In vbfelix/relper: Miscellaneous Functions for Data Preparation, Cleaning, and Visualization
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
library(relper)
library(dplyr)
library(ggplot2)
x <- rnorm(100)
y <- rexp(100)
calc_ functions compute a certain value.
calc_acf
The goal of calc_acf is to compute the auto-correlation function,
given by:
$$\frac{\sum_\limits{t = k+1}^{n}(x_t - \bar{x})(x_{t-k} - \bar{x})}{\sum_\limits{t = 1}^{n} (x_t - \bar{x})^2 },$$
where:
- $x_t$ is a time series of length $n$;
- $x_{t-k}$ is a shifted time series by $k$ units in time;
- $\bar{x}$ is the average of the time series.
calc_acf(x)
If you pass a second vector in the argument y the cross-correlation
will be computed instead:
$$\frac{n \left( \sum_\limits{t = 1}^{n}x_ty_t \right) - \left[\left(\sum_\limits{t = 1}^{n}x_t \right) \left(\sum_\limits{t = 1}^{n}y_t\right) \right]}{\sqrt{\left[n \left( \sum_\limits{t = 1}^{n}x_t^2 \right) - \left( \sum_\limits{t = 1}^{n}x_t \right)^2\right]\left[n \left( \sum_\limits{t = 1}^{n}y_t^2 \right) - \left( \sum_\limits{t = 1}^{n}y_t \right)^2\right]}},$$
where:
- $x_t$ is a time series of length $n$;
- $y_t$ is a time series of length $n$.
calc_acf(x,y)
calc_association
The goal of calc_association is to compute associations metrics.
Contingency
Contingency is a measure of the degree to which two nominal variables are associated. It has a value between 0 and 1, with 0 indicating no relationship and 1 indicating perfect association, and is calculated as follows:
$$\sqrt{\frac{X^2}{n+X^2}},$$
where:
- $X^2$ the chi-square statistic;
- $n$ is the sample size.
calc_association(mtcars$am,mtcars$vs,type = "contingency")
Cramér's V
Cramér's V is a measure of the degree to which two nominal variables are associated. It has a value between 0 and 1, with 0 indicating no relationship and 1 indicating perfect association, and is calculated as follows:
$$\sqrt{\frac{X^2}{n\min(r-1,c-1)}},$$
where:
- $X^2$ the chi-square statistic;
- $n$ is the sample size;
- $r$ is the number of rows in the contingency table;
- $c$ is the number of columns in the contingency table.
calc_association(mtcars$am,mtcars$vs,type = "cramers-v")
Phi
Phi is a measure of association between two nominal dichotomous variables that takes into account a marginal table of the variables given by:
| | y = 0 | y = 1 | Total |
|-----------|-----------|------------|-----------|
| x = 0 | $n_{00}$ | $n_{01}$ | $n_{0.}$ |
| x = 1 | $n_{10}$ | $n_{11}$ | $n_{1.}$ |
| Total | $n_{.0}$ | $n_{.1}$ | $n$ |
Then the phi coefficient is given by:
$$\frac{n_{11}n_{00} - n_{10}n_{01} }{\sqrt{n_{1.}n_{0.}n_{.1}*n_{.0}}}.$$
calc_association(mtcars$am,mtcars$vs,type = "phi")
calc_auc
The goal of calc_auc is to compute the area under a curve (AUC).
x <- seq(-3,3,l = 100)
y <- dnorm(x)
ggplot(tibble(x = x,
y = y),
aes(x,y))+
geom_point()+
plt_theme_y()
The function default compute the area considering the range of x.
#from min to max of x
range(x)
calc_auc(x,y)
df_auc <-
tibble(
x = x,
y = y)
df_auc %>%
ggplot(aes(x,y))+
geom_area(alpha = .7, fill = "chocolate2")+
geom_point()+
plt_theme_y()+
annotate("text",
x = mean(x),y = mean(y),label = round(calc_auc(x,y),3),
fontface = "bold", size = 7)+
geom_vline(xintercept = c(-3,3), linetype = "dashed")+
scale_x_continuous(breaks = -5:5)+
scale_y_continuous(expand = c(0,0))
But you can define the argument limits to get the AUC of that
respective range.
