Dimension Reduction

class: center, middle, inverse, title-slide

.title[
# Dimension Reduction
]
.author[
### Jun Kang
]
.date[
### 2022.06.15
]

---

##### DNA methylation-based classification of central nervous system tumours
![:scale 60%](data:image/png;base64,#https://media.springernature.com/full/springer-static/image/art%3A10.1038%2Fnature26000/MediaObjects/41586_2018_Article_BFnature26000_Fig1_HTML.jpg?as=webp)

.footer[Nature 555, 469–474 (2018). https://doi.org/10.1038/nature26000]

---
##### Sarcoma classification by DNA methylation profiling
![:scale 60%](data:image/png;base64,#https://media.springernature.com/full/springer-static/image/art%3A10.1038%2Fs41467-020-20603-4/MediaObjects/41467_2020_20603_Fig1_HTML.png?as=webp)

.footer[Nat Commun 12, 498 (2021). https://doi.org/10.1038/s41467-020-20603-4]

---
![](data:image/png;base64,#https://media.springernature.com/full/springer-static/image/art%3A10.1038%2Fs10038-020-00851-4/MediaObjects/10038_2020_851_Fig4_HTML.png?as=webp)
.footer[J Hum Genet 66, 85–91 (2021). https://doi.org/10.1038/s10038-020-00851-4]

???
The Genome Aggregation Database (gnomAD, left) and Biobank Japan (BBJ, right) visualized using UMAP. UMAP illustrates the ancestral diversity of gnomAD, showing many the relationships between populations on continental and subcontinental levels. For the relatively more homogeneous BBJ data, it splits data geographically into the large mainland cluster (consisting of Hokkaido, Tohoku, Kanto-Koshinetsu, Chubu-Hokuriku, Kinki, and Kyushu regions), and smaller non-mainland clusters. The gnomAD image is reproduced from [10], and the BBJ image is reproduced from [12]

---
# High dimensional data

* Population genetics
* Single cell sequencing
* Spatial transcriptomics

---
### Dimension reduction technique

* Principal component analysis (PCA)
* t-Distributed Stochastic Neighbor Embedding (t-SNE)
* Uniform Manifold Approximation and Projection (UMAP)

---
### Dimension reduction

* Change and select basis to clustering 
* Visualization in 2 dimensional space

---
# Concepts

* Dimension
* Basis or Latent features
* Graph
* Projection or Embedding
* Linearity

---
![](data:image/png;base64,#https://media.springernature.com/full/springer-static/image/art%3A10.1038%2Fs10038-020-00851-4/MediaObjects/10038_2020_851_Fig3_HTML.png?as=webp)

.footer[J Hum Genet 66, 85–91 (2021). https://doi.org/10.1038/s10038-020-00851-4]

???
PCA (left) and UMAP (right) projections of the UKB data, coloured by self-identified ethnic background. Unlike PCA, UMAP focuses on preserving local relationships and emphasizes fine-scale patterns in data. Groups in the UMAP projection are less compressed showing, for example, the relative size of the British and Irish populations in the UKB, alongside populations of other ancestries, while simultaneously showing the population structure between and within groups

---
class: center

![:scale 40%](data:image/png;base64,#https://media.springernature.com/full/springer-static/image/art%3A10.1038%2Fs10038-020-00851-4/MediaObjects/10038_2020_851_Fig5_HTML.png?as=webp)

.footer[J Hum Genet 66, 85–91 (2021). https://doi.org/10.1038/s10038-020-00851-4]

???
UMAP projection of the same genotype data from the 1000GP comparing parametrization with a small (top) and large (bottom) number of nearest neighbours. Left images are coloured by population; right images are the same points but with the simplicial complex drawn. When adding more neighbours, subclusters become less separated, as with the LWK population, for example. Looking at the connectivity maps, we see new connections between continental groups, such as the Central/South American clusters and East Asian clusters. Darker lines indicate that individuals are closer to each other in genotype space

---
# Iris
![:scale 90%](data:image/png;base64,#https://machinelearninghd.com/wp-content/uploads/2021/03/iris-dataset.png)

.footer[/ˈpedl/,  /ˈsēpəl/]
---
class: center

## Iris data

<div id="htmlwidget-8ebca687e10bfe4f5845" style="width:504px;height:504px;" class="plotly html-widget"></div>
<script type="application/json" data-for="htmlwidget-8ebca687e10bfe4f5845">{"x":{"visdat":{"5d80c8fc120":["function () ","plotlyVisDat"]},"cur_data":"5d80c8fc120","attrs":{"5d80c8fc120":{"alpha_stroke":1,"sizes":[10,100],"spans":[1,20],"type":"splom","dimensions":[{"label":"sepal_width","values":{}},{"label":"sepal_length","values":{}},{"label":"petal_width","values":{}},{"label":"petal_length","values":{}}],"color":{},"colors":["#636EFA","#EF553B","#00CC96"],"inherit":true}},"layout":{"margin":{"b":40,"l":60,"t":25,"r":10},"legend":{"title":{"text":"species"}},"hovermode":"closest","dragmode":"select","plot_bgcolor":"rgba(240,240,240,0.95)","xaxis":{"domain":[],"showline":false,"zeroline":false,"gridcolor":"#ffff","ticklen":4},"yaxis":{"domain":[],"showline":false,"zeroline":false,"gridcolor":"#ffff","ticklen":4},"xaxis2":{"showline":false,"zeroline":false,"gridcolor":"#ffff","ticklen":4},"xaxis3":{"showline":false,"zeroline":false,"gridcolor":"#ffff","ticklen":4},"xaxis4":{"showline":false,"zeroline":false,"gridcolor":"#ffff","ticklen":4},"yaxis2":{"showline":false,"zeroline":false,"gridcolor":"#ffff","ticklen":4},"yaxis3":{"showline":false,"zeroline":false,"gridcolor":"#ffff","ticklen":4},"yaxis4":{"showline":false,"zeroline":false,"gridcolor":"#ffff","ticklen":4},"showlegend":true},"source":"A","config":{"modeBarButtonsToAdd":["hoverclosest","hovercompare"],"showSendToCloud":false},"data":[{"type":"splom","dimensions":[{"label":"sepal_width","values":[3.5,3,3.2,3.1,3.6,3.9,3.4,3.4,2.9,3.1,3.7,3.4,3,3,4,4.4,3.9,3.5,3.8,3.8,3.4,3.7,3.6,3.3,3.4,3,3.4,3.5,3.4,3.2,3.1,3.4,4.1,4.2,3.1,3.2,3.5,3.6,3,3.4,3.5,2.3,3.2,3.5,3.8,3,3.8,3.2,3.7,3.3]},{"label":"sepal_length","values":[5.1,4.9,4.7,4.6,5,5.4,4.6,5,4.4,4.9,5.4,4.8,4.8,4.3,5.8,5.7,5.4,5.1,5.7,5.1,5.4,5.1,4.6,5.1,4.8,5,5,5.2,5.2,4.7,4.8,5.4,5.2,5.5,4.9,5,5.5,4.9,4.4,5.1,5,4.5,4.4,5,5.1,4.8,5.1,4.6,5.3,5]},{"label":"petal_width","values":[0.2,0.2,0.2,0.2,0.2,0.4,0.3,0.2,0.2,0.1,0.2,0.2,0.1,0.1,0.2,0.4,0.4,0.3,0.3,0.3,0.2,0.4,0.2,0.5,0.2,0.2,0.4,0.2,0.2,0.2,0.2,0.4,0.1,0.2,0.2,0.2,0.2,0.1,0.2,0.2,0.3,0.3,0.2,0.6,0.4,0.3,0.2,0.2,0.2,0.2]},{"label":"petal_length","values":[1.4,1.4,1.3,1.5,1.4,1.7,1.4,1.5,1.4,1.5,1.5,1.6,1.4,1.1,1.2,1.5,1.3,1.4,1.7,1.5,1.7,1.5,1,1.7,1.9,1.6,1.6,1.5,1.4,1.6,1.6,1.5,1.5,1.4,1.5,1.2,1.3,1.4,1.3,1.5,1.3,1.3,1.3,1.6,1.9,1.4,1.6,1.4,1.5,1.4]}],"name":"setosa","marker":{"color":"rgba(99,110,250,1)","line":{"color":"rgba(99,110,250,1)"}},"frame":null},{"type":"splom","dimensions":[{"label":"sepal_width","values":[3.2,3.2,3.1,2.3,2.8,2.8,3.3,2.4,2.9,2.7,2,3,2.2,2.9,2.9,3.1,3,2.7,2.2,2.5,3.2,2.8,2.5,2.8,2.9,3,2.8,3,2.9,2.6,2.4,2.4,2.7,2.7,3,3.4,3.1,2.3,3,2.5,2.6,3,2.6,2.3,2.7,3,2.9,2.9,2.5,2.8]},{"label":"sepal_length","values":[7,6.4,6.9,5.5,6.5,5.7,6.3,4.9,6.6,5.2,5,5.9,6,6.1,5.6,6.7,5.6,5.8,6.2,5.6,5.9,6.1,6.3,6.1,6.4,6.6,6.8,6.7,6,5.7,5.5,5.5,5.8,6,5.4,6,6.7,6.3,5.6,5.5,5.5,6.1,5.8,5,5.6,5.7,5.7,6.2,5.1,5.7]},{"label":"petal_width","values":[1.4,1.5,1.5,1.3,1.5,1.3,1.6,1,1.3,1.4,1,1.5,1,1.4,1.3,1.4,1.5,1,1.5,1.1,1.8,1.3,1.5,1.2,1.3,1.4,1.4,1.7,1.5,1,1.1,1,1.2,1.6,1.5,1.6,1.5,1.3,1.3,1.3,1.2,1.4,1.2,1,1.3,1.2,1.3,1.3,1.1,1.3]},{"label":"petal_length","values":[4.7,4.5,4.9,4,4.6,4.5,4.7,3.3,4.6,3.9,3.5,4.2,4,4.7,3.6,4.4,4.5,4.1,4.5,3.9,4.8,4,4.9,4.7,4.3,4.4,4.8,5,4.5,3.5,3.8,3.7,3.9,5.1,4.5,4.5,4.7,4.4,4.1,4,4.4,4.6,4,3.3,4.2,4.2,4.2,4.3,3,4.1]}],"name":"versicolor","marker":{"color":"rgba(239,85,59,1)","line":{"color":"rgba(239,85,59,1)"}},"frame":null},{"type":"splom","dimensions":[{"label":"sepal_width","values":[3.3,2.7,3,2.9,3,3,2.5,2.9,2.5,3.6,3.2,2.7,3,2.5,2.8,3.2,3,3.8,2.6,2.2,3.2,2.8,2.8,2.7,3.3,3.2,2.8,3,2.8,3,2.8,3.8,2.8,2.8,2.6,3,3.4,3.1,3,3.1,3.1,3.1,2.7,3.2,3.3,3,2.5,3,3.4,3]},{"label":"sepal_length","values":[6.3,5.8,7.1,6.3,6.5,7.6,4.9,7.3,6.7,7.2,6.5,6.4,6.8,5.7,5.8,6.4,6.5,7.7,7.7,6,6.9,5.6,7.7,6.3,6.7,7.2,6.2,6.1,6.4,7.2,7.4,7.9,6.4,6.3,6.1,7.7,6.3,6.4,6,6.9,6.7,6.9,5.8,6.8,6.7,6.7,6.3,6.5,6.2,5.9]},{"label":"petal_width","values":[2.5,1.9,2.1,1.8,2.2,2.1,1.7,1.8,1.8,2.5,2,1.9,2.1,2,2.4,2.3,1.8,2.2,2.3,1.5,2.3,2,2,1.8,2.1,1.8,1.8,1.8,2.1,1.6,1.9,2,2.2,1.5,1.4,2.3,2.4,1.8,1.8,2.1,2.4,2.3,1.9,2.3,2.5,2.3,1.9,2,2.3,1.8]},{"label":"petal_length","values":[6,5.1,5.9,5.6,5.8,6.6,4.5,6.3,5.8,6.1,5.1,5.3,5.5,5,5.1,5.3,5.5,6.7,6.9,5,5.7,4.9,6.7,4.9,5.7,6,4.8,4.9,5.6,5.8,6.1,6.4,5.6,5.1,5.6,6.1,5.6,5.5,4.8,5.4,5.6,5.1,5.1,5.9,5.7,5.2,5,5.2,5.4,5.1]}],"name":"virginica","marker":{"color":"rgba(0,204,150,1)","line":{"color":"rgba(0,204,150,1)"}},"frame":null}],"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.2,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script>
???
clustering 가장 좋은 2개 basis? petal 1 개?

