DATA 609 - HW 6

Ex. 1

Use a data set such as PlantGrowth in R to calculate three different distance metrics and discuss the results.

Without thinking much about it I calculated the distance using both variables, independent and dependent, (you can see the results below), but then the numbers didn’t really make sense. For the Manhattan distance it is just doubling the difference in the independent variable, weight, from one point to another. For the euclidean distance it’s summing the square of the difference in weight with the square of the same difference in weight and then taking the square root, and the same for minkowski with p=3 except it’s the difference cubed plus the same difference cubed and then the cube root. This really makes no sense. So then I removed the categorical dependent variable and recalculated. All three distance measures are exactly the same and equal to the difference in weight in this case, which I think is as it should be…

Distance calculated using both the independent and dependent variables

p <- PlantGrowth
p

p_manhattan <- as.data.frame(as.matrix(dist(p, method = "manhattan")))
rownames(p_manhattan) <- paste0("   ", as.character(c(1:30)))
kbl(p_manhattan) %>%
  kable_minimal(font_size = 10) 

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
1	0.00	2.82	2.02	3.88	0.66	0.88	2.00	0.72	2.32	1.94	1.28	0.00	0.48	1.16	3.40	0.68	3.72	1.44	0.30	1.04	4.28	1.90	2.74	2.66	2.40	2.24	1.50	3.96	3.26	2.18
2	2.82	0.00	0.80	1.06	2.16	1.94	0.82	2.10	0.50	0.88	1.54	2.82	2.34	3.98	0.58	3.50	0.90	1.38	2.52	1.78	1.46	0.92	0.08	0.16	0.42	0.58	1.32	1.14	0.44	0.64
3	2.02	0.80	0.00	1.86	1.36	1.14	0.02	1.30	0.30	0.08	0.74	2.02	1.54	3.18	1.38	2.70	1.70	0.58	1.72	0.98	2.26	0.12	0.72	0.64	0.38	0.22	0.52	1.94	1.24	0.16
4	3.88	1.06	1.86	0.00	3.22	3.00	1.88	3.16	1.56	1.94	2.60	3.88	3.40	5.04	0.48	4.56	0.16	2.44	3.58	2.84	0.40	1.98	1.14	1.22	1.48	1.64	2.38	0.08	0.62	1.70
5	0.66	2.16	1.36	3.22	0.00	0.22	1.34	0.06	1.66	1.28	0.62	0.66	0.18	1.82	2.74	1.34	3.06	0.78	0.36	0.38	3.62	1.24	2.08	2.00	1.74	1.58	0.84	3.30	2.60	1.52
6	0.88	1.94	1.14	3.00	0.22	0.00	1.12	0.16	1.44	1.06	0.40	0.88	0.40	2.04	2.52	1.56	2.84	0.56	0.58	0.16	3.40	1.02	1.86	1.78	1.52	1.36	0.62	3.08	2.38	1.30
7	2.00	0.82	0.02	1.88	1.34	1.12	0.00	1.28	0.32	0.06	0.72	2.00	1.52	3.16	1.40	2.68	1.72	0.56	1.70	0.96	2.28	0.10	0.74	0.66	0.40	0.24	0.50	1.96	1.26	0.18
8	0.72	2.10	1.30	3.16	0.06	0.16	1.28	0.00	1.60	1.22	0.56	0.72	0.24	1.88	2.68	1.40	3.00	0.72	0.42	0.32	3.56	1.18	2.02	1.94	1.68	1.52	0.78	3.24	2.54	1.46
9	2.32	0.50	0.30	1.56	1.66	1.44	0.32	1.60	0.00	0.38	1.04	2.32	1.84	3.48	1.08	3.00	1.40	0.88	2.02	1.28	1.96	0.42	0.42	0.34	0.08	0.08	0.82	1.64	0.94	0.14
10	1.94	0.88	0.08	1.94	1.28	1.06	0.06	1.22	0.38	0.00	0.66	1.94	1.46	3.10	1.46	2.62	1.78	0.50	1.64	0.90	2.34	0.04	0.80	0.72	0.46	0.30	0.44	2.02	1.32	0.24
11	1.28	1.54	0.74	2.60	0.62	0.40	0.72	0.56	1.04	0.66	0.00	1.28	0.80	2.44	2.12	1.96	2.44	0.16	0.98	0.24	3.00	0.62	1.46	1.38	1.12	0.96	0.22	2.68	1.98	0.90
12	0.00	2.82	2.02	3.88	0.66	0.88	2.00	0.72	2.32	1.94	1.28	0.00	0.48	1.16	3.40	0.68	3.72	1.44	0.30	1.04	4.28	1.90	2.74	2.66	2.40	2.24	1.50	3.96	3.26	2.18
13	0.48	2.34	1.54	3.40	0.18	0.40	1.52	0.24	1.84	1.46	0.80	0.48	0.00	1.64	2.92	1.16	3.24	0.96	0.18	0.56	3.80	1.42	2.26	2.18	1.92	1.76	1.02	3.48	2.78	1.70
14	1.16	3.98	3.18	5.04	1.82	2.04	3.16	1.88	3.48	3.10	2.44	1.16	1.64	0.00	4.56	0.48	4.88	2.60	1.46	2.20	5.44	3.06	3.90	3.82	3.56	3.40	2.66	5.12	4.42	3.34
15	3.40	0.58	1.38	0.48	2.74	2.52	1.40	2.68	1.08	1.46	2.12	3.40	2.92	4.56	0.00	4.08	0.32	1.96	3.10	2.36	0.88	1.50	0.66	0.74	1.00	1.16	1.90	0.56	0.14	1.22
16	0.68	3.50	2.70	4.56	1.34	1.56	2.68	1.40	3.00	2.62	1.96	0.68	1.16	0.48	4.08	0.00	4.40	2.12	0.98	1.72	4.96	2.58	3.42	3.34	3.08	2.92	2.18	4.64	3.94	2.86
17	3.72	0.90	1.70	0.16	3.06	2.84	1.72	3.00	1.40	1.78	2.44	3.72	3.24	4.88	0.32	4.40	0.00	2.28	3.42	2.68	0.56	1.82	0.98	1.06	1.32	1.48	2.22	0.24	0.46	1.54
18	1.44	1.38	0.58	2.44	0.78	0.56	0.56	0.72	0.88	0.50	0.16	1.44	0.96	2.60	1.96	2.12	2.28	0.00	1.14	0.40	2.84	0.46	1.30	1.22	0.96	0.80	0.06	2.52	1.82	0.74
19	0.30	2.52	1.72	3.58	0.36	0.58	1.70	0.42	2.02	1.64	0.98	0.30	0.18	1.46	3.10	0.98	3.42	1.14	0.00	0.74	3.98	1.60	2.44	2.36	2.10	1.94	1.20	3.66	2.96	1.88
20	1.04	1.78	0.98	2.84	0.38	0.16	0.96	0.32	1.28	0.90	0.24	1.04	0.56	2.20	2.36	1.72	2.68	0.40	0.74	0.00	3.24	0.86	1.70	1.62	1.36	1.20	0.46	2.92	2.22	1.14
21	4.28	1.46	2.26	0.40	3.62	3.40	2.28	3.56	1.96	2.34	3.00	4.28	3.80	5.44	0.88	4.96	0.56	2.84	3.98	3.24	0.00	2.38	1.54	1.62	1.88	2.04	2.78	0.32	1.02	2.10
22	1.90	0.92	0.12	1.98	1.24	1.02	0.10	1.18	0.42	0.04	0.62	1.90	1.42	3.06	1.50	2.58	1.82	0.46	1.60	0.86	2.38	0.00	0.84	0.76	0.50	0.34	0.40	2.06	1.36	0.28
23	2.74	0.08	0.72	1.14	2.08	1.86	0.74	2.02	0.42	0.80	1.46	2.74	2.26	3.90	0.66	3.42	0.98	1.30	2.44	1.70	1.54	0.84	0.00	0.08	0.34	0.50	1.24	1.22	0.52	0.56
24	2.66	0.16	0.64	1.22	2.00	1.78	0.66	1.94	0.34	0.72	1.38	2.66	2.18	3.82	0.74	3.34	1.06	1.22	2.36	1.62	1.62	0.76	0.08	0.00	0.26	0.42	1.16	1.30	0.60	0.48
25	2.40	0.42	0.38	1.48	1.74	1.52	0.40	1.68	0.08	0.46	1.12	2.40	1.92	3.56	1.00	3.08	1.32	0.96	2.10	1.36	1.88	0.50	0.34	0.26	0.00	0.16	0.90	1.56	0.86	0.22
26	2.24	0.58	0.22	1.64	1.58	1.36	0.24	1.52	0.08	0.30	0.96	2.24	1.76	3.40	1.16	2.92	1.48	0.80	1.94	1.20	2.04	0.34	0.50	0.42	0.16	0.00	0.74	1.72	1.02	0.06
27	1.50	1.32	0.52	2.38	0.84	0.62	0.50	0.78	0.82	0.44	0.22	1.50	1.02	2.66	1.90	2.18	2.22	0.06	1.20	0.46	2.78	0.40	1.24	1.16	0.90	0.74	0.00	2.46	1.76	0.68
28	3.96	1.14	1.94	0.08	3.30	3.08	1.96	3.24	1.64	2.02	2.68	3.96	3.48	5.12	0.56	4.64	0.24	2.52	3.66	2.92	0.32	2.06	1.22	1.30	1.56	1.72	2.46	0.00	0.70	1.78
29	3.26	0.44	1.24	0.62	2.60	2.38	1.26	2.54	0.94	1.32	1.98	3.26	2.78	4.42	0.14	3.94	0.46	1.82	2.96	2.22	1.02	1.36	0.52	0.60	0.86	1.02	1.76	0.70	0.00	1.08
30	2.18	0.64	0.16	1.70	1.52	1.30	0.18	1.46	0.14	0.24	0.90	2.18	1.70	3.34	1.22	2.86	1.54	0.74	1.88	1.14	2.10	0.28	0.56	0.48	0.22	0.06	0.68	1.78	1.08	0.00

p_euclidean <- as.matrix(dist(p, method = "euclidean"))
rownames(p_euclidean) <- paste0("   ", as.character(c(1:30)))
kbl(p_euclidean) %>%
  kable_minimal(font_size = 10) 