#from -2 to 2
calc_auc(x,y,limits = c(-2,2))
df_auc %>%
ggplot(aes(x,y))+
geom_area(data = df_auc %>%
filter(between(x,-2,2)),
alpha = .7, fill = "chocolate2")+
geom_point()+
plt_theme_y()+
annotate("text",
x = mean(x),y = mean(y),label = round(calc_auc(x,y,limits = c(-2,2)),3),
fontface = "bold", size = 7)+
geom_vline(xintercept = c(-2,2), linetype = "dashed")+
scale_x_continuous(breaks = -5:5)+
scale_y_continuous(expand = c(0,0))
#from -1 to 1
calc_auc(x,y,limits = c(-1,1))
df_auc %>%
ggplot(aes(x,y))+
geom_area(data = df_auc %>%
filter(between(x,-1,1)),
alpha = .7, fill = "chocolate2")+
geom_point()+
plt_theme_y()+
annotate("text",
x = mean(x),y = mean(y),label = round(calc_auc(x,y,limits = c(-1,1)),3),
fontface = "bold", size = 7)+
geom_vline(xintercept = c(-1,1), linetype = "dashed")+
scale_x_continuous(breaks = -5:5)+
scale_y_continuous(expand = c(0,0))
calc_combination
The goal of calc_combination is to compute the number of combinations/permutations. Given that there are a total of $n$ observations and that $r$ will be chosen.
Order matter with repetition
$$n^r.$$
calc_combination(n = 10,r = 4,order_matter = TRUE,with_repetition = TRUE)
Order matter without repetition
$$\frac{n!}{(n-r)!}.$$
calc_combination(n = 10,r = 4,order_matter = TRUE,with_repetition = FALSE)
Order does not matter with repetition
$$\frac{(n+r-1)!}{r!(n-1)!}.$$
calc_combination(n = 10,r = 4,order_matter = FALSE,with_repetition = TRUE)
Order does not matter without repetition
$$\frac{n!}{r!(n-r)!}.$$
calc_combination(n = 10,r = 4,order_matter = FALSE,with_repetition = FALSE)
calc_correlation
The goal of calc_correlation is to compute associations metrics.
Kendall
The Kendall correlation coefficient, also known as the Kendall's Tau coefficient, measures the relationship between two ranked variables.
Maurice Kendall created it, and it is especially useful for analyzing non-linear relationships or ranked data. The coefficient is calculated by counting the number of concordant pairs (ranks in the same order) and discordant pairs (ranks in opposite order) in the data.
$$\frac{n_c-n_d}{\frac{1}{2}*n(n/1)},$$
where:
- $n_c$ is the number of concordant observations;
- $n_d$ is the number of discordant observations;
- $n$ is the number of observations.
calc_correlation(mtcars$hp,mtcars$drat,type = "kendall")
Pearson
The Pearson correlation coefficient quantifies the linear relationship that exists between two continuous variables. It ranges from -1 to 1, indicating the association's strength and direction.
A value of 1 indicates a perfect positive linear relationship, a value of -1 indicates a perfect negative linear relationship, and a value of 0 indicates no linear relationship.
$$\frac{\sigma_{xy}}{\sigma_x\sigma_y},$$
where:
- $\sigma_{xy}$ is the covariance of $x$ and $y$;
- $\sigma_{x}$ is the variance of $x$;
- $\sigma_{y}$ is the variance of $y$.
calc_correlation(mtcars$hp,mtcars$drat,type = "pearson")
Spearman
The Spearman correlation coefficient assesses the strength and direction of a monotonic relationship between two variables, regardless of whether it is linear or non-linear.
It also has a value between -1 and 1, with 1 representing a perfect monotonic relationship and -1 representing a perfect inverse monotonic relationship. A value of 0 indicates that there is no monotonic relationship.
$$1- \frac{6\sum\limits_{i=1}^{n}d_i^2}{n(n^2-1)},$$
where:
- $d_i$ is the difference between the ranks of $x$ and $y$;
- $n$ is the number of observations.
calc_correlation(mtcars$hp,mtcars$drat,type = "spearman")
calc_cv
The goal of calc_cv is to compute the coefficient of variation (CV),
given by:
$$\frac{s}{\bar{x}},$$ where:
- $s$ is the sample standard deviation;
- $\bar{x}$ is the sample mean.
set.seed(123);x <- rexp(n = 100)
calc_cv(x)
If you set the argument as_perc to TRUE, the CV will be multiplied
by 100.
calc_cv(x,as_perc = TRUE)
calc_error
The goal of calc_error is to compute errors metrics.