---
## Dimension reducsion by human selection
![](data:image/png;base64,#dimensionReduction_files/figure-html/unnamed-chunk-1-1.png)

---

## Clustering? and select one dimension

![](data:image/png;base64,#dimensionReduction_files/figure-html/unnamed-chunk-2-1.png)

---
## Principal component analysis (PCA)

![](data:image/png;base64,#img/pca.jpg)
---
## Linearity

$$
PC1 = \alpha_1v_1 + \alpha_2v_2 + \alpha_3v_3 ... + \alpha_nv_n\\\
PC2 = \beta_2v_1 + \beta_2v_2 + \beta_3v_3 ... + \beta_nv_n\\
$$

---
class: center
# Projection

![:scale 50%](data:image/png;base64,#https://cdn.pixabay.com/photo/2017/09/08/15/31/marigold-2729124_1280.jpg)

---
# Orthogonality

* PCs are orthogonal each other

---
# Principal component analysis (PCA)

<div id="htmlwidget-c6fbee1687af8f208d51" style="width:504px;height:504px;" class="plotly html-widget"></div>
<script type="application/json" data-for="htmlwidget-c6fbee1687af8f208d51">{"x":{"data":[{"x":[-0.107878129475915,-0.0991256758781124,-0.112619018704969,-0.10953016321449,-0.113839088131987,-0.0988717739479014,-0.116420246442089,-0.106360699500147,-0.111209579807091,-0.104049518884768,-0.103191235785805,-0.110804228180452,-0.105674939938425,-0.125426601970102,-0.104736048615599,-0.107759601477619,-0.105157476006749,-0.104336342513111,-0.090437648059616,-0.11162536167409,-0.0911879400806897,-0.105130093152723,-0.132154698580868,-0.08663152892624,-0.106089951636316,-0.09297534775118,-0.0977057000598279,-0.103299226863694,-0.101917170819842,-0.107904742160833,-0.101943783504761,-0.0872472171845061,-0.124561903177881,-0.116522591995828,-0.100507731921964,-0.105168012927155,-0.0974196466005929,-0.120388352192299,-0.115734486880366,-0.103353222402639,-0.108915245125331,-0.0885108519101075,-0.121641449997494,-0.0935756076927842,-0.101797872652439,-0.0985913660128171,-0.113595723122182,-0.114055070287765,-0.106198712883314,-0.104978643456295],"y":[-0.0408589158360105,0.057380916914182,0.0291025357194968,0.0508490406299959,-0.0550573502828649,-0.126755849636756,-0.00405538000424925,-0.018993923226488,0.0949344889461115,0.0399215080376265,-0.088836887082972,-0.0113273650638198,0.062023476045196,0.0818415537964958,-0.158324387842385,-0.228651229919747,-0.126281898840662,-0.0416089520361787,-0.119592424851425,-0.0960003118683027,-0.0348009903479184,-0.0786593906907707,-0.0390133077633874,-0.00728257277818513,-0.0116828281608904,0.0532514185852892,-0.0206124833258478,-0.0448699264658799,-0.0266604813891562,0.0287470726224263,0.0429455070692807,-0.0360640873502077,-0.152665634531913,-0.183065709203295,0.0391714718374583,0.0175434546259828,-0.0563105198603704,-0.0504147911518509,0.0769620192674348,-0.0228864461573339,-0.0375979414063093,0.198956161780322,0.0407801045120344,-0.0402035131038843,-0.0972242988645649,0.0605234036448598,-0.0953687633671581,0.0328765709513192,-0.0849443641521261,-0.000784478149764319],"text":["PC1: -0.1078781295 PC2: -0.0408589158 Species: setosa","PC1: -0.0991256759 PC2: 0.0573809169 Species: setosa","PC1: -0.1126190187 PC2: 0.0291025357 Species: setosa","PC1: -0.1095301632 PC2: 0.0508490406 Species: setosa","PC1: -0.1138390881 PC2: -0.0550573503 Species: setosa","PC1: -0.0988717739 PC2: -0.1267558496 Species: setosa","PC1: -0.1164202464 PC2: -0.0040553800 Species: setosa","PC1: -0.1063606995 PC2: -0.0189939232 Species: setosa","PC1: -0.1112095798 PC2: 0.0949344889 Species: setosa","PC1: -0.1040495189 PC2: 0.0399215080 Species: setosa","PC1: -0.1031912358 PC2: -0.0888368871 Species: setosa","PC1: -0.1108042282 PC2: -0.0113273651 Species: setosa","PC1: -0.1056749399 PC2: 0.0620234760 Species: setosa","PC1: -0.1254266020 PC2: 0.0818415538 Species: setosa","PC1: -0.1047360486 PC2: -0.1583243878 Species: setosa","PC1: -0.1077596015 PC2: -0.2286512299 Species: setosa","PC1: -0.1051574760 PC2: -0.1262818988 Species: setosa","PC1: -0.1043363425 PC2: -0.0416089520 Species: setosa","PC1: -0.0904376481 PC2: -0.1195924249 Species: setosa","PC1: -0.1116253617 PC2: -0.0960003119 Species: setosa","PC1: -0.0911879401 PC2: -0.0348009903 Species: setosa","PC1: -0.1051300932 PC2: -0.0786593907 Species: setosa","PC1: -0.1321546986 PC2: -0.0390133078 Species: setosa","PC1: -0.0866315289 PC2: -0.0072825728 Species: setosa","PC1: -0.1060899516 PC2: -0.0116828282 Species: setosa","PC1: -0.0929753478 PC2: 0.0532514186 Species: setosa","PC1: -0.0977057001 PC2: -0.0206124833 Species: setosa","PC1: -0.1032992269 PC2: -0.0448699265 Species: setosa","PC1: -0.1019171708 PC2: -0.0266604814 Species: setosa","PC1: -0.1079047422 PC2: 0.0287470726 Species: setosa","PC1: -0.1019437835 PC2: 0.0429455071 Species: setosa","PC1: -0.0872472172 PC2: -0.0360640874 Species: setosa","PC1: -0.1245619032 PC2: -0.1526656345 Species: setosa","PC1: -0.1165225920 PC2: -0.1830657092 Species: setosa","PC1: -0.1005077319 PC2: 0.0391714718 Species: setosa","PC1: -0.1051680129 PC2: 0.0175434546 Species: setosa","PC1: -0.0974196466 PC2: -0.0563105199 Species: setosa","PC1: -0.1203883522 PC2: -0.0504147912 Species: setosa","PC1: -0.1157344869 PC2: 0.0769620193 Species: setosa","PC1: -0.1033532224 PC2: -0.0228864462 Species: setosa","PC1: -0.1089152451 PC2: -0.0375979414 Species: setosa","PC1: -0.0885108519 PC2: 0.1989561618 Species: setosa","PC1: -0.1216414500 PC2: 0.0407801045 Species: setosa","PC1: -0.0935756077 PC2: -0.0402035131 Species: setosa","PC1: -0.1017978727 PC2: -0.0972242989 Species: setosa","PC1: -0.0985913660 PC2: 0.0605234036 Species: setosa","PC1: -0.1135957231 PC2: -0.0953687634 Species: setosa","PC1: -0.1140550703 PC2: 0.0328765710 Species: setosa","PC1: -0.1061987129 PC2: -0.0849443642 Species: setosa","PC1: -0.1049786435 PC2: -0.0007844781 Species: setosa"],"type":"scatter","mode":"markers","marker":{"autocolorscale":false,"color":"rgba(248,118,109,1)","opacity":1,"size":5.66929133858268,"symbol":"circle","line":{"width":1.88976377952756,"color":"rgba(248,118,109,1)"}},"hoveron":"points","name":"setosa","legendgroup":"setosa","showlegend":true,"xaxis":"x","yaxis":"y","hoverinfo":"text","frame":null},{"x":[0.0524828655915756,0.034836938939905,0.0591135080448586,0.0194102775902364,0.0512297677863806,0.0185149515659939,0.0355606182760603,-0.0232134060447727,0.0442001893997177,0.000544281511547665,-0.00524915168358511,0.0209922400253547,0.0267757837479311,0.0342760163896913,-0.00158883672248587,0.0416996393134783,0.0166840852769652,0.00756484727480711,0.0583568003305263,0.00785579213029778,0.0351391908849099,0.0226877323824678,0.058789534811191,0.0301459240226477,0.0334709586605651,0.0416456437745337,0.059853263145526,0.064707034848725,0.0316674752255627,-0.00191770135240911,0.00623037107664125,0.00111715859912558,0.0115055701709907,0.0505447783937658,0.0106691310819484,0.0204418543955474,0.0499557028204179,0.0497557964291515,0.00331480929251001,0.0135033144731088,0.013293748010589,0.0297511093164155,0.0160304772442665,-0.0172524473887005,0.0137466794829134,0.00435192494192666,0.0108471934632942,0.0274560044655482,-0.0213244229896518,0.0122292495071461],"y":[-0.0734545078587749,-0.0506124310758209,-0.052458039148444,0.149331402596547,0.0177403878051105,0.0504991313512371,-0.0657978771208903,0.157675105297591,-0.00274302010309918,0.0880135933771394,0.225909436479499,0.00538756143087928,0.150209853920523,0.0158510706519384,0.0373670861955944,-0.0433305185914666,0.0167096671263463,0.0674216251946899,0.138082188562873,0.110875525009661,-0.0337553881191666,0.0355214781229713,0.0794428427028322,0.0354421004299749,0.00539748885566302,-0.0213470382829205,0.00657587981469414,-0.0282005960084119,0.0192305327806631,0.090116031697377,0.13297749301723,0.133846016916422,0.0661585281924006,0.0539514851417542,0.024494712988038,-0.0719742903080059,-0.0444360178887053,0.117717268353686,0.0186836903227767,0.113149487841147,0.0953346158675208,-0.00212139902673828,0.0841309978710773,0.171873539744446,0.0728380747568537,0.0154227158930754,0.0327636370706075,0.0131825347173547,0.131404528955102,0.0509730821473312],"text":["PC1: 0.0524828656 PC2: -0.0734545079 Species: versicolor","PC1: 0.0348369389 PC2: -0.0506124311 Species: versicolor","PC1: 0.0591135080 PC2: -0.0524580391 Species: versicolor","PC1: 0.0194102776 PC2: 0.1493314026 Species: versicolor","PC1: 0.0512297678 PC2: 0.0177403878 Species: versicolor","PC1: 0.0185149516 PC2: 0.0504991314 Species: versicolor","PC1: 0.0355606183 PC2: -0.0657978771 Species: versicolor","PC1: -0.0232134060 PC2: 0.1576751053 Species: versicolor","PC1: 0.0442001894 PC2: -0.0027430201 Species: versicolor","PC1: 0.0005442815 PC2: 0.0880135934 Species: versicolor","PC1: -0.0052491517 PC2: 0.2259094365 Species: versicolor","PC1: 0.0209922400 PC2: 0.0053875614 Species: versicolor","PC1: 0.0267757837 PC2: 0.1502098539 Species: versicolor","PC1: 0.0342760164 PC2: 0.0158510707 Species: versicolor","PC1: -0.0015888367 PC2: 0.0373670862 Species: versicolor","PC1: 0.0416996393 PC2: -0.0433305186 Species: versicolor","PC1: 0.0166840853 PC2: 0.0167096671 Species: versicolor","PC1: 0.0075648473 PC2: 0.0674216252 Species: versicolor","PC1: 0.0583568003 PC2: 0.1380821886 Species: versicolor","PC1: 0.0078557921 PC2: 0.1108755250 Species: versicolor","PC1: 0.0351391909 PC2: -0.0337553881 Species: versicolor","PC1: 0.0226877324 PC2: 0.0355214781 Species: versicolor","PC1: 0.0587895348 PC2: 0.0794428427 Species: versicolor","PC1: 0.0301459240 PC2: 0.0354421004 Species: versicolor","PC1: 0.0334709587 PC2: 0.0053974889 Species: versicolor","PC1: 0.0416456438 PC2: -0.0213470383 Species: versicolor","PC1: 0.0598532631 PC2: 0.0065758798 Species: versicolor","PC1: 