Plotting multivariate data

Transcription

1 Plotting multivariate data Statistics 7 ISU Multivariate data and graphics With multivariate data we want to understand the associations between multiple s, which might be considered to be understanding the shape of the data in high-dimensional space. Graphics are used to explore the data, and also to diagnose models. Typically start with plots of 1, then, then more s.

2 Count Count 3 1 One - categorical Eg Examining the number in each level of one of the label s, prior to doing classification or MANOVA Barchart Variable 3 Variable What do you think the counts are for each of the categories? Naomi Robbins () Creating More Effective Graphs Wiley Counts are,, 6, 8 PLEASE DON T USE 3D UNNECESSARILY One - real Histogram: bin the s, and represent counts as bars. Use different bin sizes. Density: Like a smoothed histogram, estimate the density, represent as a curve. Dotplot: 1D scatterplot, s represented by dots. Boxplot: Representation of the 5 number summary, min/max, Q1/Q3, median. Good for comparing distrbutions of multiple categories.

3 One - real 7 count What do you learn about tips? tip 35 count What else do you learn about tips? tip One real + one categorical tars How do the species differ in the tars1 s? Concinna Heikert. Heptapot. species tars Concinna Heikert. Heptapot. species

4 One real + one categorical LAve How does the type of music differ? Classical Type Rock LAve Do you learn anything different from the dotplot than the boxplot? Classical Type Rock Plotting Means 5 mean(tars1) mean(tars1) Means Concinna Heikert. Heptapot. species What s wrong with this plot? are points estimates, use points to represent them. What do we learn about differences between species? Concinna Heikert. Heptapot. species

5 Two s - real Scatterplot How is tip related to bill? How would you expect tip to be related to bill? tip total_bill Does aspect ratio matter? Which is the response and which is the explanatory? Two s - real Scatterplot How is aede1 related to tars1? aede tars1 Does aspect ratio matter? Which is the response and which is the explanatory? For scatterplots of multivariate data use a SQUARE aspect ratio!

6 Two real + one categorical Scatterplot 15 How is aede1 related to tars1, by species? aede species Concinna Heikert. Heptapot tars1 Two real + one categorical linoleic Scatterplot oleic Area Calabria Coastal Sardinia East Liguria Inland Sardinia North Apulia Sicily South Apulia Umbria West Liguria How is linoleic acid related to oleic acid, by area? How many colors is too many? Generally only use 3- colors in a plot, only 3- categories.

7 Two s - real D density How is aede1 related to tars1? aede aede tars tars1 Two real + one categorical Scatterplot + contour, facetted by species Concinna Heikert. Heptapot. 15 aede tars How is aede1 related to tars1, by species?

8 Two s - real D histogram, scatterplot + linear model 1 How is tip related to bill? 8 1 count tip tip total_bill total_bill Two real + one categorical Scatterplot + linear model, facetted by sex of the bill payer, and smoking status. 1 Female Male tip No Yes How is tip related to total bill, by sex and smoker? total_bill

9 BEYOND D... Scatterplot matrix Correlation matrix. How do correlations differ by type of chocolate?

10 y Calories Scatterplot matrix TotFat Chol Na Fiber Sugars x Calories TotFat Chol Na Fiber Sugars Red=milk Black=dark All s plotted pairwise, like correlation matrix. What do you learn? Scatterplot matrix V1 density V1 vs V V vs V1 V3 vs V1 V density V1 vs V3 V vs V3 V3 vs V V3 density GENERAL Pairwise dependence only. A cross-check on the correlation matrix, whether correlation is a good summary or NOT.

11 Parallel coordinates V V1 Orthogonal V V1 V1 V Parallel V1 V GENERAL Orthogonal axes become parallel. Can lay out many s on a page. Need to learn a new set of patterns of structure. Re-ordering of axes is important. Parallel coordinates Calories TotFat Chol Na Fiber Sugars Calories TotFat Fiber Chol Na Sugars Red=milk Black=dark Re-ordered so that s with high s for dark chocolate are first. What s the difference between milk and dark chocolate?