Mean Absolute Error (MAE)
MAE measures the average absolute difference between the predicted and actual values:
$$\frac{\sum\limits_{i=1}^{n}|X_i-Y_i|}{n}.$$
Mean Absolute Percentage Error (MAPE)
MAPE measures the average percentage difference between the predicted and actual values relative to the actual values:
$$\frac{\sum\limits_{i=1}^{n}\left|\frac{X_i-Y_i}{X_i}\right|}{n}.$$
Mean Squared Error (MSE)
MSE measures the average of the squared differences between the predicted and actual values:
$$\frac{\sum\limits_{i=1}^{n}(X_i-Y_i)^2}{n}.$$
Root Mean Squared Error (RMSE)
RMSE is the square root of the MSE, providing the measure of average prediction error in the same units as the target variable:
$$\sqrt{\text{MSE}}.$$
Root Mean Squared Percentage Error (RMSPE)
RMSPE is the square root of the average of the squared percentage differences between the predicted and actual values relative to the actual values:
$$\sqrt{\frac{\sum\limits_{i=1}^{n}\left(\frac{X_i-Y_i}{X_i}\right)^2}{n}}.$$
calc_kurtosis
The goal of calc_kurtosis is to compute a kurtosis coefficient.
calc_kurtosis(x = x)
Biased
The biased kurtosis coefficient, is given by:
$$\frac{\sum\limits_{i=1}^n(x_i - \bar{x})^4}{n*s_x^4},$$
where:
- $x_i$ is a numeric vector of length $n$;
- $\bar{x}$ is the mean of $x$;
- $s_x$ is the standard deviation of $x$.
calc_kurtosis(x = x,type = "biased")
Excess
The excess kurtosis coefficient, is given by:
$$\frac{\sum\limits_{i=1}^n(x_i - \bar{x})^4}{n*s_x^4}-3,$$
where:
- $x_i$ is a numeric vector of length $n$;
- $\bar{x}$ is the mean of $x$;
- $s_x$ is the standard deviation of $x$.
calc_kurtosis(x = x,type = "excess")
Percentile
The percentile kurtosis coefficient, is given by:
$$\frac{Q_3-Q_1}{P_{90}-P_{10}},$$ where:
- $Q_1$ is the first quartile;
- $Q_3$ is the third quartile;
- $P_{90}$ is the 90th percentile;
- $P_{10}$ is the 10th percentile.
calc_kurtosis(x = x,type = "percentile")
Unbiased
The unbiased kurtosis coefficient, is given by:
$$\frac{(n+1)n}{(n-1)(n-2)(n-3)}\frac{\sum\limits_{i=1}^n(x_i - \bar{x})^4}{ns_x^4} - 3\frac{(n-1)^2}{(n-2)*(n-3)},$$
where:
- $x_i$ is a numeric vector of length $n$;
- $\bar{x}$ is the mean of $x$;
- $s_x$ is the standard deviation of $x$.
calc_kurtosis(x = x,type = "unbiased")
calc_mean
The goal of calc_mean is to compute the mean.
Arithmetic
Simple arithmetic mean
$$\frac{1}{n}\sum\limits_{i=1}^{n}x_i,$$ where:
- $x_i$ is a numeric vector of length $n$.
calc_mean(x = 1:10,type = "arithmetic")
Weighted arithmetic mean
$$\frac{1}{\sum\limits_{i=1}^{n}w_i}\sum\limits_{i=1}^{n}w_ix_i,$$ where:
- $x_i$ is a numeric vector of length $n$;
- $w_i$ is a numeric vector of length $n$, with the respective weights of $x_i$.
