# lm2: Fitting Bidimensional Regression Models In BiDimRegression: Calculates the Bidimensional Regression Between Two 2D Configurations

## Description

lm2 is used to fit bidimensional linear regression models using Euclidean and Affine transformations following the approach by Tobler (1965).

## Usage

 `1` ```lm2(formula, data, transformation) ```

## Arguments

 `formula` a symbolic description of the model to be fitted in the format `A + B ~ C + D`, where `A` and `B` are dependent and `C` and `D` are independent variables `data` a data frame containing variables for the model. `transformation` the transformation to be used, either `'euclidean'`, `'affine'`, or `'projective'`.

## Value

lm2 returns an object of class "lm2". An object of class "lm" is a list containing at least the following components:

 `transformation` string with the transformation type (`euclidean`, `affine`, or `projective`) `npredictors` number of predictors used in the model: 4 for euclidean, 6 for affine, 8 for projective. `df_model, df_residual` degrees of freedom for the model and for the residuals `transformation_matrix` `3x3` transformation matrix `coeff` transformation coefficients, with `a` denoting the intercept terms. `transformed_coeff` `scale`, `angle`, and `sheer` coefficients, depends on transformation. `fitted_values` data frame containing fitted values for the original data set `residuals` data frame containing residuals for the original fit `r.squared, adj.r.squared` R-squared and adjusted R-squared. `F, p.value` F-statistics and the corresponding p-value, given the `df_model` and `df_residual` degrees of freedom. `dAIC` Akaike Information Criterion (AIC) difference between the regression model and the null model. A negative values indicates that the regression model is better. See Nakaya (1997). `distortion_index` Distortion index following Waterman and Gordon (1984), as adjusted by Friedman and Kohler (2003) `lm` an underlying linear model for `Euclidean` and `affine` transformations. `formula` formula, describing input and output columns `data` data used to fit the model `Call` function call information, incorporates the `formula`, `transformation`, and `data`.

## See Also

`anova.lm2` `BiDimRegression`

## Examples

 ```1 2 3 4 5 6``` ```lm2euc <- lm2(depV1 + depV2 ~ indepV1 + indepV2, NakayaData, 'euclidean') lm2aff <- lm2(depV1 + depV2 ~ indepV1 + indepV2, NakayaData, 'affine') lm2prj <- lm2(depV1 + depV2 ~ indepV1 + indepV2, NakayaData, 'projective') anova(lm2euc, lm2aff, lm2prj) predict(lm2euc) summary(lm2euc) ```

### Example output

```Bidimensional regression:
dAIC df1 df2      F   p.value
euclidean vs. affine     -13.29021   2  32 9.2189 0.0006891 ***
euclidean vs. projective -11.66134   4  30 5.0825 0.0030072 **
affine vs. projective      1.62887   2  30 0.9658 0.3922043
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
depV1      depV2
1  -0.10714634  0.7355034
2   0.76751919  0.1777630
3   0.71377356  1.5461024
4   0.97343384  0.8287322
5   2.32044682 -0.1858853
6   3.66895537 -0.8335199
7  -1.10962809  1.7756851
8  -1.16310065  1.2187896
9  -1.63076109  1.8346422
10  0.11495606 -0.4178982
11  0.90314950 -0.6598266
12  1.82560030 -0.8248245
13 -0.07100152 -0.8971433
14  1.60705225 -1.8426904
15 -0.19372050 -2.8581066
16  0.62693595 -2.6027024
17  2.68465593  0.8439124
18  3.51168672  0.2106835
19  2.11119272  1.9577835
Call:
lm2.formula(formula = depV1 + depV2 ~ indepV1 + indepV2, data = NakayaData,      transformation = "euclidean")

Coefficients:
Estimate Std. Error t value  Pr(>|t|)
a1  0.140769   0.095863  1.4684    0.1512
a2 -0.010582   0.095863 -0.1104    0.9127
b1  1.348680   0.064359 20.9557 < 2.2e-16 ***
b2  0.565693   0.064359  8.7897 2.848e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Transformed coefficients:
scale1 scale2  angle
1.4625 1.4625 0.3972

Distortion index:
Dependent Independent
Distortion distance, squared          5.169255       0.000
Maximal distortion distance, squared 83.680948      36.706
Distortion index, squared             0.061773       0.000

Multiple R-squared: 0.9382266 	Adjusted R-squared: 0.8906729
F-statistic: 258.1994 on 2 and 34 DF, p-value: < 2.22e-16
Difference in AIC to the null model: -101.8027*
```

BiDimRegression documentation built on May 1, 2019, 10:13 p.m.