| SOCR ≫ | BPAD Website ≫ | BPAD GitHub ≫ |
This BPAD/SOCR case-study is focused on understanding clinical data (spreadsheets), imaging data (MRI), and extraction of imaging-derived biomarkers (morphometric measures). The joint clinical and imaging data can be used for prediction, classification, segmentation, modeling, and forecasting of kidney stones and tumors using imaging and clinical data.
This kidney case-study can be used for building some simple models (e.g., correlational or linear-models using the meta-data, or imaging models computing the size, volume, diameter, surface area etc. of the kidney mask volumes, which are small, see attached 3D NII.gz kidney stone segmentation-mask volume).
Technical Details
We’ll start with importing the clinical meta-data, which is available in a .csv file format on Canvas.
Kidney_dataset <- read.csv("https://umich.instructure.com/files/29594707/download?download_frd=1",
header = TRUE)Below we describe the types of each variable in the clinical meta-data table and provide appropriate feature summaries for these quantitative and qualitative variables.
## Kidney_dataset
##
## 63 Variables 210 Observations
## --------------------------------------------------------------------------------
## case_id
## n missing distinct
## 210 0 210
##
## lowest : case_00000 case_00001 case_00002 case_00003 case_00004
## highest: case_00205 case_00206 case_00207 case_00208 case_00209
## --------------------------------------------------------------------------------
## age_at_nephrectomy
## n missing distinct Info Mean Gmd .05 .10
## 210 0 59 0.999 58.35 15.77 35.00 39.80
## .25 .50 .75 .90 .95
## 50.00 61.00 68.00 74.00 77.55
##
## lowest : 1 11 12 17 21, highest: 82 83 84 86 90
## --------------------------------------------------------------------------------
## gender
## n missing distinct
## 210 0 2
##
## Value female male
## Frequency 87 123
## Proportion 0.414 0.586
## --------------------------------------------------------------------------------
## body_mass_index
## n missing distinct Info Mean Gmd .05 .10
## 210 0 198 1 31.17 7.293 22.05 23.72
## .25 .50 .75 .90 .95
## 26.47 30.07 35.61 40.07 42.87
##
## lowest : 16.2 18.92 19.96 20.13 20.16, highest: 45.6 46.62 48.24 48.42 49.61
## --------------------------------------------------------------------------------
## comorbidities.myocardial_infarction
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 201 9
## Proportion 0.957 0.043
## --------------------------------------------------------------------------------
## comorbidities.congestive_heart_failure
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 204 6
## Proportion 0.971 0.029
## --------------------------------------------------------------------------------
## comorbidities.peripheral_vascular_disease
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 201 9
## Proportion 0.957 0.043
## --------------------------------------------------------------------------------
## comorbidities.cerebrovascular_disease
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 206 4
## Proportion 0.981 0.019
## --------------------------------------------------------------------------------
## comorbidities.dementia
## n missing distinct value
## 210 0 1 false
##
## Value false
## Frequency 210
## Proportion 1
## --------------------------------------------------------------------------------
## comorbidities.copd
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 199 11
## Proportion 0.948 0.052
## --------------------------------------------------------------------------------
## comorbidities.connective_tissue_disease
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 209 1
## Proportion 0.995 0.005
## --------------------------------------------------------------------------------
## comorbidities.peptic_ulcer_disease
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 209 1
## Proportion 0.995 0.005
## --------------------------------------------------------------------------------
## comorbidities.uncomplicated_diabetes_mellitus
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 172 38
## Proportion 0.819 0.181
## --------------------------------------------------------------------------------
## comorbidities.diabetes_mellitus_with_end_organ_damage
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 208 2