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
1	0.0000	1.9940	1.4284	2.7436	0.4667	0.6223	1.4142	0.5091	1.6405	1.3718	0.9051	0.0000	0.3394	0.8202	2.4042	0.4808	2.6304	1.0182	0.2121	0.7354	3.0264	1.3435	1.9375	1.8809	1.6971	1.5839	1.0607	2.8001	2.3052	1.5415
2	1.9940	0.0000	0.5657	0.7495	1.5274	1.3718	0.5798	1.4849	0.3536	0.6223	1.0889	1.9940	1.6546	2.8143	0.4101	2.4749	0.6364	0.9758	1.7819	1.2587	1.0324	0.6505	0.0566	0.1131	0.2970	0.4101	0.9334	0.8061	0.3111	0.4525
3	1.4284	0.5657	0.0000	1.3152	0.9617	0.8061	0.0141	0.9192	0.2121	0.0566	0.5233	1.4284	1.0889	2.2486	0.9758	1.9092	1.2021	0.4101	1.2162	0.6930	1.5981	0.0849	0.5091	0.4525	0.2687	0.1556	0.3677	1.3718	0.8768	0.1131
4	2.7436	0.7495	1.3152	0.0000	2.2769	2.1213	1.3294	2.2345	1.1031	1.3718	1.8385	2.7436	2.4042	3.5638	0.3394	3.2244	0.1131	1.7253	2.5314	2.0082	0.2828	1.4001	0.8061	0.8627	1.0465	1.1597	1.6829	0.0566	0.4384	1.2021
5	0.4667	1.5274	0.9617	2.2769	0.0000	0.1556	0.9475	0.0424	1.1738	0.9051	0.4384	0.4667	0.1273	1.2869	1.9375	0.9475	2.1637	0.5515	0.2546	0.2687	2.5597	0.8768	1.4708	1.4142	1.2304	1.1172	0.5940	2.3335	1.8385	1.0748
6	0.6223	1.3718	0.8061	2.1213	0.1556	0.0000	0.7920	0.1131	1.0182	0.7495	0.2828	0.6223	0.2828	1.4425	1.7819	1.1031	2.0082	0.3960	0.4101	0.1131	2.4042	0.7212	1.3152	1.2587	1.0748	0.9617	0.4384	2.1779	1.6829	0.9192
7	1.4142	0.5798	0.0141	1.3294	0.9475	0.7920	0.0000	0.9051	0.2263	0.0424	0.5091	1.4142	1.0748	2.2345	0.9899	1.8950	1.2162	0.3960	1.2021	0.6788	1.6122	0.0707	0.5233	0.4667	0.2828	0.1697	0.3536	1.3859	0.8910	0.1273
8	0.5091	1.4849	0.9192	2.2345	0.0424	0.1131	0.9051	0.0000	1.1314	0.8627	0.3960	0.5091	0.1697	1.3294	1.8950	0.9899	2.1213	0.5091	0.2970	0.2263	2.5173	0.8344	1.4284	1.3718	1.1879	1.0748	0.5515	2.2910	1.7961	1.0324
9	1.6405	0.3536	0.2121	1.1031	1.1738	1.0182	0.2263	1.1314	0.0000	0.2687	0.7354	1.6405	1.3011	2.4607	0.7637	2.1213	0.9899	0.6223	1.4284	0.9051	1.3859	0.2970	0.2970	0.2404	0.0566	0.0566	0.5798	1.1597	0.6647	0.0990
10	1.3718	0.6223	0.0566	1.3718	0.9051	0.7495	0.0424	0.8627	0.2687	0.0000	0.4667	1.3718	1.0324	2.1920	1.0324	1.8526	1.2587	0.3536	1.1597	0.6364	1.6546	0.0283	0.5657	0.5091	0.3253	0.2121	0.3111	1.4284	0.9334	0.1697
11	0.9051	1.0889	0.5233	1.8385	0.4384	0.2828	0.5091	0.3960	0.7354	0.4667	0.0000	0.9051	0.5657	1.7253	1.4991	1.3859	1.7253	0.1131	0.6930	0.1697	2.1213	0.4384	1.0324	0.9758	0.7920	0.6788	0.1556	1.8950	1.4001	0.6364
12	0.0000	1.9940	1.4284	2.7436	0.4667	0.6223	1.4142	0.5091	1.6405	1.3718	0.9051	0.0000	0.3394	0.8202	2.4042	0.4808	2.6304	1.0182	0.2121	0.7354	3.0264	1.3435	1.9375	1.8809	1.6971	1.5839	1.0607	2.8001	2.3052	1.5415
13	0.3394	1.6546	1.0889	2.4042	0.1273	0.2828	1.0748	0.1697	1.3011	1.0324	0.5657	0.3394	0.0000	1.1597	2.0648	0.8202	2.2910	0.6788	0.1273	0.3960	2.6870	1.0041	1.5981	1.5415	1.3576	1.2445	0.7212	2.4607	1.9658	1.2021
14	0.8202	2.8143	2.2486	3.5638	1.2869	1.4425	2.2345	1.3294	2.4607	2.1920	1.7253	0.8202	1.1597	0.0000	3.2244	0.3394	3.4507	1.8385	1.0324	1.5556	3.8467	2.1637	2.7577	2.7011	2.5173	2.4042	1.8809	3.6204	3.1254	2.3617
15	2.4042	0.4101	0.9758	0.3394	1.9375	1.7819	0.9899	1.8950	0.7637	1.0324	1.4991	2.4042	2.0648	3.2244	0.0000	2.8850	0.2263	1.3859	2.1920	1.6688	0.6223	1.0607	0.4667	0.5233	0.7071	0.8202	1.3435	0.3960	0.0990	0.8627
16	0.4808	2.4749	1.9092	3.2244	0.9475	1.1031	1.8950	0.9899	2.1213	1.8526	1.3859	0.4808	0.8202	0.3394	2.8850	0.0000	3.1113	1.4991	0.6930	1.2162	3.5072	1.8243	2.4183	2.3617	2.1779	2.0648	1.5415	3.2810	2.7860	2.0223
17	2.6304	0.6364	1.2021	0.1131	2.1637	2.0082	1.2162	2.1213	0.9899	1.2587	1.7253	2.6304	2.2910	3.4507	0.2263	3.1113	0.0000	1.6122	2.4183	1.8950	0.3960	1.2869	0.6930	0.7495	0.9334	1.0465	1.5698	0.1697	0.3253	1.0889
18	1.0182	0.9758	0.4101	1.7253	0.5515	0.3960	0.3960	0.5091	0.6223	0.3536	0.1131	1.0182	0.6788	1.8385	1.3859	1.4991	1.6122	0.0000	0.8061	0.2828	2.0082	0.3253	0.9192	0.8627	0.6788	0.5657	0.0424	1.7819	1.2869	0.5233
19	0.2121	1.7819	1.2162	2.5314	0.2546	0.4101	1.2021	0.2970	1.4284	1.1597	0.6930	0.2121	0.1273	1.0324	2.1920	0.6930	2.4183	0.8061	0.0000	0.5233	2.8143	1.1314	1.7253	1.6688	1.4849	1.3718	0.8485	2.5880	2.0930	1.3294
20	0.7354	1.2587	0.6930	2.0082	0.2687	0.1131	0.6788	0.2263	0.9051	0.6364	0.1697	0.7354	0.3960	1.5556	1.6688	1.2162	1.8950	0.2828	0.5233	0.0000	2.2910	0.6081	1.2021	1.1455	0.9617	0.8485	0.3253	2.0648	1.5698	0.8061
21	3.0264	1.0324	1.5981	0.2828	2.5597	2.4042	1.6122	2.5173	1.3859	1.6546	2.1213	3.0264	2.6870	3.8467	0.6223	3.5072	0.3960	2.0082	2.8143	2.2910	0.0000	1.6829	1.0889	1.1455	1.3294	1.4425	1.9658	0.2263	0.7212	1.4849
22	1.3435	0.6505	0.0849	1.4001	0.8768	0.7212	0.0707	0.8344	0.2970	0.0283	0.4384	1.3435	1.0041	2.1637	1.0607	1.8243	1.2869	0.3253	1.1314	0.6081	1.6829	0.0000	0.5940	0.5374	0.3536	0.2404	0.2828	1.4566	0.9617	0.1980
23	1.9375	0.0566	0.5091	0.8061	1.4708	1.3152	0.5233	1.4284	0.2970	0.5657	1.0324	1.9375	1.5981	2.7577	0.4667	2.4183	0.6930	0.9192	1.7253	1.2021	1.0889	0.5940	0.0000	0.0566	0.2404	0.3536	0.8768	0.8627	0.3677	0.3960
24	1.8809	0.1131	0.4525	0.8627	1.4142	1.2587	0.4667	1.3718	0.2404	0.5091	0.9758	1.8809	1.5415	2.7011	0.5233	2.3617	0.7495	0.8627	1.6688	1.1455	1.1455	0.5374	0.0566	0.0000	0.1838	0.2970	0.8202	0.9192	0.4243	0.3394
25	1.6971	0.2970	0.2687	1.0465	1.2304	1.0748	0.2828	1.1879	0.0566	0.3253	0.7920	1.6971	1.3576	2.5173	0.7071	2.1779	0.9334	0.6788	1.4849	0.9617	1.3294	0.3536	0.2404	0.1838	0.0000	0.1131	0.6364	1.1031	0.6081	0.1556
26	1.5839	0.4101	0.1556	1.1597	1.1172	0.9617	0.1697	1.0748	0.0566	0.2121	0.6788	1.5839	1.2445	2.4042	0.8202	2.0648	1.0465	0.5657	1.3718	0.8485	1.4425	0.2404	0.3536	0.2970	0.1131	0.0000	0.5233	1.2162	0.7212	0.0424
27	1.0607	0.9334	0.3677	1.6829	0.5940	0.4384	0.3536	0.5515	0.5798	0.3111	0.1556	1.0607	0.7212	1.8809	1.3435	1.5415	1.5698	0.0424	0.8485	0.3253	1.9658	0.2828	0.8768	0.8202	0.6364	0.5233	0.0000	1.7395	1.2445	0.4808
28	2.8001	0.8061	1.3718	0.0566	2.3335	2.1779	1.3859	2.2910	1.1597	1.4284	1.8950	2.8001	2.4607	3.6204	0.3960	3.2810	0.1697	1.7819	2.5880	2.0648	0.2263	1.4566	0.8627	0.9192	1.1031	1.2162	1.7395	0.0000	0.4950	1.2587
29	2.3052	0.3111	0.8768	0.4384	1.8385	1.6829	0.8910	1.7961	0.6647	0.9334	1.4001	2.3052	1.9658	3.1254	0.0990	2.7860	0.3253	1.2869	2.0930	1.5698	0.7212	0.9617	0.3677	0.4243	0.6081	0.7212	1.2445	0.4950	0.0000	0.7637
30	1.5415	0.4525	0.1131	1.2021	1.0748	0.9192	0.1273	1.0324	0.0990	0.1697	0.6364	1.5415	1.2021	2.3617	0.8627	2.0223	1.0889	0.5233	1.3294	0.8061	1.4849	0.1980	0.3960	0.3394	0.1556	0.0424	0.4808	1.2587	0.7637	0.0000

p_minkowski_p3 <- as.matrix(dist(p, method = "minkowski", p=3))
rownames(p_minkowski_p3) <- paste0("   ", as.character(c(1:30)))
kbl(p_minkowski_p3) %>%
  kable_minimal(font_size = 10) 