0.0647070348 PC2: -0.0282005960 Species: versicolor","PC1: 0.0316674752 PC2: 0.0192305328 Species: versicolor","PC1: -0.0019177014 PC2: 0.0901160317 Species: versicolor","PC1: 0.0062303711 PC2: 0.1329774930 Species: versicolor","PC1: 0.0011171586 PC2: 0.1338460169 Species: versicolor","PC1: 0.0115055702 PC2: 0.0661585282 Species: versicolor","PC1: 0.0505447784 PC2: 0.0539514851 Species: versicolor","PC1: 0.0106691311 PC2: 0.0244947130 Species: versicolor","PC1: 0.0204418544 PC2: -0.0719742903 Species: versicolor","PC1: 0.0499557028 PC2: -0.0444360179 Species: versicolor","PC1: 0.0497557964 PC2: 0.1177172684 Species: versicolor","PC1: 0.0033148093 PC2: 0.0186836903 Species: versicolor","PC1: 0.0135033145 PC2: 0.1131494878 Species: versicolor","PC1: 0.0132937480 PC2: 0.0953346159 Species: versicolor","PC1: 0.0297511093 PC2: -0.0021213990 Species: versicolor","PC1: 0.0160304772 PC2: 0.0841309979 Species: versicolor","PC1: -0.0172524474 PC2: 0.1718735397 Species: versicolor","PC1: 0.0137466795 PC2: 0.0728380748 Species: versicolor","PC1: 0.0043519249 PC2: 0.0154227159 Species: versicolor","PC1: 0.0108471935 PC2: 0.0327636371 Species: versicolor","PC1: 0.0274560045 PC2: 0.0131825347 Species: versicolor","PC1: -0.0213244230 PC2: 0.1314045290 Species: versicolor","PC1: 0.0122292495 PC2: 0.0509730821 Species: versicolor"],"type":"scatter","mode":"markers","marker":{"autocolorscale":false,"color":"rgba(0,186,56,1)","opacity":1,"size":5.66929133858268,"symbol":"circle","line":{"width":1.88976377952756,"color":"rgba(0,186,56,1)"}},"hoveron":"points","name":"versicolor","legendgroup":"versicolor","showlegend":true,"xaxis":"x","yaxis":"y","hoverinfo":"text","frame":null},{"x":[0.0878652326325491,0.0551551850871601,0.105046920722381,0.0686009480683305,0.0889724195854224,0.131084284812907,0.0174827273128329,0.109675697646398,0.0955876337220434,0.107643507349145,0.0649819039397037,0.0763428987016345,0.089738787371008,0.0600250325548711,0.0699106383426149,0.0757426387600303,0.0700909951900716,0.11557712191942,0.157703538614414,0.0601989737090692,0.0970657263064202,0.0465856852669593,0.138028363579451,0.0635079325824746,0.0810137166272321,0.0930934993290626,0.0559755484116904,0.0486325337117663,0.085187267612814,0.0887740374911589,0.11603559223585,0.109794225644694,0.0887290545756177,0.0530719411649235,0.0572792906977902,0.133318208262463,0.0750842620523338,0.0641300365339994,0.0440536310995459,0.0882213573952407,0.0959746151180589,0.0905906547767123,0.0551551850871601,0.0972011002383356,0.095180864478447,0.0891006076549713,0.0745281081771179,0.0724602925715432,0.0653921469625978,0.0457604305461734],"y":[-0.0740885430097093,0.0594864224029415,-0.0478372218236795,0.0039994906983625,-0.025113632739749,-0.0681292503710733,0.132912042130432,-0.0357551525032606,0.0605562530877328,-0.163512609219446,-0.0589660612016485,0.0358943094198193,-0.0356857022350479,0.0989293115880431,0.0376452840244006,-0.0575606222693542,-0.0217580248420059,-0.217617956222712,-0.00151347928852529,0.145274795929447,-0.0774971877196775,0.0486674500848119,-0.0352083100453741,0.0410108193469274,-0.0863030268353496,-0.0857800386084448,0.0269308725990966,-0.00547700692448142,0.0159478165447124,-0.0478610760546611,-0.0220697788585274,-0.223547466586997,0.0151977803445442,0.0249329951716847,0.0690575534938273,-0.0729294072071379,-0.0909555133911473,-0.0359564592888603,-0.00146599629461207,-0.0575506948445704,-0.0522527329814302,-0.0586953041478362,0.0594864224029415,-0.0738416401868786,-0.0893031716360222,-0.0329377886074678,0.0763242102031361,-0.0229026341452717,-0.0860759788620863,0.00207106353916324],"text":["PC1: 0.0878652326 PC2: -0.0740885430 Species: virginica","PC1: 0.0551551851 PC2: 0.0594864224 Species: virginica","PC1: 0.1050469207 PC2: -0.0478372218 Species: virginica","PC1: 0.0686009481 PC2: 0.0039994907 Species: virginica","PC1: 0.0889724196 PC2: -0.0251136327 Species: virginica","PC1: 0.1310842848 PC2: -0.0681292504 Species: virginica","PC1: 0.0174827273 PC2: 0.1329120421 Species: virginica","PC1: 0.1096756976 PC2: -0.0357551525 Species: virginica","PC1: 0.0955876337 PC2: 0.0605562531 Species: virginica","PC1: 0.1076435073 PC2: -0.1635126092 Species: virginica","PC1: 0.0649819039 PC2: -0.0589660612 Species: virginica","PC1: 0.0763428987 PC2: 0.0358943094 Species: virginica","PC1: 0.0897387874 PC2: -0.0356857022 Species: virginica","PC1: 0.0600250326 PC2: 0.0989293116 Species: virginica","PC1: 0.0699106383 PC2: 0.0376452840 Species: virginica","PC1: 0.0757426388 PC2: -0.0575606223 Species: virginica","PC1: 0.0700909952 PC2: -0.0217580248 Species: virginica","PC1: 0.1155771219 PC2: -0.2176179562 Species: virginica","PC1: 0.1577035386 PC2: -0.0015134793 Species: virginica","PC1: 0.0601989737 PC2: 0.1452747959 Species: virginica","PC1: 0.0970657263 PC2: -0.0774971877 Species: virginica","PC1: 0.0465856853 PC2: 0.0486674501 Species: virginica","PC1: 0.1380283636 PC2: -0.0352083100 Species: virginica","PC1: 0.0635079326 PC2: 0.0410108193 Species: virginica","PC1: 0.0810137166 PC2: -0.0863030268 Species: virginica","PC1: 0.0930934993 PC2: -0.0857800386 Species: virginica","PC1: 0.0559755484 PC2: 0.0269308726 Species: virginica","PC1: 0.0486325337 PC2: -0.0054770069 Species: virginica","PC1: 0.0851872676 PC2: 0.0159478165 Species: virginica","PC1: 0.0887740375 PC2: -0.0478610761 Species: virginica","PC1: 0.1160355922 PC2: -0.0220697789 Species: virginica","PC1: 0.1097942256 PC2: -0.2235474666 Species: virginica","PC1: 0.0887290546 PC2: 0.0151977803 Species: virginica","PC1: 0.0530719412 PC2: 0.0249329952 Species: virginica","PC1: 0.0572792907 PC2: 0.0690575535 Species: virginica","PC1: 0.1333182083 PC2: -0.0729294072 Species: virginica","PC1: 0.0750842621 PC2: -0.0909555134 Species: virginica","PC1: 0.0641300365 PC2: -0.0359564593 Species: virginica","PC1: 0.0440536311 PC2: -0.0014659963 Species: virginica","PC1: 0.0882213574 PC2: -0.0575506948 Species: virginica","PC1: 0.0959746151 PC2: -0.0522527330 Species: virginica","PC1: 0.0905906548 PC2: -0.0586953041 Species: virginica","PC1: 0.0551551851 PC2: 0.0594864224 Species: virginica","PC1: 0.0972011002 PC2: -0.0738416402 Species: virginica","PC1: 0.0951808645 PC2: -0.0893031716 Species: virginica","PC1: 0.0891006077 PC2: -0.0329377886 Species: virginica","PC1: 0.0745281082 PC2: 0.0763242102 Species: virginica","PC1: 0.0724602926 PC2: -0.0229026341 Species: virginica","PC1: 0.0653921470 PC2: -0.0860759789 Species: virginica","PC1: 0.0457604305 PC2: 0.0020710635 Species: virginica"],"type":"scatter","mode":"markers","marker":{"autocolorscale":false,"color":"rgba(97,156,255,1)","opacity":1,"size":5.66929133858268,"symbol":"circle","line":{"width":1.88976377952756,"color":"rgba(97,156,255,1)"}},"hoveron":"points","name":"virginica","legendgroup":"virginica","showlegend":true,"xaxis":"x","yaxis":"y","hoverinfo":"text","frame":null}],"layout":{"margin":{"t":23.3059360730594,"r":7.30593607305936,"b":37.2602739726027,"l":48.9497716894977},"plot_bgcolor":"rgba(235,235,235,1)","paper_bgcolor":"rgba(255,255,255,1)","font":{"color":"rgba(0,0,0,1)","family":"","size":14.6118721461187},"xaxis":{"domain":[0,1],"automargin":true,"type":"linear","autorange":false,"range":[-0.146647610440632,0.172196450474178],"tickmode":"array","ticktext":["-0.1","0.0","0.1"],"tickvals":[-0.1,0,0.1],"categoryorder":"array","categoryarray":["-0.1","0.0","0.1"],"nticks":null,"ticks":"outside","tickcolor":"rgba(51,51,51,1)","ticklen":3.65296803652968,"tickwidth":0.66417600664176,"showticklabels":true,"tickfont":{"color":"rgba(77,77,77,1)","family":"","size":11.689497716895},"tickangle":-0,"showline":false,"linecolor":null,"linewidth":0,"showgrid":true,"gridcolor":"rgba(255,255,255,1)","gridwidth":0.66417600664176,"zeroline":false,"anchor":"y","title":{"text":"PC1 (72.96%)","font":{"color":"rgba(0,0,0,1)","family":"","size":14.6118721461187}},"hoverformat":".2f"},"yaxis":{"domain":[0,1],"automargin":true,"type":"linear","autorange":false,"range":[-0.251379263239709,0.248637469799461],"tickmode":"array","ticktext":["-0.2","-0.1","0.0","0.1","0.2"],"tickvals":[-0.2,-0.1,0,0.1,0.2],"categoryorder":"array","categoryarray":["-0.2","-0.1","0.0","0.1","0.2"],"nticks":null,"ticks":"outside","tickcolor":"rgba(51,51,51,1)","ticklen":3.65296803652968,"tickwidth":0.66417600664176,"showticklabels":true,"tickfont":{"color":"rgba(77,77,77,1)","family":"","size":11.689497716895},"tickangle":-0,"showline":false,"linecolor":null,"linewidth":0,"showgrid":true,"gridcolor":"rgba(255,255,255,1)","gridwidth":0.66417600664176,"zeroline":false,"anchor":"x","title":{"text":"PC2 (22.85%)","font":{"color":"rgba(0,0,0,1)","family":"","size":14.6118721461187}},"hoverformat":".2f"},"shapes":[{"type":"rect","fillcolor":null,"line":{"color":null,"width":0,"linetype":[]},"yref":"paper","xref":"paper","x0":0,"x1":1,"y0":0,"y1":1}],"showlegend":true,"legend":{"bgcolor":"rgba(255,255,255,1)","bordercolor":"transparent","borderwidth":1.88976377952756,"font":{"color":"rgba(0,0,0,1)","family":"","size":11.689497716895},"title":{"text":"Species","font":{"color":"rgba(0,0,0,1)","family":"","size":14.6118721461187}}},"hovermode":"closest","barmode":"relative"},"config":{"doubleClick":"reset","modeBarButtonsToAdd":["hoverclosest","hovercompare"],"showSendToCloud":false},"source":"A","attrs":{"5de235f19ebb":{"x":{},"y":{},"colour":{},"type":"scatter"}},"cur_data":"5de235f19ebb","visdat":{"5de235f19ebb":["function (y) ","x"]},"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.2,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script>

