SciPy Cheatsheet¶

Installation¶

Platform	Command
Ubuntu/Debian	`sudo apt-get install python3-scipy` or `pip install scipy`
macOS	`pip3 install scipy` or `brew install python3 && pip3 install scipy`
Windows	`pip install scipy` or `conda install scipy` (recommended)
Conda (All platforms)	`conda install scipy` or `conda install -c conda-forge scipy`
Virtual Environment	`python -m venv env && source env/bin/activate && pip install scipy`
Specific Version	`pip install scipy==1.11.4`
With Dependencies	`pip install scipy[all]`
Verify Installation	`python -c "import scipy; print(scipy.__version__)"`

Basic Commands¶

Import and Setup¶

Command	Description
`import scipy`	Import the SciPy library
`import numpy as np`	Import NumPy (required for SciPy)
`from scipy import optimize`	Import optimization module
`from scipy import stats`	Import statistics module
`from scipy import linalg`	Import linear algebra module
`scipy.__version__`	Check SciPy version
`scipy.show_config()`	Display SciPy configuration

Optimization Basics¶

Command	Description
`minimize(func, x0)`	Find minimum of a scalar function
`minimize(func, x0, method='BFGS')`	Minimize using BFGS algorithm
`root(func, x0)`	Find root of a function
`curve_fit(func, xdata, ydata)`	Fit curve to data points
`fsolve(func, x0)`	Solve system of nonlinear equations
`minimize_scalar(func)`	Minimize single-variable function
`linprog(c, A_ub, b_ub)`	Linear programming optimization

Integration¶

Command	Description
`quad(func, a, b)`	Integrate function from a to b
`dblquad(func, a, b, gfun, hfun)`	Double integration
`odeint(func, y0, t)`	Solve ordinary differential equations
`solve_ivp(func, t_span, y0)`	Modern ODE solver with events
`trapz(y, x)`	Trapezoidal rule integration
`simps(y, x)`	Simpson's rule integration

Interpolation¶

Command	Description
`interp1d(x, y)`	1D interpolation function
`interp1d(x, y, kind='cubic')`	Cubic spline interpolation
`griddata(points, values, xi)`	Interpolate scattered data
`UnivariateSpline(x, y)`	Univariate spline approximation
`interp2d(x, y, z)`	2D interpolation function

Statistics¶

Command	Description
`stats.norm.rvs(loc, scale, size)`	Generate normal random variables
`stats.ttest_1samp(data, popmean)`	One-sample t-test
`stats.ttest_ind(data1, data2)`	Independent two-sample t-test
`stats.pearsonr(x, y)`	Pearson correlation coefficient
`stats.norm.pdf(x, loc, scale)`	Normal probability density function
`stats.norm.cdf(x, loc, scale)`	Cumulative distribution function
`stats.kstest(data, 'norm')`	Kolmogorov-Smirnov test
`stats.chi2_contingency(table)`	Chi-square test of independence

Linear Algebra¶

Command	Description
`linalg.solve(A, b)`	Solve linear system Ax = b
`linalg.inv(A)`	Calculate matrix inverse
`linalg.det(A)`	Calculate determinant
`linalg.eig(A)`	Compute eigenvalues and eigenvectors
`linalg.svd(A)`	Singular value decomposition
`linalg.lstsq(A, b)`	Least-squares solution
`linalg.norm(A)`	Matrix or vector norm
`linalg.qr(A)`	QR decomposition

Signal Processing¶

Command	Description
`signal.butter(N, Wn, btype)`	Butterworth filter design
`signal.filtfilt(b, a, data)`	Zero-phase digital filtering
`signal.find_peaks(data)`	Find peaks in signal
`signal.correlate(sig1, sig2)`	Cross-correlation of signals
`signal.convolve(sig1, sig2)`	Convolution of signals
`signal.spectrogram(data, fs)`	Compute spectrogram
`signal.resample(data, num)`	Resample signal to num samples

Sparse Matrices¶

Command	Description
`sparse.csr_matrix(data)`	Create compressed sparse row matrix
`sparse.csc_matrix(data)`	Create compressed sparse column matrix
`sparse.eye(n)`	Sparse identity matrix
`sparse.diags(diagonals, offsets)`	Construct sparse diagonal matrix
`matrix.toarray()`	Convert sparse to dense array
`matrix.todense()`	Convert sparse to dense matrix

Advanced Usage¶

Advanced Optimization¶

Command	Description
`minimize(func, x0, method='SLSQP', constraints=cons, bounds=bnds)`	Constrained optimization with bounds
`differential_evolution(func, bounds)`	Global optimization using genetic algorithm
`basinhopping(func, x0, minimizer_kwargs)`	Global optimization with basin-hopping
`least_squares(func, x0, bounds)`	Nonlinear least squares with bounds
`minimize(func, x0, jac=gradient_func)`	Optimization with analytical gradient
`shgo(func, bounds)`	Simplicial homology global optimization
`dual_annealing(func, bounds)`	Dual annealing global optimization