## Proportion 0.99 0.01
## --------------------------------------------------------------------------------
## comorbidities.chronic_kidney_disease
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 194 16
## Proportion 0.924 0.076
## --------------------------------------------------------------------------------
## comorbidities.hemiplegia_from_stroke
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 209 1
## Proportion 0.995 0.005
## --------------------------------------------------------------------------------
## comorbidities.leukemia
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 209 1
## Proportion 0.995 0.005
## --------------------------------------------------------------------------------
## comorbidities.malignant_lymphoma
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 208 2
## Proportion 0.99 0.01
## --------------------------------------------------------------------------------
## comorbidities.localized_solid_tumor
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 181 29
## Proportion 0.862 0.138
## --------------------------------------------------------------------------------
## comorbidities.metastatic_solid_tumor
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 198 12
## Proportion 0.943 0.057
## --------------------------------------------------------------------------------
## comorbidities.mild_liver_disease
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 206 4
## Proportion 0.981 0.019
## --------------------------------------------------------------------------------
## comorbidities.moderate_to_severe_liver_disease
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 209 1
## Proportion 0.995 0.005
## --------------------------------------------------------------------------------
## comorbidities.aids
## n missing distinct value
## 210 0 1 false
##
## Value false
## Frequency 210
## Proportion 1
## --------------------------------------------------------------------------------
## smoking_history
## n missing distinct
## 210 0 3
##
## Value current_smoker never_smoked previous_smoker
## Frequency 34 100 76
## Proportion 0.162 0.476 0.362
## --------------------------------------------------------------------------------
## age_when_quit_smoking
## n missing distinct
## 188 22 36
##
## lowest : 22 24 25 26 29
## highest: 60 64 71 73 not_applicable
## --------------------------------------------------------------------------------
## pack_years
## n missing distinct Info Mean Gmd .05 .10
## 143 67 23 0.658 9.909 16.58 0 0
## .25 .50 .75 .90 .95
## 0 0 8 34 45
##
## Value 0.0 4.2 6.3 8.4 14.7 16.8 18.9 23.1 29.4 33.6 35.7
## Frequency 101 4 2 3 2 1 6 2 7 2 1
## Proportion 0.706 0.028 0.014 0.021 0.014 0.007 0.042 0.014 0.049 0.014 0.007
##
## Value 37.8 39.9 44.1 48.3 63.0 73.5 113.4 210.0
## Frequency 1 2 3 2 1 1 1 1
## Proportion 0.007 0.014 0.021 0.014 0.007 0.007 0.007 0.007
##
## For the frequency table, variable is rounded to the nearest 2.1
## --------------------------------------------------------------------------------
## chewing_tobacco_use
## n missing distinct
## 210 0 2
##
## Value never_or_not_in_last_3mo quit_in_last_3mo
## Frequency 209 1
## Proportion 0.995 0.005
## --------------------------------------------------------------------------------
## alcohol_use
## n missing distinct
## 210 0 3
##
## Value more_than_two_daily never_or_not_in_last_3mo
## Frequency 13 95
## Proportion 0.062 0.452
##
## Value two_or_less_daily
## Frequency 102
## Proportion 0.486
## --------------------------------------------------------------------------------
## intraoperative_complications.blood_transfusion
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 207 3
## Proportion 0.986 0.014
## --------------------------------------------------------------------------------
## intraoperative_complications.injury_to_surrounding_organ
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 207 3