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
1	0.0000	1.7765	1.2725	2.4442	0.4158	0.5544	1.2599	0.4536	1.4615	1.2221	0.8063	0.0000	0.3024	0.7308	2.1419	0.4284	2.3435	0.9071	0.1890	0.6552	2.6962	1.1969	1.7261	1.6757	1.5119	1.4111	0.9449	2.4946	2.0537	1.3733
2	1.7765	0.0000	0.5040	0.6678	1.3607	1.2221	0.5166	1.3229	0.3150	0.5544	0.9701	1.7765	1.4741	2.5072	0.3654	2.2049	0.5670	0.8693	1.5875	1.1213	0.9197	0.5796	0.0504	0.1008	0.2646	0.3654	0.8315	0.7182	0.2772	0.4032
3	1.2725	0.5040	0.0000	1.1717	0.8567	0.7182	0.0126	0.8189	0.1890	0.0504	0.4662	1.2725	0.9701	2.0033	0.8693	1.7009	1.0709	0.3654	1.0835	0.6174	1.4237	0.0756	0.4536	0.4032	0.2394	0.1386	0.3276	1.2221	0.7812	0.1008
4	2.4442	0.6678	1.1717	0.0000	2.0285	1.8899	1.1843	1.9907	0.9827	1.2221	1.6379	2.4442	2.1419	3.1750	0.3024	2.8726	0.1008	1.5371	2.2553	1.7891	0.2520	1.2473	0.7182	0.7686	0.9323	1.0331	1.4993	0.0504	0.3906	1.0709
5	0.4158	1.3607	0.8567	2.0285	0.0000	0.1386	0.8441	0.0378	1.0457	0.8063	0.3906	0.4158	0.1134	1.1465	1.7261	0.8441	1.9277	0.4914	0.2268	0.2394	2.2805	0.7812	1.3103	1.2599	1.0961	0.9953	0.5292	2.0789	1.6379	0.9575
6	0.5544	1.2221	0.7182	1.8899	0.1386	0.0000	0.7056	0.1008	0.9071	0.6678	0.2520	0.5544	0.2520	1.2851	1.5875	0.9827	1.7891	0.3528	0.3654	0.1008	2.1419	0.6426	1.1717	1.1213	0.9575	0.8567	0.3906	1.9403	1.4993	0.8189
7	1.2599	0.5166	0.0126	1.1843	0.8441	0.7056	0.0000	0.8063	0.2016	0.0378	0.4536	1.2599	0.9575	1.9907	0.8819	1.6883	1.0835	0.3528	1.0709	0.6048	1.4363	0.0630	0.4662	0.4158	0.2520	0.1512	0.3150	1.2347	0.7938	0.1134
8	0.4536	1.3229	0.8189	1.9907	0.0378	0.1008	0.8063	0.0000	1.0079	0.7686	0.3528	0.4536	0.1512	1.1843	1.6883	0.8819	1.8899	0.4536	0.2646	0.2016	2.2427	0.7434	1.2725	1.2221	1.0583	0.9575	0.4914	2.0411	1.6001	0.9197
9	1.4615	0.3150	0.1890	0.9827	1.0457	0.9071	0.2016	1.0079	0.0000	0.2394	0.6552	1.4615	1.1591	2.1923	0.6804	1.8899	0.8819	0.5544	1.2725	0.8063	1.2347	0.2646	0.2646	0.2142	0.0504	0.0504	0.5166	1.0331	0.5922	0.0882
10	1.2221	0.5544	0.0504	1.2221	0.8063	0.6678	0.0378	0.7686	0.2394	0.0000	0.4158	1.2221	0.9197	1.9529	0.9197	1.6505	1.1213	0.3150	1.0331	0.5670	1.4741	0.0252	0.5040	0.4536	0.2898	0.1890	0.2772	1.2725	0.8315	0.1512
11	0.8063	0.9701	0.4662	1.6379	0.3906	0.2520	0.4536	0.3528	0.6552	0.4158	0.0000	0.8063	0.5040	1.5371	1.3355	1.2347	1.5371	0.1008	0.6174	0.1512	1.8899	0.3906	0.9197	0.8693	0.7056	0.6048	0.1386	1.6883	1.2473	0.5670
12	0.0000	1.7765	1.2725	2.4442	0.4158	0.5544	1.2599	0.4536	1.4615	1.2221	0.8063	0.0000	0.3024	0.7308	2.1419	0.4284	2.3435	0.9071	0.1890	0.6552	2.6962	1.1969	1.7261	1.6757	1.5119	1.4111	0.9449	2.4946	2.0537	1.3733
13	0.3024	1.4741	0.9701	2.1419	0.1134	0.2520	0.9575	0.1512	1.1591	0.9197	0.5040	0.3024	0.0000	1.0331	1.8395	0.7308	2.0411	0.6048	0.1134	0.3528	2.3938	0.8945	1.4237	1.3733	1.2095	1.1087	0.6426	2.1923	1.7513	1.0709
14	0.7308	2.5072	2.0033	3.1750	1.1465	1.2851	1.9907	1.1843	2.1923	1.9529	1.5371	0.7308	1.0331	0.0000	2.8726	0.3024	3.0742	1.6379	0.9197	1.3859	3.4270	1.9277	2.4568	2.4064	2.2427	2.1419	1.6757	3.2254	2.7844	2.1041
15	2.1419	0.3654	0.8693	0.3024	1.7261	1.5875	0.8819	1.6883	0.6804	0.9197	1.3355	2.1419	1.8395	2.8726	0.0000	2.5702	0.2016	1.2347	1.9529	1.4867	0.5544	0.9449	0.4158	0.4662	0.6300	0.7308	1.1969	0.3528	0.0882	0.7686
16	0.4284	2.2049	1.7009	2.8726	0.8441	0.9827	1.6883	0.8819	1.8899	1.6505	1.2347	0.4284	0.7308	0.3024	2.5702	0.0000	2.7718	1.3355	0.6174	1.0835	3.1246	1.6253	2.1545	2.1041	1.9403	1.8395	1.3733	2.9230	2.4820	1.8017
17	2.3435	0.5670	1.0709	0.1008	1.9277	1.7891	1.0835	1.8899	0.8819	1.1213	1.5371	2.3435	2.0411	3.0742	0.2016	2.7718	0.0000	1.4363	2.1545	1.6883	0.3528	1.1465	0.6174	0.6678	0.8315	0.9323	1.3985	0.1512	0.2898	0.9701
18	0.9071	0.8693	0.3654	1.5371	0.4914	0.3528	0.3528	0.4536	0.5544	0.3150	0.1008	0.9071	0.6048	1.6379	1.2347	1.3355	1.4363	0.0000	0.7182	0.2520	1.7891	0.2898	0.8189	0.7686	0.6048	0.5040	0.0378	1.5875	1.1465	0.4662
19	0.1890	1.5875	1.0835	2.2553	0.2268	0.3654	1.0709	0.2646	1.2725	1.0331	0.6174	0.1890	0.1134	0.9197	1.9529	0.6174	2.1545	0.7182	0.0000	0.4662	2.5072	1.0079	1.5371	1.4867	1.3229	1.2221	0.7560	2.3057	1.8647	1.1843
20	0.6552	1.1213	0.6174	1.7891	0.2394	0.1008	0.6048	0.2016	0.8063	0.5670	0.1512	0.6552	0.3528	1.3859	1.4867	1.0835	1.6883	0.2520	0.4662	0.0000	2.0411	0.5418	1.0709	1.0205	0.8567	0.7560	0.2898	1.8395	1.3985	0.7182
21	2.6962	0.9197	1.4237	0.2520	2.2805	2.1419	1.4363	2.2427	1.2347	1.4741	1.8899	2.6962	2.3938	3.4270	0.5544	3.1246	0.3528	1.7891	2.5072	2.0411	0.0000	1.4993	0.9701	1.0205	1.1843	1.2851	1.7513	0.2016	0.6426	1.3229
22	1.1969	0.5796	0.0756	1.2473	0.7812	0.6426	0.0630	0.7434	0.2646	0.0252	0.3906	1.1969	0.8945	1.9277	0.9449	1.6253	1.1465	0.2898	1.0079	0.5418	1.4993	0.0000	0.5292	0.4788	0.3150	0.2142	0.2520	1.2977	0.8567	0.1764
23	1.7261	0.0504	0.4536	0.7182	1.3103	1.1717	0.4662	1.2725	0.2646	0.5040	0.9197	1.7261	1.4237	2.4568	0.4158	2.1545	0.6174	0.8189	1.5371	1.0709	0.9701	0.5292	0.0000	0.0504	0.2142	0.3150	0.7812	0.7686	0.3276	0.3528
24	1.6757	0.1008	0.4032	0.7686	1.2599	1.1213	0.4158	1.2221	0.2142	0.4536	0.8693	1.6757	1.3733	2.4064	0.4662	2.1041	0.6678	0.7686	1.4867	1.0205	1.0205	0.4788	0.0504	0.0000	0.1638	0.2646	0.7308	0.8189	0.3780	0.3024
25	1.5119	0.2646	0.2394	0.9323	1.0961	0.9575	0.2520	1.0583	0.0504	0.2898	0.7056	1.5119	1.2095	2.2427	0.6300	1.9403	0.8315	0.6048	1.3229	0.8567	1.1843	0.3150	0.2142	0.1638	0.0000	0.1008	0.5670	0.9827	0.5418	0.1386
26	1.4111	0.3654	0.1386	1.0331	0.9953	0.8567	0.1512	0.9575	0.0504	0.1890	0.6048	1.4111	1.1087	2.1419	0.7308	1.8395	0.9323	0.5040	1.2221	0.7560	1.2851	0.2142	0.3150	0.2646	0.1008	0.0000	0.4662	1.0835	0.6426	0.0378
27	0.9449	0.8315	0.3276	1.4993	0.5292	0.3906	0.3150	0.4914	0.5166	0.2772	0.1386	0.9449	0.6426	1.6757	1.1969	1.3733	1.3985	0.0378	0.7560	0.2898	1.7513	0.2520	0.7812	0.7308	0.5670	0.4662	0.0000	1.5497	1.1087	0.4284
28	2.4946	0.7182	1.2221	0.0504	2.0789	1.9403	1.2347	2.0411	1.0331	1.2725	1.6883	2.4946	2.1923	3.2254	0.3528	2.9230	0.1512	1.5875	2.3057	1.8395	0.2016	1.2977	0.7686	0.8189	0.9827	1.0835	1.5497	0.0000	0.4410	1.1213
29	2.0537	0.2772	0.7812	0.3906	1.6379	1.4993	0.7938	1.6001	0.5922	0.8315	1.2473	2.0537	1.7513	2.7844	0.0882	2.4820	0.2898	1.1465	1.8647	1.3985	0.6426	0.8567	0.3276	0.3780	0.5418	0.6426	1.1087	0.4410	0.0000	0.6804
30	1.3733	0.4032	0.1008	1.0709	0.9575	0.8189	0.1134	0.9197	0.0882	0.1512	0.5670	1.3733	1.0709	2.1041	0.7686	1.8017	0.9701	0.4662	1.1843	0.7182	1.3229	0.1764	0.3528	0.3024	0.1386	0.0378	0.4284	1.1213	0.6804	0.0000

kbl(data.frame(rowSums(p_manhattan), rowSums(p_euclidean), rowSums(p_minkowski_p3))) %>%
  kable_minimal(full_width = F)

	rowSums.p_manhattan.	rowSums.p_euclidean.	rowSums.p_minkowski_p3.
1	57.86	40.91	36.45
2	41.58	29.40	26.19
3	33.62	23.77	21.18
4	63.18	44.67	39.80
5	44.42	31.41	27.98
6	41.22	29.15	25.97
7	33.58	23.74	21.15
8	43.46	30.73	27.38
9	35.26	24.93	22.21
10	33.58	23.74	21.15
11	36.90	26.09	23.25
12	57.86	40.91	36.45
13	47.66	33.70	30.02
14	88.98	62.92	56.05
15	52.30	36.98	32.95
16	75.54	53.41	47.59
17	59.34	41.96	37.38
18	35.62	25.19	22.44
19	51.26	36.25	32.29
20	39.30	27.79	24.76
21	74.22	52.48	46.76
22	33.66	23.80	21.20
23	40.30	28.50	25.39
24	39.18	27.70	24.68
25	36.06	25.50	22.72
26	34.62	24.48	21.81
27	35.26	24.93	22.21
28	65.26	46.15	41.11
29	49.50	35.00	31.18
30	34.26	24.23	21.58

Distance calculated using only the independent variable

p1 <- p[,1]

p1_manhattan <- as.data.frame(as.matrix(dist(p1, method = "manhattan")))
rownames(p1_manhattan) <- paste0("   ", as.character(c(1:30)))
kbl(p1_manhattan) %>%
  kable_minimal(font_size = 10) 

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
1	0.00	1.41	1.01	1.94	0.33	0.44	1.00	0.36	1.16	0.97	0.64	0.00	0.24	0.58	1.70	0.34	1.86	0.72	0.15	0.52	2.14	0.95	1.37	1.33	1.20	1.12	0.75	1.98	1.63	1.09
2	1.41	0.00	0.40	0.53	1.08	0.97	0.41	1.05	0.25	0.44	0.77	1.41	1.17	1.99	0.29	1.75	0.45	0.69	1.26	0.89	0.73	0.46	0.04	0.08	0.21	0.29	0.66	0.57	0.22	0.32
3	1.01	0.40	0.00	0.93	0.68	0.57	0.01	0.65	0.15	0.04	0.37	1.01	0.77	1.59	0.69	1.35	0.85	0.29	0.86	0.49	1.13	0.06	0.36	0.32	0.19	0.11	0.26	0.97	0.62	0.08
4	1.94	0.53	0.93	0.00	1.61	1.50	0.94	1.58	0.78	0.97	1.30	1.94	1.70	2.52	0.24	2.28	0.08	1.22	1.79	1.42	0.20	0.99	0.57	0.61	0.74	0.82	1.19	0.04	0.31	0.85
5	0.33	1.08	0.68	1.61	0.00	0.11	0.67	0.03	0.83	0.64	0.31	0.33	0.09	0.91	1.37	0.67	1.53	0.39	0.18	0.19	1.81	0.62	1.04	1.00	0.87	0.79	0.42	1.65	1.30	0.76
6	0.44	0.97	0.57	1.50	0.11	0.00	0.56	0.08	0.72	0.53	0.20	0.44	0.20	1.02	1.26	0.78	1.42	0.28	0.29	0.08	1.70	0.51	0.93	0.89	0.76	0.68	0.31	1.54	1.19	0.65
7	1.00	0.41	0.01	0.94	0.67	0.56	0.00	0.64	0.16	0.03	0.36	1.00	0.76	1.58	0.70	1.34	0.86	0.28	0.85	0.48	1.14	0.05	0.37	0.33	0.20	0.12	0.25	0.98	0.63	0.09
8	0.36	1.05	0.65	1.58	0.03	0.08	0.64	0.00	0.80	0.61	0.28	0.36	0.12	0.94	1.34	0.70	1.50	0.36	0.21	0.16	1.78	0.59	1.01	0.97	0.84	0.76	0.39	1.62	1.27	0.73
9	1.16	0.25	0.15	0.78	0.83	0.72	0.16	0.80	0.00	0.19	0.52	1.16	0.92	1.74	0.54	1.50	0.70	0.44	1.01	0.64	0.98	0.21	0.21	0.17	0.04	0.04	0.41	0.82	0.47	0.07
10	0.97	0.44	0.04	0.97	0.64	0.53	0.03	0.61	0.19	0.00	0.33	0.97	0.73	1.55	0.73	1.31	0.89	0.25	0.82	0.45	1.17	0.02	0.40	0.36	0.23	0.15	0.22	1.01	0.66	0.12
11	0.64	0.77	0.37	1.30	0.31	0.20	0.36	0.28	0.52	0.33	0.00	0.64	0.40	1.22	1.06	0.98	1.22	0.08	0.49	0.12	1.50	0.31	0.73	0.69	0.56	0.48	0.11	1.34	0.99	0.45
12	0.00	1.41	1.01	1.94	0.33	0.44	1.00	0.36	1.16	0.97	0.64	0.00	0.24	0.58	1.70	0.34	1.86	0.72	0.15	0.52	2.14	0.95	1.37	1.33	1.20	1.12	0.75	1.98	1.63	1.09
13	0.24	1.17	0.77	1.70	0.09	0.20	0.76	0.12	0.92	0.73	0.40	0.24	0.00	0.82	1.46	0.58	1.62	0.48	0.09	0.28	1.90	0.71	1.13	1.09	0.96	0.88	0.51	1.74	1.39	0.85
14	0.58	1.99	1.59	2.52	0.91	1.02	1.58	0.94	1.74	1.55	1.22	0.58	0.82	0.00	2.28	0.24	2.44	1.30	0.73	1.10	2.72	1.53	1.95	1.91	1.78	1.70	1.33	2.56	2.21	1.67
15	1.70	0.29	0.69	0.24	1.37	1.26	0.70	1.34	0.54	0.73	1.06	1.70	1.46	2.28	0.00	2.04	0.16	0.98	1.55	1.18	0.44	0.75	0.33	0.37	0.50	0.58	0.95	0.28	0.07	0.61
16	0.34	1.75	1.35	2.28	0.67	0.78	1.34	0.70	1.50	1.31	0.98	0.34	0.58	0.24	2.04	0.00	2.20	1.06	0.49	0.86	2.48	1.29	1.71	1.67	1.54	1.46	1.09	2.32	1.97	1.43
17	1.86	0.45	0.85	0.08	1.53	1.42	0.86	1.50	0.70	0.89	1.22	1.86	1.62	2.44	0.16	2.20	0.00	1.14	1.71	1.34	0.28	0.91	0.49	0.53	0.66	0.74	1.11	0.12	0.23	0.77
18	0.72	0.69	0.29	1.22	0.39	0.28	0.28	0.36	0.44	0.25	0.08	0.72	0.48	1.30	0.98	1.06	1.14	0.00	0.57	0.20	1.42	0.23	0.65	0.61	0.48	0.40	0.03	1.26	0.91	0.37
19	0.15	1.26	0.86	1.79	0.18	0.29	0.85	0.21	1.01	0.82	0.49	0.15	0.09	0.73	1.55	0.49	1.71	0.57	0.00	0.37	1.99	0.80	1.22	1.18	1.05	0.97	0.60	1.83	1.48	0.94
20	0.52	0.89	0.49	1.42	0.19	0.08	0.48	0.16	0.64	0.45	0.12	0.52	0.28	1.10	1.18	0.86	1.34	0.20	0.37	0.00	1.62	0.43	0.85	0.81	0.68	0.60	0.23	1.46	1.11	0.57
21	2.14	0.73	1.13	0.20	1.81	1.70	1.14	1.78	0.98	1.17	1.50	2.14	1.90	2.72	0.44	2.48	0.28	1.42	1.99	1.62	0.00	1.19	0.77	0.81	0.94	1.02	1.39	0.16	0.51	1.05
22	0.95	0.46	0.06	0.99	0.62	0.51	0.05	0.59	0.21	0.02	0.31	0.95	0.71	1.53	0.75	1.29	0.91	0.23	0.80	0.43	1.19	0.00	0.42	0.38	0.25	0.17	0.20	1.03	0.68	0.14
23	1.37	0.04	0.36	0.57	1.04	0.93	0.37	1.01	0.21	0.40	0.73	1.37	1.13	1.95	0.33	1.71	0.49	0.65	1.22	0.85	0.77	0.42	0.00	0.04	0.17	0.25	0.62	0.61	0.26	0.28
24	1.33	0.08	0.32	0.61	1.00	0.89	0.33	0.97	0.17	0.36	0.69	1.33	1.09	1.91	0.37	1.67	0.53	0.61	1.18	0.81	0.81	0.38	0.04	0.00	0.13	0.21	0.58	0.65	0.30	0.24
25	1.20	0.21	0.19	0.74	0.87	0.76	0.20	0.84	0.04	0.23	0.56	1.20	0.96	1.78	0.50	1.54	0.66	0.48	1.05	0.68	0.94	0.25	0.17	0.13	0.00	0.08	0.45	0.78	0.43	0.11
26	1.12	0.29	0.11	0.82	0.79	0.68	0.12	0.76	0.04	0.15	0.48	1.12	0.88	1.70	0.58	1.46	0.74	0.40	0.97	0.60	1.02	0.17	0.25	0.21	0.08	0.00	0.37	0.86	0.51	0.03
27	0.75	0.66	0.26	1.19	0.42	0.31	0.25	0.39	0.41	0.22	0.11	0.75	0.51	1.33	0.95	1.09	1.11	0.03	0.60	0.23	1.39	0.20	0.62	0.58	0.45	0.37	0.00	1.23	0.88	0.34
28	1.98	0.57	0.97	0.04	1.65	1.54	0.98	1.62	0.82	1.01	1.34	1.98	1.74	2.56	0.28	2.32	0.12	1.26	1.83	1.46	0.16	1.03	0.61	0.65	0.78	0.86	1.23	0.00	0.35	0.89
29	1.63	0.22	0.62	0.31	1.30	1.19	0.63	1.27	0.47	0.66	0.99	1.63	1.39	2.21	0.07	1.97	0.23	0.91	1.48	1.11	0.51	0.68	0.26	0.30	0.43	0.51	0.88	0.35	0.00	0.54
30	1.09	0.32	0.08	0.85	0.76	0.65	0.09	0.73	0.07	0.12	0.45	1.09	0.85	1.67	0.61	1.43	0.77	0.37	0.94	0.57	1.05	0.14	0.28	0.24	0.11	0.03	0.34	0.89	0.54	0.00