Advanced Integration¶

Command	Description
`solve_ivp(func, t_span, y0, method='RK45', events=event_func)`	ODE solver with event detection
`solve_ivp(func, t_span, y0, dense_output=True)`	ODE with continuous solution
`nquad(func, ranges)`	N-dimensional integration
`odeint(func, y0, t, Dfun=jacobian)`	ODE with Jacobian matrix
`solve_bvp(func, bc, x, y)`	Boundary value problem solver
`quad(func, a, b, weight='cos')`	Integration with weight functions

Advanced Interpolation¶

Command	Description
`RBFInterpolator(points, values)`	Radial basis function interpolation
`BSpline(t, c, k)`	B-spline basis functions
`make_interp_spline(x, y, k=3)`	Make interpolating B-spline
`Akima1DInterpolator(x, y)`	Akima interpolation (smooth curves)
`PchipInterpolator(x, y)`	PCHIP 1-D monotonic cubic interpolation
`RegularGridInterpolator((x, y), z)`	Interpolation on regular grid

Advanced Statistics¶

Command	Description
`stats.gaussian_kde(data)`	Kernel density estimation
`stats.multivariate_normal.rvs(mean, cov, size)`	Multivariate normal samples
`stats.bootstrap((data,), statistic, n_resamples)`	Bootstrap confidence intervals
`stats.anderson(data, dist='norm')`	Anderson-Darling test
`stats.mannwhitneyu(x, y)`	Mann-Whitney U test
`stats.kruskal(*groups)`	Kruskal-Wallis H-test
`stats.wilcoxon(x, y)`	Wilcoxon signed-rank test
`stats.spearmanr(x, y)`	Spearman rank correlation
`stats.linregress(x, y)`	Linear regression analysis

Advanced Linear Algebra¶

Command	Description
`linalg.cholesky(A)`	Cholesky decomposition
`linalg.lu(A)`	LU decomposition
`linalg.schur(A)`	Schur decomposition
`linalg.expm(A)`	Matrix exponential
`linalg.logm(A)`	Matrix logarithm
`linalg.sqrtm(A)`	Matrix square root
`linalg.solve_triangular(A, b)`	Solve triangular system
`linalg.pinv(A)`	Moore-Penrose pseudoinverse

Advanced Signal Processing¶

Command	Description
`signal.stft(data, fs)`	Short-time Fourier transform
`signal.istft(Zxx, fs)`	Inverse STFT
`signal.welch(data, fs)`	Power spectral density (Welch method)
`signal.coherence(x, y, fs)`	Coherence between signals
`signal.hilbert(data)`	Hilbert transform
`signal.savgol_filter(data, window, polyorder)`	Savitzky-Golay filter
`signal.detrend(data)`	Remove linear trend from data
`signal.wiener(data)`	Wiener filter for noise reduction

Advanced Sparse Operations¶

Command	Description
`sparse.linalg.spsolve(A, b)`	Solve sparse linear system
`sparse.linalg.eigsh(A, k)`	Find k eigenvalues of sparse matrix
`sparse.linalg.svds(A, k)`	Sparse SVD for k singular values
`sparse.linalg.cg(A, b)`	Conjugate gradient solver
`sparse.linalg.gmres(A, b)`	GMRES iterative solver
`sparse.kron(A, B)`	Kronecker product of sparse matrices

Spatial Operations¶

Command	Description
`spatial.distance.pdist(X, metric='euclidean')`	Pairwise distances between observations
`spatial.distance.cdist(XA, XB)`	Distance between two collections
`spatial.KDTree(data)`	k-d tree for fast nearest neighbor
`spatial.ConvexHull(points)`	Convex hull of points
`spatial.Delaunay(points)`	Delaunay triangulation
`spatial.Voronoi(points)`	Voronoi diagram
`spatial.distance_matrix(x, y)`	Distance matrix computation

Image Processing¶

Command	Description
`ndimage.gaussian_filter(image, sigma)`	Gaussian smoothing filter
`ndimage.median_filter(image, size)`	Median filter for noise reduction
`ndimage.rotate(image, angle)`	Rotate image by angle
`ndimage.zoom(image, zoom)`	Zoom image by factor
`ndimage.binary_erosion(image)`	Binary morphological erosion
`ndimage.binary_dilation(image)`	Binary morphological dilation
`ndimage.label(image)`	Label connected components
`ndimage.sobel(image)`	Sobel edge detection

