## Regressions Regressions |

## Synopsis

    enum                RegressionResult;
                        go_regression_stat_t;
    go_regression_stat_t* go_regression_stat_new            (void);
    void                go_regression_stat_destroy          (go_regression_stat_t *regression_stat);
    RegressionResult    go_linear_regression                (double **xss,
                                                             int dim,
                                                             const double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             double *res,
                                                             go_regression_stat_t *stat);
    RegressionResult    go_exponential_regression           (double **xss,
                                                             int dim,
                                                             const double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             double *res,
                                                             go_regression_stat_t *stat);
    RegressionResult    go_power_regression                 (double **xss,
                                                             int dim,
                                                             const double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             double *res,
                                                             go_regression_stat_t *stat);
    RegressionResult    go_logarithmic_regression           (double **xss,
                                                             int dim,
                                                             const double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             double *res,
                                                             go_regression_stat_t *stat);
    #define             LOGFIT_C_ACCURACY
    #define             LOGFIT_C_STEP_FACTOR
    #define             LOGFIT_C_RANGE_FACTOR
    RegressionResult    go_logarithmic_fit                  (double *xs,
                                                             const double *ys,
                                                             int n,
                                                             double *res);
    RegressionResult    (*GORegressionFunction)             (double *x,
                                                             double *params,
                                                             double *f);
    RegressionResult    go_non_linear_regression            (GORegressionFunction f,
                                                             double **xvals,
                                                             double *par,
                                                             double *yvals,
                                                             double *sigmas,
                                                             int x_dim,
                                                             int p_dim,
                                                             double *chi,
                                                             double *errors);
    gboolean            go_matrix_invert                    (double **A,
                                                             int n);
    double              go_matrix_determinant               (double **A,
                                                             int n);
                        go_regression_stat_tl;
    go_regression_stat_tl* go_regression_stat_newl          (void);
    void                go_regression_stat_destroyl         (go_regression_stat_tl *regression_stat);
    RegressionResult    go_linear_regressionl               (long double **xss,
                                                             int dim,
                                                             long double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             long double *res,
                                                             go_regression_stat_tl *stat);
    RegressionResult    go_exponential_regressionl          (long double **xss,
                                                             int dim,
                                                             long double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             long double *res,
                                                             go_regression_stat_tl *stat);
    RegressionResult    go_power_regressionl                (long double **xss,
                                                             int dim,
                                                             long double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             long double *res,
                                                             go_regression_stat_tl *stat);
    RegressionResult    go_logarithmic_regressionl          (long double **xss,
                                                             int dim,
                                                             long double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             long double *res,
                                                             go_regression_stat_tl *stat);
    RegressionResult    go_logarithmic_fitl                 (long double *xs,
                                                             long double *ys,
                                                             int n,
                                                             long double *res);
    RegressionResult    (*GORegressionFunctionl)            (long double *x,
                                                             long double *params,
                                                             long double *f);
    RegressionResult    go_non_linear_regressionl           (GORegressionFunctionl f,
                                                             long double **xvals,
                                                             long double *par,
                                                             long double *yvals,
                                                             long double *sigmas,
                                                             int x_dim,
                                                             int p_dim,
                                                             long double *chi,
                                                             long double *errors);
    gboolean            go_matrix_invertl                   (long double **A,
                                                             int n);

## Description

## Details

### enum RegressionResult

    typedef enum {
    	REG_ok,
    	REG_invalid_dimensions,
    	REG_invalid_data,
    	REG_not_enough_data,
    	REG_near_singular_good,	/* Probably good result */
    	REG_near_singular_bad, 	/* Probably bad result */
    	REG_singular
    } RegressionResult;

### go_regression_stat_t

    typedef struct {
            double *se;		/* SE for each parameter estimator */
            double *t;  		/* t values for each parameter estimator */
            double sqr_r;
    	double adj_sqr_r;
            double se_y; 		/* The Standard Error of Y */
            double F;
            int    df_reg;
            int    df_resid;
            int    df_total;
            double ss_reg;
            double ss_resid;
            double ss_total;
            double ms_reg;
            double ms_resid;
    	double ybar;
    	double *xbar;
    	double var; 		/* The variance of the entire regression: sum(errors^2)/(n-xdim) */
    } go_regression_stat_t;

### go_regression_stat_new ()

    go_regression_stat_t* go_regression_stat_new            (void);

Returns : |

### go_regression_stat_destroy ()

    void                go_regression_stat_destroy          (go_regression_stat_t *regression_stat);

`regression_stat` : |

### go_linear_regression ()

    RegressionResult    go_linear_regression                (double **xss,
                                                             int dim,
                                                             const double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             double *res,
                                                             go_regression_stat_t *stat);

Performs multi-dimensional linear regressions on the input points. Fits to "y = b + a1 * x1 + ... ad * xd".