p1_euclidean <- as.matrix(dist(p1, method = "euclidean"))
rownames(p1_euclidean) <- paste0("   ", as.character(c(1:30)))
kbl(p1_euclidean) %>%
  kable_minimal(font_size = 10) 

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
1	0.00	1.41	1.01	1.94	0.33	0.44	1.00	0.36	1.16	0.97	0.64	0.00	0.24	0.58	1.70	0.34	1.86	0.72	0.15	0.52	2.14	0.95	1.37	1.33	1.20	1.12	0.75	1.98	1.63	1.09
2	1.41	0.00	0.40	0.53	1.08	0.97	0.41	1.05	0.25	0.44	0.77	1.41	1.17	1.99	0.29	1.75	0.45	0.69	1.26	0.89	0.73	0.46	0.04	0.08	0.21	0.29	0.66	0.57	0.22	0.32
3	1.01	0.40	0.00	0.93	0.68	0.57	0.01	0.65	0.15	0.04	0.37	1.01	0.77	1.59	0.69	1.35	0.85	0.29	0.86	0.49	1.13	0.06	0.36	0.32	0.19	0.11	0.26	0.97	0.62	0.08
4	1.94	0.53	0.93	0.00	1.61	1.50	0.94	1.58	0.78	0.97	1.30	1.94	1.70	2.52	0.24	2.28	0.08	1.22	1.79	1.42	0.20	0.99	0.57	0.61	0.74	0.82	1.19	0.04	0.31	0.85
5	0.33	1.08	0.68	1.61	0.00	0.11	0.67	0.03	0.83	0.64	0.31	0.33	0.09	0.91	1.37	0.67	1.53	0.39	0.18	0.19	1.81	0.62	1.04	1.00	0.87	0.79	0.42	1.65	1.30	0.76
6	0.44	0.97	0.57	1.50	0.11	0.00	0.56	0.08	0.72	0.53	0.20	0.44	0.20	1.02	1.26	0.78	1.42	0.28	0.29	0.08	1.70	0.51	0.93	0.89	0.76	0.68	0.31	1.54	1.19	0.65
7	1.00	0.41	0.01	0.94	0.67	0.56	0.00	0.64	0.16	0.03	0.36	1.00	0.76	1.58	0.70	1.34	0.86	0.28	0.85	0.48	1.14	0.05	0.37	0.33	0.20	0.12	0.25	0.98	0.63	0.09
8	0.36	1.05	0.65	1.58	0.03	0.08	0.64	0.00	0.80	0.61	0.28	0.36	0.12	0.94	1.34	0.70	1.50	0.36	0.21	0.16	1.78	0.59	1.01	0.97	0.84	0.76	0.39	1.62	1.27	0.73
9	1.16	0.25	0.15	0.78	0.83	0.72	0.16	0.80	0.00	0.19	0.52	1.16	0.92	1.74	0.54	1.50	0.70	0.44	1.01	0.64	0.98	0.21	0.21	0.17	0.04	0.04	0.41	0.82	0.47	0.07
10	0.97	0.44	0.04	0.97	0.64	0.53	0.03	0.61	0.19	0.00	0.33	0.97	0.73	1.55	0.73	1.31	0.89	0.25	0.82	0.45	1.17	0.02	0.40	0.36	0.23	0.15	0.22	1.01	0.66	0.12
11	0.64	0.77	0.37	1.30	0.31	0.20	0.36	0.28	0.52	0.33	0.00	0.64	0.40	1.22	1.06	0.98	1.22	0.08	0.49	0.12	1.50	0.31	0.73	0.69	0.56	0.48	0.11	1.34	0.99	0.45
12	0.00	1.41	1.01	1.94	0.33	0.44	1.00	0.36	1.16	0.97	0.64	0.00	0.24	0.58	1.70	0.34	1.86	0.72	0.15	0.52	2.14	0.95	1.37	1.33	1.20	1.12	0.75	1.98	1.63	1.09
13	0.24	1.17	0.77	1.70	0.09	0.20	0.76	0.12	0.92	0.73	0.40	0.24	0.00	0.82	1.46	0.58	1.62	0.48	0.09	0.28	1.90	0.71	1.13	1.09	0.96	0.88	0.51	1.74	1.39	0.85
14	0.58	1.99	1.59	2.52	0.91	1.02	1.58	0.94	1.74	1.55	1.22	0.58	0.82	0.00	2.28	0.24	2.44	1.30	0.73	1.10	2.72	1.53	1.95	1.91	1.78	1.70	1.33	2.56	2.21	1.67
15	1.70	0.29	0.69	0.24	1.37	1.26	0.70	1.34	0.54	0.73	1.06	1.70	1.46	2.28	0.00	2.04	0.16	0.98	1.55	1.18	0.44	0.75	0.33	0.37	0.50	0.58	0.95	0.28	0.07	0.61
16	0.34	1.75	1.35	2.28	0.67	0.78	1.34	0.70	1.50	1.31	0.98	0.34	0.58	0.24	2.04	0.00	2.20	1.06	0.49	0.86	2.48	1.29	1.71	1.67	1.54	1.46	1.09	2.32	1.97	1.43
17	1.86	0.45	0.85	0.08	1.53	1.42	0.86	1.50	0.70	0.89	1.22	1.86	1.62	2.44	0.16	2.20	0.00	1.14	1.71	1.34	0.28	0.91	0.49	0.53	0.66	0.74	1.11	0.12	0.23	0.77
18	0.72	0.69	0.29	1.22	0.39	0.28	0.28	0.36	0.44	0.25	0.08	0.72	0.48	1.30	0.98	1.06	1.14	0.00	0.57	0.20	1.42	0.23	0.65	0.61	0.48	0.40	0.03	1.26	0.91	0.37
19	0.15	1.26	0.86	1.79	0.18	0.29	0.85	0.21	1.01	0.82	0.49	0.15	0.09	0.73	1.55	0.49	1.71	0.57	0.00	0.37	1.99	0.80	1.22	1.18	1.05	0.97	0.60	1.83	1.48	0.94
20	0.52	0.89	0.49	1.42	0.19	0.08	0.48	0.16	0.64	0.45	0.12	0.52	0.28	1.10	1.18	0.86	1.34	0.20	0.37	0.00	1.62	0.43	0.85	0.81	0.68	0.60	0.23	1.46	1.11	0.57
21	2.14	0.73	1.13	0.20	1.81	1.70	1.14	1.78	0.98	1.17	1.50	2.14	1.90	2.72	0.44	2.48	0.28	1.42	1.99	1.62	0.00	1.19	0.77	0.81	0.94	1.02	1.39	0.16	0.51	1.05
22	0.95	0.46	0.06	0.99	0.62	0.51	0.05	0.59	0.21	0.02	0.31	0.95	0.71	1.53	0.75	1.29	0.91	0.23	0.80	0.43	1.19	0.00	0.42	0.38	0.25	0.17	0.20	1.03	0.68	0.14
23	1.37	0.04	0.36	0.57	1.04	0.93	0.37	1.01	0.21	0.40	0.73	1.37	1.13	1.95	0.33	1.71	0.49	0.65	1.22	0.85	0.77	0.42	0.00	0.04	0.17	0.25	0.62	0.61	0.26	0.28
24	1.33	0.08	0.32	0.61	1.00	0.89	0.33	0.97	0.17	0.36	0.69	1.33	1.09	1.91	0.37	1.67	0.53	0.61	1.18	0.81	0.81	0.38	0.04	0.00	0.13	0.21	0.58	0.65	0.30	0.24
25	1.20	0.21	0.19	0.74	0.87	0.76	0.20	0.84	0.04	0.23	0.56	1.20	0.96	1.78	0.50	1.54	0.66	0.48	1.05	0.68	0.94	0.25	0.17	0.13	0.00	0.08	0.45	0.78	0.43	0.11
26	1.12	0.29	0.11	0.82	0.79	0.68	0.12	0.76	0.04	0.15	0.48	1.12	0.88	1.70	0.58	1.46	0.74	0.40	0.97	0.60	1.02	0.17	0.25	0.21	0.08	0.00	0.37	0.86	0.51	0.03
27	0.75	0.66	0.26	1.19	0.42	0.31	0.25	0.39	0.41	0.22	0.11	0.75	0.51	1.33	0.95	1.09	1.11	0.03	0.60	0.23	1.39	0.20	0.62	0.58	0.45	0.37	0.00	1.23	0.88	0.34
28	1.98	0.57	0.97	0.04	1.65	1.54	0.98	1.62	0.82	1.01	1.34	1.98	1.74	2.56	0.28	2.32	0.12	1.26	1.83	1.46	0.16	1.03	0.61	0.65	0.78	0.86	1.23	0.00	0.35	0.89
29	1.63	0.22	0.62	0.31	1.30	1.19	0.63	1.27	0.47	0.66	0.99	1.63	1.39	2.21	0.07	1.97	0.23	0.91	1.48	1.11	0.51	0.68	0.26	0.30	0.43	0.51	0.88	0.35	0.00	0.54
30	1.09	0.32	0.08	0.85	0.76	0.65	0.09	0.73	0.07	0.12	0.45	1.09	0.85	1.67	0.61	1.43	0.77	0.37	0.94	0.57	1.05	0.14	0.28	0.24	0.11	0.03	0.34	0.89	0.54	0.00

p1_minkowski_p3 <- as.matrix(dist(p1, method = "minkowski", p=3))
rownames(p1_minkowski_p3) <- paste0("   ", as.character(c(1:30)))
kbl(p1_minkowski_p3) %>%
  kable_minimal(font_size = 10) 