## Proportion 0.986 0.014
## --------------------------------------------------------------------------------
## intraoperative_complications.cardiac_event
## n missing distinct value
## 210 0 1 false
##
## Value false
## Frequency 210
## Proportion 1
## --------------------------------------------------------------------------------
## hospitalization
## n missing distinct
## 210 0 15
##
## lowest : 0 1 10 11 2
## highest: 6 7 8 9 died_before_discharge
## --------------------------------------------------------------------------------
## ischemia_time
## n missing distinct
## 200 10 39
##
## lowest : 0 10 11 12 13
## highest: 5 50 54 8 not_applicable
## --------------------------------------------------------------------------------
## radiographic_size
## n missing distinct Info Mean Gmd .05 .10
## 210 0 84 0.999 4.772 3.211 1.678 2.000
## .25 .50 .75 .90 .95
## 2.500 4.000 5.600 9.611 11.455
##
## lowest : 1.2 1.4 1.5 1.6 1.66, highest: 13.2 13.9 14.1 15 16.2
## --------------------------------------------------------------------------------
## pathologic_size
## n missing distinct Info Mean Gmd .05 .10
## 210 0 77 0.999 4.716 3.217 1.500 1.990
## .25 .50 .75 .90 .95
## 2.400 3.900 5.975 9.200 11.065
##
## lowest : 0.8 1 1.2 1.3 1.4 , highest: 13 13.6 15 16 22.5
## --------------------------------------------------------------------------------
## malignant
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 18 192
## Proportion 0.086 0.914
## --------------------------------------------------------------------------------
## pathology_t_stage
## n missing distinct
## 210 0 7
##
## Value 0 1a 1b 2a 2b 3 4
## Frequency 18 92 41 9 3 44 3
## Proportion 0.086 0.438 0.195 0.043 0.014 0.210 0.014
## --------------------------------------------------------------------------------
## pathology_n_stage
## n missing distinct
## 210 0 3
##
## Value 0 1 X
## Frequency 90 6 114
## Proportion 0.429 0.029 0.543
## --------------------------------------------------------------------------------
## pathology_m_stage
## n missing distinct
## 210 0 3
##
## Value 0 1 X
## Frequency 93 17 100
## Proportion 0.443 0.081 0.476
## --------------------------------------------------------------------------------
## tumor_histologic_subtype
## n missing distinct
## 210 0 12
##
## lowest : angiomyolipoma chromophobe clear_cell_papillary_rcc clear_cell_rcc mest
## highest: papillary rcc_unclassified spindle_cell_neoplasm urothelial wilms
## --------------------------------------------------------------------------------
## tumor_necrosis
## n missing distinct
## 187 23 2
##
## Value false true
## Frequency 144 43
## Proportion 0.77 0.23
## --------------------------------------------------------------------------------
## tumor_isup_grade
## n missing distinct Info Mean Gmd
## 172 38 4 0.826 2.238 0.8374
##
## Value 1.00 1.99 2.98 4.00
## Frequency 26 93 39 14
## Proportion 0.151 0.541 0.227 0.081
##
## For the frequency table, variable is rounded to the nearest 0.03
## --------------------------------------------------------------------------------
## clavien_surgical_complications
## n missing distinct
## 209 1 7
##
## Value 0 1 2 3a 3b 4 5
## Frequency 149 27 15 6 7 4 1
## Proportion 0.713 0.129 0.072 0.029 0.033 0.019 0.005
## --------------------------------------------------------------------------------
## er_visit
## n missing distinct
## 208 2 2
##
## Value false true
## Frequency 191 17
## Proportion 0.918 0.082
## --------------------------------------------------------------------------------
## readmission
## n missing distinct
## 208 2 2
##
## Value false true
## Frequency 187 21
## Proportion 0.899 0.101
## --------------------------------------------------------------------------------
## estimated_blood_loss
## n missing distinct Info Mean Gmd .05 .10
## 209 1 43 0.994 314 339.9 25 50
## .25 .50 .75 .90 .95
## 75 200 400 800 1000
##
## lowest : 0 5 10 20 25, highest: 1200 1300 1500 1800 2000
## --------------------------------------------------------------------------------
## surgery_type
## n missing distinct
## 210 0 3
##
## Value laparoscopic open robotic
## Frequency 28 60 122
## Proportion 0.133 0.286 0.581
## --------------------------------------------------------------------------------
## surgical_procedure
## n missing distinct
## 210 0 2
##
## Value partial_nephrectomy radical_nephrectomy
## Frequency 140 70
## Proportion 0.667 0.333
## --------------------------------------------------------------------------------
## surgical_approach
## n missing distinct
## 210 0 2
##
## Value Retroperitoneal Transperitoneal
## Frequency 39 171
## Proportion 0.186 0.814
## --------------------------------------------------------------------------------
## operative_time
## n missing distinct Info Mean Gmd .05 .10
## 208 2 151 1 243 106.6 120.4 139.4
## .25 .50 .75 .90 .95
## 174.8 229.0 283.2 372.3 437.8
##
## lowest : 44 47 64 91 97, highest: 530 553 586 592 613
## --------------------------------------------------------------------------------
## cytoreductive
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 193 17
