are the standardized EDR-directi- ons. The maximum numbers of eigenvalues unequal to zero are just dependent on the number of slices-1.
7. Transform ηˆi back to the original scale.
βˆi = Σˆ1/2xxηˆi (4 6)−
8. Product βˆ1,...,βˆk 1− with X.
Table 7-1~7-16 in appendix are predicted and classification results using WEKA and R.
SVM is one of the classifier functions in WEKA called SMO. J48 is one of the classifier trees in WEKA which is used the C4.5 decision tree algorithm. IBk and OneR are lazy learners and rule learners in WEKA that are also utilized frequently.
First, we find that features extracted from skeleton neurons on revised images can help us to improve accuracy by observing table 7-1~7-16. Second, if we extract RST-invariant on red channel, it can also help us to improve accuracy. By observing table 7-2, it receives 59.29
% accuracy which is bigger than table 7-4 55.75 %. In table 7-2 and table 7-4, we can conclude that if we just want to classifer neuron images and don’t need analysis of 3d structures, then we can only use ordinary image without noise removal. So, when we remove noise, we might also remove useful informations. Nonetheless, our noise removal filter can help us to visualize neuron clearly.
We useβˆ X1 ,βˆ X2 andβˆ X3 to plot 2D and 3D scatter plots of six categories. We can find that DL1, DA1 and VL2a are so close. Therefore, we combine them into one group. After that, we classify new data into about four groups. By combining groups, we can get higher accuracy. We can see at length in appendix table 7-9~7-16.
Figure 12. 2D and 3D scatter plots of six categories using βˆ X1 , βˆ X2 and βˆ X3 .
Figure 13. 3D scatter plots of combined four categories using βˆ X1 , βˆ X2 and βˆ X3 . Here we combine DL1, DA1 and VL2a into a new group and call the new group VLA1.
6. Conclusion
By observing table 7-1 ~ table 7-16, we find the accuracies after our noise removal method are sometimes lower than using raw image directly. But in table 7-6 and table 7-8, if we also consider red channel and skeleton neuron, then our predicted result have a better behaviors about 58.4071 % and 59.292 % respectively. In table 7-9 ~ table 7-16, the highest accuracy of revised data is 70.8 % and it’s lower than 77.8761 % of raw data. If we use sir on extracted features can help us to increase the accuracy but not on raw data.
Our volume filter might still not good enough. So, when we remove noise, we might also remove useful informations that make our accuracies are sometimes lower than using raw images. Nonetheless, our noise removal filter can help us to visualize neurons clearly. Besides, we can try more other features and methods to improve accuracy in the future.
Every animal’s behavior is controlled by its central nervous system. Neuroscientists believe that much of mankind’s abnormal behavior is caused by genetic errors. Modern rese- arch in this area has improved to the point where scientists can construct an olfactory nerve network. Furthermore, we can learn more about how nerve networks express which genes.
Although current research in olfactory systems have been done only on Drosophila and focus on only parts of the cells, the brain’s olfactory nerve network can already be constructed, and this technology can later be utilized to construct taste, visual, auditory, or higher-level images, or even on mammals. Hopefully these research results can be used to cure humanity’s sickn- esses one day.
7. Appendix
Table 7- 1. Classification results in R/WEKA without using sliced inverse regression and without using leave-one-out cross-validation to evaluate correctness.
SVM J48 IBk OneR
(take log + skeleton)
86
Table 7-2. Predicted results in R/WEKA without using sliced inverse regression and using leave-one-out cross-validation to evaluate correctness.
(take log + skeleton)
38
Table 7-3. Classification results in R/WEKA using sliced inverse regression and without using
(take log + skeleton)
113
Table 7-4. Predicted results in R/WEKA using sliced inverse regression and using leave-one -out cross-validation to evaluate correctness.
SVM J48 IBk OneR
(take log + skeleton)
30
Table 7-5. Classification results added red channel in R/WEKA and without using sliced inverse regression and without using leave-one-out cross-validation to evaluate correctness.
(take log + skeleton)
92
Table 7-6. Predicted results added red channel in R/WEKA and without using sliced inverse regression and using leave-one-out cross-validation to evaluate correctness.
SVM J48 IBk OneR
(take log + skeleton)
42
Table 7-7. Classification results added red channel in R/WEKA and using sliced inverse regression and without using leave-one-out cross-validation to evaluate correct- ness.
(take log + skeleton)
113
Table 7-8. Predicted results added red channel in R/WEKA and using sliced inverse regression and using leave-one-out cross-validation to evaluate correctness.
SVM J48 IBk OneR
(take log + skeleton)
34
Table 7-9. Classification results on combined groups in R/WEKA without using sliced inverse regression and without using leave-one-out cross-validation to evaluate correct- ness.
(take log + skeleton)
97
Table 7-10. Predicted results on combined groups in R/WEKA without using sliced inverse regression and using leave-one-out cross-validation to evaluate correctness.
SVM J48 IBk OneR
(take log + skeleton)
73
Table 7-11. Classification results on combined groups in R/WEKA using sliced inverse regression and without using leave-one-out cross-validation to evaluate correct- ness.
(take log + skeleton)
113
Table 7-12. Predicted results on combined groups in R/WEKA using sliced inverse regression and using leave-one -out cross-validation to evaluate correctness.
SVM J48 IBk OneR
(take log + skeleton)
32
Table 7-13. Classification results on combined groups added red channel in R/WEKA and without using sliced inverse regression and without using leave-one-out cross-validation to evaluate correctness.
SVM J48 IBk OneR
(take log + skeleton)
97
Table 7-14. Classification results on combined groups added red channel in R/WEKA and without using sliced inverse regression and using leave-one-out cross-validation to evaluate correctness.
SVM J48 IBk OneR
(take log + skeleton)
70
Table 7-15. Classification results on combined groups added red channel in R/WEKA and using sliced inverse regression and without using leave-one-out cross-validation to evaluate correctness.
SVM J48 IBk OneR
(take log + skeleton)
111 Table 7-16. Classification results on combined groups added red channel in R/WEKA and
using sliced inverse regression and using leave-one-out cross-validation to evaluate correctness.
(take log + skeleton)
49
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