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
1	0.00	1.41	1.01	1.94	0.33	0.44	1.00	0.36	1.16	0.97	0.64	0.00	0.24	0.58	1.70	0.34	1.86	0.72	0.15	0.52	2.14	0.95	1.37	1.33	1.20	1.12	0.75	1.98	1.63	1.09
2	1.41	0.00	0.40	0.53	1.08	0.97	0.41	1.05	0.25	0.44	0.77	1.41	1.17	1.99	0.29	1.75	0.45	0.69	1.26	0.89	0.73	0.46	0.04	0.08	0.21	0.29	0.66	0.57	0.22	0.32
3	1.01	0.40	0.00	0.93	0.68	0.57	0.01	0.65	0.15	0.04	0.37	1.01	0.77	1.59	0.69	1.35	0.85	0.29	0.86	0.49	1.13	0.06	0.36	0.32	0.19	0.11	0.26	0.97	0.62	0.08
4	1.94	0.53	0.93	0.00	1.61	1.50	0.94	1.58	0.78	0.97	1.30	1.94	1.70	2.52	0.24	2.28	0.08	1.22	1.79	1.42	0.20	0.99	0.57	0.61	0.74	0.82	1.19	0.04	0.31	0.85
5	0.33	1.08	0.68	1.61	0.00	0.11	0.67	0.03	0.83	0.64	0.31	0.33	0.09	0.91	1.37	0.67	1.53	0.39	0.18	0.19	1.81	0.62	1.04	1.00	0.87	0.79	0.42	1.65	1.30	0.76
6	0.44	0.97	0.57	1.50	0.11	0.00	0.56	0.08	0.72	0.53	0.20	0.44	0.20	1.02	1.26	0.78	1.42	0.28	0.29	0.08	1.70	0.51	0.93	0.89	0.76	0.68	0.31	1.54	1.19	0.65
7	1.00	0.41	0.01	0.94	0.67	0.56	0.00	0.64	0.16	0.03	0.36	1.00	0.76	1.58	0.70	1.34	0.86	0.28	0.85	0.48	1.14	0.05	0.37	0.33	0.20	0.12	0.25	0.98	0.63	0.09
8	0.36	1.05	0.65	1.58	0.03	0.08	0.64	0.00	0.80	0.61	0.28	0.36	0.12	0.94	1.34	0.70	1.50	0.36	0.21	0.16	1.78	0.59	1.01	0.97	0.84	0.76	0.39	1.62	1.27	0.73
9	1.16	0.25	0.15	0.78	0.83	0.72	0.16	0.80	0.00	0.19	0.52	1.16	0.92	1.74	0.54	1.50	0.70	0.44	1.01	0.64	0.98	0.21	0.21	0.17	0.04	0.04	0.41	0.82	0.47	0.07
10	0.97	0.44	0.04	0.97	0.64	0.53	0.03	0.61	0.19	0.00	0.33	0.97	0.73	1.55	0.73	1.31	0.89	0.25	0.82	0.45	1.17	0.02	0.40	0.36	0.23	0.15	0.22	1.01	0.66	0.12
11	0.64	0.77	0.37	1.30	0.31	0.20	0.36	0.28	0.52	0.33	0.00	0.64	0.40	1.22	1.06	0.98	1.22	0.08	0.49	0.12	1.50	0.31	0.73	0.69	0.56	0.48	0.11	1.34	0.99	0.45
12	0.00	1.41	1.01	1.94	0.33	0.44	1.00	0.36	1.16	0.97	0.64	0.00	0.24	0.58	1.70	0.34	1.86	0.72	0.15	0.52	2.14	0.95	1.37	1.33	1.20	1.12	0.75	1.98	1.63	1.09
13	0.24	1.17	0.77	1.70	0.09	0.20	0.76	0.12	0.92	0.73	0.40	0.24	0.00	0.82	1.46	0.58	1.62	0.48	0.09	0.28	1.90	0.71	1.13	1.09	0.96	0.88	0.51	1.74	1.39	0.85
14	0.58	1.99	1.59	2.52	0.91	1.02	1.58	0.94	1.74	1.55	1.22	0.58	0.82	0.00	2.28	0.24	2.44	1.30	0.73	1.10	2.72	1.53	1.95	1.91	1.78	1.70	1.33	2.56	2.21	1.67
15	1.70	0.29	0.69	0.24	1.37	1.26	0.70	1.34	0.54	0.73	1.06	1.70	1.46	2.28	0.00	2.04	0.16	0.98	1.55	1.18	0.44	0.75	0.33	0.37	0.50	0.58	0.95	0.28	0.07	0.61
16	0.34	1.75	1.35	2.28	0.67	0.78	1.34	0.70	1.50	1.31	0.98	0.34	0.58	0.24	2.04	0.00	2.20	1.06	0.49	0.86	2.48	1.29	1.71	1.67	1.54	1.46	1.09	2.32	1.97	1.43
17	1.86	0.45	0.85	0.08	1.53	1.42	0.86	1.50	0.70	0.89	1.22	1.86	1.62	2.44	0.16	2.20	0.00	1.14	1.71	1.34	0.28	0.91	0.49	0.53	0.66	0.74	1.11	0.12	0.23	0.77
18	0.72	0.69	0.29	1.22	0.39	0.28	0.28	0.36	0.44	0.25	0.08	0.72	0.48	1.30	0.98	1.06	1.14	0.00	0.57	0.20	1.42	0.23	0.65	0.61	0.48	0.40	0.03	1.26	0.91	0.37
19	0.15	1.26	0.86	1.79	0.18	0.29	0.85	0.21	1.01	0.82	0.49	0.15	0.09	0.73	1.55	0.49	1.71	0.57	0.00	0.37	1.99	0.80	1.22	1.18	1.05	0.97	0.60	1.83	1.48	0.94
20	0.52	0.89	0.49	1.42	0.19	0.08	0.48	0.16	0.64	0.45	0.12	0.52	0.28	1.10	1.18	0.86	1.34	0.20	0.37	0.00	1.62	0.43	0.85	0.81	0.68	0.60	0.23	1.46	1.11	0.57
21	2.14	0.73	1.13	0.20	1.81	1.70	1.14	1.78	0.98	1.17	1.50	2.14	1.90	2.72	0.44	2.48	0.28	1.42	1.99	1.62	0.00	1.19	0.77	0.81	0.94	1.02	1.39	0.16	0.51	1.05
22	0.95	0.46	0.06	0.99	0.62	0.51	0.05	0.59	0.21	0.02	0.31	0.95	0.71	1.53	0.75	1.29	0.91	0.23	0.80	0.43	1.19	0.00	0.42	0.38	0.25	0.17	0.20	1.03	0.68	0.14
23	1.37	0.04	0.36	0.57	1.04	0.93	0.37	1.01	0.21	0.40	0.73	1.37	1.13	1.95	0.33	1.71	0.49	0.65	1.22	0.85	0.77	0.42	0.00	0.04	0.17	0.25	0.62	0.61	0.26	0.28
24	1.33	0.08	0.32	0.61	1.00	0.89	0.33	0.97	0.17	0.36	0.69	1.33	1.09	1.91	0.37	1.67	0.53	0.61	1.18	0.81	0.81	0.38	0.04	0.00	0.13	0.21	0.58	0.65	0.30	0.24
25	1.20	0.21	0.19	0.74	0.87	0.76	0.20	0.84	0.04	0.23	0.56	1.20	0.96	1.78	0.50	1.54	0.66	0.48	1.05	0.68	0.94	0.25	0.17	0.13	0.00	0.08	0.45	0.78	0.43	0.11
26	1.12	0.29	0.11	0.82	0.79	0.68	0.12	0.76	0.04	0.15	0.48	1.12	0.88	1.70	0.58	1.46	0.74	0.40	0.97	0.60	1.02	0.17	0.25	0.21	0.08	0.00	0.37	0.86	0.51	0.03
27	0.75	0.66	0.26	1.19	0.42	0.31	0.25	0.39	0.41	0.22	0.11	0.75	0.51	1.33	0.95	1.09	1.11	0.03	0.60	0.23	1.39	0.20	0.62	0.58	0.45	0.37	0.00	1.23	0.88	0.34
28	1.98	0.57	0.97	0.04	1.65	1.54	0.98	1.62	0.82	1.01	1.34	1.98	1.74	2.56	0.28	2.32	0.12	1.26	1.83	1.46	0.16	1.03	0.61	0.65	0.78	0.86	1.23	0.00	0.35	0.89
29	1.63	0.22	0.62	0.31	1.30	1.19	0.63	1.27	0.47	0.66	0.99	1.63	1.39	2.21	0.07	1.97	0.23	0.91	1.48	1.11	0.51	0.68	0.26	0.30	0.43	0.51	0.88	0.35	0.00	0.54
30	1.09	0.32	0.08	0.85	0.76	0.65	0.09	0.73	0.07	0.12	0.45	1.09	0.85	1.67	0.61	1.43	0.77	0.37	0.94	0.57	1.05	0.14	0.28	0.24	0.11	0.03	0.34	0.89	0.54	0.00

kbl(data.frame(rowSums(p1_manhattan), rowSums(p1_euclidean), rowSums(p1_minkowski_p3))) %>%
  kable_minimal(full_width = F)

	rowSums.p1_manhattan.	rowSums.p1_euclidean.	rowSums.p1_minkowski_p3.
1	28.93	28.93	28.93
2	20.79	20.79	20.79
3	16.81	16.81	16.81
4	31.59	31.59	31.59
5	22.21	22.21	22.21
6	20.61	20.61	20.61
7	16.79	16.79	16.79
8	21.73	21.73	21.73
9	17.63	17.63	17.63
10	16.79	16.79	16.79
11	18.45	18.45	18.45
12	28.93	28.93	28.93
13	23.83	23.83	23.83
14	44.49	44.49	44.49
15	26.15	26.15	26.15
16	37.77	37.77	37.77
17	29.67	29.67	29.67
18	17.81	17.81	17.81
19	25.63	25.63	25.63
20	19.65	19.65	19.65
21	37.11	37.11	37.11
22	16.83	16.83	16.83
23	20.15	20.15	20.15
24	19.59	19.59	19.59
25	18.03	18.03	18.03
26	17.31	17.31	17.31
27	17.63	17.63	17.63
28	32.63	32.63	32.63
29	24.75	24.75	24.75
30	17.13	17.13	17.13

Ex. 2

Now use a higher-dimensional data set mtcars to carry out the same three distance metrics in the previous question and discuss the results.

cars <- mtcars
cars

c_manhattan <- as.data.frame(as.matrix(dist(cars, method = "manhattan")))
kbl(c_manhattan) %>%
  kable_paper(bootstrap_options = "striped", full_width = F, font_size = 11)