## Proportion 0.919 0.081
## --------------------------------------------------------------------------------
## positive_resection_margins
## n missing distinct
## 210 0 2
##
## Value false true
## Frequency 199 11
## Proportion 0.948 0.052
## --------------------------------------------------------------------------------
## last_preop_egfr.value
## n missing distinct
## 153 57 49
##
## lowest : >=90 17 38 41 42 , highest: 86 87 88 89 90
## --------------------------------------------------------------------------------
## last_preop_egfr.days_before_nephrectomy
## n missing distinct Info Mean Gmd .05 .10
## 153 57 54 0.998 23.39 23.93 1.0 2.0
## .25 .50 .75 .90 .95
## 6.0 15.0 36.0 58.2 69.0
##
## lowest : 1 2 3 4 5, highest: 75 76 77 83 90
## --------------------------------------------------------------------------------
## first_postop_egfr.value
## n missing distinct
## 157 53 59
##
## lowest : >=90 25 27 28 29 , highest: 88 89 9 90 age<16
## --------------------------------------------------------------------------------
## first_postop_egfr.days_before_nephrectomy
## n missing distinct Info Mean Gmd .05 .10
## 157 53 112 1 437.6 526.5 78.0 82.2
## .25 .50 .75 .90 .95
## 103.0 154.0 500.0 1162.2 2078.4
##
## lowest : 76 77 78 79 80, highest: 2317 2380 2393 2473 2503
## --------------------------------------------------------------------------------
## last_postop_egfr.value
## n missing distinct
## 157 53 60
##
## lowest : >=90 27 28 30 32 , highest: 88 89 9 90 age<16
## --------------------------------------------------------------------------------
## last_postop_egfr.days_before_nephrectomy
## n missing distinct Info Mean Gmd .05 .10
## 157 53 147 1 1048 818.1 149.8 197.6
## .25 .50 .75 .90 .95
## 442.0 932.0 1487.0 2256.8 2426.0
##
## lowest : 76 77 93 100 114, highest: 2499 2503 2584 2604 3070
## --------------------------------------------------------------------------------
## vital_status
## n missing distinct
## 210 0 2
##
## Value censored dead
## Frequency 189 21
## Proportion 0.9 0.1
## --------------------------------------------------------------------------------
## vital_days_after_surgery
## n missing distinct Info Mean Gmd .05 .10
## 210 0 185 1 956.8 875.2 15.45 23.80
## .25 .50 .75 .90 .95
## 271.25 790.50 1427.50 2288.10 2431.60
##
## lowest : 0 2 8 9 11, highest: 2539 2607 2611 2900 3070
## --------------------------------------------------------------------------------
## voxel_spacing.x_spacing
## n missing distinct Info Mean Gmd .05 .10
## 210 0 125 0.999 0.7966 0.1216 0.6565 0.6832
## .25 .50 .75 .90 .95
## 0.7183 0.7812 0.8682 0.9631 0.9766
##
## lowest : 0.4375 0.451172 0.518 0.546875 0.550781
## highest: 0.970703 0.976562 0.976562 0.984 1.041
## --------------------------------------------------------------------------------
## voxel_spacing.y_spacing
## n missing distinct Info Mean Gmd .05 .10
## 210 0 125 0.999 0.7966 0.1216 0.6565 0.6832
## .25 .50 .75 .90 .95
## 0.7183 0.7812 0.8682 0.9631 0.9766
##
## lowest : 0.4375 0.451172 0.518 0.546875 0.550781
## highest: 0.970703 0.976562 0.976562 0.984 1.041
## --------------------------------------------------------------------------------
## voxel_spacing.z_spacing
## n missing distinct Info Mean Gmd .05 .10
## 210 0 10 0.92 3.162 1.93 0.50 0.50
## .25 .50 .75 .90 .95
## 1.25 3.00 5.00 5.00 5.00
##
## Value 0.500 0.995 1.220 1.490 1.985 2.480 2.975 3.740 3.965 5.000
## Frequency 31 18 5 2 4 23 38 1 2 86
## Proportion 0.148 0.086 0.024 0.010 0.019 0.110 0.181 0.005 0.010 0.410
##
## For the frequency table, variable is rounded to the nearest 0.045
## --------------------------------------------------------------------------------At the initial stages, it may be useful to ignore the comorbidity indicator variables (too many Boolean characteristics of comorbid conditions). A deeper analysis may need to explore these interdependencies.
Next, we perform some simple exploratory data analytics (EDA). These graphs are just examples of univariate and multivariate relations.