Configuration¶

NumPy/SciPy Configuration¶

# Display build and configuration information
import scipy
scipy.show_config()

# Set NumPy print options (affects SciPy output)
import numpy as np
np.set_printoptions(precision=4, suppress=True, linewidth=100)

# Configure warning filters
import warnings
warnings.filterwarnings('ignore', category=RuntimeWarning)

Optimization Configuration¶

# Configure optimization options
from scipy.optimize import minimize

options = {
    'maxiter': 1000,      # Maximum iterations
    'disp': True,         # Display convergence messages
    'ftol': 1e-8,        # Function tolerance
    'gtol': 1e-8         # Gradient tolerance
}

result = minimize(func, x0, method='BFGS', options=options)

Integration Tolerances¶

from scipy.integrate import quad, solve_ivp

# Configure integration accuracy
result, error = quad(func, a, b, 
                    epsabs=1e-10,  # Absolute error tolerance
                    epsrel=1e-10,  # Relative error tolerance
                    limit=100)     # Subdivision limit

# Configure ODE solver
sol = solve_ivp(func, t_span, y0,
               method='RK45',
               rtol=1e-6,      # Relative tolerance
               atol=1e-9,      # Absolute tolerance
               max_step=0.1)   # Maximum step size

Sparse Matrix Format Selection¶

from scipy import sparse

# Choose format based on use case
# CSR: efficient row slicing, matrix-vector products
A_csr = sparse.csr_matrix(data)

# CSC: efficient column slicing, matrix-vector products
A_csc = sparse.csc_matrix(data)

# COO: efficient construction, conversion
A_coo = sparse.coo_matrix(data)

# LIL: efficient incremental construction
A_lil = sparse.lil_matrix((1000, 1000))

Random Number Generation¶

from scipy import stats
import numpy as np

# Set random seed for reproducibility
np.random.seed(42)

# Use RandomState for thread-safe operations
rng = np.random.RandomState(42)
data = stats.norm.rvs(loc=0, scale=1, size=1000, random_state=rng)

Common Use Cases¶

Use Case 1: Curve Fitting and Model Selection¶

import numpy as np
from scipy.optimize import curve_fit
from scipy import stats
import matplotlib.pyplot as plt

# Generate noisy data
x = np.linspace(0, 10, 100)
y_true = 2.5 * np.exp(-0.5 * x) + 1.0
y_noisy = y_true + 0.2 * np.random.normal(size=len(x))

# Define model
def exponential_model(x, a, b, c):
    return a * np.exp(-b * x) + c

# Fit curve
params, covariance = curve_fit(exponential_model, x, y_noisy)
y_fit = exponential_model(x, *params)

# Calculate R-squared
residuals = y_noisy - y_fit
ss_res = np.sum(residuals**2)
ss_tot = np.sum((y_noisy - np.mean(y_noisy))**2)
r_squared = 1 - (ss_res / ss_tot)

print(f"Parameters: a={params[0]:.3f}, b={params[1]:.3f}, c={params[2]:.3f}")
print(f"R-squared: {r_squared:.4f}")

Use Case 2: Signal Filtering and Analysis¶

import numpy as np
from scipy import signal
from scipy.fft import fft, fftfreq

# Create noisy signal
fs = 1000  # Sampling frequency
t = np.linspace(0, 1, fs)
clean_signal = np.sin(2 * np.pi * 50 * t) + np.sin(2 * np.pi * 120 * t)
noisy_signal = clean_signal + 0.5 * np.random.normal(size=len(t))

# Design and apply Butterworth filter
sos = signal.butter(10, 100, btype='low', fs=fs, output='sos')
filtered_signal = signal.sosfilt(sos, noisy_signal)

# Find peaks
peaks, properties = signal.find_peaks(filtered_signal, 
                                     height=0.5, 
                                     distance=20)

# Compute power spectral density
f, Pxx = signal.welch(filtered_signal, fs, nperseg=256)

print(f"Found {len(peaks)} peaks")
print(f"Dominant frequency: {f[np.argmax(Pxx)]:.2f} Hz")

Use Case 3: Statistical Hypothesis Testing¶

import numpy as np
from scipy import stats

# Generate two sample datasets
np.random.seed(42)
group1 = stats.norm.rvs(loc=100, scale=15, size=50)
group2 = stats.norm.rvs(loc=105, scale=15, size=50)

# Test for normality
_, p_norm1 = stats.shapiro(group1)
_, p_norm2 = stats.shapiro(group2)

# Test for equal variances
_, p_var = stats.levene(group1, group2)

# Perform appropriate t-test
if p_var > 0.05:
    # Equal variances
    t_stat, p_value = stats.ttest_ind(group1, group2)
    test_type = "Independent t-test (equal variances)"
else:
    # Unequal variances (Welch's t-test)
    t_stat, p_value = stats.ttest_ind(group1, group2, equal_var=False)
    test_type = "Welch's t-test (unequal variances)"