	Mazda RX4	Mazda RX4 Wag	Datsun 710	Hornet 4 Drive	Hornet Sportabout	Valiant	Duster 360	Merc 240D	Merc 230	Merc 280	Merc 280C	Merc 450SE	Merc 450SL	Merc 450SLC	Cadillac Fleetwood	Lincoln Continental	Chrysler Imperial	Fiat 128	Honda Civic	Toyota Corolla	Toyota Corona	Dodge Challenger	AMC Javelin	Camaro Z28	Pontiac Firebird	Fiat X1-9	Porsche 914-2	Lotus Europa	Ford Pantera L	Ferrari Dino	Maserati Bora	Volvo 142E
Mazda RX4	0.000	0.815	79.30	108.80	275.43	84.64	347.96	75.02	48.99	27.08	29.08	198.62	197.58	200.13	426.72	424.66	414.65	146.310	160.79	157.34	65.31	211.95	198.21	339.14	315.44	140.605	69.95	84.98	356.03	85.69	382.2	47.91
Mazda RX4 Wag	0.815	0.000	79.00	107.98	274.62	83.83	348.26	74.20	48.17	26.27	28.27	197.81	196.76	199.31	425.90	423.85	413.84	146.005	160.49	157.04	65.00	211.44	197.39	339.44	314.62	140.300	70.28	84.91	356.33	86.20	382.5	47.28
Datsun 710	79.300	78.995	0.00	174.90	349.51	141.54	427.16	75.72	41.99	100.70	102.08	273.94	272.50	274.25	502.88	501.14	491.94	67.110	83.78	78.14	21.09	286.33	271.73	418.34	389.45	61.405	23.17	45.10	435.33	134.89	461.4	32.13
Hornet 4 Drive	108.795	107.980	174.90	0.00	176.41	42.65	254.19	167.50	141.97	111.81	112.61	100.70	99.27	101.02	329.64	327.91	319.00	240.345	258.67	251.38	154.94	113.09	98.63	246.41	216.22	235.720	173.47	185.83	267.73	193.62	293.1	145.31
Hornet Sportabout	275.430	274.615	349.51	176.41	0.00	213.21	77.77	341.77	316.24	252.95	253.95	93.59	92.55	95.10	155.29	153.23	143.38	416.620	431.11	427.65	331.21	70.82	84.78	89.99	41.01	410.915	340.90	349.27	109.76	227.66	234.6	317.90
Valiant	84.640	83.825	141.54	42.65	213.21	0.00	289.74	133.02	107.05	83.60	82.20	136.24	134.80	136.55	364.90	363.30	354.56	206.930	225.31	217.10	120.44	148.01	134.24	281.96	253.97	202.365	140.11	162.48	303.77	166.87	328.6	119.95
Duster 360	347.960	348.265	427.16	254.19	77.77	289.74	0.00	419.42	393.89	326.60	325.80	154.50	155.26	153.61	160.00	137.94	98.78	494.270	508.75	505.31	408.87	141.73	155.56	12.22	118.52	488.565	417.91	426.68	35.25	298.95	158.3	395.55
Merc 240D	75.020	74.205	75.72	167.50	341.77	133.02	419.42	0.00	43.67	93.28	94.08	266.20	264.76	266.51	495.14	493.40	484.19	83.910	92.30	92.08	67.25	278.59	263.99	410.68	381.71	79.345	65.09	115.46	429.95	133.39	455.6	78.93
Merc 230	48.990	48.175	41.99	141.97	316.24	107.05	393.89	43.67	0.00	67.29	68.09	240.67	239.23	240.98	469.61	467.87	458.67	107.240	123.62	117.42	29.80	253.06	238.46	385.07	356.19	102.675	38.42	81.09	403.92	104.38	430.1	41.06
Merc 280	27.080	26.265	100.70	111.81	252.95	83.60	326.60	93.28	67.29	0.00	2.00	175.38	173.94	175.69	402.32	400.58	391.38	167.670	182.16	178.71	84.70	189.77	175.18	317.78	292.89	161.965	96.51	103.18	337.17	83.87	362.8	68.95
Merc 280C	29.080	28.265	102.08	112.61	253.95	82.20	325.80	94.08	68.09	2.00	0.00	174.58	173.14	174.89	401.52	399.78	390.57	168.470	183.72	179.50	85.50	188.97	174.38	316.98	294.89	162.765	98.51	105.18	336.37	85.87	362.0	70.35
Merc 450SE	198.620	197.805	273.94	100.70	93.59	136.24	154.50	266.20	240.67	175.38	174.58	0.00	1.44	2.09	230.10	228.04	218.35	341.050	355.54	352.08	255.65	75.49	61.22	146.18	133.59	335.345	266.09	274.46	168.75	150.85	193.4	242.33
Merc 450SL	197.580	196.765	272.50	99.27	92.55	134.80	155.26	264.76	239.23	173.94	173.14	1.44	0.00	2.55	231.14	229.08	219.75	339.610	354.10	350.64	254.21	76.25	61.98	147.16	132.78	333.905	265.05	273.42	169.51	149.81	194.1	240.89
Merc 450SLC	200.130	199.315	274.25	101.02	95.10	136.55	153.61	266.51	240.98	175.69	174.89	2.09	2.55	0.00	228.63	226.89	218.00	341.360	355.85	352.39	255.96	75.20	60.33	145.41	135.22	335.655	267.60	275.97	169.06	152.36	192.5	242.64
Cadillac Fleetwood	426.720	425.905	502.88	329.64	155.29	364.90	160.00	495.14	469.61	402.32	401.52	230.10	231.14	228.63	0.00	22.40	62.26	569.990	584.48	581.02	484.58	219.11	232.51	169.68	115.28	564.285	496.19	504.56	195.25	378.95	318.3	471.27
Lincoln Continental	424.664	423.849	501.14	327.91	153.23	363.30	137.94	493.40	467.87	400.58	399.78	228.04	229.08	226.89	22.40	0.00	40.01	568.254	582.74	579.29	482.85	217.19	230.46	147.62	113.23	562.549	494.13	502.50	173.19	376.89	296.2	469.53
Chrysler Imperial	414.655	413.840	491.94	319.00	143.38	354.56	98.78	484.19	458.67	391.38	390.57	218.35	219.75	218.00	62.26	40.01	0.00	559.045	573.53	570.08	473.64	207.65	220.61	110.42	103.52	553.340	484.12	492.49	133.19	366.88	256.2	460.32
Fiat 128	146.310	146.005	67.11	240.34	416.62	206.93	494.27	83.91	107.24	167.67	168.47	341.05	339.61	341.36	569.99	568.25	559.04	0.000	22.39	11.04	86.48	353.44	338.83	485.45	456.56	6.235	79.18	70.97	501.98	202.00	528.5	98.78
Honda Civic	160.795	160.490	83.78	258.67	431.11	225.31	508.75	92.30	123.62	182.16	183.72	355.54	354.10	355.85	584.48	582.74	573.53	22.385	0.00	24.41	104.87	367.93	353.32	499.94	471.05	22.950	92.84	84.28	516.18	216.49	543.0	113.36
Toyota Corolla	157.345	157.040	78.14	251.38	427.65	217.10	505.31	92.08	117.42	178.71	179.50	352.08	350.64	352.39	581.02	579.29	570.08	11.035	24.41	0.00	96.66	364.48	349.87	496.49	467.60	16.740	89.81	81.27	512.74	213.03	539.5	109.75
Toyota Corona	65.305	65.000	21.09	154.94	331.21	120.44	408.87	67.25	29.80	84.70	85.50	255.65	254.21	255.96	484.58	482.85	473.64	86.485	104.87	96.66	0.00	268.04	253.43	400.11	371.16	81.920	20.07	58.03	421.33	120.59	447.1	18.14
Dodge Challenger	211.950	211.435	286.33	113.09	70.82	148.01	141.73	278.59	253.06	189.77	188.97	75.49	76.25	75.20	219.11	217.19	207.65	353.440	367.93	364.48	268.04	0.00	15.21	133.95	111.53	347.735	277.42	285.85	156.48	214.18	214.6	254.72
AMC Javelin	198.205	197.390	271.73	98.63	84.78	134.24	155.56	263.99	238.46	175.18	174.38	61.22	61.98	60.33	232.51	230.46	220.61	338.835	353.32	349.87	253.43	15.21	0.00	147.78	125.73	333.130	263.68	272.04	170.74	200.44	200.4	240.12
Camaro Z28	339.140	339.445	418.34	246.41	89.99	281.96	12.22	410.68	385.07	317.78	316.98	146.18	147.16	145.41	169.68	147.62	110.42	485.450	499.94	496.49	400.11	133.95	147.78	0.00	130.19	479.745	409.09	417.86	27.57	289.67	149.0	386.73
Pontiac Firebird	315.435	314.620	389.45	216.22	41.01	253.97	118.52	381.71	356.19	292.89	294.89	133.59	132.78	135.22	115.28	113.23	103.52	456.565	471.05	467.60	371.16	111.53	125.73	130.19	0.00	450.860	380.90	389.27	150.76	267.67	275.4	357.85
Fiat X1-9	140.605	140.300	61.41	235.72	410.92	202.37	488.56	79.34	102.67	161.97	162.76	335.35	333.90	335.65	564.28	562.55	553.34	6.235	22.95	16.74	81.92	347.74	333.13	479.75	450.86	0.000	73.36	70.93	496.27	196.29	522.8	93.08
Porsche 914-2	69.950	70.285	23.17	173.47	340.90	140.11	417.91	65.09	38.42	96.51	98.51	266.09	265.05	267.60	496.19	494.13	484.12	79.180	92.84	89.81	20.07	277.42	263.68	409.09	380.90	73.355	0.00	54.09	423.34	123.64	450.1	28.16
Lotus Europa	84.977	84.912	45.10	185.83	349.27	162.48	426.68	115.46	81.09	103.18	105.18	274.46	273.42	275.97	504.56	502.50	492.49	70.967	84.28	81.27	58.03	285.85	272.04	417.86	389.27	70.932	54.09	0.00	433.01	132.41	458.9	43.21
Ford Pantera L	356.030	356.335	435.33	267.73	109.76	303.77	35.25	429.95	403.92	337.17	336.37	168.75	169.51	169.06	195.25	173.19	133.19	501.980	516.18	512.74	421.33	156.48	170.74	27.57	150.76	496.275	423.34	433.01	0.00	304.90	127.0	403.20
Ferrari Dino	85.690	86.205	134.89	193.62	227.66	166.87	298.95	133.39	104.38	83.87	85.87	150.85	149.81	152.36	378.95	376.89	366.88	202.000	216.49	213.03	120.59	214.18	200.44	289.67	267.67	196.295	123.64	132.41	304.90	0.00	326.5	103.30
Maserati Bora	382.170	382.475	461.37	293.06	234.64	328.61	158.27	455.63	430.10	362.81	362.01	193.37	194.13	192.48	318.27	296.21	256.20	528.480	542.97	539.51	447.07	214.60	200.43	148.97	275.38	522.775	450.12	458.89	126.98	326.48	0.0	429.76
Volvo 142E	47.910	47.285	32.13	145.31	317.90	119.95	395.55	78.93	41.06	68.95	70.35	242.33	240.89	242.64	471.27	469.53	460.32	98.780	113.36	109.75	18.14	254.72	240.12	386.73	357.85	93.075	28.16	43.21	403.20	103.30	429.8	0.00

c_euclidean <- as.matrix(dist(cars, method = "euclidean"))
kbl(c_euclidean) %>%
  kable_paper(bootstrap_options = "striped", full_width = F, font_size = 11)