`xss` : | x-vectors (i.e. independent data)
`dim` : | number of x-vectors.
`ys` : | y-vector. (Dependent data.)
`n` : | number of data points.
`affine` : | if true, a non-zero constant is allowed.
`res` : | output place for constant[0] and slope1[1], slope2[2],... There will be dim+1 results.
`stat` : |
Returns : | RegressionResult as above.

### go_exponential_regression ()

    RegressionResult    go_exponential_regression           (double **xss,
                                                             int dim,
                                                             const double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             double *res,
                                                             go_regression_stat_t *stat);

Performs one-dimensional linear regressions on the input points. Fits to "y = b * m1^x1 * ... * md^xd " or equivalently to "log y = log b + x1 * log m1 + ... + xd * log md".

`xss` : | x-vectors (i.e. independent data)
`dim` : | number of x-vectors
`ys` : | y-vector (dependent data)
`n` : | number of data points
`affine` : | if `TRUE`, a non-one multiplier is allowed
`res` : | output place for constant[0] and root1[1], root2[2],... There will be dim+1 results.
`stat` : |
Returns : | RegressionResult as above.

### go_power_regression ()

    RegressionResult    go_power_regression                 (double **xss,
                                                             int dim,
                                                             const double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             double *res,
                                                             go_regression_stat_t *stat);

Performs one-dimensional linear regressions on the input points. Fits to "y = b * x1^m1 * ... * xd^md " or equivalently to "log y = log b + m1 * log x1 + ... + md * log xd".

`xss` : | x-vectors (i.e. independent data)
`dim` : | number of x-vectors
`ys` : | y-vector (dependent data)
`n` : | number of data points
`affine` : | if `TRUE`, a non-one multiplier is allowed
`res` : | output place for constant[0] and root1[1], root2[2],... There will be dim+1 results.
`stat` : |
Returns : | RegressionResult as above.

### go_logarithmic_regression ()

    RegressionResult    go_logarithmic_regression           (double **xss,
                                                             int dim,
                                                             const double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             double *res,
                                                             go_regression_stat_t *stat);

This is almost a copy of linear_regression and produces multi-dimensional linear regressions on the input points after transforming xss to ln(xss). Fits to "y = b + a1 * z1 + ... ad * zd" with "zi = ln (xi)". Problems with arrays in the calling function: see comment to gnumeric_linest, which is also valid for gnumeric_logreg.

(Errors: less than two points, all points on a vertical line, non-positive x data.)

`xss` : | x-vectors (i.e. independent data)
`dim` : | number of x-vectors
`ys` : | y-vector (dependent data)
`n` : | number of data points
`affine` : | if `TRUE`, a non-zero constant is allowed
`res` : | output place for constant[0] and factor1[1], factor2[2],... There will be dim+1 results.
`stat` : |
Returns : | RegressionResult as above.

### LOGFIT_C_ACCURACY

    #define LOGFIT_C_ACCURACY 0.000001

### LOGFIT_C_STEP_FACTOR

    #define LOGFIT_C_STEP_FACTOR 0.05

### LOGFIT_C_RANGE_FACTOR

    #define LOGFIT_C_RANGE_FACTOR 100

### go_logarithmic_fit ()

    RegressionResult    go_logarithmic_fit                  (double *xs,
                                                             const double *ys,
                                                             int n,
                                                             double *res);

Performs a two-dimensional non-linear fitting on the input points. Fits to "y = a + b * ln (sign * (x - c))", with sign in {-1, +1}. The graph is a logarithmic curve moved horizontally by c and possibly mirrored across the y-axis (if sign = -1).

Fits c (and sign) by iterative trials, but seems to be fast enough even for automatic recomputation.

Adapts c until a local minimum of squared residuals is reached. For each new c tried out the corresponding a and b are calculated by linear regression. If no local minimum is found, an error is returned. If there is more than one local minimum, the one found is not necessarily the smallest (i.e., there might be cases in which the returned fit is not the best possible). If the shape of the point cloud is to different from ``logarithmic'', either sign can not be determined (error returned) or no local minimum will be found.

(Requires: at least 3 different x values, at least 3 different y values.)

`xs` : | x-vector (i.e. independent data)
`ys` : | y-vector (dependent data)
`n` : | number of data points
`res` : | output place for sign[0], a[1], b[2], c[3], and sum of squared residuals[4].
Returns : | RegressionResult as above.