radiographic_size by surgery_type# pl <- ggplot(Kidney_dataset, aes(x=surgery_type, y=radiographic_size)) +
# geom_violin() + ggtitle("Distribution of radiographic_size by surgery_type") +
# labs(x = "Surgery Type", y = "Tumor Size") + theme(legend.position = "bottom") # +
# # geom_text(stat = 'count', aes(label = ..count..), vjust = 1)
# ggplotly(pl)
plot_ly(Kidney_dataset, x = ~surgery_type, y = ~radiographic_size, split = ~surgery_type,
type = 'violin', box = list(visible = T), meanline = list(visible = T)) %>%
layout(title="Distribution of radiographic_size by surgery_type",
xaxis=list(title="Surgery Type"), yaxis=list(title="Tumor Size"),
legend = list(title="Surgery Type", orientation = 'h'))vital_days_after_surgery by surgical_approach
and genderplot_ly(Kidney_dataset, type = 'violin') %>%
add_trace(x = ~surgical_approach[Kidney_dataset$gender == 'female'],
y = ~vital_days_after_surgery[Kidney_dataset$gender == 'female'],
legendgroup = 'Yes', scalegroup = 'Yes', name = 'Female',
side='negative', box=list(visible=T), meanline=list(visible=T),
color = I("blue"), points = 'all', jitter = 0.5, pointpos = -1) %>%
add_trace(x = ~surgical_approach[Kidney_dataset$gender == 'male'],
y = ~vital_days_after_surgery[Kidney_dataset$gender == 'male'],
legendgroup = 'Yes', scalegroup = 'Yes', name = 'Male',
side='positive', box=list(visible=T), meanline=list(visible=T),
color = I("yellow"), points = 'all', jitter = 0.5, pointpos = +1) %>%
layout(title="Distribution of vital_days_after_surgery by surgical_approach and `ender",
xaxis=list(title="Surgical Approach"), yaxis=list(title="Vital days after surgery"),
legend = list(title="Surgical Approach", orientation = 'h')) #, vital_days_after_surgery and
body_mass_index2D, 3D, and higher dimensional distributions provide more more holistic depiction of the underlying phenomena, see pub 1 and pub 2.
If the marginal components \(\{X_i\}\) are independent, computing the joint multivariate distribution involves simply the multiplication of the individual marginal probability distributions, i.e., \[f_{{\bf{X}}}({\bf{x}})=\prod_{i=1}^n{f_{X_i}(x_i)}\ .\]
When there are inter-dependencies between variables, computing the joint probability distribution \(f_{{\bf{X}}}({\bf{x}})\) from the corresponding marginal univariate distribution models, \(f_{X_i}(x_i)\), requires linking the marginal distributions via connecting copula function models. For a pair of density functions, \(f_X (x)\) and \(f_Y (y)\), and their corresponding cumulative probability distribution functions, \(F_X (x)\) and \(F_Y (y)\), the joint bivariate cumulative distribution function model, \(F_{XY} (x,y)\) can be expressed via a distribution copula \(C(∙,∙)\) function. Similarly, the bivariate copula density function model, \(f_{XY} (x,y)\) can be explicated in terms of some appropriate probability density copula function, \(c(∙,∙)\).
\[F_{XY} (x,y)=F_X (x)\times F_Y (y)\times
\underbrace{C(F_X (x),F_Y (y))}_{distribution\ copula},\]
\[f_{XY} (x,y)=f_X (x)\times f_Y (y)\times
\underbrace{c(f_X (x),f_Y (y))}_{density\ copula}.\]
Here is an example illustrating the joint bivariate distribution of
vital_days_after_surgery and
body_mass_index.
library(extraDistr)
library(pracma)
# marginal domains, # summary(Kidney_dataset$vital_days_after_surgery)
len <- 300
t <- 1:len
x_dom <- seq(0, 3070, length=len) # range of vital_days_after_surgery
y_dom <- seq(16, 50, length=len) # range of body_mass_index
rho <- 0.3 # initial dependence (correlation), -1 < rho = Corr(t,phi) < 1
# Base function definitions
# f(t), Kidney_dataset$vital_days_after_surgery
pdf_x <- density(Kidney_dataset$vital_days_after_surgery, bw = "sj")
