| Type: | Package |
| Title: | Fitting Differential Equations to Time Series Data |
| Version: | 0.3.0 |
| Description: | Use numerical optimization to fit ordinary differential equations (ODEs) to time series data to examine the dynamic relationships between variables or the characteristics of a dynamical system. It can now be used to estimate the parameters of ODEs up to second order, and can also apply to multilevel systems. See https://github.com/yueqinhu/defit for details. |
| License: | GPL (≥ 3) |
| Encoding: | UTF-8 |
| LazyData: | true |
| RoxygenNote: | 7.2.3 |
| Suggests: | testthat (≥ 3.0.0) |
| Config/testthat/edition: | 3 |
| Imports: | deSolve, ggplot2, stats, R6 |
| Depends: | R (≥ 3.50) |
| NeedsCompilation: | no |
| Packaged: | 2024-10-18 10:42:11 UTC; yueqinhu |
| Author: | Yueqin Hu [aut, cre], Qingshan Liu [aut], Minglan Li [aut] |
| Maintainer: | Yueqin Hu <yueqinhu@bnu.edu.cn> |
| Repository: | CRAN |
| Date/Publication: | 2024-10-18 11:10:13 UTC |
Calculate R-squared RMSE SE and so on.
Description
Calculate R-squared RMSE SE and so on.
Usage
Info_func(solve_data, userdata, predict, var_model, table)
Arguments
solve_data |
a list data, that is a solve of differential equations. |
userdata |
a data frame containing all model variables. The "time" column must be included. |
predict |
predict data. |
var_model |
a dataframe containing equations. |
table |
outtable. |
Value
a list
Initialize model Judgement variables and so on.
Description
Initialize model Judgement variables and so on.
Usage
Init_func(userdata, model, guess, method, plot)
Arguments
userdata |
a data frame containing all model variables. The "time" column must be included. |
model |
a string of model. |
guess |
a list or a string. Guess the coefficients or initial values. |
method |
a list or a string. The available options are 'Nelder-Mead','L-BFGS-B','SANN' and 'BFGS'. |
plot |
TRUE or FALSE. |
Value
a list
Draw the diagram of differential equation
Description
Draw the diagram of differential equation
Usage
Plot_func(
userdata,
predict,
modelDF,
var_model,
field_model,
order_model,
multi_model
)
Arguments
userdata |
a data frame containing all model variables. The "time" column must be included. |
predict |
predict data. |
modelDF |
a dataframe of full model. |
var_model |
a dataframe containing equations. |
field_model |
the user's data columns. |
order_model |
N-order differential equation. |
multi_model |
TRUE or FALSE |
Value
plot
Center the data according to model
Description
Center the data according to model
Usage
Scale_within(userdata, model = NA, center = FALSE, scale = FALSE)
Arguments
userdata |
users' data |
model |
a string specifying the model to be used. The "=~" operator is used to define variables, with the name of the variable user defined on the left and the name of the variable in the data on the right. The '~' operator specifies a differential equation, with the dependent variable on the left and the independent variables on the right. See also ‘Details’. |
center |
TRUE or FALSE |
scale |
TRUE or FALSE |
Value
dataframe
Examples
#eg1.