calc_mean(x = 1:10,type = "arithmetic",weight = 1:10)
Trimmed arithmetic mean
calc_mean(x = 1:10,type = "arithmetic",trim = .4)
Geometric
$$\sqrt[n]{\prod\limits_{i=1}^{n}x_i} = \sqrt[n]{x_1\times x_2 \times...\times x_n},$$
where:
- $x_i$ is a numeric vector of length $n$.
calc_mean(x = 1:10,type = "geometric")
Harmonic
$$\frac{n}{\sum\limits_{i=1}^{n}\frac{1}{x_i}},$$ where:
- $x_i$ is a numeric vector of length $n$.
calc_mean(x = 1:10,type = "harmonic")
calc_modality
The goal of calc_modality is to compute the number of modes.
calc_modality(x = c("a","a","b","b"))
calc_mode
The goal of calc_mode is to compute the mode.
set.seed(123);cat_var <- sample(letters,100,replace = TRUE)
table(cat_var)
We can see that the letter "y" appears the most, indicating that it is the variable's mode.
calc_mode(cat_var)
calc_peak_density
The goal of calc_peak_density is to compute the peak density value of
a numeric value.
pd_plot <-
ggplot2::ggplot(data = dplyr::tibble(x = x),
ggplot2::aes(x))+
ggplot2::geom_density()+
relper::plt_theme_y()+
ggplot2::scale_x_continuous(breaks = 0:20, expand = c(0,0))+
plt_flip_y_title+
labs(y = "Density (x)")
pd_plot
Assume we want to know what the density's peak value is.
calc_peak_density(x)
x_peak <- relper::calc_peak_density(x)
pd_plot+
ggplot2::geom_vline(
xintercept = relper::calc_peak_density(x),
col = "royalblue4",
size = 1
)+
ggplot2::scale_x_continuous(
breaks = 0:20,
expand = c(0,0),
sec.axis = sec_axis(~.,
breaks = x_peak,
labels = format_num(x_peak,digits = 3)
)
)
calc_perc
The goal of calc_perc is to compute the percentage.
#without main_var
calc_perc(mtcars,grp_var = c(cyl,vs))
#main_var within grp_var
calc_perc(mtcars,grp_var = c(cyl,vs),main_var = vs)
#main_var not within grp_var
calc_perc(mtcars,grp_var = c(cyl),main_var = vs)
calc_skewness
The goal of calc_skewness is to compute a skewness coefficient.
calc_skewness(x = x)
Where different types of coefficients are provided, they are:
Bowley
The Bowley skewness coefficient, is given by:
$$\frac{Q_3+Q_1-2Q_2}{Q_3-Q_1},$$ where:
- $Q_1$ is the first quartile;
- $Q_2$ is the second quartile;
- $Q_3$ is the third quartile.
calc_skewness(x = x,type = "bowley")
Fisher-Pearson
The Fisher-Pearson skewness coefficient, is given by:
$$\frac{\sum_\limits{i=1}^{n}(x_i - \bar{x})^3}{n*(s_x)^3},$$
where:
- $\bar{x}$ is the mean of $x$;
- $x_i$ is a numeric vector of length $n$;
- $s_x$ is the standard deviation of $x$.
calc_skewness(x = x,type = "fisher_pearson")
Kelly
The Kelly skewness coefficient, is given by:
$$\frac{P_{90}+P_{10}-2Q_2}{P_{90}-P_{10}},$$ where:
- $P_{90}$ is the 90th percentile;
- $Q_2$ is the second quartile, i.e., $P_{50}$;
- $P_{10}$ is the 10th percentile;
calc_skewness(x = x,type = "kelly")
Pearson median
The Pearson median skewness coefficent, or second skewness coefficient,
is given by:
$$\frac{3(\bar{x}- \tilde{x})}{s_x},$$
where:
- $\bar{x}$ is the mean of $x$;
- $\tilde{x}$ is the median of $x$;
- $s_x$ is the standard deviation of $x$.
calc_skewness(x = x,type = "pearson_median")
Rao
The Rao skewness coefficient, is given by:
$$\frac{n/(n-1)}{\sqrt{(n-2)/n}},$$
where:
- $\bar{x}$ is the mean of $x$;
- $\tilde{x}$ is the median of $x$;
- $n$ is the length of $x$.
calc_skewness(x = x,type = "rao")
vbfelix/relper documentation built on May 10, 2024, 10:50 p.m.