	Mazda RX4	Mazda RX4 Wag	Datsun 710	Hornet 4 Drive	Hornet Sportabout	Valiant	Duster 360	Merc 240D	Merc 230	Merc 280	Merc 280C	Merc 450SE	Merc 450SL	Merc 450SLC	Cadillac Fleetwood	Lincoln Continental	Chrysler Imperial	Fiat 128	Honda Civic	Toyota Corolla	Toyota Corona	Dodge Challenger	AMC Javelin	Camaro Z28	Pontiac Firebird	Fiat X1-9	Porsche 914-2	Lotus Europa	Ford Pantera L	Ferrari Dino	Maserati Bora	Volvo 142E
Mazda RX4	0.0000	0.6153	54.91	98.11	210.34	65.47	241.41	50.15	25.47	15.364	15.672	135.4307	135.4014	135.480	326.34	318.05	304.72	93.268	102.83	100.604	42.307	163.12	149.60	233.22	248.68	92.505	44.403	65.73	245.42	66.77	265.65	39.19
Mazda RX4 Wag	0.6153	0.0000	54.89	98.10	210.34	65.44	241.41	50.11	25.33	15.296	15.584	135.4255	135.3960	135.472	326.34	318.04	304.72	93.253	102.82	100.589	42.266	163.11	149.60	233.22	248.68	92.494	44.407	65.74	245.43	66.78	265.65	39.16
Datsun 710	54.9086	54.8915	0.00	150.99	265.08	117.75	294.48	49.66	33.18	66.936	67.026	189.1955	189.1632	189.234	381.09	372.80	359.30	40.993	52.77	47.654	12.966	217.78	204.32	286.00	303.36	39.882	13.136	25.09	297.29	90.24	309.77	20.69
Hornet 4 Drive	98.1125	98.0959	150.99	0.00	121.03	33.55	169.43	121.27	118.24	91.422	91.461	72.4964	72.4314	72.572	234.44	227.97	218.15	184.969	191.55	192.671	138.530	72.44	61.36	163.66	156.22	184.447	139.158	163.24	180.11	130.55	229.34	137.04
Hornet Sportabout	210.3374	210.3359	265.08	121.03	0.00	152.12	70.18	241.51	233.49	199.334	199.341	84.3888	84.3684	84.433	116.28	108.06	97.20	302.038	310.03	309.558	252.333	48.98	61.43	70.97	40.01	301.567	254.145	272.36	89.59	215.07	170.71	248.01
Valiant	65.4718	65.4392	117.75	33.55	152.12	0.00	194.61	89.59	85.01	60.291	60.266	90.6970	90.6770	90.709	266.63	259.63	248.77	152.115	158.96	159.830	105.288	103.43	91.04	187.85	188.53	151.438	106.059	130.82	203.02	106.57	242.44	104.19
Duster 360	241.4076	241.4089	294.48	169.43	70.18	194.61	0.00	281.30	265.88	227.900	227.881	106.4084	106.4321	106.401	119.02	104.51	81.43	333.979	344.05	341.022	282.051	103.90	110.31	10.08	80.81	333.484	285.199	296.46	21.27	226.20	107.72	275.14
Merc 240D	50.1533	50.1146	49.66	121.27	241.51	89.59	281.30	0.00	33.69	64.775	64.890	175.1620	175.1190	175.212	355.66	348.99	338.20	68.611	72.00	76.281	44.085	192.86	180.55	273.84	277.46	67.916	39.447	72.90	287.52	113.30	313.86	53.68
Merc 230	25.4683	25.3285	33.18	118.24	233.49	85.01	265.88	33.69	0.00	39.299	39.387	159.8180	159.7761	159.850	349.28	341.32	328.43	69.313	78.54	76.773	21.096	185.83	172.53	257.75	271.39	68.556	22.118	50.11	269.98	80.66	288.88	24.69
Merc 280	15.3642	15.2957	66.94	91.42	199.33	60.29	227.90	64.78	39.30	0.000	1.523	122.3642	122.3444	122.394	315.39	306.68	292.71	106.505	116.73	113.629	54.364	152.89	139.15	219.55	238.17	105.741	57.646	74.14	231.41	56.84	250.59	48.81
Merc 280C	15.6725	15.5838	67.03	91.46	199.34	60.27	227.88	64.89	39.39	1.523	0.000	122.3461	122.3355	122.359	315.36	306.64	292.70	106.683	116.87	113.812	54.426	152.87	139.12	219.53	238.18	105.856	57.847	74.38	231.40	56.90	250.58	48.89
Merc 450SE	135.4307	135.4255	189.20	72.50	84.39	90.70	106.41	175.16	159.82	122.364	122.346	0.0000	0.9826	1.373	197.88	187.60	171.66	228.325	238.01	235.518	176.602	51.80	41.21	98.72	124.34	227.763	179.503	193.31	112.82	131.03	157.16	170.45
Merc 450SL	135.4014	135.3960	189.16	72.43	84.37	90.68	106.43	175.12	159.78	122.344	122.335	0.9826	0.0000	2.138	197.92	187.63	171.67	228.259	237.96	235.448	176.573	51.82	41.24	98.76	124.32	227.717	179.455	193.24	112.83	131.01	157.18	170.42
Merc 450SLC	135.4795	135.4723	189.23	72.57	84.43	90.71	106.40	175.21	159.85	122.394	122.359	1.3726	2.1383	0.000	197.85	187.57	171.66	228.405	238.08	235.602	176.631	51.80	41.19	98.70	124.37	227.818	179.572	193.40	112.83	131.07	157.17	170.48
Cadillac Fleetwood	326.3396	326.3355	381.09	234.44	116.28	266.63	119.02	355.66	349.28	315.390	315.356	197.8843	197.9154	197.853	0.00	15.62	40.84	417.769	425.33	425.345	368.320	163.63	176.86	128.46	78.54	417.249	370.096	388.54	134.81	328.54	214.94	364.10
Lincoln Continental	318.0470	318.0429	372.80	227.97	108.06	259.63	104.51	348.99	341.32	306.676	306.641	187.5997	187.6331	187.567	15.62	0.00	25.37	410.021	417.97	417.543	360.027	156.28	169.09	114.09	72.69	409.500	362.014	379.47	119.72	317.71	199.34	355.40
Chrysler Imperial	304.7203	304.7169	359.30	218.15	97.20	248.77	81.43	338.20	328.43	292.715	292.699	171.6601	171.6743	171.656	40.84	25.37	0.00	397.228	405.82	404.634	346.572	145.92	157.81	91.29	68.20	396.760	348.847	364.60	95.38	300.16	174.29	341.29
Fiat 128	93.2680	93.2531	40.99	184.97	302.04	152.12	333.98	68.61	69.31	106.505	106.683	228.3248	228.2592	228.405	417.77	410.02	397.23	0.000	14.56	7.832	52.880	254.24	241.12	325.66	339.59	5.147	49.064	49.91	337.16	128.40	349.53	61.33
Honda Civic	102.8308	102.8239	52.77	191.55	310.03	158.96	344.05	72.00	78.54	116.728	116.871	238.0142	237.9588	238.083	425.33	417.97	405.82	14.559	0.00	14.348	63.899	261.85	248.96	335.89	347.07	14.781	59.459	64.05	347.83	141.70	362.16	73.38
Toyota Corolla	100.6040	100.5888	47.65	192.67	309.56	159.83	341.02	76.28	76.77	113.629	113.812	235.5184	235.4482	235.602	425.34	417.54	404.63	7.832	14.35	0.000	59.845	261.83	248.69	332.66	347.17	10.392	56.324	53.88	343.99	133.47	355.26	67.72
Toyota Corona	42.3075	42.2659	12.97	138.53	252.33	105.29	282.05	44.09	21.10	54.364	54.426	176.6021	176.5727	176.631	368.32	360.03	346.57	52.880	63.90	59.845	0.000	205.03	191.56	273.63	290.62	51.841	8.654	31.25	285.13	82.24	299.19	12.25
Dodge Challenger	163.1151	163.1134	217.78	72.44	48.98	103.43	103.90	192.86	185.83	152.893	152.872	51.8009	51.8243	51.801	163.63	156.28	145.92	254.237	261.85	261.834	205.035	0.00	14.02	100.30	85.81	253.662	206.645	226.50	118.75	174.93	185.91	201.37
AMC Javelin	149.6047	149.6015	204.32	61.36	61.43	91.04	110.31	180.55	172.53	139.146	139.118	41.2080	41.2412	41.193	176.86	169.09	157.81	241.120	248.96	248.692	191.558	14.02	0.00	105.61	99.28	240.527	193.308	212.76	123.38	161.11	185.16	187.70
Camaro Z28	233.2229	233.2249	286.00	163.66	70.97	187.85	10.08	273.84	257.75	219.552	219.528	98.7203	98.7567	98.704	128.46	114.09	91.29	325.664	335.89	332.659	273.632	100.30	105.61	0.00	86.27	325.149	276.892	287.62	19.36	216.75	102.59	266.53
Pontiac Firebird	248.6780	248.6762	303.36	156.22	40.01	188.53	80.81	277.46	271.39	238.173	238.181	124.3369	124.3204	124.373	78.54	72.69	68.20	339.586	347.07	347.167	290.624	85.81	99.28	86.27	0.00	339.140	292.165	311.39	101.74	255.06	188.32	286.75
Fiat X1-9	92.5048	92.4940	39.88	184.45	301.57	151.44	333.48	67.92	68.56	105.741	105.856	227.7628	227.7173	227.818	417.25	409.50	396.76	5.147	14.78	10.392	51.841	253.66	240.53	325.15	339.14	0.000	48.377	49.84	336.70	127.82	349.12	60.41
Porsche 914-2	44.4034	44.4074	13.14	139.16	254.15	106.06	285.20	39.45	22.12	57.646	57.847	179.5034	179.4551	179.572	370.10	362.01	348.85	49.064	59.46	56.324	8.654	206.65	193.31	276.89	292.16	48.377	0.000	33.77	288.59	87.91	303.92	18.76
Lotus Europa	65.7328	65.7363	25.09	163.24	272.36	130.82	296.46	72.90	50.11	74.144	74.382	193.3074	193.2408	193.397	388.54	379.47	364.60	49.911	64.05	53.885	31.254	226.50	212.76	287.62	311.39	49.841	33.768	0.00	297.54	80.46	303.28	27.81
Ford Pantera L	245.4247	245.4294	297.29	180.11	89.59	203.02	21.27	287.52	269.98	231.408	231.402	112.8182	112.8297	112.833	134.81	119.72	95.38	337.164	347.83	343.992	285.129	118.75	123.38	19.36	101.74	336.702	288.585	297.54	0.00	224.46	86.94	277.48
Ferrari Dino	66.7661	66.7764	90.24	130.55	215.07	106.57	226.20	113.30	80.66	56.837	56.899	131.0272	131.0078	131.070	328.54	317.71	300.16	128.395	141.70	133.471	82.236	174.93	161.11	216.75	255.06	127.821	87.911	80.46	224.46	0.00	223.53	70.48
Maserati Bora	265.6454	265.6491	309.77	229.34	170.71	242.44	107.72	313.86	288.88	250.587	250.577	157.1633	157.1769	157.168	214.94	199.34	174.29	349.534	362.16	355.260	299.187	185.91	185.16	102.59	188.32	349.120	303.922	303.28	86.94	223.53	0.00	289.12
Volvo 142E	39.1894	39.1626	20.69	137.04	248.01	104.19	275.14	53.68	24.69	48.805	48.889	170.4501	170.4225	170.484	364.10	355.40	341.29	61.330	73.38	67.719	12.251	201.37	187.70	266.53	286.75	60.412	18.756	27.81	277.48	70.48	289.12	0.00

c_minkowski_p3 <- as.matrix(dist(cars, method = "minkowski", p=3))
kbl(c_minkowski_p3) %>%
  kable_paper(bootstrap_options = "striped", full_width = F, font_size = 11)