data('example3')
multi_model <- '
X =~ current
time =~ myTime
X(2) ~ X(1) + X + (1 + X(1) + X | year)
'
scale_mydata <- Scale_within(example3[(example3["year"] >= 2015)&(example3["year"] <= 2018),]
,multi_model
,center=TRUE)
Solver of bivariate first-order differential equation
Description
Solver of bivariate first-order differential equation
Usage
Solver_BiFirst_func(userdata, var_model, guess, method)
Arguments
userdata |
a data frame containing all model variables. The "time" column must be included. |
var_model |
a dataframe containing equations. |
guess |
a list or a string. Guess the coefficients or initial values. |
method |
a list or a string. The available options are 'Nelder-Mead','L-BFGS-B','SANN' and 'BFGS'. |
Value
a list
Solver of Multilevel bivariate first-order differential equation
Description
Solver of Multilevel bivariate first-order differential equation
Usage
Solver_MultiBiFirst_func(init_list)
Arguments
init_list |
a list of init_func |
Value
a list
Solver of Multilevel univariate second-order differential equation
Description
Solver of Multilevel univariate second-order differential equation
Usage
Solver_MultiUniSecond_func(init_list)
Arguments
init_list |
a list of init_func |
Value
a list
Solver of univariate second-order differential equation
Description
Solver of univariate second-order differential equation
Usage
Solver_UniSecond_func(userdata, var_model, guess, method)
Arguments
userdata |
a data frame containing all model variables. The "time" column must be included. |
var_model |
a dataframe containing equations |
guess |
a list or a string. Guess the coefficients or initial values. |
method |
a list or a string. The available options are 'Nelder-Mead','L-BFGS-B','SANN' and 'BFGS'. |
Value
a list
Calculating the Derivatives
Description
Calculating the Derivatives
Usage
calcDerivatives(data, column, groupby, time = NA, order = 2, window = 5)
Arguments
data |
a data frame. |
column |
names of variables in the long format that correspond to multiple variables in the wide format. |
groupby |
Character vector. Only used if the data is in a data.frame. |
time |
A variable name in the data frame containing sampling time information. |
order |
integer scalar. |
window |
integer scalar.Must be an odd number |
Details
examples
#eg1.
derivatives1 <- calcDerivatives(data=example3,column='expected',groupby='year',order=2,window=5,interval=1)
#eg2.
derivatives2 <- calcDerivatives(data=example3,column=c('expected','current'),groupby='year',time='myTime',order=2,window=5,interval=1)
#eg3.
derivativese3 <- calcDerivatives(data=example3,column=c('expected','current'),groupby='year',order=2,window=5,interval=1)
Value
a data frame contains derivatives.
Fitting Differential Equations to Time Series Data
Description
Use numerical optimization to fit ordinary differential equations (ODEs) to time series data to examine the dynamic relationships between variables or the characteristics of a dynamical system. It can now be used to estimate the parameters of ODEs up to second order, and can also apply to multilevel systems.
Usage
defit(
data,
model,
guess = NULL,
method = NULL,
plot = FALSE,
fixed_first = TRUE
)
Arguments
data |
a data frame containing all model variables. The "time" column must be included. |
model |
a string specifying the model to be used. The "=~" operator is used to define variables, with the name of the variable user defined on the left and the name of the variable in the data on the right. The '~' operator specifies a differential equation, with the dependent variable on the left and the independent variables on the right. See also ‘Details’. |
guess |
an optional vector that allows the user to give starting values for the model parameters, including the model coefficients and variable initial states. |
method |
an optional string indicating which optimizer to use. The default method is subject to the specific model. The available options are 'Nelder-Mead','L-BFGS-B','SANN' and 'BFGS'. |
plot |
an optional TRUE or FALSE that TRUE will draw the plot of the raw data and the predicted values. |
fixed_first |
an optional TRUE or FALSE that TRUE will estimate the multilevel model parameters using a two-step approach. |
Details
We suggest choosing the method by default. The guess values contain the coefficient of the model and initial values (the values of t0). Different models have different number of values.
Time(param) sequence for which output is wanted; the first value of times must be the initial time.
