| Title: | 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. See <https://github.com/yueqinhu/defit> for details. |
| Authors: | Yueqin Hu [aut, cre], Qingshan Liu [aut], Minglan Li [aut] |
| Maintainer: | Yueqin Hu <[email protected]> |
| License: | GPL (>= 3) |
| Version: | 0.3.0 |
| Built: | 2024-12-18 06:28:03 UTC |
| Source: | CRAN |
Calculating the Derivatives
calcDerivatives(data, column, groupby, time = NA, order = 2, window = 5)calcDerivatives(data, column, groupby, time = NA, order = 2, window = 5)
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 |
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)
a data frame contains derivatives.
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.
defit( data, model, guess = NULL, method = NULL, plot = FALSE, fixed_first = TRUE )defit( data, model, guess = NULL, method = NULL, plot = FALSE, fixed_first = TRUE )
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. |
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$table get the result
# names(result1) get all names of object
#--------------
# eg3. An example of the multilevel univariate second-order differential equation
data('example3')
model3 <- '
X =~ current
time =~ myTime
X(2) ~ X + X(1) + (1 + X + X(1) | year)
'
example3_use <- example3[(example3["year"] >= 2015)&(example3["year"] <= 2018),] # Note: select a subset of the data as an example.
example3_c <- Scale_within(example3_use, model3) # note: centering X variable by year
result3 <- defit(data=example3_c,model = model3,plot=FALSE)
#--------------
# eg4. An example of the multilevel bivariate first-order differential equations
data('example3')
model4 <- '
X =~ current
Y =~ expected
time =~ myTime
X(1) ~ X + Y + (1 + X + Y | year)
Y(1) ~ X + Y + (1 + X + Y | year)
'
example4_use <- example3[(example3["year"] >= 2015)&(example3["year"] <= 2018),] # Note: select a subset of the data as an example.
example4_c <- Scale_within(example4_use, model4) # centering X and Y variable by year
result4 <- defit(data=example4_c,model = model4,plot=FALSE)
object: directly type the defit object will print all results. The function summary is used to print the summary of all results, and the exact values of each result can be extracted by the "$" 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.
#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$plot()#eg2. An example of bivariate first-order differential equation model2 <- ' # define variable X =~ myX Y =~ myY # define time time =~ myTime # define differential equation X(1) ~ X + Y Y(1) ~ X + Y ' result2 <- defit(data = example2, model = model2,method='Nelder-Mead') # Note: the method argument will override the default "L-BFGS-B" method result2 # #extract details and values # result2$summary() # result2$userdata # result2$parameter$par # result2$equation # result2$table # result2$plot()
A dataset containing the myX and time of almost 100 example. The variables are as follows:
example1example1
A data frame with 100 rows and 3 variables:
sequence of observations
timestamp of observations; the first value of the time variable must be the initial time.
the observed scores
A dataset containing the myX, myY and time of almost 15 example. The variables are as follows:
example2example2
A data frame with 15 rows and 3 variables:
timestamp of observations; the first value of the time variable must be the initial time.
the observed scores of variable X
the observed scores of variable Y
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.
example3example3
A data frame with 540 rows and 6 variables:
sequence of observations
months of data
from 1978 to 2022
the Index of Current Economic Conditions (ICC)
the Index of Consumer Expectations (ICE)
months converted to time series
University of Michigan, Survey Research Center, Surveys of Consumers. https://data.sca.isr.umich.edu/
A dataset containing the myX, myY and time of almost 30 example. The variables are as follows:
example4example4
A data frame with 15 rows and 3 variables:
timestamp of observations; the first value of the time variable must be the initial time.
the observed scores of variable X
the observed scores of variable Y
Calculate R-squared RMSE SE and so on.
Info_func(solve_data, userdata, predict, var_model, table)Info_func(solve_data, userdata, predict, var_model, table)
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. |
a list
Initialize model Judgement variables and so on.
Init_func(userdata, model, guess, method, plot)Init_func(userdata, model, guess, method, plot)
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. |
a list
Draw the diagram of differential equation
Plot_func( userdata, predict, modelDF, var_model, field_model, order_model, multi_model )Plot_func( userdata, predict, modelDF, var_model, field_model, order_model, multi_model )
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 |
plot
Center the data according to model
Scale_within(userdata, model = NA, center = FALSE, scale = FALSE)Scale_within(userdata, model = NA, center = FALSE, scale = FALSE)
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 |
dataframe
#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)#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
Solver_BiFirst_func(userdata, var_model, guess, method)Solver_BiFirst_func(userdata, var_model, guess, method)
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'. |
a list
Solver of Multilevel bivariate first-order differential equation
Solver_MultiBiFirst_func(init_list)Solver_MultiBiFirst_func(init_list)
init_list |
a list of init_func |
a list
Solver of Multilevel univariate second-order differential equation
Solver_MultiUniSecond_func(init_list)Solver_MultiUniSecond_func(init_list)
init_list |
a list of init_func |
a list
Solver of univariate second-order differential equation
Solver_UniSecond_func(userdata, var_model, guess, method)Solver_UniSecond_func(userdata, var_model, guess, method)
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'. |
a list