	Mazda RX4	Mazda RX4 Wag	Datsun 710	Hornet 4 Drive	Hornet Sportabout	Valiant	Duster 360	Merc 240D	Merc 230	Merc 280	Merc 280C	Merc 450SE	Merc 450SL	Merc 450SLC	Cadillac Fleetwood	Lincoln Continental	Chrysler Imperial	Fiat 128	Honda Civic	Toyota Corolla	Toyota Corona	Dodge Challenger	AMC Javelin	Camaro Z28	Pontiac Firebird	Fiat X1-9	Porsche 914-2	Lotus Europa	Ford Pantera L	Ferrari Dino	Maserati Bora	Volvo 142E
Mazda RX4	0.0000	0.5771	52.60	98.00	202.26	65.02	218.70	48.35	22.07	13.839	13.900	123.7681	123.7671	123.770	314.91	304.23	287.16	85.458	92.64	92.675	40.37	158.85	145.02	210.44	241.58	85.129	41.13	64.97	219.81	65.27	242.12	39.01
Mazda RX4 Wag	0.5771	0.0000	52.60	98.00	202.26	65.02	218.70	48.35	22.02	13.832	13.886	123.7681	123.7671	123.770	314.91	304.23	287.16	85.457	92.64	92.674	40.37	158.85	145.02	210.44	241.58	85.129	41.13	64.97	219.81	65.27	242.12	39.00
Datsun 710	52.6050	52.6043	0.00	150.07	254.86	117.05	269.23	44.44	32.83	62.040	62.047	175.2629	175.2618	175.265	367.51	356.82	339.60	35.758	46.88	41.901	12.26	211.39	197.60	260.54	294.14	35.347	12.41	21.96	268.47	84.45	277.47	18.47
Hornet 4 Drive	98.0020	98.0016	150.07	0.00	110.13	33.05	152.15	114.20	117.28	90.491	90.493	70.3917	70.3878	70.399	220.08	211.06	197.95	180.193	184.24	187.784	137.94	65.44	54.47	147.97	146.40	179.884	137.82	162.91	164.57	119.76	225.53	137.00
Hornet Sportabout	202.2631	202.2631	254.86	110.13	0.00	141.00	70.01	223.39	222.70	193.658	193.658	84.2065	84.2061	84.208	112.73	102.11	87.87	286.664	291.78	294.135	242.62	44.77	57.62	70.08	40.00	286.367	243.09	266.04	89.03	215.00	162.63	240.67
Valiant	65.0185	65.0169	117.05	33.05	141.00	0.00	173.30	82.42	84.25	57.988	57.987	82.0816	82.0812	82.082	252.35	242.78	228.25	147.263	151.52	154.851	104.92	96.39	83.60	167.48	178.66	146.934	104.80	129.95	181.92	94.92	232.74	104.00
Duster 360	218.7023	218.7023	269.23	152.15	70.01	173.30	0.00	251.11	240.49	207.531	207.531	95.5220	95.5227	95.522	113.68	100.89	80.18	303.661	311.32	310.558	257.38	97.66	101.09	10.00	74.12	303.390	259.26	275.42	19.66	217.45	97.77	252.86
Merc 240D	48.3525	48.3499	44.44	114.20	223.39	82.42	251.11	0.00	33.07	61.821	61.834	155.9849	155.9827	155.988	334.27	325.03	310.63	68.042	71.08	75.652	39.52	178.72	166.00	244.03	260.59	67.707	34.99	64.66	255.97	113.01	288.54	49.44
Merc 230	22.0659	22.0220	32.83	117.28	222.70	84.25	240.49	33.07	0.00	34.583	34.596	145.4169	145.4151	145.419	335.20	324.76	308.10	64.215	70.87	71.597	20.73	178.96	165.26	232.28	261.72	63.872	20.77	46.71	241.67	80.03	261.77	21.96
Merc 280	13.8393	13.8315	62.04	90.49	193.66	57.99	207.53	61.82	34.58	0.000	1.436	113.2359	113.2355	113.237	306.37	295.41	277.80	96.195	104.33	103.133	49.97	150.69	136.75	199.03	233.26	95.874	51.79	72.65	207.79	53.39	228.32	47.02
Merc 280C	13.9004	13.8861	62.05	90.49	193.66	57.99	207.53	61.83	34.60	1.436	0.000	113.2354	113.2353	113.236	306.37	295.41	277.80	96.224	104.34	103.164	49.98	150.69	136.75	199.03	233.26	95.886	51.82	72.69	207.79	53.39	228.32	47.03
Merc 450SE	123.7681	123.7681	175.26	70.39	84.21	82.08	95.52	155.98	145.42	113.236	113.235	0.0000	0.9191	1.253	196.34	184.62	165.73	209.102	216.24	216.186	163.20	46.75	36.70	88.07	124.20	208.815	164.68	183.75	100.60	130.80	155.22	159.63
Merc 450SL	123.7671	123.7671	175.26	70.39	84.21	82.08	95.52	155.98	145.42	113.236	113.235	0.9191	0.0000	2.105	196.34	184.62	165.73	209.097	216.24	216.180	163.20	46.75	36.70	88.08	124.20	208.812	164.68	183.74	100.60	130.80	155.22	159.63
Merc 450SLC	123.7703	123.7702	175.26	70.40	84.21	82.08	95.52	155.99	145.42	113.237	113.236	1.2532	2.1048	0.000	196.34	184.62	165.73	209.110	216.25	216.194	163.20	46.75	36.70	88.07	124.20	208.818	164.68	183.76	100.60	130.81	155.22	159.63
Cadillac Fleetwood	314.9128	314.9128	367.51	220.08	112.73	252.35	113.68	334.27	335.20	306.373	306.372	196.3371	196.3381	196.336	0.00	13.97	36.46	399.027	403.78	406.539	355.26	156.31	169.94	123.42	73.74	398.723	355.66	378.74	125.51	327.09	193.07	353.38
Lincoln Continental	304.2321	304.2320	356.82	211.06	102.11	242.78	100.89	325.03	324.76	295.408	295.406	184.6224	184.6235	184.621	13.97	0.00	22.54	388.762	393.85	396.228	344.58	146.40	159.68	110.74	65.48	388.460	345.13	367.56	112.21	315.22	179.13	342.42
Chrysler Imperial	287.1633	287.1633	339.60	197.95	87.87	228.25	80.18	310.63	308.10	277.796	277.796	165.7312	165.7314	165.731	36.46	22.54	0.00	372.243	377.96	379.607	327.39	132.53	144.66	90.14	61.31	371.952	328.23	349.34	90.63	295.64	156.64	324.70
Fiat 128	85.4576	85.4570	35.76	180.19	286.66	147.26	303.66	68.04	64.22	96.195	96.224	209.1024	209.0974	209.110	399.03	388.76	372.24	0.000	14.06	7.626	46.73	242.73	229.16	295.15	325.44	5.103	44.46	47.66	303.51	116.68	312.29	53.89
Honda Civic	92.6398	92.6396	46.88	184.24	291.78	151.52	311.32	71.08	70.87	104.326	104.345	216.2448	216.2412	216.250	403.78	393.85	377.96	14.064	0.00	13.278	56.40	247.55	234.19	303.05	330.10	14.114	52.91	61.65	312.09	129.96	324.30	65.33
Toyota Corolla	92.6746	92.6740	41.90	187.78	294.13	154.85	310.56	75.65	71.60	103.133	103.164	216.1856	216.1799	216.194	406.54	396.23	379.61	7.626	13.28	0.000	53.41	250.25	236.65	301.97	332.96	9.215	51.58	49.93	310.08	120.22	316.95	59.57
Toyota Corona	40.3718	40.3681	12.26	137.94	242.62	104.92	257.38	39.52	20.73	49.974	49.978	163.1983	163.1975	163.200	355.26	344.58	327.39	46.730	56.40	53.406	0.00	199.16	185.36	248.76	281.91	46.325	7.08	27.35	256.97	78.85	268.71	12.02
Dodge Challenger	158.8524	158.8524	211.39	65.44	44.77	96.39	97.66	178.72	178.96	150.691	150.690	46.7467	46.7475	46.747	156.31	146.40	132.53	242.729	247.55	250.247	199.16	0.00	14.00	96.20	82.77	242.420	199.45	223.26	114.92	173.18	185.05	197.59
AMC Javelin	145.0249	145.0249	197.60	54.47	57.62	83.60	101.09	166.00	165.26	136.753	136.752	36.6995	36.7013	36.699	169.94	159.68	144.66	229.160	234.19	236.654	185.36	14.00	0.00	98.47	96.56	228.849	185.72	209.31	116.60	159.21	185.00	183.69
Camaro Z28	210.4438	210.4439	260.54	147.97	70.08	167.48	10.00	244.03	232.28	199.033	199.032	88.0736	88.0751	88.073	123.42	110.74	90.14	295.149	303.05	301.969	248.76	96.20	98.47	0.00	77.65	294.879	250.79	266.22	19.02	207.69	94.61	243.99
Pontiac Firebird	241.5790	241.5790	294.14	146.40	40.00	178.66	74.12	260.59	261.72	233.265	233.265	124.2032	124.2029	124.204	73.74	65.48	61.31	325.436	330.10	332.961	281.91	82.77	96.56	77.65	0.00	325.138	282.20	305.76	93.70	255.00	171.75	280.23
Fiat X1-9	85.1290	85.1287	35.35	179.88	286.37	146.93	303.39	67.71	63.87	95.874	95.886	208.8145	208.8122	208.818	398.72	388.46	371.95	5.103	14.11	9.215	46.33	242.42	228.85	294.88	325.14	0.000	44.15	47.63	303.25	116.55	312.13	53.58
Porsche 914-2	41.1287	41.1288	12.41	137.82	243.09	104.80	259.26	34.99	20.77	51.792	51.823	164.6791	164.6763	164.684	355.66	345.13	328.23	44.463	52.91	51.577	7.08	199.45	185.72	250.79	282.20	44.155	0.00	29.90	259.41	84.72	273.37	18.11
Lotus Europa	64.9693	64.9694	21.96	162.91	266.04	129.95	275.42	64.66	46.71	72.654	72.692	183.7476	183.7427	183.755	378.74	367.56	349.34	47.660	61.65	49.932	27.35	223.26	209.31	266.22	305.76	47.627	29.90	0.00	272.36	71.39	269.96	26.29
Ford Pantera L	219.8090	219.8091	268.47	164.57	89.03	181.92	19.66	255.97	241.67	207.793	207.793	100.5964	100.5967	100.597	125.51	112.21	90.63	303.505	312.09	310.084	256.97	114.92	116.60	19.02	93.70	303.252	259.41	272.36	0.00	211.40	78.46	251.41
Ferrari Dino	65.2661	65.2663	84.45	119.76	215.00	94.92	217.45	113.01	80.03	53.390	53.392	130.8042	130.8038	130.805	327.09	315.22	295.64	116.685	129.96	120.223	78.85	173.18	159.21	207.69	255.00	116.548	84.72	71.39	211.40	0.00	199.10	67.05
Maserati Bora	242.1232	242.1232	277.47	225.53	162.63	232.74	97.77	288.54	261.77	228.320	228.319	155.2239	155.2241	155.224	193.07	179.13	156.64	312.291	324.30	316.955	268.71	185.05	185.00	94.61	171.75	312.127	273.37	269.96	78.46	199.10	0.00	259.01
Volvo 142E	39.0061	39.0048	18.47	137.00	240.67	104.00	252.86	49.44	21.96	47.022	47.027	159.6292	159.6284	159.631	353.38	342.42	324.70	53.893	65.33	59.570	12.02	197.59	183.69	243.99	280.23	53.578	18.11	26.29	251.41	67.05	259.01	0.00

kbl(data.frame(rowSums(c_manhattan), rowSums(c_euclidean), rowSums(c_minkowski_p3))) %>%
  kable_minimal(full_width = F)

	rowSums.c_manhattan.	rowSums.c_euclidean.	rowSums.c_minkowski_p3.
Mazda RX4	5701	4086	3837
Mazda RX4 Wag	5688	4086	3837
Datsun 710	6698	4798	4507
Hornet 4 Drive	5751	4317	4126
Hornet Sportabout	7149	5424	5200
Valiant	5764	4163	3927
Duster 360	8637	5990	5482
Merc 240D	6816	4750	4385
Merc 230	6207	4336	4051
Merc 280	5614	4030	3787
Merc 280C	5626	4032	3787
Merc 450SE	6044	4320	4036
Merc 450SL	6027	4320	4037
Merc 450SLC	6054	4322	4037
Cadillac Fleetwood	10794	8158	7838
Lincoln Continental	10637	7877	7509
Chrysler Imperial	10284	7486	7042
Fiat 128	8206	5678	5281
Honda Civic	8650	5960	5505
Toyota Corolla	8521	5880	5473
Toyota Corona	6389	4562	4282
Dodge Challenger	6330	4729	4536
AMC Javelin	6124	4540	4336
Camaro Z28	8478	5841	5338
Pontiac Firebird	8115	6254	6020
Fiat X1-9	8064	5664	5274
Porsche 914-2	6573	4607	4306
Lotus Europa	6920	5034	4798
Ford Pantera L	9026	6180	5610
Ferrari Dino	6088	4658	4483
Maserati Bora	10514	7339	6733
Volvo 142E	6172	4513	4268

The results above make mathematical sense if you treat all of the variables as numeric, however, I am not sure if vs and am should be treated as numeric since they are actually categorical? But for example let’s just look at the Mazda RX4 and the Hornet 4 Drive…

Maz_Horn <- cars[c(1,4),]
Maz_Horn

If we take absolute value of the difference between each variable and just add them up we get the Manhattan distance…

Mazda <- Maz_Horn[1,]
Hornet <- Maz_Horn[2,]
differences <- Mazda-Hornet
rownames(differences) <- 'differences'

differences

sum(abs(Mazda-Hornet))

## [1] 108.8

If we take the difference between each variable square each one then add them up and finally take the square root we get the euclidean distance…

differences_sq <- (Mazda-Hornet)^2
rownames(differences_sq) <- 'differences_squared'
differences_sq

sum(differences_sq)

## [1] 9626

sqrt(sum(differences_sq))

## [1] 98.11

And finally the minkowski distance is the difference between each variable cubed then summed up and then take the cube root…

differences_cubed <- (Mazda-Hornet)^3
rownames(differences_cubed) <- 'differences_cubed'
differences_cubed

sum(abs(differences_cubed))

## [1] 941249

sum(abs(differences_cubed))^(1/3)

## [1] 98

In each case above the numeric codes for the categorical variables, vs and am, are included in the calculations that match the output of the dist function.

Ex. 3

Use the built in data set mtcars to carry out hierarchy clustering using two different distance metrics and compare if they get the same results. Discuss the results.

manhattan_clusters <- hclust(dist(cars, method = "manhattan"))
plot(manhattan_clusters)

euclidean_clusters <- hclust(dist(cars, method = "euclidean"))
plot(euclidean_clusters)

minkowski_clusters <- hclust(dist(cars, method = "minkowski", p=3))
plot(minkowski_clusters)

All 3 distance measures give similar but not exactly the same results. The euclidean and minkowski models are more similar than either of those are to the manhattan model however. This makes sense since the calculations and numeric values for the euclidean and minkowski distances are very similar as well.

Ex. 4

Load the well-known Fisher’s iris flower data set that consists of 150 samples for three species (50 samples each species). The four measures or features are the lengths and widths of sepals and petals. Use kNN clustering to analyze this iris data set by selecting 120 samples for training and 30 samples for testing.

set.seed(42)
index <- createDataPartition(iris$Species, p = 0.80, list = FALSE)
train <- iris[index, ]
test <- iris[-index, ]

knn_iris <- train(Species ~ ., data = train, 
                 method = "knn")
knn_iris

## k-Nearest Neighbors 
## 
## 120 samples
##   4 predictor
##   3 classes: 'setosa', 'versicolor', 'virginica' 
## 
## No pre-processing
## Resampling: Bootstrapped (25 reps) 
## Summary of sample sizes: 120, 120, 120, 120, 120, 120, ... 
## Resampling results across tuning parameters:
## 
##   k  Accuracy  Kappa 
##   5  0.9710    0.9560
##   7  0.9755    0.9627
##   9  0.9743    0.9610
## 
## Accuracy was used to select the optimal model using the largest value.
## The final value used for the model was k = 7.

pred_iris <- predict(knn_iris, newdata = test)
confusionMatrix(pred_iris, test$Species)

## Confusion Matrix and Statistics
## 
##             Reference
## Prediction   setosa versicolor virginica
##   setosa         10          0         0
##   versicolor      0          9         1
##   virginica       0          1         9
## 
## Overall Statistics
##                                           
##                Accuracy : 0.933           
##                  95% CI : (0.779, 0.992)  
##     No Information Rate : 0.333           
##     P-Value [Acc > NIR] : 0.00000000000875
##                                           
##                   Kappa : 0.9             
##                                           
##  Mcnemar's Test P-Value : NA              
## 
## Statistics by Class:
## 
##                      Class: setosa Class: versicolor Class: virginica
## Sensitivity                  1.000             0.900            0.900
## Specificity                  1.000             0.950            0.950
## Pos Pred Value               1.000             0.900            0.900
## Neg Pred Value               1.000             0.950            0.950
## Prevalence                   0.333             0.333            0.333
## Detection Rate               0.333             0.300            0.300
## Detection Prevalence         0.333             0.333            0.333
## Balanced Accuracy            1.000             0.925            0.925

Ex. 5

Use the iris data set to carry out k-means clustering. Compare the results to the actual classes and estimate the clustering accuracy.

set.seed(42)
kmeans_iris <- kmeans(iris[,c(1:4)], centers = 3, nstart = 25)
kmeans_iris

## K-means clustering with 3 clusters of sizes 50, 62, 38
## 
## Cluster means:
##   Sepal.Length Sepal.Width Petal.Length Petal.Width
## 1        5.006       3.428        1.462       0.246
## 2        5.902       2.748        4.394       1.434
## 3        6.850       3.074        5.742       2.071
## 
## Clustering vector:
##   [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
##  [38] 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
##  [75] 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 3 3 3 3 2 3 3 3 3
## [112] 3 3 2 2 3 3 3 3 2 3 2 3 2 3 3 2 2 3 3 3 3 3 2 3 3 3 3 2 3 3 3 2 3 3 3 2 3
## [149] 3 2
## 
## Within cluster sum of squares by cluster:
## [1] 15.15 39.82 23.88
##  (between_SS / total_SS =  88.4 %)
## 
## Available components:
## 
## [1] "cluster"      "centers"      "totss"        "withinss"     "tot.withinss"
## [6] "betweenss"    "size"         "iter"         "ifault"

conf_mat <- table(kmeans_iris$cluster, iris$Species)
conf_mat

##    
##     setosa versicolor virginica
##   1     50          0         0
##   2      0         48        14
##   3      0          2        36

accuracy <- sum(diag(conf_mat))/sum(conf_mat)
accuracy

## [1] 0.8933