# eg1. An example of the univariate second-order differential equation (damped oscillator model)
data('example1')
model1 <- '
X =~ myX
time =~ myTime
X(2) ~ X + X(1)
'
result1 <- defit(data = example1, model = model1)
# result1$table get the result
# names(result1) get all names of object
#--------------
# eg3. An example of the multilevel univariate second-order differential equation
data('example3')
model3 <- '
X =~ current
time =~ myTime
X(2) ~ X + X(1) + (1 + X + X(1) | year)
'
example3_use <- example3[(example3["year"] >= 2015)&(example3["year"] <= 2018),] # Note: select a subset of the data as an example.
example3_c <- Scale_within(example3_use, model3) # note: centering X variable by year
result3 <- defit(data=example3_c,model = model3,plot=FALSE)
#--------------
# eg4. An example of the multilevel bivariate first-order differential equations
data('example3')
model4 <- '
X =~ current
Y =~ expected
time =~ myTime
X(1) ~ X + Y + (1 + X + Y | year)
Y(1) ~ X + Y + (1 + X + Y | year)
'
example4_use <- example3[(example3["year"] >= 2015)&(example3["year"] <= 2018),] # Note: select a subset of the data as an example.
example4_c <- Scale_within(example4_use, model4) # centering X and Y variable by year
result4 <- defit(data=example4_c,model = model4,plot=FALSE)
Value
object: directly type the defit object will print all results. The function summary is used to print the summary of all results, and the exact values of each result can be extracted by the "$" operator.
userdata: the data that contains a sequence 'seq' starting from 1, the original time variable 'time', and all other variables user defined.
parameter: the best set of parameters found, including parameter values, gradient, convergence, message and hessian matrix.
predict: a dataframe of model predicted variable states at each time point.
r_squared: r_squared is the square of the correlation between the observed values and the predicted values, representing the proportion of variance explained by the model.
RMSE: RMSE (Root Mean Squared Error) is the standard deviation of the residuals.
SE: a symmetric matrix giving standard error of the model parameters.
equation: a string prints the estimated differential equations and initial states.
table: a summary table of parameter estimates and their corresponding SEs.
convergence: a message returns the result of the optimization convergence check.
Examples
#eg2. An example of bivariate first-order differential equation
model2 <- '
# define variable
X =~ myX
Y =~ myY
# define time
time =~ myTime
# define differential equation
X(1) ~ X + Y
Y(1) ~ X + Y
'
result2 <- defit(data = example2, model = model2,method='Nelder-Mead')
# Note: the method argument will override the default "L-BFGS-B" method
result2
# #extract details and values
# result2$summary()
# result2$userdata
# result2$parameter$par
# result2$equation
# result2$table
# result2$plot()
Univariate second-order differential equation
Description
A dataset containing the myX and time of almost 100 example. The variables are as follows:
Usage
example1
Format
A data frame with 100 rows and 3 variables:
- seq
sequence of observations
- myTime
timestamp of observations; the first value of the time variable must be the initial time.
- myX
the observed scores
Bivariate first-order differential equation
Description
A dataset containing the myX, myY and time of almost 15 example. The variables are as follows:
Usage
example2
Format
A data frame with 15 rows and 3 variables:
- myTime
timestamp of observations; the first value of the time variable must be the initial time.
- myX
the observed scores of variable X
- myY
the observed scores of variable Y
University of Michigan consumer sentiment index
Description
The Surveys of Consumers are conducted by the Survey Research Center at the University of Michigan. Founded in 1946 by George Katona, the surveys have long stressed the important influence of consumer spending and saving decisions in determining the course of the national economy.
Usage
example3
Format
A data frame with 540 rows and 6 variables:
- seq
sequence of observations
- month
months of data
- year
from 1978 to 2022
- current
the Index of Current Economic Conditions (ICC)
- expected
the Index of Consumer Expectations (ICE)
- myTime
months converted to time series
Source
University of Michigan, Survey Research Center, Surveys of Consumers. https://data.sca.isr.umich.edu/
Bivariate first-order differential equation
Description
A dataset containing the myX, myY and time of almost 30 example. The variables are as follows:
Usage
example4
Format
A data frame with 15 rows and 3 variables:
- myTime
timestamp of observations; the first value of the time variable must be the initial time.
- myX
the observed scores of variable X
- myY
the observed scores of variable Y