### GORegressionFunction ()

    RegressionResult    (*GORegressionFunction)             (double *x,
                                                             double *params,
                                                             double *f);

`x` : |
`params` : |
`f` : |
Returns : |

### go_non_linear_regression ()

    RegressionResult    go_non_linear_regression            (GORegressionFunction f,
                                                             double **xvals,
                                                             double *par,
                                                             double *yvals,
                                                             double *sigmas,
                                                             int x_dim,
                                                             int p_dim,
                                                             double *chi,
                                                             double *errors);

`f` : |
`xvals` : |
`par` : |
`yvals` : |
`sigmas` : |
`x_dim` : |
`p_dim` : |
`chi` : |
`errors` : |
Returns : |

### go_matrix_invert ()

    gboolean            go_matrix_invert                    (double **A,
                                                             int n);

`A` : |
`n` : |
Returns : |

### go_matrix_determinant ()

    double              go_matrix_determinant               (double **A,
                                                             int n);

`A` : |
`n` : |
Returns : |

### go_regression_stat_tl

    typedef struct {
            long double *se; /*SE for each parameter estimator*/
            long double *t;  /*t values for each parameter estimator*/
            long double sqr_r;
    	long double adj_sqr_r;
            long double se_y; /* The Standard Error of Y */
            long double F;
            int        df_reg;
            int        df_resid;
            int        df_total;
            long double ss_reg;
            long double ss_resid;
            long double ss_total;
            long double ms_reg;
            long double ms_resid;
    	long double ybar;
    	long double *xbar;
    	long double var; /* The variance of the entire regression:
    			sum(errors^2)/(n-xdim) */
    } go_regression_stat_tl;

### go_regression_stat_newl ()

    go_regression_stat_tl* go_regression_stat_newl          (void);

Returns : |

### go_regression_stat_destroyl ()

    void                go_regression_stat_destroyl         (go_regression_stat_tl *regression_stat);

`regression_stat` : |

### go_linear_regressionl ()

    RegressionResult    go_linear_regressionl               (long double **xss,
                                                             int dim,
                                                             long double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             long double *res,
                                                             go_regression_stat_tl *stat);

`xss` : |
`dim` : |
`ys` : |
`n` : |
`affine` : |
`res` : |
`stat` : |
Returns : |

### go_exponential_regressionl ()

    RegressionResult    go_exponential_regressionl          (long double **xss,
                                                             int dim,
                                                             long double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             long double *res,
                                                             go_regression_stat_tl *stat);

`xss` : |
`dim` : |
`ys` : |
`n` : |
`affine` : |
`res` : |
`stat` : |
Returns : |

### go_power_regressionl ()

    RegressionResult    go_power_regressionl                (long double **xss,
                                                             int dim,
                                                             long double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             long double *res,
                                                             go_regression_stat_tl *stat);

`xss` : |
`dim` : |
`ys` : |
`n` : |
`affine` : |
`res` : |
`stat` : |
Returns : |

### go_logarithmic_regressionl ()

    RegressionResult    go_logarithmic_regressionl          (long double **xss,
                                                             int dim,
                                                             long double *ys,
                                                             int n,
                                                             gboolean affine,
                                                             long double *res,
                                                             go_regression_stat_tl *stat);

`xss` : |
`dim` : |
`ys` : |
`n` : |
`affine` : |
`res` : |
`stat` : |
Returns : |

### go_logarithmic_fitl ()

    RegressionResult    go_logarithmic_fitl                 (long double *xs,
                                                             long double *ys,
                                                             int n,
                                                             long double *res);

`xs` : |
`ys` : |
`n` : |
`res` : |
Returns : |

### GORegressionFunctionl ()

    RegressionResult    (*GORegressionFunctionl)            (long double *x,
                                                             long double *params,
                                                             long double *f);

`x` : |
`params` : |
`f` : |
Returns : |

### go_non_linear_regressionl ()

    RegressionResult    go_non_linear_regressionl           (GORegressionFunctionl f,
                                                             long double **xvals,
                                                             long double *par,
                                                             long double *yvals,
                                                             long double *sigmas,
                                                             int x_dim,
                                                             int p_dim,
                                                             long double *chi,
                                                             long double *errors);

`f` : |
`xvals` : |
`par` : |
`yvals` : |
`sigmas` : |
`x_dim` : |
`p_dim` : |
`chi` : |
`errors` : |
Returns : |

### go_matrix_invertl ()

    gboolean            go_matrix_invertl                   (long double **A,
                                                             int n);

`A` : |
`n` : |
Returns : |