f <- approxfun(pdf_x, rule=1:2) # density as an abstract function
fMax <- max(abs(pdf_x$y))
F1 <- function (x) { # CDF as an abstract function
# message(paste('input to F1 =',x))
if (x < 1) { val = 0 }
else if (x > len) { val = 1 }
else { val=integral(f, xmin=x_dom[1]-200, xmax=x_dom[x],
reltol=1e-12, method="Kronrod")
}
# val=integral(f, xmin=x_dom[1]-200, xmax=x, reltol=1e-12, method="Kronrod")
return(val)
} # F1(100)
# g(phi), Kidney_dataset$body_mass_index
pdf_y <- density(Kidney_dataset$body_mass_index)
g <- approxfun(pdf_y, rule=1:2) # density as an abstract function
gMax <- max(abs(pdf_y$y))
G1 <- function (y) {
if (y < 1) { val = 0 }
else if (y > len) { val = 1 }
else { val = integral(g, y_dom[1], y_dom[y], method = "Kronrod") }
return(val)
} # G1(100)
# Gauss-copula
# C_{XY}^{Gauss} (x,y)=\frac{1}{\sqrt{1-ρ^2}} \times \exp \left (-\frac{ρ^2 (a^2+b^2 )-2abρ)}{2(1-ρ^2 )} \right ),
# where the pairwise correlation $-1 < \rho < 1$ and
# a(x)= \sqrt{2}\times erf^{-1} (2\times F_X (x)-1),
# b(y)= \sqrt{2}\times erf^{-1} (2\times F_Y (y)-1),
# and the error function is
# $$erf^{-1}(s)= \frac{2}{\sqrt{\pi}}\int_{0}^{\infty}{e^{-t^2}dt}.$$
CGauss <- function (t=len/2, s=len/2) {
if (t < 1) { A= -1e+9 }
else if (t >= len) { A= 1e+9 }
else { A = sqrt(2) * erfinv(2*(F1(t))-1) }
if (s < 1) { B= -1e+9 }
else if (s > len) { B= 1e+9 }
else { B = sqrt(2) * erfinv(2*(G1(s))-1) }
# message(paste('A=', A, 'B=', B))
val = (1/(sqrt(1-rho^2))) *
exp(-(rho^2*(A^2+B^2) - 2*A*B*rho)/(2*(1-rho^2)))
if (is.finite(val)) return (val)
else return (0)
} # CGauss(1, 17)
# F(t)
fJoint <- function (t=len/2, s=len/2, rho = rho) {
return ( f(x_dom[t]) * g(y_dom[s]) * CGauss(t, s) )
} # fJoint(1,18)
### Part 1: Classical Time Dynamics: $\phi=0$.
# plot(s, f(s)/fMax)
# plot(s, g(s)/gMax)
# # plot(s, 2*g(rep(0, length(s)))/gMax)
# # plot(s, log(F(t=s, phi=0, rho=0.1)))
# plot(s, log(F(t=s, phi=rep(0, length(s)), rho=0.1)))
# plot(s, CGauss(s))
library(plotly)
y1 <- 1000*f(x_dom) # f: first marginal
y2 <- 10* g(y_dom) # g: second marginal
y3 <- c() # Gaussian Copula (. , s=sMax/2)
for (i in 1:len) y3[i]=CGauss(i, 55) # or y_dom[len/2]
y4 <- c() # joint density f(t,t)
for (i in 1:len) y4[i] <- 10000*fJoint(t=i, s=55, rho)
plot_ly() %>%
add_trace(x=~t, y=~y1, type='scatter', mode='markers+lines',
name="f: first marginal") %>%
add_trace(x=~t, y=~y2, type='scatter', mode='markers+lines',
name="g: second marginal", opacity=1) %>%
add_trace(x=~t, y=~y3, type='scatter', mode='markers+lines',
name="C_Gauss(t,s): Gaussian Copula", opacity=0.5) %>%
add_trace(x=~t, y=~y4, type='scatter',
mode='markers+lines',
name="f(t,s, rho)): Joint Density") %>%
layout(title = "Copula-based Modeling of Joint Distribution",
xaxis = list(title = "time"),
yaxis = list(title = "Functional Value"), # showlegend = FALSE,
legend = list(orientation = 'h'),
scene = list(aspectmode = "manual",
aspectratio = list(x=1, y=1)))Next we can display the raw and smoothed versions of the joint distribution as flat 2D images.
#### Path Kynamics over curves \gamma:\mathbb{R}\to \mathbb{C}
### Part 2: Path Kynamics over curves \gamma:\mathbb{R}\to \mathbb{C}
### Kime tube gamma(k1,k2) on torus ########## CARTESIAN
#### Parametric curve on torus
# define kime-surface
xsurf <- x_dom[t] # longitudinal trajectory
ysurf <- y_dom[t] # phase trajectory
zsurf <- mat<-matrix(0, nrow=len, ncol=len) # joint density f(t,s)
for (i in 1:len) {
for (j in 1:len) {
zsurf[i,j] <- 100000*fJoint(t=i, s=j, rho)
}
if (i %% 30 ==0) message(paste0('Progress: ... ', round((100*i)/len, 0), '%'))
}
# image(zsurf)
plot_ly(z= zsurf, type="heatmap") # %>% hide_colorbar()