# Calculate effect size (Cohen's d)
pooled_std = np.sqrt((np.std(group1)**2 + np.std(group2)**2) / 2)
cohens_d = (np.mean(group1) - np.mean(group2)) / pooled_std

print(f"Test: {test_type}")
print(f"t-statistic: {t_stat:.4f}, p-value: {p_value:.4f}")
print(f"Cohen's d: {cohens_d:.4f}")

Use Case 4: Optimization with Constraints¶

import numpy as np
from scipy.optimize import minimize

# Portfolio optimization: maximize return, minimize risk
def portfolio_objective(weights, returns, cov_matrix, risk_aversion=1.0):
    portfolio_return = np.sum(returns * weights)
    portfolio_risk = np.sqrt(np.dot(weights.T, np.dot(cov_matrix, weights)))
    return -(portfolio_return - risk_aversion * portfolio_risk)

# Sample data
n_assets = 4
returns = np.array([0.10, 0.12, 0.15, 0.08])
cov_matrix = np.array([[0.005, -0.001, 0.001, 0.000],
                       [-0.001, 0.008, 0.002, 0.001],
                       [0.001, 0.002, 0.012, 0.003],
                       [0.000, 0.001, 0.003, 0.004]])

# Constraints: weights sum to 1
constraints = {'type': 'eq', 'fun': lambda x: np.sum(x) - 1}

# Bounds: each weight between 0 and 1
bounds = tuple((0, 1) for _ in range(n_assets))

# Initial guess
x0 = np.array([1/n_assets] * n_assets)

# Optimize
result = minimize(portfolio_objective, x0, 
                 args=(returns, cov_matrix),
                 method='SLSQP',
                 bounds=bounds,
                 constraints=constraints)

optimal_weights = result.x
print("Optimal portfolio weights:")
for i, weight in enumerate(optimal_weights):
    print(f"Asset {i+1}: {weight*100:.2f}%")

Use Case 5: Image Processing Pipeline¶

import numpy as np
from scipy import ndimage
from scipy import signal

# Load or create image (grayscale)
image = np.random.rand(256, 256) * 255

# Step 1: Denoise with Gaussian filter
denoised = ndimage.gaussian_filter(image, sigma=2)

# Step 2: Edge detection
edges_x = ndimage.sobel(denoised, axis=0)
edges_y = ndimage.sobel(denoised, axis=1)
edges = np.hypot(edges_x, edges_y)

# Step 3: Threshold to create binary image
threshold = np.mean(edges) + np.std(edges)
binary = edges > threshold

# Step 4: Morphological operations
struct = ndimage.generate_binary_structure(2, 2)
cleaned = ndimage.binary_opening(binary, structure=struct)
cleaned = ndimage.binary_closing(cleaned, structure=struct)

# Step 5: Label connected components
labeled, num_features = ndimage.label(cleaned)

# Step 6: Calculate properties
sizes = ndimage.sum(cleaned, labeled, range(num_features + 1))
centers = ndimage.center_of_mass(cleaned, labeled, range(1, num_features + 1))

print(f"Found {num_features} objects")
print(f"Average object size: {np.mean(sizes[1:]):.2f} pixels")

Best Practices¶

Choose the Right Method: Select optimization algorithms based on problem characteristics (gradient-based for smooth functions, global optimizers for multimodal problems)
Vectorize Operations: Use NumPy array operations instead of loops for better performance; SciPy functions are optimized for array inputs
Handle Numerical Stability: Use appropriate tolerances (rtol, atol) for integration and optimization; be aware of condition numbers in linear algebra operations
Leverage Sparse Matrices: For large matrices with many zeros, use sparse matrix formats (csr_matrix, csc_matrix) to save memory and improve computation speed
Provide Good Initial Guesses: Optimization and root-finding algorithms converge faster with reasonable starting points; use domain knowledge when possible
Use Appropriate Statistical Tests: Verify assumptions (normality, equal variance) before applying parametric tests; use non-parametric alternatives when assumptions are violated
Set Random Seeds: Ensure reproducibility in stochastic algorithms by setting random seeds with np.random.seed() or using RandomState objects
Profile Performance: Use %timeit in Jupyter or cProfile to identify bottlenecks; consider Numba or Cython for critical sections if SciPy functions are insufficient
Check Convergence: Always examine optimization results (result.success, result.message) and integration errors before trusting outputs
Document Units and Scales: Clearly document physical units and data scales; normalize data when appropriate to improve numerical stability