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knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
library(relper) library(dplyr) library(ggplot2) x <- rnorm(100) y <- rexp(100)
calc_ functions compute a certain value.
calc_acf
The goal of calc_acf is to compute the auto-correlation function,
given by:
$$\frac{\sum_\limits{t = k+1}^{n}(x_t - \bar{x})(x_{t-k} - \bar{x})}{\sum_\limits{t = 1}^{n} (x_t - \bar{x})^2 },$$ where:
- $x_t$ is a time series of length $n$;
- $x_{t-k}$ is a shifted time series by $k$ units in time;
- $\bar{x}$ is the average of the time series.
calc_acf(x)
If you pass a second vector in the argument y the cross-correlation
will be computed instead:
$$\frac{n \left( \sum_\limits{t = 1}^{n}x_ty_t \right) - \left[\left(\sum_\limits{t = 1}^{n}x_t \right) \left(\sum_\limits{t = 1}^{n}y_t\right) \right]}{\sqrt{\left[n \left( \sum_\limits{t = 1}^{n}x_t^2 \right) - \left( \sum_\limits{t = 1}^{n}x_t \right)^2\right]\left[n \left( \sum_\limits{t = 1}^{n}y_t^2 \right) - \left( \sum_\limits{t = 1}^{n}y_t \right)^2\right]}},$$ where:
- $x_t$ is a time series of length $n$;
- $y_t$ is a time series of length $n$.
calc_acf(x,y)
calc_association
The goal of calc_association is to compute associations metrics.
Contingency
Contingency is a measure of the degree to which two nominal variables are associated. It has a value between 0 and 1, with 0 indicating no relationship and 1 indicating perfect association, and is calculated as follows:
$$\sqrt{\frac{X^2}{n+X^2}},$$
where:
- $X^2$ the chi-square statistic;
- $n$ is the sample size.
calc_association(mtcars$am,mtcars$vs,type = "contingency")
Cramér's V
Cramér's V is a measure of the degree to which two nominal variables are associated. It has a value between 0 and 1, with 0 indicating no relationship and 1 indicating perfect association, and is calculated as follows:
$$\sqrt{\frac{X^2}{n\min(r-1,c-1)}},$$
where:
- $X^2$ the chi-square statistic;
- $n$ is the sample size;
- $r$ is the number of rows in the contingency table;
- $c$ is the number of columns in the contingency table.
calc_association(mtcars$am,mtcars$vs,type = "cramers-v")
Phi
Phi is a measure of association between two nominal dichotomous variables that takes into account a marginal table of the variables given by:
| | y = 0 | y = 1 | Total | |-----------|-----------|------------|-----------| | x = 0 | $n_{00}$ | $n_{01}$ | $n_{0.}$ | | x = 1 | $n_{10}$ | $n_{11}$ | $n_{1.}$ | | Total | $n_{.0}$ | $n_{.1}$ | $n$ |
Then the phi coefficient is given by:
$$\frac{n_{11}n_{00} - n_{10}n_{01} }{\sqrt{n_{1.}n_{0.}n_{.1}*n_{.0}}}.$$
calc_association(mtcars$am,mtcars$vs,type = "phi")
calc_auc
The goal of calc_auc is to compute the area under a curve (AUC).
x <- seq(-3,3,l = 100) y <- dnorm(x)
ggplot(tibble(x = x, y = y), aes(x,y))+ geom_point()+ plt_theme_y()
The function default compute the area considering the range of x.
#from min to max of x range(x) calc_auc(x,y)
df_auc <- tibble( x = x, y = y) df_auc %>% ggplot(aes(x,y))+ geom_area(alpha = .7, fill = "chocolate2")+ geom_point()+ plt_theme_y()+ annotate("text", x = mean(x),y = mean(y),label = round(calc_auc(x,y),3), fontface = "bold", size = 7)+ geom_vline(xintercept = c(-3,3), linetype = "dashed")+ scale_x_continuous(breaks = -5:5)+ scale_y_continuous(expand = c(0,0))
But you can define the argument limits to get the AUC of that
respective range.