---
# Latent variables (features)
* Linear
* Non-linear

---
# Non-linear

![:scale 70%](data:image/png;base64,#http://rasbt.github.io/mlxtend/user_guide/feature_extraction/RBFKernelPCA_files/linear_vs_nonlinear.png)

.footer[http://rasbt.github.io/mlxtend/user_guide/feature_extraction/RBFKernelPCA/]
---
## Non-linear

* Feature expansion: Lower dimension to higher dimension
* Gaussian Kernel (Radial basis function kernel)

---
![:scale 50%](data:image/png;base64,#https://gregorygundersen.com/image/kerneltrick/idea.png)

The "lifting trick". (a) A binary classification problem that is not linearly separable in `$\mathbb{R}^2$` 
(b) A lifting of the data into `$\mathbb{R}^3$` using a polynomial kernel, `$\varphi([x_1 \;\; x_2]) = [x_1^2 \;\; x_2^2 \;\; \sqrt{2} x_1 x_2]$`

Polynomial kernal `$(x_1 + x_2)^2 = x_1^2 + x_2^2 + 2x_1x_2$`

.footer[https://gregorygundersen.com/blog/2019/12/10/kernel-trick/]

---
## Kernel basis functions (Eigenfunctions)

![](data:image/png;base64,#img/Eigenfunctions.png)

.footer[Electronic Journal of Statistics. 10. 423-463. 10.1214/16-EJS1112.]

---
![:scale 70%](data:image/png;base64,#img/lec08_1.png)
.footer[https://www.cs.cornell.edu/courses/cs4786/2020sp/lectures/lec08.pdf]

???
Basis가 manifold 개념으로 
---
![:scale 70%](data:image/png;base64,#img/lec08_2.png)

---
![:scale 70%](data:image/png;base64,#img/lec08_3.png)

---
![:scale 70%](data:image/png;base64,#img/lec08_4.png)

---
![:scale 70%](data:image/png;base64,#img/lec08_5.png)

---
## t-Distributed Stochastic Neighbor Embedding (t-SNE)

.footer[https://lvdmaaten.github.io/publications/papers/JMLR_2008.pdf]

---
## Graph (Neighbor) Embedding

![](data:image/png;base64,#https://gearons.org/assets/img/mage_example.png)

.footer[https://gearons.org/blog/2016/MAGE/]

---
# Probablistic/Stochastic Neighbor Embedding (SNE)

---
## Similarity scores (probablistic/stochastic)

![:scale 60%](data:image/png;base64,#img/tsne1.png)

.footer[https://towardsdatascience.com/t-sne-machine-learning-algorithm-a-great-tool-for-dimensionality-reduction-in-python-ec01552f1a1e]

---
## Similarity matrix (High dimension)

![:scale 60%](data:image/png;base64,#img/tsne2.png)
.footer[https://towardsdatascience.com/t-sne-machine-learning-algorithm-a-great-tool-for-dimensionality-reduction-in-python-ec01552f1a1e]

---
## Similarity matrix (low dimension initial)

![:scale 60%](data:image/png;base64,#img/tsne3.png)
.footer[https://towardsdatascience.com/t-sne-machine-learning-algorithm-a-great-tool-for-dimensionality-reduction-in-python-ec01552f1a1e]

---
##  Minimize the Kullback–Leibler divergence (KL divergence) through gradient descent.