12 Tennis data 1 Australian Open tennis tournament statistics for women % 1st serves in, Aces, Double faults, Unforced errors, Win on 1st serve %, Win on nd serve %, Winners, Receiving points won, Break point conversions, Fastest serve speed, Return games won, Server points won (1 s) Scatterplot matrix 7 6 Pct.1st.S Corr: -.3 Aces Corr: Corr:.51 Corr: Corr:.888 Corr: -.1 Corr:.8 Weak associations Outliers 8 DF Corr:.75 UE 6 Corr:.17 Corr:.37 Can t plot many s 8 6 W.1st.S 6 8

13 Parallel coordinates 6 - Pct.1st.S Aces DF UE W.1st.S W.nd.S Winners RPW BPC FSS RGW SPW Plot a lot more s Some outliers Some weak associations can be seen, both positive and negative 6 - Pct.1st.S Aces DF UE W.1st.S W.nd.S Winners RPW BPC FSS RGW SPW Scale matters: mean/sd, /1, individual/ global Enables correlation to be seen better, and outliers

14 Pct.1st.S Aces DF UE W.1st.S W.nd.S Winners RPW BPC FSS RGW SPW Scale matters: mean/sd, /1, individual/ global Emphasizes the univariate distributions Pct.1st.S Aces DF UE W.1st.S W.nd.S Winners RPW BPC FSS RGW SPW Scale matters: mean/sd, /1, individual/ global

15 6 - Pct.1st.S Aces DF UE W.1st.S W.nd.S Winners RPW BPC FSS RGW SPW Order matters: place s that are highly correlated close to each other 6 - Pct.1st.S RPW BPC RGW Aces W.1st.S FSS DF UE W.nd.S Winners SPW Order matters: place s that are highly correlated close to each other Less line crossing, easier to digest positive correlation, and then negative correlation

16 Normal data - X1 X X3 X X5 X6 Nothing interesting! All a little moderate correlation. Modelling is going to be easy! Clustered data - X1 X X3 X X5 X6 See the criss-crossing, gaps between lines. Will need to extract the clusters before doing any other modeling, otherwise pretty regular data

17 (Less) Clustered data - X1 X X3 X X5 X6 Still see the criss-crossing, gaps between lines, but less prominent. Will need to deal with the multi-modality Outliers in the data X1 X X3 X X5 X6 Small group of observations that are outliers on X1-X3. Need to do something with these cases, remove with justification, or fix

18 Strong association - - X1 X X3 X X5 X6 Very flat lines indicate strong positive association between all s. Strong negative association - X1 X X3 X X5 X6 Crossed lines in first three s indicate X is strongly negatively correlated with other vars.

19 Strong negative association - fixed - X1 X X3 X X5 X6 X is multiplied by -1, then it is positively associated with other s Pct.1st.S Aces DF UE W.1st.S W.nd.S Winners RPW BPC FSS RGW SPW Tennis data again: Aces, DF, UE, W.1st.S, Winners, FSS reversed (multiplied by -1) Mostly now the outliers are visible, but not so much multivariate outliers, mostly only in one or few s

20 Another data set: olive oils - - palmitic palmitoleic stearic oleic linoleic linolenic arachidic eicosenoic Outliers, clustering, some positive and some negative associations, some discreteness Another data set: olive oils - - palmitic palmitoleic stearic oleic linoleic linolenic arachidic eicosenoic Outliers, clustering, some positive and some negative associations, some discreteness

21 Another data set: olive oils - - palmitic palmitoleic stearic oleic linoleic linolenic arachidic eicosenoic Outliers, clustering, some positive and some negative associations, some discreteness Another data set: olive oils - - palmitic palmitoleic stearic oleic linoleic linolenic arachidic eicosenoic Outliers, clustering, some positive and some negative associations, some discreteness

22 Another data set: olive oils - - palmitic palmitoleic stearic oleic linoleic linolenic arachidic eicosenoic Outliers, clustering, some positive and some negative associations, some discreteness Another data set: olive oils - - palmitic palmitoleic stearic oleic linoleic linolenic arachidic eicosenoic Outliers, clustering, some positive and some negative associations, some discreteness