#from -2 to 2 calc_auc(x,y,limits = c(-2,2))
df_auc %>% ggplot(aes(x,y))+ geom_area(data = df_auc %>% filter(between(x,-2,2)), alpha = .7, fill = "chocolate2")+ geom_point()+ plt_theme_y()+ annotate("text", x = mean(x),y = mean(y),label = round(calc_auc(x,y,limits = c(-2,2)),3), fontface = "bold", size = 7)+ geom_vline(xintercept = c(-2,2), linetype = "dashed")+ scale_x_continuous(breaks = -5:5)+ scale_y_continuous(expand = c(0,0))
#from -1 to 1 calc_auc(x,y,limits = c(-1,1))
df_auc %>% ggplot(aes(x,y))+ geom_area(data = df_auc %>% filter(between(x,-1,1)), alpha = .7, fill = "chocolate2")+ geom_point()+ plt_theme_y()+ annotate("text", x = mean(x),y = mean(y),label = round(calc_auc(x,y,limits = c(-1,1)),3), fontface = "bold", size = 7)+ geom_vline(xintercept = c(-1,1), linetype = "dashed")+ scale_x_continuous(breaks = -5:5)+ scale_y_continuous(expand = c(0,0))
calc_combination
The goal of calc_combination is to compute the number of combinations/permutations. Given that there are a total of $n$ observations and that $r$ will be chosen.
Order matter with repetition
$$n^r.$$
calc_combination(n = 10,r = 4,order_matter = TRUE,with_repetition = TRUE)
Order matter without repetition
$$\frac{n!}{(n-r)!}.$$
calc_combination(n = 10,r = 4,order_matter = TRUE,with_repetition = FALSE)
Order does not matter with repetition
$$\frac{(n+r-1)!}{r!(n-1)!}.$$
calc_combination(n = 10,r = 4,order_matter = FALSE,with_repetition = TRUE)
Order does not matter without repetition
$$\frac{n!}{r!(n-r)!}.$$
calc_combination(n = 10,r = 4,order_matter = FALSE,with_repetition = FALSE)
calc_correlation
The goal of calc_correlation is to compute associations metrics.
Kendall
The Kendall correlation coefficient, also known as the Kendall's Tau coefficient, measures the relationship between two ranked variables.
Maurice Kendall created it, and it is especially useful for analyzing non-linear relationships or ranked data. The coefficient is calculated by counting the number of concordant pairs (ranks in the same order) and discordant pairs (ranks in opposite order) in the data.
$$\frac{n_c-n_d}{\frac{1}{2}*n(n/1)},$$ where:
- $n_c$ is the number of concordant observations;
- $n_d$ is the number of discordant observations;
- $n$ is the number of observations.
calc_correlation(mtcars$hp,mtcars$drat,type = "kendall")
Pearson
The Pearson correlation coefficient quantifies the linear relationship that exists between two continuous variables. It ranges from -1 to 1, indicating the association's strength and direction.
A value of 1 indicates a perfect positive linear relationship, a value of -1 indicates a perfect negative linear relationship, and a value of 0 indicates no linear relationship.
$$\frac{\sigma_{xy}}{\sigma_x\sigma_y},$$ where:
- $\sigma_{xy}$ is the covariance of $x$ and $y$;
- $\sigma_{x}$ is the variance of $x$;
- $\sigma_{y}$ is the variance of $y$.
calc_correlation(mtcars$hp,mtcars$drat,type = "pearson")
Spearman
The Spearman correlation coefficient assesses the strength and direction of a monotonic relationship between two variables, regardless of whether it is linear or non-linear.
It also has a value between -1 and 1, with 1 representing a perfect monotonic relationship and -1 representing a perfect inverse monotonic relationship. A value of 0 indicates that there is no monotonic relationship.
$$1- \frac{6\sum\limits_{i=1}^{n}d_i^2}{n(n^2-1)},$$
where:
- $d_i$ is the difference between the ranks of $x$ and $y$;
- $n$ is the number of observations.
calc_correlation(mtcars$hp,mtcars$drat,type = "spearman")
calc_cv
The goal of calc_cv is to compute the coefficient of variation (CV),
given by:
$$\frac{s}{\bar{x}},$$ where:
- $s$ is the sample standard deviation;
- $\bar{x}$ is the sample mean.
set.seed(123);x <- rexp(n = 100) calc_cv(x)
If you set the argument as_perc to TRUE, the CV will be multiplied
by 100.
calc_cv(x,as_perc = TRUE)
calc_error
The goal of calc_error is to compute errors metrics.