* Learning rate
* Iteration number

---
## Perplexity

![:scale 35%](data:image/png;base64,#img/tsne4.png)

---
## Perplexity

![](data:image/png;base64,#img/tsne1.gif)
.footer[https://towardsdatascience.com/t-sne-machine-learning-algorithm-a-great-tool-for-dimensionality-reduction-in-python-ec01552f1a1e]

---
![](data:image/png;base64,#https://distill.pub/2016/misread-tsne/)

.footer[https://distill.pub/2016/misread-tsne/]
---
class: center

### Simplicial complex
![:scale 50%](data:image/png;base64,#https://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Simplicial_complex_example.svg/1920px-Simplicial_complex_example.svg.png)

---
## Test data set of a noisy sine wave

![](data:image/png;base64,#https://umap-learn.readthedocs.io/en/latest/_images/how_umap_works_raw_data.png)

.footer[https://umap-learn.readthedocs.io/en/latest/how_umap_works.html]

---
# Graph with combined edge weights
![](data:image/png;base64,#https://umap-learn.readthedocs.io/en/latest/_images/how_umap_works_umap_graph.png)
---

## Minimize cross entropy

$$
\sum_{e\in E} w_h(e) \log\left(\frac{w_h(e)}{w_l(e)}\right) + (1 - w_h(e)) \log\left(\frac{1 - w_h(e)}{1 - w_l(e)}\right)
$$

---
## Parameters of UMAP

* Number of neighbors
* Minimal distance
* Learning rate
* Number of epoch

---
# Principal component analysis (PCA)