23 Tours Red=milk Grey=dark Are the two clusters different? How? Are there any outliers? And points from one group that seem to belong to the other group? Tours A tour is a movie of low-dimensional projections of high-dimensional space. Let A =[a 1 a ]= a 11 a 1 be a -dimensional projection matrix, where a 1 and a are both of length 1, and orthogonal to each other, then for data matrix X n p XA = a 11 X 11 + a 1 X a p1 X 1p. a 11 X n1 + a 1 X n a p1 X np is a -dimensional projection of the data.. a p1 a p a 1 X 11 + a X a p X 1p. a 1 X n1 + a X n a p X np The s (call them coefficients) in the projection matrix are varied producing the views in the movie. n

24 Visual representation of coefficients (axes) Tours Projection coefficients and scaling A = scaling for each.58/1.15/1.15/39.75/39.185/15.363/15.31/1.63/1.793/8.116/8.11/68./68 Multiply the data matrix by this to get the particular view of the data. Types of Tours Grand tour: The coefficients are randomly varied, giving a random walk over the space of all projections. Guided tour: The coefficients are chosen to maximize some criterion of interestingness, eg clusters, outliers, separation of classes. Manual tour: The user controls the coefficients of one of the s. Good for exploring the importance of a single on the structure visible in the plot.

25 Clusters Tours Non-linear dependence Non-linear dependence + noise s Class differences, outliers, nonlinear dependence Tours Something hidden

26 Icons aede1 head tars tars1 aede aede3 1 Concinna Concinna Each case is plotted separately. Each is mapped a feature of the icon 3 Heptapot. 33 Heptapot. 67 Heikert. 68 Heikert. Icons Example: Monthly ozone for 6 years over central America. What do you see?

27 Multiple linked plots Interactive graphics (direct manipulation graphics) allow the user to brush plot elements and convey this selection to other plots. Te rminology Brushing (one-to-one) Brushing (categorical ) Brushing (one point to one line) Transient brushing Persistent painting Identification Scaling

28 This 1-1 Linking 18 RW species.sex Blue Female Blue Male Orange Female Orange Male OR This? 1 15 FL What do you learn differently? This 1-1 Linking 1 Area Calabria 1 Coastal Sardinia East Liguria linoleic 1 8 Inland Sardinia North Apulia Sicily South Apulia Umbria West Liguria OR This? oleic What do you learn differently?

29 This Categorical Variable Linking Longitudinal measurements on wages with workforce experience: 888 subjects. lnw OR This? exper Example Exploring dark chocolates that look more like milk chocolates.

30 One more DON t 57 Heikert. 73 Heikert. 7 Heikert. 61 Heikert. 7 Heikert. 65 Heikert. 7 Heikert. Heikert. 8 Heikert. 66 Heikert. 63 Heikert. 67 Heikert. 56 Heikert. 53 Heikert. 6 Heikert. 58 Heikert. 6 Heikert. 69 Heikert. 68 Heikert. 5 Heikert. 5 Heikert. 55 Heikert. 5 Heikert. 9 Heikert. 51 Heikert. 6 Heikert. 6 Heikert. 5 Heikert. 59 Heikert. 71 Heikert. 7 Heikert. Concinna 15 Concinna 8 Concinna 1 Concinna 1 Concinna 1 Concinna 7 Concinna 11 Concinna 16 Concinna 17 Concinna 18 Concinna Concinna 9 Concinna 13 Concinna 3 Concinna 19 Concinna 38 Heptapot. 33 Heptapot. 37 Heptapot. 35 Heptapot. Heptapot. Heptapot. 6 Heptapot. 3 Heptapot. 36 Heptapot. Heptapot. Heptapot. 1 Heptapot. 1 Concinna Concinna 5 Concinna 6 Concinna 1 Concinna 3 Heptapot. 8 Heptapot. 3 Heptapot. 9 Heptapot. 3 Heptapot. 3 Heptapot. 31 Heptapot. 39 Heptapot. 7 Heptapot. 5 Heptapot. Heatmaps: each cell of the data matrix is colored according to. Rows and columns sorted to get most similar together. How many clusters? Describe the clusters? aede head tars1 aede3 tars aede1 What s wrong with this plot? This work is licensed under the Creative Commons Attribution-Noncommercial 3. United States License. To view a copy of this license, visit creativecommons.org/licenses/by-nc/3./us/ or send a letter to Creative Commons, 171 Second Street, Suite 3, San Francisco, California, 915, USA.