Mean Absolute Error (MAE)
MAE measures the average absolute difference between the predicted and actual values:
$$\frac{\sum\limits_{i=1}^{n}|X_i-Y_i|}{n}.$$
Mean Absolute Percentage Error (MAPE)
MAPE measures the average percentage difference between the predicted and actual values relative to the actual values:
$$\frac{\sum\limits_{i=1}^{n}\left|\frac{X_i-Y_i}{X_i}\right|}{n}.$$
Mean Squared Error (MSE)
MSE measures the average of the squared differences between the predicted and actual values:
$$\frac{\sum\limits_{i=1}^{n}(X_i-Y_i)^2}{n}.$$
Root Mean Squared Error (RMSE)
RMSE is the square root of the MSE, providing the measure of average prediction error in the same units as the target variable:
$$\sqrt{\text{MSE}}.$$
Root Mean Squared Percentage Error (RMSPE)
RMSPE is the square root of the average of the squared percentage differences between the predicted and actual values relative to the actual values:
$$\sqrt{\frac{\sum\limits_{i=1}^{n}\left(\frac{X_i-Y_i}{X_i}\right)^2}{n}}.$$
calc_kurtosis
The goal of calc_kurtosis is to compute a kurtosis coefficient.
calc_kurtosis(x = x)
Biased
The biased kurtosis coefficient, is given by:
$$\frac{\sum\limits_{i=1}^n(x_i - \bar{x})^4}{n*s_x^4},$$
where:
- $x_i$ is a numeric vector of length $n$;
- $\bar{x}$ is the mean of $x$;
- $s_x$ is the standard deviation of $x$.
calc_kurtosis(x = x,type = "biased")
Excess
The excess kurtosis coefficient, is given by:
$$\frac{\sum\limits_{i=1}^n(x_i - \bar{x})^4}{n*s_x^4}-3,$$
where:
- $x_i$ is a numeric vector of length $n$;
- $\bar{x}$ is the mean of $x$;
- $s_x$ is the standard deviation of $x$.
calc_kurtosis(x = x,type = "excess")
Percentile
The percentile kurtosis coefficient, is given by:
$$\frac{Q_3-Q_1}{P_{90}-P_{10}},$$ where:
- $Q_1$ is the first quartile;
- $Q_3$ is the third quartile;
- $P_{90}$ is the 90th percentile;
- $P_{10}$ is the 10th percentile.
calc_kurtosis(x = x,type = "percentile")
Unbiased
The unbiased kurtosis coefficient, is given by:
$$\frac{(n+1)n}{(n-1)(n-2)(n-3)}\frac{\sum\limits_{i=1}^n(x_i - \bar{x})^4}{ns_x^4} - 3\frac{(n-1)^2}{(n-2)*(n-3)},$$
where:
- $x_i$ is a numeric vector of length $n$;
- $\bar{x}$ is the mean of $x$;
- $s_x$ is the standard deviation of $x$.
calc_kurtosis(x = x,type = "unbiased")
calc_mean
The goal of calc_mean is to compute the mean.
Arithmetic
Simple arithmetic mean
$$\frac{1}{n}\sum\limits_{i=1}^{n}x_i,$$ where:
- $x_i$ is a numeric vector of length $n$.
calc_mean(x = 1:10,type = "arithmetic")
Weighted arithmetic mean
$$\frac{1}{\sum\limits_{i=1}^{n}w_i}\sum\limits_{i=1}^{n}w_ix_i,$$ where:
- $x_i$ is a numeric vector of length $n$;
- $w_i$ is a numeric vector of length $n$, with the respective weights of $x_i$.