<div id="htmlwidget-5b28950959e20acec5b0" style="width:504px;height:504px;" class="plotly html-widget"></div>
<script type="application/json" data-for="htmlwidget-5b28950959e20acec5b0">{"x":{"data":[{"x":[-0.107878129475915,-0.0991256758781124,-0.112619018704969,-0.10953016321449,-0.113839088131987,-0.0988717739479014,-0.116420246442089,-0.106360699500147,-0.111209579807091,-0.104049518884768,-0.103191235785805,-0.110804228180452,-0.105674939938425,-0.125426601970102,-0.104736048615599,-0.107759601477619,-0.105157476006749,-0.104336342513111,-0.090437648059616,-0.11162536167409,-0.0911879400806897,-0.105130093152723,-0.132154698580868,-0.08663152892624,-0.106089951636316,-0.09297534775118,-0.0977057000598279,-0.103299226863694,-0.101917170819842,-0.107904742160833,-0.101943783504761,-0.0872472171845061,-0.124561903177881,-0.116522591995828,-0.100507731921964,-0.105168012927155,-0.0974196466005929,-0.120388352192299,-0.115734486880366,-0.103353222402639,-0.108915245125331,-0.0885108519101075,-0.121641449997494,-0.0935756076927842,-0.101797872652439,-0.0985913660128171,-0.113595723122182,-0.114055070287765,-0.106198712883314,-0.104978643456295],"y":[-0.0408589158360105,0.057380916914182,0.0291025357194968,0.0508490406299959,-0.0550573502828649,-0.126755849636756,-0.00405538000424925,-0.018993923226488,0.0949344889461115,0.0399215080376265,-0.088836887082972,-0.0113273650638198,0.062023476045196,0.0818415537964958,-0.158324387842385,-0.228651229919747,-0.126281898840662,-0.0416089520361787,-0.119592424851425,-0.0960003118683027,-0.0348009903479184,-0.0786593906907707,-0.0390133077633874,-0.00728257277818513,-0.0116828281608904,0.0532514185852892,-0.0206124833258478,-0.0448699264658799,-0.0266604813891562,0.0287470726224263,0.0429455070692807,-0.0360640873502077,-0.152665634531913,-0.183065709203295,0.0391714718374583,0.0175434546259828,-0.0563105198603704,-0.0504147911518509,0.0769620192674348,-0.0228864461573339,-0.0375979414063093,0.198956161780322,0.0407801045120344,-0.0402035131038843,-0.0972242988645649,0.0605234036448598,-0.0953687633671581,0.0328765709513192,-0.0849443641521261,-0.000784478149764319],"text":["PC1: -0.1078781295 PC2: -0.0408589158 Species: setosa","PC1: -0.0991256759 PC2: 0.0573809169 Species: setosa","PC1: -0.1126190187 PC2: 0.0291025357 Species: setosa","PC1: -0.1095301632 PC2: 0.0508490406 Species: setosa","PC1: -0.1138390881 PC2: -0.0550573503 Species: setosa","PC1: -0.0988717739 PC2: -0.1267558496 Species: setosa","PC1: -0.1164202464 PC2: -0.0040553800 Species: setosa","PC1: -0.1063606995 PC2: -0.0189939232 Species: setosa","PC1: -0.1112095798 PC2: 0.0949344889 Species: setosa","PC1: -0.1040495189 PC2: 0.0399215080 Species: setosa","PC1: -0.1031912358 PC2: -0.0888368871 Species: setosa","PC1: -0.1108042282 PC2: -0.0113273651 Species: setosa","PC1: -0.1056749399 PC2: 0.0620234760 Species: setosa","PC1: -0.1254266020 PC2: 0.0818415538 Species: setosa","PC1: -0.1047360486 PC2: -0.1583243878 Species: setosa","PC1: -0.1077596015 PC2: -0.2286512299 Species: setosa","PC1: -0.1051574760 PC2: -0.1262818988 Species: setosa","PC1: -0.1043363425 PC2: -0.0416089520 Species: setosa","PC1: -0.0904376481 PC2: -0.1195924249 Species: setosa","PC1: -0.1116253617 PC2: -0.0960003119 Species: setosa","PC1: -0.0911879401 PC2: -0.0348009903 Species: setosa","PC1: -0.1051300932 PC2: -0.0786593907 Species: setosa","PC1: -0.1321546986 PC2: -0.0390133078 Species: setosa","PC1: -0.0866315289 PC2: -0.0072825728 Species: setosa","PC1: -0.1060899516 PC2: -0.0116828282 Species: setosa","PC1: -0.0929753478 PC2: 0.0532514186 Species: setosa","PC1: -0.0977057001 PC2: -0.0206124833 Species: setosa","PC1: -0.1032992269 PC2: -0.0448699265 Species: setosa","PC1: -0.1019171708 PC2: -0.0266604814 Species: setosa","PC1: -0.1079047422 PC2: 0.0287470726 Species: setosa","PC1: -0.1019437835 PC2: 0.0429455071 Species: setosa","PC1: -0.0872472172 PC2: -0.0360640874 Species: setosa","PC1: -0.1245619032 PC2: -0.1526656345 Species: setosa","PC1: -0.1165225920 PC2: -0.1830657092 Species: setosa","PC1: -0.1005077319 PC2: 0.0391714718 Species: setosa","PC1: -0.1051680129 PC2: 0.0175434546 Species: setosa","PC1: -0.0974196466 PC2: -0.0563105199 Species: setosa","PC1: -0.1203883522 PC2: -0.0504147912 Species: setosa","PC1: -0.1157344869 PC2: 0.0769620193 Species: setosa","PC1: -0.1033532224 PC2: -0.0228864462 Species: setosa","PC1: -0.1089152451 PC2: -0.0375979414 Species: setosa","PC1: -0.0885108519 PC2: 0.1989561618 Species: setosa","PC1: -0.1216414500 PC2: 0.0407801045 Species: setosa","PC1: -0.0935756077 PC2: -0.0402035131 Species: setosa","PC1: -0.1017978727 PC2: -0.0972242989 Species: setosa","PC1: -0.0985913660 PC2: 0.0605234036 Species: setosa","PC1: -0.1135957231 PC2: -0.0953687634 Species: setosa","PC1: -0.1140550703 PC2: 0.0328765710 Species: setosa","PC1: -0.1061987129 PC2: -0.0849443642 Species: setosa","PC1: -0.1049786435 PC2: -0.0007844781 Species: setosa"],"type":"scatter","mode":"markers","marker":{"autocolorscale":false,"color":"rgba(248,118,109,1)","opacity":1,"size":5.66929133858268,"symbol":"circle","line":{"width":1.88976377952756,"color":"rgba(248,118,109,1)"}},"hoveron":"points","name":"setosa","legendgroup":"setosa","showlegend":true,"xaxis":"x","yaxis":"y","hoverinfo":"text","frame":null},{"x":[0.0524828655915756,0.034836938939905,0.0591135080448586,0.0194102775902364,0.0512297677863806,0.0185149515659939,0.0355606182760603,-0.0232134060447727,0.0442001893997177,0.000544281511547665,-0.00524915168358511,0.0209922400253547,0.0267757837479311,0.0342760163896913,-0.00158883672248587,0.0416996393134783,0.0166840852769652,0.00756484727480711,0.0583568003305263,0.00785579213029778,0.0351391908849099,0.0226877323824678,0.058789534811191,0.0301459240226477,0.0334709586605651,0.0416456437745337,0.059853263145526,0.064707034848725,0.0316674752255627,-0.00191770135240911,0.00623037107664125,0.00111715859912558,0.0115055701709907,0.0505447783937658,0.0106691310819484,0.0204418543955474,0.0499557028204179,0.0497557964291515,0.00331480929251001,0.0135033144731088,0.013293748010589,0.0297511093164155,0.0160304772442665,-0.0172524473887005,0.0137466794829134,0.00435192494192666,0.0108471934632942,0.0274560044655482,-0.0213244229896518,0.0122292495071461],"y":[-0.0734545078587749,-0.0506124310758209,-0.052458039148444,0.149331402596547,0.0177403878051105,0.0504991313512371,-0.0657978771208903,0.157675105297591,-0.00274302010309918,0.0880135933771394,0.225909436479499,0.00538756143087928,0.150209853920523,0.0158510706519384,0.0373670861955944,-0.0433305185914666,0.0167096671263463,0.0674216251946899,0.138082188562873,0.110875525009661,-0.0337553881191666,0.0355214781229713,0.0794428427028322,0.0354421004299749,0.00539748885566302,-0.0213470382829205,0.00657587981469414,-0.0282005960084119,0.0192305327806631,0.090116031697377,0.13297749301723,0.133846016916422,0.0661585281924006,0.0539514851417542,0.024494712988038,-0.0719742903080059,-0.0444360178887053,0.117717268353686,0.0186836903227767,0.113149487841147,0.0953346158675208,-0.00212139902673828,0.0841309978710773,0.171873539744446,0.0728380747568537,0.0154227158930754,0.0327636370706075,0.0131825347173547,0.131404528955102,0.0509730821473312],"text":["PC1: 0.0524828656 PC2: -0.0734545079 Species: versicolor","PC1: 0.0348369389 PC2: -0.0506124311 Species: versicolor","PC1: 0.0591135080 PC2: -0.0524580391 Species: versicolor","PC1: 0.0194102776 PC2: 0.1493314026 Species: versicolor","PC1: 0.0512297678 PC2: 0.0177403878 Species: versicolor","PC1: 0.0185149516 PC2: 0.0504991314 Species: versicolor","PC1: 0.0355606183 PC2: -0.0657978771 Species: versicolor","PC1: -0.0232134060 PC2: 0.1576751053 Species: versicolor","PC1: 0.0442001894 PC2: -0.0027430201 Species: versicolor","PC1: 0.0005442815 PC2: 0.0880135934 Species: versicolor","PC1: -0.0052491517 PC2: 0.2259094365 Species: versicolor","PC1: 0.0209922400 PC2: 0.0053875614 Species: versicolor","PC1: 0.0267757837 PC2: 0.1502098539 Species: versicolor","PC1: 0.0342760164 PC2: 0.0158510707 Species: versicolor","PC1: -0.0015888367 PC2: 0.0373670862 Species: versicolor","PC1: 0.0416996393 PC2: -0.0433305186 Species: versicolor","PC1: 0.0166840853 PC2: 0.0167096671 Species: versicolor","PC1: 0.0075648473 PC2: 0.0674216252 Species: versicolor","PC1: 0.0583568003 PC2: 0.1380821886 Species: versicolor","PC1: 0.0078557921 PC2: 0.1108755250 Species: versicolor","PC1: 0.0351391909 PC2: -0.0337553881 Species: versicolor","PC1: 0.0226877324 PC2: 0.0355214781 Species: versicolor","PC1: 0.0587895348 PC2: 0.0794428427 Species: versicolor","PC1: 0.0301459240 PC2: 0.0354421004 Species: versicolor","PC1: 0.0334709587 PC2: 0.0053974889 Species: versicolor","PC1: 0.0416456438 PC2: -0.0213470383 Species: versicolor","PC1: 0.0598532631 PC2: 0.0065758798 Species: versicolor","PC1: 0.0647070348 PC2: -0.0282005960 Species: versicolor","PC1: 0.0316674752 PC2: 0.0192305328 Species: versicolor","PC1: -0.0019177014 PC2: 0.0901160317 Species: versicolor","PC1: 0.0062303711 PC2: 0.1329774930 Species: versicolor","PC1: 0.0011171586 PC2: 0.1338460169 Species: versicolor","PC1: 0.0115055702 PC2: 0.0661585282 Species: versicolor","PC1: 0.0505447784 PC2: 0.0539514851 Species: versicolor","PC1: 0.0106691311 PC2: 0.0244947130 Species: versicolor","PC1: 0.0204418544 PC2: -0.0719742903 Species: versicolor","PC1: 0.0499557028 PC2: -0.0444360179 Species: versicolor","PC1: 0.0497557964 PC2: 0.1177172684 Species: versicolor","PC1: 0.0033148093 PC2: 0.0186836903 Species: versicolor","PC1: 0.0135033145 PC2: 0.1131494878 Species: versicolor","PC1: 0.0132937480 PC2: 0.0953346159 Species: versicolor","PC1: 0.0297511093 PC2: -0.0021213990 Species: versicolor","PC1: 0.0160304772 PC2: 0.0841309979 Species: versicolor","PC1: -0.0172524474 PC2: 0.1718735397 Species: versicolor","PC1: 0.0137466795 PC2: 0.0728380748 Species: versicolor","PC1: 0.0043519249 PC2: 0.0154227159 Species: versicolor","PC1: 0.0108471935 PC2: 0.0327636371 Species: versicolor","PC1: 0.0274560045 PC2: 0.0131825347 Species: versicolor","PC1: -0.0213244230 PC2: 0.1314045290 Species: versicolor","PC1: 0.0122292495 PC2: 0.0509730821 Species: versicolor"],"type":"scatter","mode":"markers","marker":{"autocolorscale":false,"color":"rgba(0,186,56,1)","opacity":1,"size":5.66929133858268,"symbol":"circle","line":{"width":1.88976377952756,"color":"rgba(0,186,56,1)"}},"hoveron":"points","name":"versicolor","legendgroup":"versicolor","showlegend":true,"xaxis":"x","yaxis":"y","hoverinfo":"text","frame":null},{"x":[0.0878652326325491,0.0551551850871601,0.105046920722381,0.0686009480683305,0.0889724195854224,0.131084284812907,0.0174827273128329,0.109675697646398,0.0955876337220434,0.107643507349145,0.0649819039397037,0.0763428987016345,0.089738787371008,0.0600250325548711,0.0699106383426149,0.0757426387600303,0.0700909951900716,0.11557712191942,0.157703538614414,0.0601989737090692,0.0970657263064202,0.0465856852669593,0.138028363579451,0.0635079325824746,0.0810137166272321,0.0930934993290626,0.0559755484116904,0.0486325337117663,0.085187267612814,0.0887740374911589,0.11603559223585,0.109794225644694,0.0887290545756177,0.0530719411649235,0.0572792906977902,0.133318208262463,0.0750842620523338,0.0641300365339994,0.0440536310995459,0.0882213573952407,0.0959746151180589,0.0905906547767123,0.0551551850871601,0.0972011002383356,0.095180864478447,0.0891006076549713,0.0745281081771179,0.0724602925715432,0.0653921469625978,0.0457604305461734],"y":[-0.0740885430097093,0.0594864224029415,-0.0478372218236795,0.0039994906983625,-0.025113632739749,-0.0681292503710733,0.132912042130432,-0.0357551525032606,0.0605562530877328,-0.163512609219446,-0.0589660612016485,0.0358943094198193,-0.0356857022350479,0.0989293115880431,0.0376452840244006,-0.0575606222693542,-0.0217580248420059,-0.217617956222712,-0.00151347928852529,0.145274795929447,-0.0774971877196775,0.0486674500848119,-0.0352083100453741,0.0410108193469274,-0.0863030268353496,-0.0857800386084448,0.0269308725990966,-0.00547700692448142,0.0159478165447124,-0.0478610760546611,-0.0220697788585274,-0.223547466586997,0.0151977803445442,0.0249329951716847,0.0690575534938273,-0.0729294072071379,-0.0909555133911473,-0.0359564592888603,-0.00146599629461207,-0.0575506948445704,-0.0522527329814302,-0.0586953041478362,0.0594864224029415,-0.0738416401868786,-0.0893031716360222,-0.0329377886074678,0.0763242102031361,-0.0229026341452717,-0.0860759788620863,0.00207106353916324],"text":["PC1: 0.0878652326 PC2: -0.0740885430 Species: virginica","PC1: 0.0551551851 PC2: 0.0594864224 Species: virginica","PC1: 0.1050469207 PC2: -0.0478372218 Species: virginica","PC1: 0.0686009481 PC2: 0.0039994907 Species: virginica","PC1: 0.0889724196 PC2: -0.0251136327 Species: virginica","PC1: 0.1310842848 PC2: -0.0681292504 Species: virginica","PC1: 0.0174827273 PC2: 0.1329120421 Species: virginica","PC1: 0.1096756976 PC2: -0.0357551525 Species: virginica","PC1: 0.0955876337 PC2: 0.0605562531 Species: virginica","PC1: 0.1076435073 PC2: -0.1635126092 Species: virginica","PC1: 0.0649819039 PC2: -0.0589660612 Species: virginica","PC1: 0.0763428987 PC2: 0.0358943094 Species: virginica","PC1: 0.0897387874 PC2: -0.0356857022 Species: virginica","PC1: 0.0600250326 PC2: 0.0989293116 Species: virginica","PC1: 0.0699106383 PC2: 0.0376452840 Species: virginica","PC1: 0.0757426388 PC2: -0.0575606223 Species: virginica","PC1: 0.0700909952 PC2: -0.0217580248 Species: virginica","PC1: 0.1155771219 PC2: -0.2176179562 Species: virginica","PC1: 0.1577035386 PC2: -0.0015134793 Species: virginica","PC1: 0.0601989737 PC2: 0.1452747959 Species: virginica","PC1: 0.0970657263 PC2: -0.0774971877 Species: virginica","PC1: 0.0465856853 PC2: 0.0486674501 Species: virginica","PC1: 0.1380283636 PC2: -0.0352083100 Species: virginica","PC1: 0.0635079326 PC2: 0.0410108193 Species: virginica","PC1: 0.0810137166 PC2: -0.0863030268 Species: virginica","PC1: 0.0930934993 PC2: -0.0857800386 Species: virginica","PC1: 0.0559755484 PC2: 0.0269308726 Species: virginica","PC1: 0.0486325337 PC2: -0.0054770069 Species: virginica","PC1: 0.0851872676 PC2: 0.0159478165 Species: virginica","PC1: 0.0887740375 PC2: -0.0478610761 Species: virginica","PC1: 0.1160355922 PC2: -0.0220697789 Species: virginica","PC1: 0.1097942256 PC2: -0.2235474666 Species: virginica","PC1: 0.0887290546 PC2: 0.0151977803 Species: virginica","PC1: 0.0530719412 PC2: 0.0249329952 Species: virginica","PC1: 0.0572792907 PC2: 0.0690575535 Species: virginica","PC1: 0.1333182083 PC2: -0.0729294072 Species: virginica","PC1: 0.0750842621 PC2: -0.0909555134 Species: virginica","PC1: 0.0641300365 PC2: -0.0359564593 Species: virginica","PC1: 0.0440536311 PC2: -0.0014659963 Species: virginica","PC1: 0.0882213574 PC2: -0.0575506948 Species: virginica","PC1: 0.0959746151 PC2: -0.0522527330 Species: virginica","PC1: 0.0905906548 PC2: -0.0586953041 Species: virginica","PC1: 0.0551551851 PC2: 0.0594864224 Species: virginica","PC1: 0.0972011002 PC2: -0.0738416402 Species: virginica","PC1: 0.0951808645 PC2: -0.0893031716 Species: virginica","PC1: 0.0891006077 PC2: -0.0329377886 Species: virginica","PC1: 0.0745281082 PC2: 0.0763242102 Species: virginica","PC1: 0.0724602926 PC2: -0.0229026341 Species: virginica","PC1: 0.0653921470 PC2: -0.0860759789 Species: virginica","PC1: 0.0457604305 PC2: 0.0020710635 Species: virginica"],"type":"scatter","mode":"markers","marker":{"autocolorscale":false,"color":"rgba(97,156,255,1)","opacity":1,"size":5.66929133858268,"symbol":"circle","line":{"width":1.88976377952756,"color":"rgba(97,156,255,1)"}},"hoveron":"points","name":"virginica","legendgroup":"virginica","showlegend":true,"xaxis":"x","yaxis":"y","hoverinfo":"text","frame":null}],"layout":{"margin":{"t":23.3059360730594,"r":7.30593607305936,"b":37.2602739726027,"l":48.9497716894977},"plot_bgcolor":"rgba(235,235,235,1)","paper_bgcolor":"rgba(255,255,255,1)","font":{"color":"rgba(0,0,0,1)","family":"","size":14.6118721461187},"xaxis":{"domain":[0,1],"automargin":true,"type":"linear","autorange":false,"range":[-0.146647610440632,0.172196450474178],"tickmode":"array","ticktext":["-0.1","0.0","0.1"],"tickvals":[-0.1,0,0.1],"categoryorder":"array","categoryarray":["-0.1","0.0","0.1"],"nticks":null,"ticks":"outside","tickcolor":"rgba(51,51,51,1)","ticklen":3.65296803652968,"tickwidth":0.66417600664176,"showticklabels":true,"tickfont":{"color":"rgba(77,77,77,1)","family":"","size":11.689497716895},"tickangle":-0,"showline":false,"linecolor":null,"linewidth":0,"showgrid":true,"gridcolor":"rgba(255,255,255,1)","gridwidth":0.66417600664176,"zeroline":false,"anchor":"y","title":{"text":"PC1 (72.96%)","font":{"color":"rgba(0,0,0,1)","family":"","size":14.6118721461187}},"hoverformat":".2f"},"yaxis":{"domain":[0,1],"automargin":true,"type":"linear","autorange":false,"range":[-0.251379263239709,0.248637469799461],"tickmode":"array","ticktext":["-0.2","-0.1","0.0","0.1","0.2"],"tickvals":[-0.2,-0.1,0,0.1,0.2],"categoryorder":"array","categoryarray":["-0.2","-0.1","0.0","0.1","0.2"],"nticks":null,"ticks":"outside","tickcolor":"rgba(51,51,51,1)","ticklen":3.65296803652968,"tickwidth":0.66417600664176,"showticklabels":true,"tickfont":{"color":"rgba(77,77,77,1)","family":"","size":11.689497716895},"tickangle":-0,"showline":false,"linecolor":null,"linewidth":0,"showgrid":true,"gridcolor":"rgba(255,255,255,1)","gridwidth":0.66417600664176,"zeroline":false,"anchor":"x","title":{"text":"PC2 (22.85%)","font":{"color":"rgba(0,0,0,1)","family":"","size":14.6118721461187}},"hoverformat":".2f"},"shapes":[{"type":"rect","fillcolor":null,"line":{"color":null,"width":0,"linetype":[]},"yref":"paper","xref":"paper","x0":0,"x1":1,"y0":0,"y1":1}],"showlegend":true,"legend":{"bgcolor":"rgba(255,255,255,1)","bordercolor":"transparent","borderwidth":1.88976377952756,"font":{"color":"rgba(0,0,0,1)","family":"","size":11.689497716895},"title":{"text":"Species","font":{"color":"rgba(0,0,0,1)","family":"","size":14.6118721461187}}},"hovermode":"closest","barmode":"relative"},"config":{"doubleClick":"reset","modeBarButtonsToAdd":["hoverclosest","hovercompare"],"showSendToCloud":false},"source":"A","attrs":{"5de25e099765":{"x":{},"y":{},"colour":{},"type":"scatter"}},"cur_data":"5de25e099765","visdat":{"5de25e099765":["function (y) ","x"]},"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.2,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script>
---
## t-Distributed Stochastic Neighbor Embedding (t-SNE)
<div id="htmlwidget-5372c4c1f4c48bc2d971" style="width:504px;height:504px;" class="plotly html-widget"></div>
<script type="application/json" data-for="htmlwidget-5372c4c1f4c48bc2d971">{"x":{"visdat":{"5dfa675428c2":["function () ","plotlyVisDat"]},"cur_data":"5dfa675428c2","attrs":{"5dfa675428c2":{"x":{},"y":{},"mode":"markers","split":{},"alpha_stroke":1,"sizes":[10,100],"spans":[1,20],"type":"scatter"}},"layout":{"margin":{"b":40,"l":60,"t":25,"r":10},"plot_bgcolor":"#e5ecf6","xaxis":{"domain":[0,1],"automargin":true,"title":"X1"},"yaxis":{"domain":[0,1],"automargin":true,"title":"X2"},"hovermode":"closest","showlegend":true},"source":"A","config":{"modeBarButtonsToAdd":["hoverclosest","hovercompare"],"showSendToCloud":false},"data":[{"x":[16.8501799015544,12.2419707431669,12.7389042452187,10.985510463194,16.768915148956,20.0327597804375,13.3327111055609,15.5019859352189,9.36038293192939,12.9193030266519,19.5785533163529,14.1410636200185,11.6697102259401,9.68325586746443,20.4422418925818,20.5249555076342,19.6778869137288,16.9892805019045,20.3148266095808,18.2903039025338,18.1685100375904,17.8525468008885,14.7219931077464,15.1744447377824,13.4613480938305,12.3024254997649,15.4250848618059,17.5182061735687,17.1206683929765,11.816009576516,11.6045096890443,18.1569624409735,19.6623469948494,20.3728120126149,12.7224766895046,14.3458055364902,18.9723188223222,16.0466705918501,9.79079209946312,16.2609412034265,16.1106514577366,8.24306137783272,10.5822264615583,16.1492557210385,18.7019493616602,11.2965877290528,18.551748137556,11.858260802025,19.2331369307418,14.8701719851948],"y":[1.84930251229252,-0.98035224582759,-1.43057959018034,-1.67343652761118,2.27502690666543,3.54030919530399,-1.00443074677809,0.742817336970747,-1.85480473080563,-0.626330906978764,3.04607405965107,-0.171044223049305,-1.40904549082229,-1.76787673841143,5.33760768270938,5.55379902850898,4.18856139384122,1.30324686369754,4.16174528168001,3.00016383140553,1.22022188827954,2.42785853051804,1.53741180320254,-0.324524336557616,0.127938111490308,0.0171948616384038,0.0769984778607891,1.84450971060063,1.07611662586767,-0.55449264805197,-0.337188022898399,1.72368711034696,4.21009619844426,5.14136020868614,-0.340912298700595,0.722985539699719,3.2257025798927,1.88574480444901,-1.73383966540554,0.82278051353481,1.60850533110901,-2.1937102672093,-1.24357438035054,0.184482043880684,1.58675186451862,-0.951011288694009,2.54589489917869,-1.6492085358001,2.67154323373487,0.565333371818932],"mode":"markers","type":"scatter","name":"setosa","marker":{"color":"rgba(31,119,180,1)","line":{"color":"rgba(31,119,180,1)"}},"error_y":{"color":"rgba(31,119,180,1)"},"error_x":{"color":"rgba(31,119,180,1)"},"line":{"color":"rgba(31,119,180,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":[-13.9726621806717,-11.4698960211568,-13.5042330654504,0.812003791143721,-9.13644975380658,-2.64229121698425,-11.4114104874551,2.45755515380508,-10.8453183345102,0.696248144957637,2.39248263752558,-5.92144114352387,-0.825556516591418,-6.61672351023486,-2.58168889883295,-12.6373930936329,-3.3147194079663,-2.62750280527957,-2.17023589186079,-0.204501671534027,-7.29508197695227,-6.09429927292112,-5.1793278218552,-5.83566667273255,-9.6985984385342,-11.5287008185127,-11.590449639667,-12.2678308764778,-5.80830384131789,-1.62394660220301,0.720635135127647,0.781767206135576,-2.62109939060596,-4.29197356312827,-1.94014837073205,-10.1657994115687,-12.6037875452486,-3.43853963413825,-3.35010103569791,0.134427085400139,-0.390821932847921,-7.39653154553446,-1.93969595011147,2.45073174439119,-1.33052498616765,-4.07690598428019,-3.40645217802746,-7.64299745729801,1.83644867090675,-2.61134606730351],"y":[4.47149307254141,3.64295202787613,3.88946583131701,-4.20057288344161,1.58322753422763,-2.15867582978764,3.08947957328452,-4.16974478941733,2.78074448148681,-3.47336901875243,-4.95367668856865,0.934279103512032,-2.54358520413485,-0.14018494044832,0.555500684568635,4.23422470794149,-1.67954135978246,-0.60469627191171,-3.6986388064092,-2.53434954083106,-0.0952831168792512,1.27571537113877,-2.88075376055263,-0.55383625108999,2.51680200480912,3.73394940600104,2.37522950590247,2.16514827309596,-0.119588573811361,-0.106783751395041,-3.23157724049568,-2.90010619120645,-0.18070137050701,-3.68062767681828,-2.56250206778827,2.75774434241393,3.70355524670786,-2.13714841843161,0.00744559656421135,-3.42013272003091,-3.72762896059915,0.806331115554069,-1.22476557591246,-4.19190121323686,-2.30454677230443,0.237903852031468,-0.533765452105793,1.47637283892431,-3.12641824025328,-0.910129560640096],"mode":"markers","type":"scatter","name":"versicolor","marker":{"color":"rgba(255,127,14,1)","line":{"color":"rgba(255,127,14,1)"}},"error_y":{"color":"rgba(255,127,14,1)"},"error_x":{"color":"rgba(255,127,14,1)"},"line":{"color":"rgba(255,127,14,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":[-10.999578076126,-2.73843858758863,-15.1552172160797,-8.22686645571603,-10.7270885188366,-16.8727788453452,1.93637802536688,-15.8501807553325,-8.39706820316962,-16.7887100237356,-12.3266919192206,-7.51170765685842,-13.3221177071496,-1.15238431856086,-3.3276152377495,-11.9109895311544,-11.2514019709774,-17.1421728778153,-16.7241283069556,-1.17749920100024,-14.9191246146933,-2.14706915245854,-16.8450347130457,-6.62278216962039,-14.278964590398,-15.9030520881871,-6.58096657228682,-7.35295027196389,-8.36619049598412,-15.1843006822994,-15.7557331773719,-17.1621481844786,-8.37374764815468,-7.37877299198302,-3.83120061761282,-16.7740749849264,-12.6556839829033,-11.2320448562818,-6.55633043197337,-14.1702659629644,-13.4040900967413,-14.0250370183616,-2.73843858758863,-14.6972745931041,-14.4593814265351,-12.5726133947267,-4.89348122782081,-11.1822552305401,-11.8818609877039,-5.79884138635734],"y":[-1.84481450199189,-4.74297186775122,1.25870371472882,-2.6503353904018,-1.85432332924525,2.30442493294227,-5.17401341475556,1.16170079973933,-3.56352825130246,3.50897301409844,1.76397611427226,-2.79182093811264,0.62679773803892,-5.27620603537689,-4.70491664757465,0.106633057788801,-0.661729570877822,4.76068331542446,0.671296098647586,-5.23687292921264,1.89771165423024,-4.52040939822506,1.76274434413481,-1.95613079397846,1.63777702819867,2.88669373598337,-1.39549905288859,-0.814215049749454,-3.11333140603788,2.61441414401856,1.44558714842553,4.84310376832861,-3.21953210283181,-1.44152892284173,-4.67784801615363,3.17651391587278,-0.174059590806448,-0.534526431448862,-0.997528657234443,2.25098086496849,0.0887436493272334,2.78149222374265,-4.74297186775122,0.641487588212665,1.08934066178333,0.724219378889676,-3.56412546350662,0.0612589373011845,0.0886401285343607,-2.66839980727708],"mode":"markers","type":"scatter","name":"virginica","marker":{"color":"rgba(44,160,44,1)","line":{"color":"rgba(44,160,44,1)"}},"error_y":{"color":"rgba(44,160,44,1)"},"error_x":{"color":"rgba(44,160,44,1)"},"line":{"color":"rgba(44,160,44,1)"},"xaxis":"x","yaxis":"y","frame":null}],"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.2,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script>