calc_mean(x = 1:10,type = "arithmetic",weight = 1:10)
Trimmed arithmetic mean
calc_mean(x = 1:10,type = "arithmetic",trim = .4)
Geometric
$$\sqrt[n]{\prod\limits_{i=1}^{n}x_i} = \sqrt[n]{x_1\times x_2 \times...\times x_n},$$
where:
- $x_i$ is a numeric vector of length $n$.
calc_mean(x = 1:10,type = "geometric")
Harmonic
$$\frac{n}{\sum\limits_{i=1}^{n}\frac{1}{x_i}},$$ where:
- $x_i$ is a numeric vector of length $n$.
calc_mean(x = 1:10,type = "harmonic")
calc_modality
The goal of calc_modality is to compute the number of modes.
calc_modality(x = c("a","a","b","b"))
calc_mode
The goal of calc_mode is to compute the mode.
set.seed(123);cat_var <- sample(letters,100,replace = TRUE) table(cat_var)
We can see that the letter "y" appears the most, indicating that it is the variable's mode.
calc_mode(cat_var)
calc_peak_density
The goal of calc_peak_density is to compute the peak density value of
a numeric value.
pd_plot <- ggplot2::ggplot(data = dplyr::tibble(x = x), ggplot2::aes(x))+ ggplot2::geom_density()+ relper::plt_theme_y()+ ggplot2::scale_x_continuous(breaks = 0:20, expand = c(0,0))+ plt_flip_y_title+ labs(y = "Density (x)") pd_plot
Assume we want to know what the density's peak value is.
calc_peak_density(x)
x_peak <- relper::calc_peak_density(x) pd_plot+ ggplot2::geom_vline( xintercept = relper::calc_peak_density(x), col = "royalblue4", size = 1 )+ ggplot2::scale_x_continuous( breaks = 0:20, expand = c(0,0), sec.axis = sec_axis(~., breaks = x_peak, labels = format_num(x_peak,digits = 3) ) )
calc_perc
The goal of calc_perc is to compute the percentage.
#without main_var calc_perc(mtcars,grp_var = c(cyl,vs)) #main_var within grp_var calc_perc(mtcars,grp_var = c(cyl,vs),main_var = vs) #main_var not within grp_var calc_perc(mtcars,grp_var = c(cyl),main_var = vs)
calc_skewness
The goal of calc_skewness is to compute a skewness coefficient.
calc_skewness(x = x)
Where different types of coefficients are provided, they are:
Bowley
The Bowley skewness coefficient, is given by:
$$\frac{Q_3+Q_1-2Q_2}{Q_3-Q_1},$$ where:
- $Q_1$ is the first quartile;
- $Q_2$ is the second quartile;
- $Q_3$ is the third quartile.
calc_skewness(x = x,type = "bowley")
Fisher-Pearson
The Fisher-Pearson skewness coefficient, is given by:
$$\frac{\sum_\limits{i=1}^{n}(x_i - \bar{x})^3}{n*(s_x)^3},$$
where:
- $\bar{x}$ is the mean of $x$;
- $x_i$ is a numeric vector of length $n$;
- $s_x$ is the standard deviation of $x$.
calc_skewness(x = x,type = "fisher_pearson")
Kelly
The Kelly skewness coefficient, is given by:
$$\frac{P_{90}+P_{10}-2Q_2}{P_{90}-P_{10}},$$ where:
- $P_{90}$ is the 90th percentile;
- $Q_2$ is the second quartile, i.e., $P_{50}$;
- $P_{10}$ is the 10th percentile;
calc_skewness(x = x,type = "kelly")
Pearson median
The Pearson median skewness coefficent, or second skewness coefficient, is given by:
$$\frac{3(\bar{x}- \tilde{x})}{s_x},$$
where:
- $\bar{x}$ is the mean of $x$;
- $\tilde{x}$ is the median of $x$;
- $s_x$ is the standard deviation of $x$.
calc_skewness(x = x,type = "pearson_median")
Rao
The Rao skewness coefficient, is given by:
$$\frac{n/(n-1)}{\sqrt{(n-2)/n}},$$
where:
- $\bar{x}$ is the mean of $x$;
- $\tilde{x}$ is the median of $x$;
- $n$ is the length of $x$.
calc_skewness(x = x,type = "rao")
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