---
## Uniform Manifold Approximation and Projection (UMAP)
<div id="htmlwidget-4d4c9e76eef7e11d196c" style="width:504px;height:504px;" class="plotly html-widget"></div>
<script type="application/json" data-for="htmlwidget-4d4c9e76eef7e11d196c">{"x":{"visdat":{"5dfa527a48de":["function () ","plotlyVisDat"]},"cur_data":"5dfa527a48de","attrs":{"5dfa527a48de":{"x":{},"y":{},"mode":"markers","color":{},"colors":["#636EFA","#EF553B","#00CC96"],"alpha_stroke":1,"sizes":[10,100],"spans":[1,20],"type":"scatter"}},"layout":{"margin":{"b":40,"l":60,"t":25,"r":10},"plot_bgcolor":"#e5ecf6","legend":{"title":{"text":"species"}},"xaxis":{"domain":[0,1],"automargin":true,"title":"0"},"yaxis":{"domain":[0,1],"automargin":true,"title":"1"},"hovermode":"closest","showlegend":true},"source":"A","config":{"modeBarButtonsToAdd":["hoverclosest","hovercompare"],"showSendToCloud":false},"data":[{"x":[14.1094414550233,14.4223654299417,14.0590117833588,14.102451872616,13.8905065087832,13.4972559339589,14.2694584658572,13.9762123213454,14.2308159001444,14.1723493555274,13.672172704001,13.7959295250419,14.211077996373,14.1161503128154,13.574082871022,13.7366084289184,13.5334727358218,13.9531112815975,13.6703489853726,13.5243734967834,14.1099434443916,13.3648386648515,14.4780007221307,13.9813757433587,13.7898935493089,14.3761826497945,13.7826429952326,14.0723413554278,14.2654944199357,13.9367871516638,14.0299296681174,14.166270575612,13.2641570687291,13.5842860508107,14.2710973838049,14.2800341982498,13.9954954122164,13.7675842904171,14.0844034850202,14.2248886358269,13.9782022480829,14.2685594447652,14.0311786946254,13.549320601842,13.2341082469795,14.2235269949246,13.3304969300943,13.9129069598983,13.6236120839166,14.1542314597312],"y":[1.26607130134418,2.72462237033804,3.15935262008049,3.30659687813219,1.41773654883342,0.406710611912152,3.24965711602071,1.71686366283467,3.62849973633835,2.9450421140796,0.435700001038976,2.20244548227176,3.06464845631136,3.70799899311738,0.0154878127033378,-0.0050234992162923,0.208666093936591,1.15967362078967,0.135221028554338,0.657726439899678,0.666625903156541,0.995819182337478,3.00189418455493,1.62625843749195,2.2832123863359,2.64934621050245,1.72746906067343,0.897642265205909,1.17120924842537,2.95656850004151,2.84981487913661,0.760996495504225,0.215608635077746,-0.0337583667685146,2.62113112711125,2.09827082303026,0.409856422394411,1.25731558623595,3.77534776630315,1.48949707077447,1.3725096176679,3.72224134682271,3.40930671682027,1.53661430106274,0.746439384178485,3.2219943116182,0.561418055710387,3.44630256025355,0.457194528875844,1.78407265422713],"mode":"markers","type":"scatter","name":"setosa","marker":{"color":"rgba(99,110,250,1)","line":{"color":"rgba(99,110,250,1)"}},"textfont":{"color":"rgba(99,110,250,1)"},"error_y":{"color":"rgba(99,110,250,1)"},"error_x":{"color":"rgba(99,110,250,1)"},"line":{"color":"rgba(99,110,250,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":[-8.92104108749944,-8.32259220070141,-9.03912793863039,-5.74687571116214,-8.41555243661672,-6.8794717263054,-8.36284297343028,-4.94453352756343,-8.60661881024847,-5.58686540121026,-4.91834800247389,-6.98386778746745,-5.80817757670336,-7.75270955479317,-5.58296282245835,-8.62084237338164,-6.95764473359637,-6.06937019867667,-7.61477329364515,-5.53449734241766,-8.58520141517385,-6.59575985435986,-7.85839193769464,-7.69512741289055,-8.02636819603129,-8.48908412736873,-8.84455575248302,-9.161670653089,-7.42556149343719,-5.3955346842377,-5.39953631387134,-5.28997974807751,-5.9941254358151,-8.34158817408855,-6.85239688635334,-8.19642244320631,-8.8915740255409,-7.51812952724357,-6.56550692708416,-5.62739680529151,-6.33128893799207,-7.87460609705015,-5.95407548968669,-4.83306546995591,-6.31779955802757,-6.66498888317268,-6.62520138499611,-7.55209708934216,-4.99021438161059,-6.32382989570942],"y":[-2.17411320584811,-2.30338781524474,-2.23642541683121,-3.49953087567009,-2.3790431640103,-3.52627665455701,-1.77463867595533,-3.33803542249732,-2.35808721401989,-3.54034690433297,-3.33965843874768,-3.37293044560223,-3.36427036407459,-2.2647820776741,-3.35503358279377,-2.61149452307316,-3.58590484298982,-3.6176654363741,-1.60310065280726,-3.43312769918333,-1.03310700098452,-3.15496214302927,-0.563206597011251,-2.32386247554311,-2.56771363649448,-2.49605850413401,-2.34554645610516,-1.84367372143713,-2.52196568737069,-3.1466479623215,-3.42343468303636,-3.26034118715343,-3.39861830874761,-0.356912169802162,-3.76214297442855,-1.76042542774419,-2.2589664043765,-1.96698657809919,-3.87544145584574,-3.61741238262617,-3.59337153341351,-2.28846214066385,-3.56978951086265,-3.27191410683165,-3.68945492281846,-3.78762018744977,-3.90714383748888,-2.59347048966166,-3.24641802280508,-3.7883168177555],"mode":"markers","type":"scatter","name":"versicolor","marker":{"color":"rgba(239,85,59,1)","line":{"color":"rgba(239,85,59,1)"}},"textfont":{"color":"rgba(239,85,59,1)"},"error_y":{"color":"rgba(239,85,59,1)"},"error_x":{"color":"rgba(239,85,59,1)"},"line":{"color":"rgba(239,85,59,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":[-6.492649465477,-8.85849436999165,-5.72945977614929,-7.42929363122723,-6.97997179649041,-5.14423685286739,-6.43817173476566,-5.20490282759006,-7.28211971237484,-5.63998015992432,-7.05796843838168,-7.54242522030379,-6.6657877649148,-8.99472461463404,-8.98872368152295,-6.95177460930055,-7.11159069087345,-5.0297527225358,-4.85304931856229,-7.96171842765041,-6.19405261205089,-9.08024282578279,-5.02724765551659,-8.16832583774833,-6.29954831178672,-5.49147952265297,-8.26261026492656,-8.5201114660746,-7.48264599880316,-5.45658028291126,-5.29691653868345,-5.01587577493042,-7.38940487145161,-8.06606318649196,-7.99157550225361,-5.14274286898943,-6.51287044944184,-7.3517069622381,-8.5289104273116,-6.50096877231978,-6.33498766418785,-6.74743979620352,-8.93353400678537,-6.06803070382063,-6.14918745493662,-6.74061737818622,-7.9111768795641,-7.08476905981592,-6.99667357896455,-8.65610152383237],"y":[1.98400146007112,-0.429787952867125,1.76032691095235,1.24276677444845,1.92575212566098,1.82410809303274,-3.45757627192922,1.67513608746367,1.32451062323267,2.13067520126187,1.73738590519195,1.05769998241251,1.71354133325289,-0.432794967599388,-0.412111945972905,2.1337483267712,1.15473355180238,1.76649643957531,1.82954740445107,-0.518813151659527,1.98576738016042,-0.484385128057418,1.45501785743853,-0.42537982896493,1.84747584739033,1.6991959659695,-0.766800244089985,-0.854455337142873,1.41867362045718,1.56399359786608,1.64067568641908,1.92995405355126,1.76347790326698,-0.372693329880442,0.231656759903622,1.66416763905122,2.1802789344634,1.50246022224411,-0.932664705879596,1.7756273592904,2.33621460831582,1.95874481927024,-0.428579506974185,2.1527506096454,2.1610591003334,1.96295016387108,-0.226590122931372,1.57257934815686,2.35333161923308,-0.664526767684013],"mode":"markers","type":"scatter","name":"virginica","marker":{"color":"rgba(0,204,150,1)","line":{"color":"rgba(0,204,150,1)"}},"textfont":{"color":"rgba(0,204,150,1)"},"error_y":{"color":"rgba(0,204,150,1)"},"error_x":{"color":"rgba(0,204,150,1)"},"line":{"color":"rgba(0,204,150,1)"},"xaxis":"x","yaxis":"y","frame":null}],"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.2,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script>

.footer[http://mlss2018.net.ar/slides/Pfau-1.pdf]